**Lemma 2.** (Saadaoui & Ozguler, 2005) *A nonzero polynomial R s*[ ] *, such that*  (0) 0 *, has r real negative roots without counting the multiplicities if and only if the signature of the polynomial*  2 2 ( ) '( ) *s s is* 2*r. All roots of are real, negative, and distinct if and only if*  2 2 ( ) '( ) *s sH .*

We now describe a slight extension of the constant stabilizing gain algorithm of (Saadaoui & Ozguler,2003). Given a plant

$$G\_0(s) = \frac{p\_0(s)}{q\_0(s)}\tag{3}$$

Stabilizing PID Controllers for a Class of Time Delay Systems 145

 0 1 01 1

012 0 1

2 2 2( 1) ( )

*l l*

1,1 0, 1, , 1 *j i for j l* .

012 1

max max ( ) ( ) 0 ( ) 1. *j j j jj H v v forwhichF v and i SF v V*

min min ( ) ( ) 0 ( ) 1. *j j j jj H v v forwhichF v and i SF v V*

The algorithm above is easily specialized to determine all stabilizing proportional

**Remark 1.** By Step 2 of Algorithm 1, a necessary condition for the existence of a

negative roots with odd multiplicities. When solving a constant stabilization problem, this

*p q is non-empty if and only if for at least one signum sequence I satisfying step 2,*

for the plant 0 *G s*( ) . This is achieved by replacing *r* in step 2 of the

has at least 0 ()1

for each sequence of signums I

2

real

*r p r*

2 2 ( 1)

,, , , ,, , .

> *l l*

.

*l l*

*iii i for even r m*

*iii i for odd r m*

*ii i for odd r m ii i for even r m*

*p q* is empty or not and outputs its

The following algorithm determines whether 0 0 (,) *r*

1. Consider all the sequences of signums

2. Choose all the sequences that satisfy

that satisfy step 3.

algorithm by *n* , the degree of

lower bound is 0 ( )1

*The set* 0 0 (,) *r*

> max min *holds.*

4. 0 0 (,) *r*

controllers *C s*( )

0 0 (,) *r*

*r p*

**Algorithm 1.**

*where*

*and*

elements when it is not empty (Saadaoui & Ozguler, 2005) :

3. For each sequence of signums *I i j* that satisfy step 2, let

*p q* is equal to the union of intervals max min (,)

(, ) *s* .

.

*p q* is that the odd part of 0 00 [ ( ) ( )] ( ) *q s psps*

2

*r p r*

where 0 0 *p* , [] *q Rs* are coprime with *m p* deg 0 less than or equal to *n q* deg 0 , the set

$$\mathcal{O}\_r(p\_0, q\_0) = \{ \alpha \in \mathbb{R} : \sigma[\varphi(s, \alpha)] = \sigma[q\_0(s) + \alpha p\_0(s)] = r \}$$

is the set of all real such that (, ) *s* has signature equal to *r*. Let (,) *h g* and ( ,) *f e* be the even-odd components of 0 *q s*( ) and 0 *p* ( ), *s* respectively, so that

$$\begin{aligned} q\_0(\mathbf{s}) &= h(\mathbf{s}^2) + \text{sg}(\mathbf{s}^2) \\\\ p\_0(\mathbf{s}) &= f(\mathbf{s}^2) + \text{se}(\mathbf{s}^2) \ . \end{aligned}$$

Let (,) *H G* be the even-odd components of 0 0 *q sp s* () ( ) . Also let 2 0 0 *Fs p sp s* ( ) () ( ) . By a simple computation, it follows that ( 2 *s* is replaced by *u* ):

$$H(u) = h(u)f(u) - ug(u)e(u)$$

$$G(u) = g(u)f(u) - h(u)e(u)\tag{4}$$

$$F(u) = f^2(u) - ue^2(u)$$

With this setting, we have

$$[q\_0(\mathbf{s}) + a p\_0(\mathbf{s})] p\_0(-\mathbf{s}) = [H(\mathbf{s}^2) + a F(\mathbf{s}^2)] + s G(\mathbf{s}^2)$$

If 0 *G* and if they exist, let the *real negative roots with odd multiplicities of G u*( ) be *vv v* 1 2 , , *l* with the ordering 1 2 *l vv v* , with 0 *v* : 0 and 1 : *l v* for notational convenience.

The following algorithm determines whether 0 0 (,) *r p q* is empty or not and outputs its elements when it is not empty (Saadaoui & Ozguler, 2005) :

#### **Algorithm 1.**

1. Consider all the sequences of signums

$$I = \begin{cases} \{i\_0, i\_1, \dots, i\_l\} & \text{for odd } r - m\_\prime\\ \{i\_0, i\_1, \dots, i\_{l+1}\} & \text{for even } r - m\_\prime \end{cases}$$

*where*

144 PID Controller Design Approaches – Theory, Tuning and Application to Frontier Areas

*where* 0 ( 0) (0 ) : ( 1) (0) *m m Sb b , m*0 *is the multiplicity of u* = 0 *as a root of b*(*u*)*, and* ( 0)(0) *m b*

*r real negative roots without counting the multiplicities if and only if the signature of the polynomial*

We now describe a slight extension of the constant stabilizing gain algorithm of (Saadaoui &

0 ( ) ( ) ( ) *p s G s*

2 2 0 *q s h s sg s* () ( ) ( ) ,

2 2 0 *p* () ( ) ( ) *s f s se s* .

*s* is replaced by *u* ):

*Gu gu f u hueu* ( ) ( ) ( ) ( )( ) (4)

2 2 *F u f u ue u* () () ()

0 00 [ ( ) ( )] ( ) [ ( ) ( )] ( ) *q s p s p s H s F s sG s*

If 0 *G* and if they exist, let the *real negative roots with odd multiplicities of G u*( ) be *vv v* 1 2 , , *l* with the ordering 1 2 *l vv v* , with 0 *v* : 0 and 1 : *l v* for notational

*H u h u f u ug u e u* ( ) ( ) ( ) ( )( )

0

where 0 0 *p* , [] *q Rs* are coprime with *m p* deg 0 less than or equal to *n q* deg 0 , the set

*r*( , ) : [ ( , )] [ ( ) ( )] *p*0 0 *q R s qs ps r* 0 0

( ) '( ) *s s is* 2*r. All roots of are real, negative, and distinct if and only if*

22 2

  *R s*[ ] *, such that*

*q s* (3)

has signature equal to *r*. Let (,) *h g* and ( ,) *f e* be the

0 0 *Fs p sp s* ( ) () ( ) . By a

(0) 0 *, has*

The following result, determines the number of real negative roots of a real polynomial.

*denotes the value at u* = 0 *of the m*0*-th derivative of b*(*u*)*.*

0

such that

(, ) *s*

Let (,) *H G* be the even-odd components of 0 0 *q sp s* () ( ) . Also let 2

even-odd components of 0 *q s*( ) and 0 *p* ( ), *s* respectively, so that

simple computation, it follows that ( 2

With this setting, we have

convenience.

2 2

2 2

is the set of all real

Ozguler,2003). Given a plant

 ( ) '( ) *s sH .*

**Lemma 2.** (Saadaoui & Ozguler, 2005) *A nonzero polynomial*

$$\{\mathbf{i}\_j \in \{-1, 1\} \text{ for } j = 0, 1, \dots, l+1\}$$

2. Choose all the sequences that satisfy

$$r - \sigma(p\_0) = \begin{vmatrix} i\_0 - 2i\_1 + 2i\_2 \cdots + 2(-1)^l i\_l & \text{for odd } r - m \\ i\_0 - 2i\_1 + 2i\_2 \cdots + (-1)^{l+1} i\_{l+1} & \text{for even } r - m \end{vmatrix}$$

3. For each sequence of signums *I i j* that satisfy step 2, let

$$\alpha\_{\text{max}} = \max \left\{ -\frac{H}{V}(\upsilon\_j) \right\} \quad \forall \upsilon\_j \text{ for which} \\ F(\upsilon\_j) \neq 0 \text{ and } i\_j \\ SF(\upsilon\_j) = 1. \dots$$

*and*

$$\alpha\_{\min} = \min \left\{ -\frac{H}{V}(v\_j) \right\} \quad \forall v\_j \text{ for which} \\ F(v\_j) \neq 0 \text{ and } i\_j \\ SF(v\_j) = -1.$$

*The set* 0 0 (,) *r p q is non-empty if and only if for at least one signum sequence I satisfying step 2,*  max min *holds.*

4. 0 0 (,) *r p q* is equal to the union of intervals max min (,) for each sequence of signums I that satisfy step 3.

The algorithm above is easily specialized to determine all stabilizing proportional controllers *C s*( ) for the plant 0 *G s*( ) . This is achieved by replacing *r* in step 2 of the algorithm by *n* , the degree of (, ) *s* .

**Remark 1.** By Step 2 of Algorithm 1, a necessary condition for the existence of a 0 0 (,) *r p q* is that the odd part of 0 00 [ ( ) ( )] ( ) *q s psps* has at least 0 ()1 2 *r p r* real

negative roots with odd multiplicities. When solving a constant stabilization problem, this lower bound is 0 ( )1 2 *r p r* .

Stabilizing PID Controllers for a Class of Time Delay Systems 147

The characteristic function of the closed loop system formed by the PID controller and the

1() () ( ) *Ls*

 *. If*

*d pi*

( )*s* is equivalent to stability of 1

*d p i*

'( )*s* is equivalent to calculating the admissible ( , ) *p d k k* values for the original

The D-decomposition method (Hohenbicher & Ackermann, 2003; Gryazina & Polyak, 2006) is used to determine the stabilizing regions in the ( , ) *p d k k* plane. The D-decomposition method is based on the fact that roots of the quasi-polynomial (9) change continuously when the coefficients are changed continuously. Hence, a stable quasi-polynomial can become unstable if and only if at least one of its roots crosses the imaginary axis. Using this fact, the (,) *p d k k* plane can be partitioned into regions having the same number of roots of (9) in the left half plane. Stability can be checked by choosing a point inside a region and applying

Evaluating the characteristic function at the imaginary axis is equivalent to replacing *s* by

>

*jw R w LwI w wI w wLR w wR w Lw k R w LwI w wI w Lw wLR w wR w Lw wk*

'( ) ( ) ( ) '( ) ( ) '( ) sin( ) ( ( ) ( ) '( ))sin( ) ( ( ) '( )cos( ) 2

*s sQ s k s k s k e* (7)

*s sQ s e k s k s k* (8)

*d p*

'( )*s* is given by,

*s Ls Q s sQ s e k s k* (9)

( )*s is stable, then*

 ( )*s* and 1 () () *Ls*

*p d*

(11)

(12)

(10)

 *s se* have

( )*s* , see (Kharitonov

( )*s* is

'( )*s* . By Lemma 3,

'( )*s is*

*, with main term* 0 0 *r h , and* 1 0 *r*

has no finite roots, the quasi-polynomial 1

will be used to study stability of the closed loop system. Now, using Lemma 3, if

*such that* 1 2 *r*

Since the term *Ls e*

stable then

stabilizing

problem.

*j*

where,

and,

, 0 , in (9) we get,

time delay system (2) is given by,

the same roots, therefore stability of

*also a stable quasi-polynomial*.

2

et al., 2005). In the sequel, the quasi-polynomial,

2 () () ( ) *Ls*

'( )*s* is also a stable quasi-polynomial, where

Note that only two parameters ( , ) *p d k k* appear in the expression of

'( ) ( 1) ( ) '( ) 2 *Ls*

classical methods such as the Nyquist criterion.

*Q j R jI* ( ) () ()

*Q j R jI* '( ) '( ) '( )

### **2.1 Determining the admissible ranges of** *p k* **using padé approximation**

In this part, our aim is to find all admissible values of *p k* . Replacing the time delay by a Padé approximation ( ) ( ) *Ls p s e p s* where *p*() , *s H* we get the following closed-loop characteristic polynomial

$$\begin{aligned} \phi\_0(\mathbf{s}) &= \mathbf{s}q(\mathbf{s})p(\mathbf{s}) + (k\_d \mathbf{s}^2 + k\_p \mathbf{s} + k\_i)p(-\mathbf{s})\\ &= q\_0(\mathbf{s}) + (k\_d \mathbf{s}^2 + k\_p \mathbf{s} + k\_i)p\_0(\mathbf{s}) \end{aligned} \tag{5}$$

Where

0 *q s sq s p s* () ()() , 0 *p* () ( ) *s ps* .

Multiplying 0 ( )*s* by 0 *p* ( ) *s* , we obtain

$$\begin{aligned} \nu\_0(\mathbf{s}) &= \phi\_0(\mathbf{s}) \nu\_0(-\mathbf{s}) \\ &= (H(\mathbf{.}) + s^2 k\_d F(\mathbf{.}) + k\_i F(\mathbf{.})) + \mathbf{s} (G(\mathbf{.}) + k\_p F(\mathbf{.})) \end{aligned} \tag{6}$$

where *H* , *G* and *F* are given by (4). Note that 0 *p* ( ) () *s ps H* , therefore by Remark 1 the odd part () () *Gu kFu p* of 0 ( )*s* must have all its roots real, negative, and distinct. At this step, two parameters *dk* and *i k* are eliminated and an auxiliary problem with only one parameter will be solved. Let

$$\varphi\_1(\mathbf{s}) = \left(\mathbf{G}(\mathbf{s}^2) + \mathbf{s}\mathbf{G}'(\mathbf{s}^2)\right) + k\_p \left(\mathbf{F}(\mathbf{s}^2) + \mathbf{F}'(\mathbf{s}^2)\right),$$

using Lemma 2, finding values of *p k* such that ( ) ( ) *Gu kFu p* has all its roots real, negative and distinct is equivalent to stabilizing the new constructed polynomial 1 ( )*s* . This new constructed problem can be solved using Algorithm 1 as only one parameter appears.

### **2.2 An alternative method for computing the admissible ranges of** *p k*

In this part an alternative method is used to get the admissible values of *p k* without the need of using Padé approximation. Moreover this method allows the calculation of the admissible ranges of the parameters ( , ) *p d k k* , where *p k* denotes the proportional gain and *dk* the derivative gain. The following result will be used in determining the admissible values of *p k* (alternatively *dk* ).

**Lemma 3**: (Kharitonov et al., 2005) *Consider the quasi-polynomial,*

$$\Psi'(s) = \sum\_{i=0}^{n} \sum\_{l=1}^{r} h\_{il} \mathbf{s}^{n-i} e^{\mathbf{r}\_l s}$$

146 PID Controller Design Approaches – Theory, Tuning and Application to Frontier Areas

In this part, our aim is to find all admissible values of *p k* . Replacing the time delay by a

2

*d pi*

*d pi*

() ()() ( )( ) () ( ) ()

*s sq s p s k s k s k p s q s ks ks k p s*

2 0 0

0 *q s sq s p s* () ()() ,

0 *p* () ( ) *s ps* .

( (.) (.) (.)) ( (.) (.)) *di p*

2 2 22 1( ) ( ( ) '( )) ( ( ) '( )) *p*

*H s k F kF sG k F*

where *H* , *G* and *F* are given by (4). Note that 0 *p* ( ) () *s ps H* , therefore by Remark 1 the

step, two parameters *dk* and *i k* are eliminated and an auxiliary problem with only one

using Lemma 2, finding values of *p k* such that ( ) ( ) *Gu kFu p* has all its roots real, negative

In this part an alternative method is used to get the admissible values of *p k* without the need of using Padé approximation. Moreover this method allows the calculation of the admissible ranges of the parameters ( , ) *p d k k* , where *p k* denotes the proportional gain and *dk* the derivative gain. The following result will be used in determining the admissible

> 0 1 ( ) *l n r n i s il*

*i l s hs e*

*s G s sG s k F s F s*

constructed problem can be solved using Algorithm 1 as only one parameter appears.

2

and distinct is equivalent to stabilizing the new constructed polynomial 1

**2.2 An alternative method for computing the admissible ranges of** *p k*

**Lemma 3**: (Kharitonov et al., 2005) *Consider the quasi-polynomial,*

() () ( )

*s sp s*

where *p*() , *s H* we get the following closed-loop

(5)

( )*s* must have all its roots real, negative, and distinct. At this

(6)

( )*s* . This new

**2.1 Determining the admissible ranges of** *p k* **using padé approximation**

( ) *Ls p s*

*p s*

0

( )*s* by 0 *p* ( ) *s* , we obtain

0 00

Padé approximation ( )

characteristic polynomial

Where

Multiplying 0

odd part () () *Gu kFu p* of 0

parameter will be solved. Let

values of *p k* (alternatively *dk* ).

*e*

*such that* 1 2 *r , with main term* 0 0 *r h , and* 1 0 *r . If*  ( )*s is stable, then*  '( )*s is also a stable quasi-polynomial*.

The characteristic function of the closed loop system formed by the PID controller and the time delay system (2) is given by,

$$\mathcal{P}\_1(\mathbf{s}) = s\mathbf{Q}(\mathbf{s}) + (k\_d s^2 + k\_p s + k\_i)e^{-Ls} \tag{7}$$

Since the term *Ls e* has no finite roots, the quasi-polynomial 1 ( )*s* and 1 () () *Ls s se* have the same roots, therefore stability of ( )*s* is equivalent to stability of 1 ( )*s* , see (Kharitonov et al., 2005). In the sequel, the quasi-polynomial,

$$\mathcal{P}(\mathbf{s}) = \mathbf{s}Q(\mathbf{s})e^{\mathbf{L}\mathbf{s}} + \left(k\_d \mathbf{s}^2 + k\_p \mathbf{s} + k\_i\right) \tag{8}$$

will be used to study stability of the closed loop system. Now, using Lemma 3, if ( )*s* is stable then '( )*s* is also a stable quasi-polynomial, where '( )*s* is given by,

$$\Psi^{\prime}(s) = \left( (Ls+1)Q(s) + sQ^{\prime}(s) \right) e^{Ls} + \mathcal{D}\_d s + k\_p \tag{9}$$

Note that only two parameters ( , ) *p d k k* appear in the expression of '( )*s* . By Lemma 3, stabilizing '( )*s* is equivalent to calculating the admissible ( , ) *p d k k* values for the original problem.

The D-decomposition method (Hohenbicher & Ackermann, 2003; Gryazina & Polyak, 2006) is used to determine the stabilizing regions in the ( , ) *p d k k* plane. The D-decomposition method is based on the fact that roots of the quasi-polynomial (9) change continuously when the coefficients are changed continuously. Hence, a stable quasi-polynomial can become unstable if and only if at least one of its roots crosses the imaginary axis. Using this fact, the (,) *p d k k* plane can be partitioned into regions having the same number of roots of (9) in the left half plane. Stability can be checked by choosing a point inside a region and applying classical methods such as the Nyquist criterion.

Evaluating the characteristic function at the imaginary axis is equivalent to replacing *s* by *j* , 0 , in (9) we get,

$$\begin{aligned} \Psi'(jw) &= \left[R(w) - LwI(w) - wI'(w)\right] - \left[wLR(w) + wR'(w)\right]\sin(Lw) + k\_p\\ \left[\left(R(w) - LwI(w) - wI'(w)\right)\sin(Lw) + \left(wLR(w) + wR'(w)\cos(Lw) + 2wk\_d\right)\right] \end{aligned} \tag{10}$$

where,

$$Q(j\alpha) = R(\alpha) + jI(\alpha) \tag{11}$$

and,

$$Q'(j\alpha) = R'(\alpha) + jI'(\alpha) \tag{12}$$

Applying the D-decomposition method implies equating the real and imaginary parts of (10) to zero. Two cases must be considered:

 **Case 1**: 0 which leads to the following equation,

$$k\_p = -Q(0) \tag{13}$$

Stabilizing PID Controllers for a Class of Time Delay Systems 149

which partition the plane into root-invariant regions among which the stabilizing regions, if any, have to be determined. Alternatively, one may fix *dk* first and repeat the above

> cos( ) sin( ) ( ) () sin( ) cos( ) sin( ) ( )

*L L k k LI L L k k LR*

*p d*

 the above two equations partition the ( ,) *p i k k* plane into root-invariant regions. Hence stabilizing regions, if any, can be found. It is interesting to note here that depending on which parameter we fix first, either *pk* or *dk* , the obtained stabilizing region is different. This can be explained by the fact that in the first case the matrix is always singular and in

Determining the total set of stabilizing PID controllers is a first step in the design process. Once this set of stabilizing PID controllers is found, it is natural to search within this set, controllers that meet extra performance specifications. Four performance measures will be

Optimization is done using genetic algorithms. The first step in this design procedure, which consists of determining the total set of stabilizing PID controllers, is very important. It enhances the application of the genetic algorithm by fixing the search space, unlike other works on optimizing PID controllers using GAs (Chen et al., 2009; Chen & Chang, 2006; Jan et al., 2008) where the ranges of ( ,, ) *p i d k kk* are set arbitrary. Moreover, it improves the optimization time and increases the chances of obtaining the global optimum. Optimization

*ISE e t dt* , where e(t) is the error at time t.

0 ( )

*f t*

2 cos *i d*

*i* . For each singular frequency

(18)

*i* , an equation of a straight

(20)

(21)

(22)

(19)

 ( )sin( ) ( )cos( ) 0 *p kI L R L*

2 ( )sin( ) ( )cos( ) *i d i i i i ii i k kR L I L*

2

 2 ( ) ( ) cos( ) ( )sin( ) *i d k kI L R L*

 ( ) ( )s ( ) ( ) s( ) *p k I in L R co L*

and are denoted by , 1,2, *i*

procedure. In this case we have,

solving this system we get,

the second case it is always non-singular.

**3.2 PID controller design using GAs**

Maximum percent overshoot (OS).

Integral square error (ISE) 2

For 0,

considered:

 Settling time (ST). Rise time (RT).

line in the ( ,) *d i k k* plane is defined by,

 **Case 2**: 0 in this case we get,

$$\begin{aligned} k\_p &= -[R(w) - LwI(w) - wI'(w)]\cos(Lw) \\ &+ [wLR(w) + wR'(w)]\sin(Lw) \end{aligned} \tag{14}$$

$$\begin{aligned} k\_d &= \left[ \left( R(\alpha) - L\alpha l(\alpha) - \alpha l'(\alpha) \right) \sin(L\alpha) \right. \\ &+ \left( \alpha L R(\alpha) + \alpha R'(\alpha) \right) \cos(L\alpha) \right] \text{(2\cdot\alpha)}^{-1} \end{aligned} \tag{15}$$

By sweeping over values of 0 , the ( , ) *p d k k* plane can be partitioned into root invariant regions. Using (13), (14) and (15) stabilizing regions in the ( , ) *p d k k* plane can be determined.

#### **3. PID controller design**

In this section, first the set of all stabilizing PID controllers are determined, this forms the set of admissible solutions for optimization. Next, GAs are used to optimize these PID parameters.

#### **3.1 Stabilizing regions**

The admissible ( , ) *p d k k* values are calculated in section 2, now, we go back to the original problem by considering (7). Our method consists of fixing one parameter *p k* or *dk* and determining the stabilizing regions in the plane of the remaining two parameters. By sweeping over the admissible values of the first parameter ( *p k* or *dk* ) the complete set of the stabilizing regions is found. Once again the D-decomposition method is used.

For 0 , we get the following equation,

$$k\_i = 0\tag{16}$$

For 0, two cases will be investigated. First, let us fix *p k* and find the stabilizing regions in the ( ,) *d i k k* plane. Using (7), replacing *s* by *j* and equating the real and imaginary parts to zero we get,

$$
\begin{bmatrix}
\cos(L\alpha) & -\alpha^2 \cos(L\alpha) \\
\sin(L\alpha) & -\alpha^2 \sin(L\alpha)
\end{bmatrix}
\begin{bmatrix}
k\_\perp \\
k\_\perp
\end{bmatrix} = \begin{bmatrix}
k\_\parallel \alpha \sin(L\alpha) + \alpha l \text{(}\alpha\text{)} \\
\end{bmatrix} \tag{17}
$$

Note that the matrix at the left-hand of (17) is singular. The singular frequencies (Hohenbicher & Ackermann, 2003) are determined as the solutions of equation (18),

$$k\_p + I(\alpha)\sin(L\alpha) - R(\alpha)\cos(L\alpha) = 0\tag{18}$$

and are denoted by , 1,2, *i i* . For each singular frequency *i* , an equation of a straight line in the ( ,) *d i k k* plane is defined by,

$$k\_{\dot{l}} = \alpha\_{\dot{l}}^2 k\_d + \alpha\_{\dot{l}} \mathbb{R}(\alpha\_{\dot{l}}) \sin(La\_{\dot{l}}) + \alpha\_{\dot{l}} l (\alpha\_{\dot{l}}) \cos(La\_{\dot{l}}) \tag{19}$$

which partition the plane into root-invariant regions among which the stabilizing regions, if any, have to be determined. Alternatively, one may fix *dk* first and repeat the above procedure. In this case we have,

$$
\begin{bmatrix}
\cos(Lo) & -o\sin(Lo) \\
\sin(Lo) & o\cos(Lo)
\end{bmatrix}
\begin{bmatrix}
k\_i \\
k\_\rho \\
k\_\rho
\end{bmatrix} =
\begin{bmatrix}
k\_d\rho^2\cos(Lo) + al\langle o\rangle \\
k\_d\rho^2\sin(Lo) - o\mathbb{R}\langle o\rangle
\end{bmatrix}
\tag{20}
$$

solving this system we get,

148 PID Controller Design Approaches – Theory, Tuning and Application to Frontier Areas

Applying the D-decomposition method implies equating the real and imaginary parts of

regions. Using (13), (14) and (15) stabilizing regions in the ( , ) *p d k k* plane can be determined.

In this section, first the set of all stabilizing PID controllers are determined, this forms the set of admissible solutions for optimization. Next, GAs are used to optimize these PID

The admissible ( , ) *p d k k* values are calculated in section 2, now, we go back to the original problem by considering (7). Our method consists of fixing one parameter *p k* or *dk* and determining the stabilizing regions in the plane of the remaining two parameters. By sweeping over the admissible values of the first parameter ( *p k* or *dk* ) the complete set of the

0 *i k* (16)

cos( ) cos( ) sin( ) ( ) cos( ) ( ) sin( ) sin( ) *i p d p*

*L Lk kLI k k LR L L*

Note that the matrix at the left-hand of (17) is singular. The singular frequencies

(17)

two cases will be investigated. First, let us fix *p k* and find the stabilizing regions

stabilizing regions is found. Once again the D-decomposition method is used.

2 2

(Hohenbicher & Ackermann, 2003) are determined as the solutions of equation (18),

*p k R w LwI w wI w Lw wLR w wR w Lw*

> ( ( ) ( ) '( ))sin( ) ( ( ) '( )) cos( ) *d R LI I L LR R L*

( ) ( ) '( ) cos( ) ( ) '( ) sin( )

(0) *p k Q* (13)

(15)

(14)

 1 (2 )

, the ( , ) *p d k k* plane can be partitioned into root invariant

and equating the real and imaginary parts

which leads to the following equation,

(10) to zero. Two cases must be considered:

By sweeping over values of 0

**3. PID controller design**

**3.1 Stabilizing regions**

, we get the following equation,

in the ( ,) *d i k k* plane. Using (7), replacing *s* by *j*

in this case we get,

*k*

 **Case 1**: 0

 **Case 2**: 0

parameters.

For 0

For 0,

to zero we get,

$$k\_i(\alpha) = \alpha^2 k\_d + \alpha l(\alpha) \cos(L\alpha) - \alpha R(\alpha) \sin(L\alpha) \tag{21}$$

$$k\_p(\alpha) = -I(\alpha)\text{s}\,\text{in}(L\alpha) - R(\alpha)\text{cos}(L\alpha) \tag{22}$$

For 0, the above two equations partition the ( ,) *p i k k* plane into root-invariant regions. Hence stabilizing regions, if any, can be found. It is interesting to note here that depending on which parameter we fix first, either *pk* or *dk* , the obtained stabilizing region is different. This can be explained by the fact that in the first case the matrix is always singular and in the second case it is always non-singular.

#### **3.2 PID controller design using GAs**

Determining the total set of stabilizing PID controllers is a first step in the design process. Once this set of stabilizing PID controllers is found, it is natural to search within this set, controllers that meet extra performance specifications. Four performance measures will be considered:

Optimization is done using genetic algorithms. The first step in this design procedure, which consists of determining the total set of stabilizing PID controllers, is very important. It enhances the application of the genetic algorithm by fixing the search space, unlike other works on optimizing PID controllers using GAs (Chen et al., 2009; Chen & Chang, 2006; Jan et al., 2008) where the ranges of ( ,, ) *p i d k kk* are set arbitrary. Moreover, it improves the optimization time and increases the chances of obtaining the global optimum. Optimization

Stabilizing PID Controllers for a Class of Time Delay Systems 151

the admissible values of (, ) *p d k k* are calculated, see Fig. 1. The first approach proposed consists of fixing *pk* and using (16), (18) and (19) to determine stabilizing regions in the

By sweeping over the admissible values of *p k* , the complete stabilizing regions of

0 0.5 1 1.5 2 2.5 3 3.5

Fig. 2. Complete set of stabilizing PID controllers for example1.

*Ki*

*Kd*

plane of the remaining two parameters.

Fig. 1. Admissible values of(, ) *p d k k* .

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

*Kp*

( ,, ) *p i d k kk* values are found as shown in Fig.2.

of the PID controllers is done by minimizing each of the four cost functions: maximum percent overshoot, settling time, rise time and integral square error. GAs tries to find the global optimum by evaluating many points of the solution space in each generation. There are mainly three operations: reproduction, crossover and mutation, used to form new generations until a termination condition is reached (Goldberg, 1989; Haupt & Haupt, 2004; Melanie, 1998). In what follows, we briefly describe the parameters and tools used in the implementation of the genetic algorithm:

To avoid local minimum and explore new parts of the search space, mutation process is applied. It consists of randomly modifying individuals in the generation.

### **4. Examples**

In this section three illustrative examples are given.

#### **Example 1:**

Consider stabilizing the second order integrating system,

$$G(s) = \frac{11.32}{s^2 + 11.32s}e^{-s} \tag{23}$$

by a PID controller. This transfer function represents a system composed of master-slave parts. The slave part is a model of a mobile robot which can move in one direction (Seuret et al., 2006). This robot is controlled through a communication network which introduces delays in the control loop. See (Seuret et al., 2006) for more details about this system. First, 150 PID Controller Design Approaches – Theory, Tuning and Application to Frontier Areas

of the PID controllers is done by minimizing each of the four cost functions: maximum percent overshoot, settling time, rise time and integral square error. GAs tries to find the global optimum by evaluating many points of the solution space in each generation. There are mainly three operations: reproduction, crossover and mutation, used to form new generations until a termination condition is reached (Goldberg, 1989; Haupt & Haupt, 2004; Melanie, 1998). In what follows, we briefly describe the parameters and tools used in the

1. Since stability is an essential property for any control system, the search space consists of the stabilizing values of the PID controller which are determined in section 3.1. Population size is chosen to be 150 individuals per population. In the beginning the initial population is randomly chosen within the stabilizing values of the PID controller. These individuals are candidate solutions to the problem. The number of generations is

2. Evaluation of a generation is done by calculating the cost function for each individual. Since minimization problems are considered, the fitness function will be the inverse of

3. Genetic operators: reproduction, crossover and mutation are used to form new generations. Individuals are chosen for reproduction according to their fitness value. Fittest ones have larger probability of selection. The Roulette Wheel selection method is

4. In order to improve next generations, crossover which is a process of combining parts of the parent individuals to produce new offsprings is done. Crossover value used is

To avoid local minimum and explore new parts of the search space, mutation process is

1. Mutation probability is always set to a small value so that the search algorithm is not

2. Steps 2-5 are repeated until a termination condition is reached. The maximum

11.32 ( ) 11.32 *s G s e s s*

by a PID controller. This transfer function represents a system composed of master-slave parts. The slave part is a model of a mobile robot which can move in one direction (Seuret et al., 2006). This robot is controlled through a communication network which introduces delays in the control loop. See (Seuret et al., 2006) for more details about this system. First,

(23)

turned to a random search algorithm. The value chosen in our case is 0.01.

applied. It consists of randomly modifying individuals in the generation.

generation termination condition is adopted.

In this section three illustrative examples are given.

2

Consider stabilizing the second order integrating system,

implementation of the genetic algorithm:

chosen as 100.

the cost function.

used.

0.8.

**4. Examples**

**Example 1:**

the admissible values of (, ) *p d k k* are calculated, see Fig. 1. The first approach proposed consists of fixing *pk* and using (16), (18) and (19) to determine stabilizing regions in the plane of the remaining two parameters.

Fig. 1. Admissible values of(, ) *p d k k* .

By sweeping over the admissible values of *p k* , the complete stabilizing regions of ( ,, ) *p i d k kk* values are found as shown in Fig.2.

Fig. 2. Complete set of stabilizing PID controllers for example1.

Stabilizing PID Controllers for a Class of Time Delay Systems 153

Using this stabilizing region of (,) *p i k k* values, we apply the genetic algorithm described in section 3.2, to minimize each of the performance indices: maximum percent overshoot (Opt1), settling time (Opt2), rise time (Opt3) and integral square error (opt4). Although a simple controller, PI controller, is used for an integrating system with delay, the results obtained are satisfactory as shown by the step response of the closed loop system in

**Step Response**

**Opt1 Opt2 Opt3 Opt4**

0 5 10 15 20 25 30 35 40 45 50

**Time(sec)**

*s Gs e s*

by a PID controller. For comparison reasons, we apply three classical methods. These methods are: Ziegler-Nichols method (ZN), Chien Hornes & Reswich method (CHR) and finally Cohen-Coon method (CC). After obtaining the stabilizing regions of ( ,, ) *p i d k kk* values, we apply the genetic algorithm to minimize the four performance indices described in section 3.2. Step responses of the closed loop system obtained with classical methods are

(24)

Fig. 5.

0

**Example 2:**

given in Fig 6.

Fig. 5. Step responses of the closed loop system.

1 2.8 ( ) 3 1

Consider stabilizing the first order system given in (Silva et al., 2005),

0.2

0.4

0.6

0.8

1

**Amplitude**

1.2

1.4

1.6

1.8

2

Alternatively, we can fix *dk* and use (16), (21) and (22) to find the stabilizing regions of ( ,, ) *p i d k kk* values as shown in Fig.3.

Fig. 3. Complete set of stabilizing PID controllers for example1.

Fixing 0 *dk* is equivalent to using a PI controller. For 0 *dk* , we obtain the stabilizing region of ( ,) *p i k k* values as given in Fig. 4.

Fig. 4. Complete set of stabilizing PI controllers for example1.

Using this stabilizing region of (,) *p i k k* values, we apply the genetic algorithm described in section 3.2, to minimize each of the performance indices: maximum percent overshoot (Opt1), settling time (Opt2), rise time (Opt3) and integral square error (opt4). Although a simple controller, PI controller, is used for an integrating system with delay, the results obtained are satisfactory as shown by the step response of the closed loop system in Fig. 5.

Fig. 5. Step responses of the closed loop system.

#### **Example 2:**

152 PID Controller Design Approaches – Theory, Tuning and Application to Frontier Areas

Alternatively, we can fix *dk* and use (16), (21) and (22) to find the stabilizing regions of

Fixing 0 *dk* is equivalent to using a PI controller. For 0 *dk* , we obtain the stabilizing

( ,, ) *p i d k kk* values as shown in Fig.3.

Fig. 3. Complete set of stabilizing PID controllers for example1.

Fig. 4. Complete set of stabilizing PI controllers for example1.

region of ( ,) *p i k k* values as given in Fig. 4.

Consider stabilizing the first order system given in (Silva et al., 2005),

$$G(s) = \frac{1}{3s+1}e^{-2.8s} \tag{24}$$

by a PID controller. For comparison reasons, we apply three classical methods. These methods are: Ziegler-Nichols method (ZN), Chien Hornes & Reswich method (CHR) and finally Cohen-Coon method (CC). After obtaining the stabilizing regions of ( ,, ) *p i d k kk* values, we apply the genetic algorithm to minimize the four performance indices described in section 3.2. Step responses of the closed loop system obtained with classical methods are given in Fig 6.

Stabilizing PID Controllers for a Class of Time Delay Systems 155

Step Response

Opt1 Opt2 Opt3 Opt4

0 5 10 15 20 25 30 35 40 45 50

Time(sec)

Fig. 7. Step responses of the closed loop system for example 2.

0

0.2

0.4

0.6

A

mplitude

0.8

1

1.2

1.4

1.6

Fig. 6. Step responses of the closed loop system for example 2.

Determining the set of stabilizing controllers and using GAs to optimize these stabilizing values, gives the step responses shown in Fig 7.

Table 1 summarizes the different results obtained using our method and classical methods. For each performance criteria we can see that our method gives better results. However it should be stressed here that minimizing one performance criteria can deteriorate other performance indices, for example we minimize overshoot at the expense of a larger settling time as shown in the results below. Optimizing the integral square error (opt4) gives better results as an overall performance. Another advantage of our method is to answer the question: what is the best performance we can get with such a controller? This is possible as optimization is done over all the admissible stabilizing values of the controller.

154 PID Controller Design Approaches – Theory, Tuning and Application to Frontier Areas

Step Response

ZN CHR CC

0 5 10 15 20 25 30 35 40 45 50

Time(sec)

Determining the set of stabilizing controllers and using GAs to optimize these stabilizing

Table 1 summarizes the different results obtained using our method and classical methods. For each performance criteria we can see that our method gives better results. However it should be stressed here that minimizing one performance criteria can deteriorate other performance indices, for example we minimize overshoot at the expense of a larger settling time as shown in the results below. Optimizing the integral square error (opt4) gives better results as an overall performance. Another advantage of our method is to answer the question: what is the best performance we can get with such a controller? This is possible as optimization is done over all the admissible stabilizing

Fig. 6. Step responses of the closed loop system for example 2.

values, gives the step responses shown in Fig 7.

values of the controller.

0

0.2

0.4

0.6

A

mplitude

0.8

1

1.2

1.4

1.6

Fig. 7. Step responses of the closed loop system for example 2.

Stabilizing PID Controllers for a Class of Time Delay Systems 157

percent overshoot, settling time, rise time and integral square error, were minimized. Application of the proposed method to uncertain time delay system is under investigation.

Chen, C. K.; Koo, H. H.; Yan, J. J. & Liao, T. L. (2009). GA-based PID active queue

Chen, C. K. & Chang, S. H. (2006). Genetic algorithms based optimization design of a PID

Datta, A.; Ho, M. T. & Bhattacharyya, S. P. (2000). *Structure and synthesis of PID controller,*

Goldberg, D. E. (1989). *Genetic algorithms in search, optimization and machine learnin,* Addison-

Gryaznia, E. N. & Polyak, B. T. (2006). Stability regions in the parameter space: D-

Haupt, R. L. & Haupt, S. E. (2004) *Practical genetic algorithms,* John Wiley & Sons, ISBN

Hohenbicher, N. & Ackermann, J. (2003). Computing stable regions in parameter spaces for

Jan, R. M.; Tseng, C. S. & Lin, R. J. (2008). Robust PID control design for permanent magnet

Normey-Rico, J. E. & Camacho, E. F. (2007). *Control of dead time process*. Springer,

Kharitonov, V. L.; Niculescu, S.; Moreno, J. & Michiels, W. (2005). Static output stabilization:

Saadaoui, K. & Ozguler, A. B. (2003). On the set of all stabilizing first order controllers, Proc.

Saadaoui, K. & Ozguler, A. B. (2005). A new method for the computation of all stabilizing controllers of a given order*. International Journal of Control,* Vol. 78, pp. 14-28. Silva, G. J.; Datta, A. & Bhattacharyya, S. P. (2001). PI stabilization of first order systems with

Silva, G. J.; Datta, A. & Bhattacharyya, S. P. (2002). New results on the synthesis of PID controllers. *IEEE Transactions on Automatic Control*, Vol. 47, 2002, pp. 241-252. Silva, G. J.; Datta, A. & Bhattacharyya, S. P. (2005) *PID controllers for time delay systems,*

Seuret, A.; Michaut, F.; Richard, J. P. & Divoux, T. (2006). Network control using GPS

synchronization, Proc. American Control Conference, Minneapolis, Minnesota,

Melanie, M. (1998) *An introduction to genetic algorithms,* MIT Press, ISBN<PHONE_NUMBER>.

a class of quasi-polynomials, Proc. 4th IFAC Workshop on Time Delay Systems

synchronous motor: A genetic approach. *Electric Power Systems Research*, Vol. 78,

Necessary conditions for multiple delay controllers. *IEEE Transactions on Automatic*

Springer-Verlag, ISBN 1-85233-614-5, London, Great Britain.

decomposition revisited. *Automatica*, Vol. 42, pp. 13-26.

American Control Conference, Denver, CO, USA, 2003*.*

time delay. *Automatica*, Vol. 37, pp. 2225-2031.

Birkhauser, ISBN<PHONE_NUMBER>662, Boston USA.

*Systems with Applications*, Vol. 36, pp. 1903-1913.

*and Network*, Vol. 6, pp. 95-99.

Wesley, ISBN<PHONE_NUMBER>.

TDS'03, Recquencourt, France*.*

*Control*, Vol. 50, pp. 82-86.

0471188735.

pp. 1161-1168.

London.

USA*.*

management control design for a class of TCP communication networks, *Expert*

controller for an active magnetic bearing. *International Journal of Computer Science*

**6. References**

Table 1. Results obtained by different methods.

#### **Example 3:**

Consider stabilizing the second order plant given in (Wang, 2007a) by

$$G(s) = \frac{k}{(1 + T\_1s)(1 + T\_2s)}e^{-Ls}s$$

by a PID controller.

Let 1 *k* and 0.2 *L* applying the procedure developed in this work, we get (the four cases given in (Wang, 2007a) will be investigated):

Although a necessary condition is used to determine the admissible ranges of *p k* , in the four cases above the intervals obtained are the same as the ones given in (Wang, 2007a).

#### **5. Conclusion**

In this work, a new method is given for calculating all stabilizing values of a PID controller which is applied to a special class of time delay systems. The proposed approach is based on determining first the admissible ranges of one of the controller's parameters. This step is solved using two different approaches. Next, the D-decomposition method is applied to obtain the stabilizing regions in the controller's parameter space. The genetic algorithm optimization method is then applied to find among those stabilizing controllers those that satisfy further performance specifications. Four time domain measures, which are maximum percent overshoot, settling time, rise time and integral square error, were minimized. Application of the proposed method to uncertain time delay system is under investigation.

### **6. References**

156 PID Controller Design Approaches – Theory, Tuning and Application to Frontier Areas

1 2 ( ) (1 )(1 ) *k Ls G s e Ts Ts* ,

Let 1 *k* and 0.2 *L* applying the procedure developed in this work, we get (the four cases

Case 1: For 1 *T* 0.2 and 2 *T* 0.5 the admissible range of *p k* values is obtained as

Case 2: For 1 *T* 1 and 2 *T* 0.5 the admissible range of *p k* values is obtained as

Case 3: For 1 *T* 1 and 2 *T* 0.5 the admissible range of *p k* values is obtained as

Case 4: For 1 *T* 1 and 2 *T* 0.5 the admissible range of *p k* values is obtained as

Although a necessary condition is used to determine the admissible ranges of *p k* , in the four

In this work, a new method is given for calculating all stabilizing values of a PID controller which is applied to a special class of time delay systems. The proposed approach is based on determining first the admissible ranges of one of the controller's parameters. This step is solved using two different approaches. Next, the D-decomposition method is applied to obtain the stabilizing regions in the controller's parameter space. The genetic algorithm optimization method is then applied to find among those stabilizing controllers those that satisfy further performance specifications. Four time domain measures, which are maximum

cases above the intervals obtained are the same as the ones given in (Wang, 2007a).

Method (Kp,Ki,Kd) OS (%) ST (s) RT (s) ISE ZN (1.28, 0.22, 1.8) 7.9 24.9 5.9 3.17 CHR (0.64, 0.21, 0.9) 9.1 24 8 3.75 CC (1.69, 0.39, 1.49) 51 32.9 5.8 3.33 Opt 1 (0.52, 0.14, 0.28) 0 19 9 4.77 Opt 2 (0.84, 0.22, 0.39) 4.9 13 7 3.73 Opt 3 (1.69, 0.35, 0.55) 58 48 6 4.32 Opt 4 (1.22, 0.35, 1.67) 21.7 21.9 5.9 2.95

Table 1. Results obtained by different methods.

given in (Wang, 2007a) will be investigated):

Consider stabilizing the second order plant given in (Wang, 2007a) by

1 5.56 *p k* with a Padé approximation of order 3.

1 0.58 *p k* with a Padé approximation of order 2.

5.21 1 *p k* with a Padé approximation of order 3.

9.79 1 *p k* with a Padé approximation of order 3.

**Example 3:**

by a PID controller.

**5. Conclusion**

**8**

*Malaysia*

**Practical Control Method for Two-Mass Rotary**

Motion control systems play an important role in industrial engineering applications such as advanced manufacturing systems, semiconductor manufacturing system, computer numerical control (CNC) machining and robot systems. In general, positioning system can be classified into two types, namely point-to-point (PTP) positioning systems and continuous path (CP) control system (Crowder R.M, 1998). PTP positioning systems, either of one-mass or multi-mass systems, is used to move an object from one point to another point either in angular or linear position. For example, in application with one-mass system, such as CNC machines, PTP positioning is used to accurately locate the spindle at one or more specific locations to perform operations, such as drilling, reaming, boring, tapping, and punching. In multi-mass systems application, such as in spot-welding robot, which has a long arm for linear system or long shaft in rotary system, PTP positioning is used to locate

PTP positioning system requires high accuracy with a high speed, fast response with no or small overshoot and to be robust to parameter variations and uncertainties. Therefore, the most important requirements in PTP positioning systems are the final accuracy and transition time whereas the transient path is considered as the second important. However, it is not easy to achieve high precision performances because of non-linearities and uncertainties exist in the motion control systems. One significant nonlinearity is friction that causes steady state error and/or limit cycles near the reference position (Armstrong et al., 1994). Another source of nonlinearity in motion control system is saturation of the actuator and/or electronic power amplifier. Saturation causes slow motion and may effect the stability of the performances (Slotine & Li, 1991). In PTP applications, the system performance is expected to be the same or as close as its performance when the system is in normal condition. Thus, robustness is also an important requirement in order to maintain

In order to satisfy the design requirements, a good controller is required. Many types of controllers have been proposed and evaluated for positioning systems. The use of proportional-integral-derivative (PID) controllers are the most popular controller used in industrial control systems including motion control systems due to their simplicity and also

**1. Introduction**

the manipulator from one location to another.

the stability of the positioning systems.

**Point-To-Point Positioning Systems**

Fitri Yakub, Rini Akmeliawati and Aminudin Abu *Malaysia-Japan International Institute of Technology (MJIIT), Universiti Teknologi Malaysia International Campus (UTM IC)*

## **Practical Control Method for Two-Mass Rotary Point-To-Point Positioning Systems**

Fitri Yakub, Rini Akmeliawati and Aminudin Abu *Malaysia-Japan International Institute of Technology (MJIIT), Universiti Teknologi Malaysia International Campus (UTM IC) Malaysia*

### **1. Introduction**

158 PID Controller Design Approaches – Theory, Tuning and Application to Frontier Areas

Wang, D. J. (2007). Further results on the synthesis of PID controllers. *IEEE Transactions on*

Wang, D. J. (2007). Stabilizing regions of PID controllers for n-th order all pole plants with

Zhong, Q. C. (2006). *Robust control of time delay systems*, Springer-Verlag, ISBN 978-

dead time. *IET Control Theory & Applications*, Vol. 1, pp. 1068-1074.

*Automatic Control,* Vol. 52, pp. 1127-1132.

1846282645.

Motion control systems play an important role in industrial engineering applications such as advanced manufacturing systems, semiconductor manufacturing system, computer numerical control (CNC) machining and robot systems. In general, positioning system can be classified into two types, namely point-to-point (PTP) positioning systems and continuous path (CP) control system (Crowder R.M, 1998). PTP positioning systems, either of one-mass or multi-mass systems, is used to move an object from one point to another point either in angular or linear position. For example, in application with one-mass system, such as CNC machines, PTP positioning is used to accurately locate the spindle at one or more specific locations to perform operations, such as drilling, reaming, boring, tapping, and punching. In multi-mass systems application, such as in spot-welding robot, which has a long arm for linear system or long shaft in rotary system, PTP positioning is used to locate the manipulator from one location to another.

PTP positioning system requires high accuracy with a high speed, fast response with no or small overshoot and to be robust to parameter variations and uncertainties. Therefore, the most important requirements in PTP positioning systems are the final accuracy and transition time whereas the transient path is considered as the second important. However, it is not easy to achieve high precision performances because of non-linearities and uncertainties exist in the motion control systems. One significant nonlinearity is friction that causes steady state error and/or limit cycles near the reference position (Armstrong et al., 1994). Another source of nonlinearity in motion control system is saturation of the actuator and/or electronic power amplifier. Saturation causes slow motion and may effect the stability of the performances (Slotine & Li, 1991). In PTP applications, the system performance is expected to be the same or as close as its performance when the system is in normal condition. Thus, robustness is also an important requirement in order to maintain the stability of the positioning systems.

In order to satisfy the design requirements, a good controller is required. Many types of controllers have been proposed and evaluated for positioning systems. The use of proportional-integral-derivative (PID) controllers are the most popular controller used in industrial control systems including motion control systems due to their simplicity and also

Practical Control Method for Two-Mass Rotary Point-To-Point Positioning Systems 161

compensator. When the signal *up* is not zero, the compensator is used to drive the value of

In Figure 2, the object motion is divided into two phases, the reaching phase and the following phase. During the reaching phase, the compensator forces the object motion to reach the NCT as fast as possible. Then, in the following phase, the compensator controls the object motion to follow the NCT and end at the origin. The object motion stops at the origin, which represents the end of the positioning motion. Thus, the NCT governs the positioning

Error, e

o

FP

RP

signal *up* to zero.

Fig. 1. Structure of NCTF control system

response performance.

Error rate, e'

Fig. 2. NCT and object motion

NCT

Object motion

RP: Reaching phase FP: Following phase

satisfactory performances (Yonezawa et al., 1990). However, it is difficult to achieve a fast response with no or small overshoot simultaneously.

In practical applications, an engineer does not need deep knowledge or be an expert in control systems theory while designing controllers. Thus, easiness of controller design process, simplicity of the controller structure and no requirement of exact object model and its parameters are very important and preferable in real applications. To achieve these, nominal characteristic trajectory following (NCTF) controller for one-mass rotary systems had been proposed as a practical controller for PTP positioning systems in (Wahyudi, 2002). The controller design procedure is simple and easily implemented since it is only based on a simple open-loop experiment. In addition, an exact object model and its parameters does not required while designing the controller. Thus, this controller is easy to design, adjustable and understands.

The NCTF controller had been proposed as a practical controller for PTP positioning systems. However, the NCTF controller is designed based on one-mass rotary positioning systems. Positioning system was considered as a one-mass system when a rigid coupling with high stiffness is used. The existing NCTF controller does not work for two-mass rotary system because of the vibration happened due to mechanical resonance of the plant such as flexible coupling or a long shaft with low stiffness are used. This vibration gives the unstable performance response of the plant. Therefore, enhancement and improvement design of NCT and a compensator are required to make the NCTF controller suitable for two-mass rotary positioning systems, which have long shaft between the actuator and load. This chapter is an attempt to address the problem of NCTF for two-mass system or multimass systems.

### **2. Basic concept of NCTF controller**

The structure of the NCTF control system is shown in Figure 1 (Mohd Fitri Mohd Yakub et al., 2010) consists of:

Therefore, the controller is called as the NCTF controller.

As shown in Figure 1, the controller output is signal *u*. This signal is used to drive the object. The input to the controller are error, *e*, and object motion, *l* . In principle, the controller compares the object motion input, *l* with the error-rate, *e* , provided by predetermined NCT, at certain error. The difference between the actual error-rate of the object and that of the NCT is denoted as signal *up*, which is the output of the NCT. If the object motion perfectly follows the NCT, the value of signal *up* is zero. Thus, no action is performed by the compensator. When the signal *up* is not zero, the compensator is used to drive the value of signal *up* to zero.

Fig. 1. Structure of NCTF control system

160 PID Controller Design Approaches – Theory, Tuning and Application to Frontier Areas

satisfactory performances (Yonezawa et al., 1990). However, it is difficult to achieve a fast

In practical applications, an engineer does not need deep knowledge or be an expert in control systems theory while designing controllers. Thus, easiness of controller design process, simplicity of the controller structure and no requirement of exact object model and its parameters are very important and preferable in real applications. To achieve these, nominal characteristic trajectory following (NCTF) controller for one-mass rotary systems had been proposed as a practical controller for PTP positioning systems in (Wahyudi, 2002). The controller design procedure is simple and easily implemented since it is only based on a simple open-loop experiment. In addition, an exact object model and its parameters does not required while designing the controller. Thus, this controller is easy to design, adjustable

The NCTF controller had been proposed as a practical controller for PTP positioning systems. However, the NCTF controller is designed based on one-mass rotary positioning systems. Positioning system was considered as a one-mass system when a rigid coupling with high stiffness is used. The existing NCTF controller does not work for two-mass rotary system because of the vibration happened due to mechanical resonance of the plant such as flexible coupling or a long shaft with low stiffness are used. This vibration gives the unstable performance response of the plant. Therefore, enhancement and improvement design of NCT and a compensator are required to make the NCTF controller suitable for two-mass rotary positioning systems, which have long shaft between the actuator and load. This chapter is an attempt to address the problem of NCTF for two-mass system or multi-

The structure of the NCTF control system is shown in Figure 1 (Mohd Fitri Mohd Yakub et

a. A nominal characteristic trajectory (NCT) that is constructed based on measured *l*

b. A compensator which is used to force the object motion to reach the NCT as fast as possible, control the object motion to follow the NCT, and end it at the origin of the

As shown in Figure 1, the controller output is signal *u*. This signal is used to drive the object.

NCT, at certain error. The difference between the actual error-rate of the object and that of the NCT is denoted as signal *up*, which is the output of the NCT. If the object motion perfectly follows the NCT, the value of signal *up* is zero. Thus, no action is performed by the

with the error-rate, *e* , provided by predetermined

. In principle, the controller

 , which were obtained by a simple open-loop experiment. Thus, the NCT provides information of characteristics of the system, which can be used to design a

,

response with no or small overshoot simultaneously.

and understands.

mass systems.

and *l*

al., 2010) consists of:

compensator.

**2. Basic concept of NCTF controller**

phase-plane *e e* 0, 0 as shown in Figure 2.

Therefore, the controller is called as the NCTF controller.

compares the object motion input, *l*

The input to the controller are error, *e*, and object motion, *l*

 In Figure 2, the object motion is divided into two phases, the reaching phase and the following phase. During the reaching phase, the compensator forces the object motion to reach the NCT as fast as possible. Then, in the following phase, the compensator controls the object motion to follow the NCT and end at the origin. The object motion stops at the origin, which represents the end of the positioning motion. Thus, the NCT governs the positioning response performance.

Fig. 2. NCT and object motion

Practical Control Method for Two-Mass Rotary Point-To-Point Positioning Systems 163

( 1)/2

( 1)/2

( 1)/2 1 ( ) ( ) *M*

*j M i i j M*

*j M i i j M*

where *θ* represents the object displacement in rad, *θ*' the object velocity in rad/s, *θav* the averaged object displacement, *θ'av* the averaged object velocity and *M* is the number of data points of the object responses used in the averaging process. The averaged velocity and displacement responses are used to determine the NCT. Since the main problem of the PTP motion control is to stop an object at a certain position, a deceleration process (curve in area *A* of Figure 4) is used. The phase plane of the NCT has a horizontal axis of error, *e*, and a vertical axis of error-rate, *e* , as shown in Figure 4(b). The horizontal axis shows the error from the desired position. Therefore, the values are taken from the displacement data of the open-loop responses. The error is the displacement data when it started entering deceleration process subtracted by the maximum displacement. The error-rate is the velocity data within the deceleration process. In Figure 4(a), *h* is the maximum velocity which depends on the input step height, and is gradually decreasing until zero. Thus, the pair data of error and error-rate construct the NCT, i.e. the phase-

0 1 2 3 4 5 6 7 8 9 10 -80

Time

From the response, the maximum velocity of the actuator is obtained. Important characteristics of the actuator can be identified from the data within the deceleration range. In the deceleration process, the actuator moves freely from its maximum velocity until it

*av*

*av*

plane diagram of *e* and *e* .

stops.

Fig. 3. Input and actual object response

Input, Displacement, Velocity

( 1)/2 1 ( ) ( ) *M*

(1)

(2)

stepwise loaddisplacement loadvelocity

The electric motor is assumed as the actuator in this discussion. To drive the object to reach the NCT, the actuator needs to reach its maximum velocity. The characteristic of the actuator when it stops from its maximum velocity influences the final accuracy of the PTP positioning operation. Thus, this characteristic is required to design the controller. In order to obtain the required characteristics, conducting a simple open-loop experiment is a simple and practical way.

In summary, the structure of the NCTF control system for PTP positioning systems shown in Figure 1 only works under the following two conditions (Wahyudi et al., 2003):

### **3. Practical controller design**

The NCTF controller consists of NCT, which is constructed based on a simple open-loop experiment of the object, and PI compensator, which is designed based on the obtained NCT. Therefore, the design of NCTF controller can be described by the following steps:

The NCT includes information of the actual object parameters. Therefore, the compensator can be designed by using only the NCT information. Due to the fact that the NCT and the compensator are constructed from a simple open-loop experiment of the object, the exact model including the friction characteristic and the conscious identification task of the object parameters are not required to design the NCTF controller. The controller adjustment is easy and the aims of its control parameters are simple and clear.

### **3.1 NCT constructions**

As mentioned earlier, in order to construct the NCT, a simple open-loop experiment has to be conducted. In the experiment, an actuator of the object is driven with a stepwise input and, load displacement and load velocity responses of the object are measured. Figure 3 shows the stepwise input, load velocity and load displacement responses of the object. In this case, the object vibrates due to its mechanical resonance. In order to eliminate the influence of the vibration on the NCT, the object response must be averaged.

In Figure 4(a), moving average filter is used to get the averaged response because of its simplicity (Oppenheim & Schafer, 1999). The moving average filter operates by averaging a number of points from the object response to produces each point in the averaged response. Mathematically, it can be expressed as follows:

162 PID Controller Design Approaches – Theory, Tuning and Application to Frontier Areas

The electric motor is assumed as the actuator in this discussion. To drive the object to reach the NCT, the actuator needs to reach its maximum velocity. The characteristic of the actuator when it stops from its maximum velocity influences the final accuracy of the PTP positioning operation. Thus, this characteristic is required to design the controller. In order to obtain the required characteristics, conducting a simple open-loop experiment is a simple

In summary, the structure of the NCTF control system for PTP positioning systems shown

The NCTF controller consists of NCT, which is constructed based on a simple open-loop experiment of the object, and PI compensator, which is designed based on the obtained NCT. Therefore, the design of NCTF controller can be described by the following steps:

i. The object is driven with an open loop stepwise input and its load displacement and

ii. Construct the NCT by using the object responses obtained during the deceleration process. Since the NCT is constructed based on the actual responses of the object, it contains nonlinear characteristics such as friction and saturation. The NCTF controller

iii. Design the compensator based on the open-loop responses and the NCT information. In order to consider the real characteristic of the mechanism in designing compensator, a

The NCT includes information of the actual object parameters. Therefore, the compensator can be designed by using only the NCT information. Due to the fact that the NCT and the compensator are constructed from a simple open-loop experiment of the object, the exact model including the friction characteristic and the conscious identification task of the object parameters are not required to design the NCTF controller. The controller adjustment is easy

As mentioned earlier, in order to construct the NCT, a simple open-loop experiment has to be conducted. In the experiment, an actuator of the object is driven with a stepwise input and, load displacement and load velocity responses of the object are measured. Figure 3 shows the stepwise input, load velocity and load displacement responses of the object. In this case, the object vibrates due to its mechanical resonance. In order to eliminate the

In Figure 4(a), moving average filter is used to get the averaged response because of its simplicity (Oppenheim & Schafer, 1999). The moving average filter operates by averaging a number of points from the object response to produces each point in the averaged response.

influence of the vibration on the NCT, the object response must be averaged.

in Figure 1 only works under the following two conditions (Wahyudi et al., 2003):

*r* .

i. A DC or an AC servomotor is used as an actuator of the object.

*r* is constant and 0

is expected to avoid impertinent behaviour by using the NCT.

load velocity responses are measured.

practical stability analysis is determined.

Mathematically, it can be expressed as follows:

and the aims of its control parameters are simple and clear.

and practical way.

ii. The reference input,

**3.1 NCT constructions**

**3. Practical controller design**

$$\theta\_{av}(i) = \frac{1}{M} \sum\_{j=-(M-1)/2}^{(M-1)/2} \theta(i+j) \tag{1}$$

$$\dot{\theta}\_{av}(i) = \frac{1}{M} \sum\_{j=-(M-1)/2}^{(M-1)/2} \dot{\theta}(i+j) \tag{2}$$

where *θ* represents the object displacement in rad, *θ*' the object velocity in rad/s, *θav* the averaged object displacement, *θ'av* the averaged object velocity and *M* is the number of data points of the object responses used in the averaging process. The averaged velocity and displacement responses are used to determine the NCT. Since the main problem of the PTP motion control is to stop an object at a certain position, a deceleration process (curve in area *A* of Figure 4) is used. The phase plane of the NCT has a horizontal axis of error, *e*, and a vertical axis of error-rate, *e* , as shown in Figure 4(b). The horizontal axis shows the error from the desired position. Therefore, the values are taken from the displacement data of the open-loop responses. The error is the displacement data when it started entering deceleration process subtracted by the maximum displacement. The error-rate is the velocity data within the deceleration process. In Figure 4(a), *h* is the maximum velocity which depends on the input step height, and is gradually decreasing until zero. Thus, the pair data of error and error-rate construct the NCT, i.e. the phaseplane diagram of *e* and *e* .

Fig. 3. Input and actual object response

From the response, the maximum velocity of the actuator is obtained. Important characteristics of the actuator can be identified from the data within the deceleration range. In the deceleration process, the actuator moves freely from its maximum velocity until it stops.

Practical Control Method for Two-Mass Rotary Point-To-Point Positioning Systems 165

An exact modeling, including friction and conscious identification processes, is not required in the NCTF controller design. The compensator is derived from the parameter *m* and *h* of the NCT. Since the DC motor is used as the actuator, the simplified object can be presented

2

( ) ( ) () ( ) 2 *f l o*

0 *l l l l*

inclination *m* of the NCT near the origin is related to the simplified object parameter

*de m de*

*l l d d*

*de de*

The NCT inclination near the origin *m* is constructed when the input to the object is

2

2

*l s*

model of Eq. (2) can be described in the state space representation as follow:

*d*

2

*Us ss s s*

where *θl (s)* represents the displacement of the object, *U(s)*, the input to the actuator and *K, ζ, α2* and *ωf* are simplified object parameters. The NCT is determined based on the averaged object response which does not include the vibration. So, it can be assumed that the averaged object response is a response to the stepwise inputs of the averaged object model

where *θl(s)* represents the displacement of the object, *U(s)* is an input to the actuator, and, *K* and *α2* are simplified object parameters. This simplified model is reasonable since it is assumed that a DC or AC servomotor is used to drive the positioning systems. The simplified object

*K Us ss* 2 2

2 2

01 0

*dt K*

Characteristic of the positioning systems near the reference position is very important because system stability and positioning accuracy depend on it characteristic. The

2

2 2

*ff f*

*u*

(3)

(4)

(5)

(6)

*r* is constant in the case of PTP positioning systems. Then,

(1)

(2)

*2*. The

**3.2 Simplified object parameters**

as follows:

NCT in Figure 4(b) shows,

As considering *r e*

zero *u* 0 . By letting 0 *u* in Eq. (3),

 and

Eq. (5) is translated to the following form,

as a following fourth-order system (Dorf, & Bishop, 2008):

*s Gs K*

In the deceleration process, non-linearity characteristics such as friction and saturation of the object have taken its effect. Therefore, the data can be translated into the form that represents the system characteristics including friction and saturation. From the curve in area *A* and *h* in Figure 4(a), the NCT in Figure 4(b) is determined.

a) Input and averaged object response

b) Nominal characteristic trajectory

Fig. 4. Construction of the NCT**.**

There are two important parameters in the NCT as shown in Figure 4(b), the maximum error indicated by *h* and the inclination of the NCT near the origin indicated by *m*. This parameter is related to the dynamics of the object. Therefore, these parameters can be used to design the compensator.

#### **3.2 Simplified object parameters**

164 PID Controller Design Approaches – Theory, Tuning and Application to Frontier Areas

In the deceleration process, non-linearity characteristics such as friction and saturation of the object have taken its effect. Therefore, the data can be translated into the form that represents the system characteristics including friction and saturation. From the curve in

A

stepwise

h

A

averaged displacement averaged velocity

A

h

0 1 2 3 4 5 6 7 8 9 10 -40

Time

a) Input and averaged object response

o

b) Nominal characteristic trajectory

There are two important parameters in the NCT as shown in Figure 4(b), the maximum error indicated by *h* and the inclination of the NCT near the origin indicated by *m*. This parameter is related to the dynamics of the object. Therefore, these parameters can be used

A

Error, e

m

area *A* and *h* in Figure 4(a), the NCT in Figure 4(b) is determined.

h

Fig. 4. Construction of the NCT**.**

to design the compensator.

Error rate, e'

0

U

h

r

20

Input, Displacement, Velocity

40

60

80

An exact modeling, including friction and conscious identification processes, is not required in the NCTF controller design. The compensator is derived from the parameter *m* and *h* of the NCT. Since the DC motor is used as the actuator, the simplified object can be presented as a following fourth-order system (Dorf, & Bishop, 2008):

$$\mathcal{L}G\_o(\mathbf{s}) = \frac{\theta\_l(\mathbf{s})}{\mathcal{U}(\mathbf{s})} = \mathcal{K} \frac{a\_2}{s(s+a\_2)} \frac{o\_f^2}{s^2 + 2\zeta\_f a\_f \mathbf{s} + o\_f^2} \tag{1}$$

where *θl (s)* represents the displacement of the object, *U(s)*, the input to the actuator and *K, ζ, α2* and *ωf* are simplified object parameters. The NCT is determined based on the averaged object response which does not include the vibration. So, it can be assumed that the averaged object response is a response to the stepwise inputs of the averaged object model as follows:

$$\frac{\partial\_l(s)}{\partial L(s)} = K \frac{a\_2}{s(s + a\_2)}\tag{2}$$

where *θl(s)* represents the displacement of the object, *U(s)* is an input to the actuator, and, *K* and *α2* are simplified object parameters. This simplified model is reasonable since it is assumed that a DC or AC servomotor is used to drive the positioning systems. The simplified object model of Eq. (2) can be described in the state space representation as follow:

$$
\frac{d}{dt} \begin{bmatrix} \theta\_l \\ \dot{\theta}\_l \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 0 & -\alpha\_2 \end{bmatrix} \begin{bmatrix} \theta\_l \\ \dot{\theta}\_l \end{bmatrix} + \begin{bmatrix} 0 \\ \alpha\_2 \, K \end{bmatrix} u \tag{3}
$$

Characteristic of the positioning systems near the reference position is very important because system stability and positioning accuracy depend on it characteristic. The inclination *m* of the NCT near the origin is related to the simplified object parameter*2*. The NCT in Figure 4(b) shows,

$$\frac{d\dot{e}}{de} = m\tag{4}$$

The NCT inclination near the origin *m* is constructed when the input to the object is zero *u* 0 . By letting 0 *u* in Eq. (3),

$$\frac{d\dot{\theta}\_l}{d\theta\_l} = -\alpha\_2\tag{5}$$

As considering *r e* and *r* is constant in the case of PTP positioning systems. Then, Eq. (5) is translated to the following form,

$$\frac{d\dot{e}}{de} = -\alpha\_2\tag{6}$$

Hence, from Eqs. (4) and (6), we obtain,

$$a\_2 = -m \tag{7}$$

Practical Control Method for Two-Mass Rotary Point-To-Point Positioning Systems 167

The PI compensator is designed based on the simplified object parameters, which are also related to the NCT, i.e *h* and *m*. Due to its simplicity, the following PI compensator is

> *i c p K Gs K*

According to the signal *up* given by Eq. (11) and the simplified object model in Eq. (2), the NCTF control system in Figure 1 can be represented by block diagram as shown in Figure 6. Using the PI compensator parameters *Kp* and *Ki*, and the simplified object model in Eq. (2), the transfer function near the origin of the closed-loop system for the NCTF controller in

> 2 2 ( ) ( ) ( )

2 2

( ) *p i*

(a)

(b) Fig. 6. NCTF control system near the origin: (a) Linearized block diagram, (b) Linearized

2

2 2

*s s*

*n n n n*

2 ( ) 2

*s G s*

2 2

*s KK s KK*

*KK s KK*

*p i*

*s G s*

*l r*

2

*G s*

*s s*

*s*

(12)

(13)

(14)

(15)

adopted.

Figure 1 is,

where,

closed-loop transfer function

*G(s)* given by Eq. (13) can be rewritten in the following form;

The value of the maximum error rate *h* of the NCT is related to the steady-state velocity due to input actuator *up*. Based on the final value theorem, the steady-state velocity is *Kur*. Then,

$$\mathbf{h} = -\mathbf{K}\,\mathbf{u}\_r\tag{8}$$

Thus, *K* can be expressed as,

$$K = \frac{h}{\mu\_r} \tag{9}$$

The friction characteristic influences the NCT inclination near the origin *m* and the maximum error rate of *h* of the NCT. Therefore, the simplified object model includes the effect of friction when they are determined with Eqs. (8) and (9).

#### **3.3 Compensator design**

The following proportional integral (PI) and notch filter (NF) compensator is proposed for two-mass systems:

$$\mathbf{G}\_c(\mathbf{s}) = \frac{(\mathbf{K}\_p \mathbf{s} + \mathbf{K}\_i)}{s} \left( \frac{\mathbf{K}\_{dc} \left( \mathbf{s}^2 + 2\boldsymbol{\zeta}\_f \boldsymbol{o}\_f \mathbf{s} + \boldsymbol{o}\_f^2 \right)}{\left( \mathbf{s}^2 + 2\boldsymbol{\zeta}\_o \boldsymbol{o}\_o \mathbf{s} + \boldsymbol{o}\_o^2 \right)} \right) \tag{10}$$

where *Kp* is the proportional gain, *Ki* is the integral gain, *Kdc* is the filter gain, *ζf* and *ωf* represent zeros of NF, while *ζo* and *ωo* represent poles of NF. The PI compensator is adopted for its simplicity to force the object motion to reach the NCT as fast as possible and control the object motion to follow the NCT and stop at the origin.

Figure 5 shows the block diagram of the continuous close-loop NCTF control system with the simplified object near the NCT origin where the NCT is linear and has an inclination *α2= m*. The signal *up* near the NCT origin in Figure 5 can be expressed as the following equation:

$$
\mu\_p = \dot{e} + \alpha\_2 \ e = \alpha\_2 \ e - \dot{\theta}\_l \tag{11}
$$

Fig. 5. Simplified NCTF control system

The PI compensator is designed based on the simplified object parameters, which are also related to the NCT, i.e *h* and *m*. Due to its simplicity, the following PI compensator is adopted.

$$\mathcal{G}\_c\left(s\right) = \mathcal{K}\_p + \frac{\mathcal{K}\_i}{s} \tag{12}$$

According to the signal *up* given by Eq. (11) and the simplified object model in Eq. (2), the NCTF control system in Figure 1 can be represented by block diagram as shown in Figure 6. Using the PI compensator parameters *Kp* and *Ki*, and the simplified object model in Eq. (2), the transfer function near the origin of the closed-loop system for the NCTF controller in Figure 1 is,

$$\frac{\theta\_l(s)}{\theta\_r(s)} = \frac{\alpha\_2}{s + \alpha\_2} G(s) \tag{13}$$

where,

166 PID Controller Design Approaches – Theory, Tuning and Application to Frontier Areas

The value of the maximum error rate *h* of the NCT is related to the steady-state velocity due to input actuator *up*. Based on the final value theorem, the steady-state velocity is *Kur*. Then,

> *r h K*

The friction characteristic influences the NCT inclination near the origin *m* and the maximum error rate of *h* of the NCT. Therefore, the simplified object model includes the

The following proportional integral (PI) and notch filter (NF) compensator is proposed for

( ) (2 ) ( ) (2 ) *p i dc f f f*

where *Kp* is the proportional gain, *Ki* is the integral gain, *Kdc* is the filter gain, *ζf* and *ωf* represent zeros of NF, while *ζo* and *ωo* represent poles of NF. The PI compensator is adopted for its simplicity to force the object motion to reach the NCT as fast as possible and control

Figure 5 shows the block diagram of the continuous close-loop NCTF control system with the simplified object near the NCT origin where the NCT is linear and has an inclination *α2= m*. The signal *up* near the NCT origin in Figure 5 can be expressed as the following equation:

> *ue e e p* 2 2

*s s s*

*Ks K K s s*

2 2 2 2

 *l*

*oo o*

(11)

effect of friction when they are determined with Eqs. (8) and (9).

*c*

*G s*

the object motion to follow the NCT and stop at the origin.

Fig. 5. Simplified NCTF control system

2 *m* (7)

*r h Ku* (8)

*u* (9)

(10)

Hence, from Eqs. (4) and (6), we obtain,

Thus, *K* can be expressed as,

**3.3 Compensator design**

two-mass systems:

$$\text{G(s)} = \frac{a\_2 K K\_p s + a\_2 K K\_i}{s^2 + a\_2 K K\_p s + a\_2 K K\_i} \tag{14}$$

(b)

Fig. 6. NCTF control system near the origin: (a) Linearized block diagram, (b) Linearized closed-loop transfer function

*G(s)* given by Eq. (13) can be rewritten in the following form;

$$\text{G(s)} = \frac{2\zeta\rho\_n s + \alpha\_n^2}{s^2 + 2\zeta\rho\_n s + \alpha\_n^2} \tag{15}$$

where,

$$a\_2 \text{KK}\_p = 2\zeta a\_n \tag{16}$$

Practical Control Method for Two-Mass Rotary Point-To-Point Positioning Systems 169

prac 20% of prac 40% of prac 60% of prac 80% of prac

prac=28.8, n=4 prac=15.2, n=8 prac=8, n=14

 prac= 28.8, n= 4 prac= 15.2, n = 8 prac=8, n =14

2 4 6 8 10 12 14

Natural frequency, n

Figure 8 shows the positioning performance response due to 0.5 deg step input. It is shown

In many systems, the mechanical couplings between the motor, load, and sensor are not perfectly rigid, but instead act like springs. Here, the motor response may overshoot or even oscillate at the resonance frequency resulting in longer settling time. The most effective way

According to standard frequency analysis, resonance is characterized by a pair of poles in the complex frequency plane. The imaginary component indicates the resonant frequency,

80% of prac

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.05 0.1 0.15 0.2 0.25 0

time, sec

Fig. 7. Practical stability limit respecting a various margin of safety

that the best performance was in between 60% to 80% margin of safety.

to deal with this torsional resonance is by using an anti resonance notch filter.

0

0

0.1

0.1 0.2 0.3 0.4 0.5

0.2

0.3

load displacem

0.5 deg input

 ent,

0.4

0.5

0.6

0.7

5

10

15

20

Damping ratio,

25

30

35

40 45

$$
\alpha\_2 K K\_i = \alpha\_n^2 \tag{17}
$$

In the case of *G(s) = 1*, Eq. (13) become;

$$\frac{\theta\_l(s)}{\theta\_r(s)} = \frac{\alpha\_2}{s + \alpha\_2} \tag{18}$$

In PTP positioning systems, Eq. (18) can be rewritten in time domain as follow:

$$
\dot{e} + \alpha\_2 e = 0 \tag{19}
$$

From Eq. (11), it can be said that Eq. (19) represents the condition for *up = 0*. Since *up* indicates the difference between the object motion and the NCT, it can be concluded that the object motion follows the NCT perfectly if *G(s) = 1*. Therefore, the PI compensator parameters should be designed based on *n* and so that *G(s) = 1.*

The PI compensator parameters, *Kp* and *Ki* can be expressed as a function of the natural frequency and the damping ratio as follows:

$$K\_p = \frac{2\zeta\alpha\rho\_n}{\alpha\_2 K} = \frac{2\zeta\alpha\rho\_n\mu\_r}{m\hbar} \tag{20}$$

$$K\_i = \frac{\alpha\_n^2}{\alpha\_2 K} = \frac{\alpha\_n^2 \mu\_r}{mh} \tag{21}$$

A higher *ωn* and a larger *ζ* are preferable in the compensator design. However, while choosing *ζ* and *ωn*, the designer must consider the stability of the control system. In continuous system, a linear stability limit can be calculated independently of the actual mechanism characteristic. However, the stability limit is too limited because of neglected Coulomb friction is known to increase the stability of the system, allowing for the use of higher gains than those predicted by a linear analysis (Townsend WT & Kenneth Salisbury J, 1987).

The higher gains are expected to produce a higher positioning performance. Thus the practical stability limit is necessary for selecting the higher gain in a design procedure. The selection of *ωn* and *ζ* are chosen to have 40% of the values of *ζprac*, so that the margin safety of design is 60% (Guilherme Jorge Maeda & Kaiji Sato, 2007). Figure 7 shows the various margin safeties based on practical stability limit, *ζprac*. During the design parameter selection, the designer may be tempted to use large values of *ωn* and *ζ* in order to improve the performance. However, excessively large values of *ωn* will cause the controller to behave as a pure integral controller, which may lead to instability. Therefore, the choice of *ωn* should start with small values and progress to larger one and not the other hand.

168 PID Controller Design Approaches – Theory, Tuning and Application to Frontier Areas

2 2

( ) ( ) *l r s s s*

2 *e e*

*n* and

2 2 2 *n nr p u K K mh*

2 *n nr i u K K mh*

A higher *ωn* and a larger *ζ* are preferable in the compensator design. However, while choosing *ζ* and *ωn*, the designer must consider the stability of the control system. In continuous system, a linear stability limit can be calculated independently of the actual mechanism characteristic. However, the stability limit is too limited because of neglected Coulomb friction is known to increase the stability of the system, allowing for the use of higher gains than those predicted by a linear analysis (Townsend WT & Kenneth Salisbury J,

The higher gains are expected to produce a higher positioning performance. Thus the practical stability limit is necessary for selecting the higher gain in a design procedure. The selection of *ωn* and *ζ* are chosen to have 40% of the values of *ζprac*, so that the margin safety of design is 60% (Guilherme Jorge Maeda & Kaiji Sato, 2007). Figure 7 shows the various margin safeties based on practical stability limit, *ζprac*. During the design parameter selection, the designer may be tempted to use large values of *ωn* and *ζ* in order to improve the performance. However, excessively large values of *ωn* will cause the controller to behave as a pure integral controller, which may lead to instability. Therefore, the choice of *ωn* should

start with small values and progress to larger one and not the other hand.

From Eq. (11), it can be said that Eq. (19) represents the condition for *up = 0*. Since *up* indicates the difference between the object motion and the NCT, it can be concluded that the object motion follows the NCT perfectly if *G(s) = 1*. Therefore, the PI compensator

The PI compensator parameters, *Kp* and *Ki* can be expressed as a function of the natural

2 2

In PTP positioning systems, Eq. (18) can be rewritten in time domain as follow:

 *KKp n*

> 2*KKi n*

2

2 2

so that *G(s) = 1.*

(16)

(17)

(18)

0 (19)

(20)

(21)

where,

1987).

In the case of *G(s) = 1*, Eq. (13) become;

parameters should be designed based on

frequency and the damping ratio as follows:

Fig. 7. Practical stability limit respecting a various margin of safety

Figure 8 shows the positioning performance response due to 0.5 deg step input. It is shown that the best performance was in between 60% to 80% margin of safety.

In many systems, the mechanical couplings between the motor, load, and sensor are not perfectly rigid, but instead act like springs. Here, the motor response may overshoot or even oscillate at the resonance frequency resulting in longer settling time. The most effective way to deal with this torsional resonance is by using an anti resonance notch filter.

According to standard frequency analysis, resonance is characterized by a pair of poles in the complex frequency plane. The imaginary component indicates the resonant frequency,

Practical Control Method for Two-Mass Rotary Point-To-Point Positioning Systems 171

Figure 9(a) shows where the poles and zeros of the system without the controller are located on the s-plane, which gives unstable responses. In Figure 9(b), NF is added to the system, the poles marked *A* are the ones due to the mechanical resonance. These are cancelled by the complex zeros marked by *B*. Although it is assumed that the NF completely cancels the resonance poles, perfect cancellation is not required. As long as the notches zeros are close enough to the original

Root Locus

a) Root locus of the system

Real Axis

Root Locus

A

A

Real Axis

b) Poles and Zeros cancellation of the notch filter

Imaginary Axis

0

100

200

300

B

B

Im aginary Ax is

0

50

100

150

200

poles, they can adequately reduce their effect, thereby improving system response.

Fig. 8. Response of the compensator due to 0.5 deg step input

while the real component determines the damping level. The larger the magnitude of the real part, the greater the damping (William East & Brian Lantz, 2005).

A NF consists of a pair of complex zeros and a pair of complex poles. The purpose of the complex zeros defined by *ωf* and *ζf* is to cancel the resonance poles of the system. The complex poles defined by *ωo* and *ζo*, on the other hand, create an additional resonance and to improve the stability, by increasing gain margin of the plant. If the magnitude of the real value of the poles is large enough, it will result in a well-damped response. The ratio between *ζf* and *ζo* will determine how deep the notch in order to eliminate the resonant frequency of the plant. Parameter *Kdc* will be affected in steady state condition when the transfer function of the NF becomes one.

170 PID Controller Design Approaches – Theory, Tuning and Application to Frontier Areas

60% of prac

prac=21.6, n=4 prac=11.4, n=8 prac=6, n=14

 prac= 21.6, n= 4 prac= 11.4, n = 8 prac= 6, n = 14

> prac=7.2, n=4 prac=3.8, n=8 prac=2, n=14

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.05 0.1 0.15 0.2 0.25 0.3

time, sec

20% of prac

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

time, sec

while the real component determines the damping level. The larger the magnitude of the

A NF consists of a pair of complex zeros and a pair of complex poles. The purpose of the complex zeros defined by *ωf* and *ζf* is to cancel the resonance poles of the system. The complex poles defined by *ωo* and *ζo*, on the other hand, create an additional resonance and to improve the stability, by increasing gain margin of the plant. If the magnitude of the real value of the poles is large enough, it will result in a well-damped response. The ratio between *ζf* and *ζo* will determine how deep the notch in order to eliminate the resonant frequency of the plant. Parameter *Kdc* will be affected in steady state condition when the

0

0 0.1 0.2 0.3 0.4 0.5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

transfer function of the NF becomes one.

Fig. 8. Response of the compensator due to 0.5 deg step input

real part, the greater the damping (William East & Brian Lantz, 2005).

load displacement,

0.5 deg input

0.1

0.2

0.3

load displacem

 ent,

0.5 deg input

0.4

0.5

0.6

0.7

Figure 9(a) shows where the poles and zeros of the system without the controller are located on the s-plane, which gives unstable responses. In Figure 9(b), NF is added to the system, the poles marked *A* are the ones due to the mechanical resonance. These are cancelled by the complex zeros marked by *B*. Although it is assumed that the NF completely cancels the resonance poles, perfect cancellation is not required. As long as the notches zeros are close enough to the original poles, they can adequately reduce their effect, thereby improving system response.

b) Poles and Zeros cancellation of the notch filter

Practical Control Method for Two-Mass Rotary Point-To-Point Positioning Systems 173

Bode Diagram

101

without NF

omega f = 40, omega o = 80 omega f = 40, omega o = 20

104 -360

Bode Diagram

100 101 102 103 104

Frequency (rad/sec)

d) Bode diagram of the compensated system with difference ratio of *ωf* and *ωo.*

Frequency (rad/sec)

102

103

Plant Plant, wo=330 Plant, wo=1000

Phase (deg)

Fig. 9. System response

Phase (deg)

Magnitude (dB)

0 50 10-1 100

Magnitude (dB)

c) Bode diagram of the compensated system with difference ratio of *ζf* and *ζo*

172 PID Controller Design Approaches – Theory, Tuning and Application to Frontier Areas

Bode Diagram

plant plant w ith nf

System: sysL1 Frequency (rad/sec): 281 Magnitude (dB): -76

100 101 102 103 104

Frequency (rad/sec)

Bode Diagram

withput NF

zheta f = 0.1, zheta o = 0.9 zheta f = 0.1, zheta o = 0.02

102

Frequency (rad/sec)

104 -360

c) Bode diagram of the compensated system with difference ratio of *ζf* and *ζo*

103

Phase (deg)

100

101

Magnitude (dB)

0

50

Phase (deg)

Magnitude (dB)

d) Bode diagram of the compensated system with difference ratio of *ωf* and *ωo.* Fig. 9. System response

Practical Control Method for Two-Mass Rotary Point-To-Point Positioning Systems 175

, *m l*

where *Vm* is input voltage, *Eemf* is electromagnetic field, *Lm* is motor inductance, *Rm* is motor resistance and *im* is current.  SI units are applicable for all

The improved NCTF controller is applied to an experimental two-mass rotary positioning systems as shown in Figure 11. It is a two-mass rotary PTP positioning system, which is used in various sequences in industrial applications. It consists of a direct current (DC) motor with encoder, a driver, a low stiffness shaft, load inertia and a rotary encoder. The reading of the encoder is used as a feedback to the controller. The motor is driven by control signals, which are sent to a DC motor driver powered by a power supply. This basic configuration is considered as a normal condition of the system. To measure a load position as a feedback signal of the system, a rotary encoder with 10000 pulses per revolution (ppr) made by Nemicon is used. The object consists of a direct current motor, a driver, inertia load

*d d dt dt*

motor, flexible shaft, coupling and inertia load mass.

Fig. 11. Experimental two-mass rotary positioning systems

notations.

*m l*

(25)

### **4. Modeling and experimental setup of the systems**

Figure 10 shows the simplified diagram of a two-mass rotary positioning system. It consists of mechanical and electromechanical components. The electromechanical (electrical and mechanical) component is the DC motor, which performs the conversion of electrical energy to mechanical energy, and the rest are mechanical components. Two masses, having the moments of inertia, *Jm* and *Jl*, are coupled by low stiffness shaft which has the torsion stiffness, *Ks* and a damping. For the case that the system can be accurately modeled without considering the major nonlinear effects by the speed dependent friction, dead time and time delay, a linear model for two-mass mechanical system can be obtain using the conventional torque balance rule (Robert L.Woods & Kent L.Lawrence, 1997):

Fig. 10. Schematic diagram of two-mass system

The electrical part of the DC motor is derived by using Kirchoff Voltage Law (KCL),

$$N\_m(t) - E\_{emf}\left(t\right) = L\_m \frac{di\_m(t)}{dt} + R\_m i\_m\left(t\right) \tag{22}$$

Next, modeling on the mechanical parts of the system is done by applying Newton's second law of motion to the motor shaft,

$$J\_I \left(\frac{do\eta}{dt}\right) = T\_s - T\_I - B\_l o\eta \tag{23}$$

$$T\_s = K\_s(\theta\_m - \theta\_l) + B\_s(\alpha\_m - \alpha\_l) \tag{24}$$

with,

174 PID Controller Design Approaches – Theory, Tuning and Application to Frontier Areas

Figure 10 shows the simplified diagram of a two-mass rotary positioning system. It consists of mechanical and electromechanical components. The electromechanical (electrical and mechanical) component is the DC motor, which performs the conversion of electrical energy to mechanical energy, and the rest are mechanical components. Two masses, having the moments of inertia, *Jm* and *Jl*, are coupled by low stiffness shaft which has the torsion stiffness, *Ks* and a damping. For the case that the system can be accurately modeled without considering the major nonlinear effects by the speed dependent friction, dead time and time delay, a linear model for two-mass mechanical system can be obtain using the conventional

**4. Modeling and experimental setup of the systems**

torque balance rule (Robert L.Woods & Kent L.Lawrence, 1997):

Fig. 10. Schematic diagram of two-mass system

law of motion to the motor shaft,

with,

The electrical part of the DC motor is derived by using Kirchoff Voltage Law (KCL),

*l*

 *m m emf m m m di t V t E t L Ri t*

Next, modeling on the mechanical parts of the system is done by applying Newton's second

*l s l ll d J TTB dt*

( )( ) *TK B s sm l s m l*

*dt*

(22)

(23)

(24)

$$\frac{d\theta\_m}{dt} = \alpha\_m \prime \frac{d\theta\_l}{dt} = \alpha\_l \tag{25}$$

where *Vm* is input voltage, *Eemf* is electromagnetic field, *Lm* is motor inductance, *Rm* is motor resistance and *im* is current.  SI units are applicable for all notations.

The improved NCTF controller is applied to an experimental two-mass rotary positioning systems as shown in Figure 11. It is a two-mass rotary PTP positioning system, which is used in various sequences in industrial applications. It consists of a direct current (DC) motor with encoder, a driver, a low stiffness shaft, load inertia and a rotary encoder. The reading of the encoder is used as a feedback to the controller. The motor is driven by control signals, which are sent to a DC motor driver powered by a power supply. This basic configuration is considered as a normal condition of the system. To measure a load position as a feedback signal of the system, a rotary encoder with 10000 pulses per revolution (ppr) made by Nemicon is used. The object consists of a direct current motor, a driver, inertia load motor, flexible shaft, coupling and inertia load mass.

Fig. 11. Experimental two-mass rotary positioning systems

Practical Control Method for Two-Mass Rotary Point-To-Point Positioning Systems 177

realizing high precision positioning systems for two-mass positioning systems compared with conventional PID in term of controller performances. In this section, the continuous time, a linearized mathematical model of the lab scale two-mass rotary system is developed. In order to identify and estimate the parameters, which constitute the two-mass rotary system transfer function, system identification is utilized. Table 1 shows the parameter

> Parameter Value Unit Motor inertia, *Jm* 17.16e-6 Kgm2 Inertia load, *Jl* 24.17e-6 Kgm2 Stiffness, *Kc* 0.039 Nm/rad Motor resistance, *R* 5.5 Ω Motor inductance, *L* 0.85e-3 H Torque constant of the motor, *Kt* 0.041 Nm/A Motor voltage constant, *Kb* 0.041 Vs/rad Frictional torque, *τt* 0.0027 Nm Motor viscous friction, *Bm* 8.35e-6 Nms/rad Load viscous friction, *Bl* 8.35e-6 Nms/rad

Conventional PID controllers were designed based on a Ziegler Nichols (Z-N) and Tyres Luyben (T-L) closed-loop method, using proportional control only. The proportional gain is increased until a sustained oscillation output occur which giving the sustained oscillation, *Ku*, and the oscillation period, *Tu* are recorded. The tuning parameter can be found in Table 2

Controller Kp Ti Td

The significance of this research lies in the fact that a simple and easy controller can be designed for high precision positioning system which is very practical. According to Figure 4, the inclination, *m* and maximum error rate, *h* of the NCT are 81.169 and 61.6 respectively. Based on the practical stability limit from Figure 7 and its responses on Figure 8, design parameters for *ζ* and *ωn* are chosen as 9.5 and 10.5 in order to evaluate the performance of NCTF controller. The object will vibrate due to a mechanical resonance and its vibrating

From Figure 9, parameters for nominator of NF (*ζf* and *ωf*) are selected as 0.7 and 40 Hz while denominator (*ζo* and *ωo*) of NF is selected as 0.9 and 100 Hz. Selection of NF parameters are also based on Ruith-Huwirt stability criterion. In order to obtain an always

stable continuous closed-loop system, the following constraint needs to be satisfied.

Ku/1.7 Tu/2 Tu/8 Ku/2.2 Tux2.2 Tu/6.3

values of the model for each mechanism.

Table 1. Model parameters

(Astrom K. & Hagglund T, 1995).

frequency is 40 Hz.

Table 2. Controller tuning rule parameters

**5. Experimental results and discussions**

Ziegler Nichlos Tyres Luyben

When a load mass has a heavy movable mass, the flexible shaft between the motor mass and load mass work as a spring elements which lead to a vibration. Figure 12 shows step responses to 0.5 deg step input to the experimental rotary system in a normal object condition without the controller. The object will vibrate due to a mechanical resonance and its vibrating frequency is 40 Hz.

Fig. 12. Vibration responses due to 0.5 deg step response

The detailed block diagram of the model of the object used only for making simulations is shown in Figure 13. In the detailed model of the object, friction and saturation are taken into consideration (De Wit C et al, 1993). The significance of this research lies in the fact that a simple and practical controller can be designed for high precision positioning systems. By improving the NCTF controller, the controller will be more reliable and practical for

Fig. 13. Dynamic model of the object

realizing high precision positioning systems for two-mass positioning systems compared with conventional PID in term of controller performances. In this section, the continuous time, a linearized mathematical model of the lab scale two-mass rotary system is developed. In order to identify and estimate the parameters, which constitute the two-mass rotary system transfer function, system identification is utilized. Table 1 shows the parameter values of the model for each mechanism.

Table 1. Model parameters

176 PID Controller Design Approaches – Theory, Tuning and Application to Frontier Areas

When a load mass has a heavy movable mass, the flexible shaft between the motor mass and load mass work as a spring elements which lead to a vibration. Figure 12 shows step responses to 0.5 deg step input to the experimental rotary system in a normal object condition without the controller. The object will vibrate due to a mechanical resonance and

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

time, sec

time, sec

The detailed block diagram of the model of the object used only for making simulations is shown in Figure 13. In the detailed model of the object, friction and saturation are taken into consideration (De Wit C et al, 1993). The significance of this research lies in the fact that a simple and practical controller can be designed for high precision positioning systems. By improving the NCTF controller, the controller will be more reliable and practical for

its vibrating frequency is 40 Hz.

Fig. 13. Dynamic model of the object

Fig. 12. Vibration responses due to 0.5 deg step response

0

20

40

Control signal,

load displacement,

0.5 deg

V

### **5. Experimental results and discussions**

Conventional PID controllers were designed based on a Ziegler Nichols (Z-N) and Tyres Luyben (T-L) closed-loop method, using proportional control only. The proportional gain is increased until a sustained oscillation output occur which giving the sustained oscillation, *Ku*, and the oscillation period, *Tu* are recorded. The tuning parameter can be found in Table 2 (Astrom K. & Hagglund T, 1995).

Table 2. Controller tuning rule parameters

The significance of this research lies in the fact that a simple and easy controller can be designed for high precision positioning system which is very practical. According to Figure 4, the inclination, *m* and maximum error rate, *h* of the NCT are 81.169 and 61.6 respectively. Based on the practical stability limit from Figure 7 and its responses on Figure 8, design parameters for *ζ* and *ωn* are chosen as 9.5 and 10.5 in order to evaluate the performance of NCTF controller. The object will vibrate due to a mechanical resonance and its vibrating frequency is 40 Hz.

From Figure 9, parameters for nominator of NF (*ζf* and *ωf*) are selected as 0.7 and 40 Hz while denominator (*ζo* and *ωo*) of NF is selected as 0.9 and 100 Hz. Selection of NF parameters are also based on Ruith-Huwirt stability criterion.  Experimental positioning performance comparison

Table 4. Object parameter comparison

Jl 0.5 deg Z-N

2 x Jl 0.5 deg Z-N

5 x Jl 0.5 deg Z-N

10 x Jl 0.5 deg

Input

1 deg Input

5 deg input

1 deg Z-N

5 deg Z-N

1 deg Z-N

5 deg Z-N

1 deg Z-N

5 deg Z-N

In order to evaluate the effectiveness of improved NCTF controller designed for a two-mass system, the controller is compared with PID controller, which are tuned using Ziegler-Nichols and Tyres-Luyben methods. The PI compensator parameters are calculated from the simplified object parameters (*K* and *α2*) and the design parameters (*ωn* and *ζ*). Table 3 shows the parameters of the compensator of the improved NCTF controller and PID controllers. In order to evaluate the robustness of the improved NCTF control system, the experiments were conducted in two conditions: with normal load and increasing the load inertia as shown in Figure. 14. Normal object is the two-mass experimental rotary positioning with nominal object parameter. Table 4 shows different object parameters for nominal and increased load inertia.

**Normal load**

(b)

Fig. 14. Nominal and increased object inertia: (a) Nominal object, (b) Increased object inertia

Table 3. Controller parameters

Table 4. Object parameter comparison

178 PID Controller Design Approaches – Theory, Tuning and Application to Frontier Areas

2 2 *o o*

In order to evaluate the effectiveness of improved NCTF controller designed for a two-mass system, the controller is compared with PID controller, which are tuned using Ziegler-Nichols and Tyres-Luyben methods. The PI compensator parameters are calculated from the simplified object parameters (*K* and *α2*) and the design parameters (*ωn* and *ζ*). Table 3 shows the parameters of the compensator of the improved NCTF controller and PID controllers. In order to evaluate the robustness of the improved NCTF control system, the experiments were conducted in two conditions: with normal load and increasing the load inertia as shown in Figure. 14. Normal object is the two-mass experimental rotary positioning with nominal object parameter. Table 4 shows different object parameters for nominal and

(a)

(b) Fig. 14.

NCTF Z-N PID T-L PID

Table 3.   Step response comparison, nominal object condition

load displaement,

5 deg

0.2 0.4 0.6

load displacement,

load displacement,

0.5 deg

1 deg

Figure 15 shows the improved NCTF controller for a two-mass rotary positioning system in nominal object and increased object inertia. Figure 16 shows step responses to 0.5, 1 and 5 deg step input when the improved NCTF controller are used to control a nominal object. The positioning performance is evaluated based on percentage overshoot, settling time and positioning accuracy. Figure 17 shows step responses to 0.5, 1 and 5 deg step input to control the increased object with load inertia of twice the nominal object (2 x Jl). Figure 18 and Figure 19 show step responses to 0.5, 1 and 5 deg step input to control the increased object inertia with five and ten times of the nominal object one, i.e 5 x Jl and 10 x Jl. Table 5 presents the experimental positioning performance. An average of 20 similar experiments was conducted.

With nominal object, the improved NCTF controller gives the smallest percentage of overshoot, fastest settling time and a better positioning accuracy than both PID controllers. Unlike the improved NCTF controller for the two-mass rotary system, the PID controllers results in a vibrating response which is 40 Hz. This vibration caused by mechanical resonance of the system. For increased object inertia, the improved NCTF controller demonstrates good positioning accuracy and stable responses. On the other hand, PID based on Ziegler-Nichols rule resulted in unstable response, faster than Tyres-Luyben method for increased object inertia. Therefore, improved NCTF controller is much more robust to inertia variations compared with the two PID controllers. Moreover, for the experimental results, the improved NCTF controller resulted in positioning accuracy near the sensor resolution.

Fig. 15. Step response of improved NCTF controller

180 PID Controller Design Approaches – Theory, Tuning and Application to Frontier Areas

Figure 15 shows the improved NCTF controller for a two-mass rotary positioning system in nominal object and increased object inertia. Figure 16 shows step responses to 0.5, 1 and 5 deg step input when the improved NCTF controller are used to control a nominal object. The positioning performance is evaluated based on percentage overshoot, settling time and positioning accuracy. Figure 17 shows step responses to 0.5, 1 and 5 deg step input to control the increased object with load inertia of twice the nominal object (2 x Jl). Figure 18 and Figure 19 show step responses to 0.5, 1 and 5 deg step input to control the increased object inertia with five and ten times of the nominal object one, i.e 5 x Jl and 10 x Jl. Table 5 presents the experimental positioning performance. An average of 20 similar experiments

With nominal object, the improved NCTF controller gives the smallest percentage of overshoot, fastest settling time and a better positioning accuracy than both PID controllers. Unlike the improved NCTF controller for the two-mass rotary system, the PID controllers results in a vibrating response which is 40 Hz. This vibration caused by mechanical resonance of the system. For increased object inertia, the improved NCTF controller demonstrates good positioning accuracy and stable responses. On the other hand, PID based on Ziegler-Nichols rule resulted in unstable response, faster than Tyres-Luyben method for increased object inertia. Therefore, improved NCTF controller is much more robust to inertia variations compared with the two PID controllers.  Step response of improved NCTF controller

Fig. 16. Step response comparison, nominal object condition

Fig. 17. Step response comparison, increased object inertia (2 x Jl)

Practical Control Method for Two-Mass Rotary Point-To-Point Positioning Systems 183

This paper has discussed and described the improved practical design of the NCTF controller for two-mass rotary positioning systems. First, the NCT determination was discussed. In order to eliminate the influence of the vibration on the NCT, the vibrating responses are averaged using simple moving averaged filter. Then, the PI with NF compensator is designed based on the NCT information and the object responses. The PI is adopted for its simplicity to force the object motion to reach the NCT as fast as possible and control the object motion to follow the NCT and stop at the origin. The NF is adopted to

Through experiments using two-mass rotary experimental positioning systems, the effectiveness and robustness of the improved NCTF controller was evaluated. The experimental results confirmed that the improved NCTF controller for a two-mass system has better performance in term of percentage of overshoot, settling time and positioning accuracy than PID controllers. The improved NCTF controller for two-mass system resulted in an insignificant vibration. Moreover, experimental resulted prove that the improved NCTF controller was much more robust to inertia variations than PID controllers designed based on Ziegler-Nichols and Tyres-Luyben methods. Therefore, the improved NCTF controller for two-mass rotary positioning systems that includes the NF will have a better performance than the PID controllers with NF. Applications of the NCTF concept to multi-

This research is supported by Ministry of Higher Education Malaysia under vot 78606 and Malaysia-Japan International Institute of Technology (MJIIT), Universiti Teknologi Malaysia

Armstrong-Helouvry B., Dupont P, and De Wit C. (1994). A survey of models, analysis tools

Astrom K. & Hagglund T, PID controllers: Theory, design and tuning. *Instrument Society of*

Guilherme Jorge Maeda & Kaiji Sato, Practical control method for ultra-precision

Mohd Fitri Mohd Yakub, Wahyudi Martono and Rini Akmeliawati, "Performance

Crowder R.M. (1998). *Electric drives and theier controls*, Oxford: Oxford University Press De Wit C., Olsson H., Astrom K.J & Lischinssky, Dynamic friction models and control

Dorf, R.C., & Bishop, R.H. (2008). *Modern Control System*. Pearson Prentice Hall.

*Applications (ISIEA2010)* 3–5 October 2010, Penang, pp. 550-555.

and compensation method for the control of machines with friction. *Automatica*,

design, *Proceedings American Control Conference,* San Francisco, USA, 1993, pp. 1920-

positioning using a ballscrew mechanism. *Precision Engineering Journal,* Vol.

Evaluation of Improved Practical Control Method for Two-Mass PTP Positioning Systems", *Proceedings of the 2010 IEEE Symposium on Industrial Electronics &*

eliminate the vibration frequency caused by mechanical resonance of the system.

mass positioning systems can be done for further study.

International Campus (UTM IC), Kuala Lumpur, Malaysia.

Vol. 30, No. 7, pp. 1083-1138.

32(2008)309-318, November 2007.

*America*, 1995.

1926.

**6. Conclusion**

**7. Acknowledgment**

**8. References**

Fig. 18. Step response comparison, increased object inertia (5 x Jl)

Fig. 19. Step response comparison, increased object inertia (10 x Jl)

### **6.  Step response comparison, increased object inertia (5 x Jl)

Fig. 19. Step response comparison, increased object inertia (10 x Jl)

load displaement,

load displacement,

load displacement,

0.5 deg

1 deg

5 deg

10

0.5 1 1.5 2

0.5

1

This paper has discussed and described the improved practical design of the NCTF controller for two-mass rotary positioning systems. First, the NCT determination was discussed. In order to eliminate the influence of the vibration on the NCT, the vibrating responses are averaged using simple moving averaged filter. Then, the PI with NF compensator is designed based on the NCT information and the object responses. The PI is adopted for its simplicity to force the object motion to reach the NCT as fast as possible and control the object motion to follow the NCT and stop at the origin. The NF is adopted to eliminate the vibration frequency caused by mechanical resonance of the system.

Through experiments using two-mass rotary experimental positioning systems, the effectiveness and robustness of the improved NCTF controller was evaluated. The experimental results confirmed that the improved NCTF controller for a two-mass system has better performance in term of percentage of overshoot, settling time and positioning accuracy than PID controllers. The improved NCTF controller for two-mass system resulted in an insignificant vibration. Moreover, experimental resulted prove that the improved NCTF controller was much more robust to inertia variations than PID controllers designed based on Ziegler-Nichols and Tyres-Luyben methods. Therefore, the improved NCTF controller for two-mass rotary positioning systems that includes the NF will have a better performance than the PID controllers with NF. Applications of the NCTF concept to multimass positioning systems can be done for further study.

### **7. Acknowledgment**

This research is supported by Ministry of Higher Education Malaysia under vot 78606 and Malaysia-Japan International Institute of Technology (MJIIT), Universiti Teknologi Malaysia International Campus (UTM IC), Kuala Lumpur, Malaysia.

### **8. References**

Dorf, R.C., & Bishop, R.H. (2008). *Modern Control System*. Pearson Prentice Hall.

**Part 3**

**Robust PID Controller Design**

**Part 3**

**Robust PID Controller Design**

184 PID Controller Design Approaches – Theory, Tuning and Application to Frontier Areas

Oppenheim A.V. & Schafer R.W, Discrete Time Signal Processing. *Englewood Cliffs*, Prentice

Robert L.Woods and Kent L.Lawrence, Modelling and Simulation of Dynamic Systems,

Wahyudi. (2002). *New practical control of PTP positioning systems*. Ph.D. Thesis. Tokyo

Wahyudi, Sato, K., & Shimokohbe, A. (2003) Characteristics of practical control for point-to-

Yonezawa H., Hirata H. and Sasai H. (1990). Positioning table with high accuracy and

point positioning systems: effect of design parameters and actuator saturation.

Slotine, J. J. & Li, W. P. (1991). Applied Nonlinear Control, Prentice-Hall, Inc., New Jersey Townsend WT and Kenneth Salisbury J, The effect of coulomb friction and stiction on force

control*, IEEE Int Conf Robot Automat* 1987;4:883-9.

William East & Brian Lantz, Notch Filter Design, August 29, 2005.

Hall, 1999.

Prentice Hall Inc, 1997.

Institute of Technology.

*Precision Engineering*, 27(2), 157-169.

speed. *Annal CIRP*. Vol. 39, pp. 433-436.

**9**

**Performance Robustness**

Donghai Li1, Mingda Li2, Feng Xu1,

*Tsinghua University, Beijing*

*P. R. China*

**Criterion of PID Controllers**

Min Zhang2, Weijie Wang3 and Jing Wang2

*University of Science and Technology Beijing, Beijing 3Beijing ABB Bailey Engineering Company, Beijing*

*2Metallurgical Engineering Research Institute,*

*1State Key Lab of Power Systems Dept of Thermal Engineering,*

PID is one of the earliest and most popular controllers. The improved PID and classical PID have been applied in various kinds of industry control fields, as its tuning methods are developing. After the PID controller was first proposed by Norm Minorsky in 1922, the various PID tuning methods were developing and the advanced and intelligent controls were proposed. In the past few decades, Z-N method which is for first-order-plus-timedelay model was proposed by Ziegler and Nichols (Ziegler & Nivhols, 1943), CHR method about generalized passive systems was proposed by Chien, Hrones and Reswick (Chien et al., 1952), and so many tuning methods were developed such as pole assignment and zeropole elimination method by Wittenmark and Astrom, internal model control (IMC) by Chien (Chien & Fruehauf, 1990). The gain and phase margin (GPM) method was proposed by Åström and Hägglund (Åström & Hägglund, 1984), the tuning formulae were simplified by

In classical feedback control system design, the PID controller was designed according to

1. The system is time variant and uncertain because of the complex dynamic of industrial

So there are two inevitable problems in control system designing. One is how to design robust PID controller to make the closed-loop system stable when the parameters are uncertain in a certain range. The other is the performance robustness which must be considered seriously when designing PID controllers. The performance robustness is that

2. The process is inevitably affected by environment and the uncertainty is introduced.

4. The error exists with the dynamic parameter measurement and identification.

precise model. But the actual industrial models has some features as follows:

**1. Introduction**

W K Ho (Ho et al., 1995).

equipment.

3. The dynamic will drift during operation.

## **Performance Robustness Criterion of PID Controllers**

Donghai Li1, Mingda Li2, Feng Xu1, Min Zhang2, Weijie Wang3 and Jing Wang2 *1State Key Lab of Power Systems Dept of Thermal Engineering, Tsinghua University, Beijing 2Metallurgical Engineering Research Institute, University of Science and Technology Beijing, Beijing 3Beijing ABB Bailey Engineering Company, Beijing P. R. China*

### **1. Introduction**

PID is one of the earliest and most popular controllers. The improved PID and classical PID have been applied in various kinds of industry control fields, as its tuning methods are developing. After the PID controller was first proposed by Norm Minorsky in 1922, the various PID tuning methods were developing and the advanced and intelligent controls were proposed. In the past few decades, Z-N method which is for first-order-plus-timedelay model was proposed by Ziegler and Nichols (Ziegler & Nivhols, 1943), CHR method about generalized passive systems was proposed by Chien, Hrones and Reswick (Chien et al., 1952), and so many tuning methods were developed such as pole assignment and zeropole elimination method by Wittenmark and Astrom, internal model control (IMC) by Chien (Chien & Fruehauf, 1990). The gain and phase margin (GPM) method was proposed by Åström and Hägglund (Åström & Hägglund, 1984), the tuning formulae were simplified by W K Ho (Ho et al., 1995).

In classical feedback control system design, the PID controller was designed according to precise model. But the actual industrial models has some features as follows:

So there are two inevitable problems in control system designing. One is how to design robust PID controller to make the closed-loop system stable when the parameters are uncertain in a certain range. The other is the performance robustness which must be considered seriously when designing PID controllers. The performance robustness is that

Performance Robustness Criterion of PID Controllers 189

In power system, Monte-Carlo method was applied in reliability assessment of generation and transmission system, the software was design and the application was successful (Ding

> ( ) ( ) ( ) *N s Ls Gs e D s*

PID G r y u

> 1 ( ) (1 ) ( ) *p d i us K Tses T s*

() ( )() *i p d K us K Kses s*

The parameters Kp, Ki, Kd are positive number, and all of the PID controllers compose a

The PID tuning methods are used on the nominal controlled models, and the closed-loop

performance index. Because the controlled models are a group of transfer function, the

Obviously, it is a collection of two-dimension vector an area in plane plot. The distance between this area and origin reflects the quality of control system, and the size of this area shows the dispersion of performance index, that is the performance robustness of control

In this system, N(s) and D(s) are coprime polynomials, and D(s)'s order is greater than or equal N(s)'s order, L is rational number greater than or equal to zero. The controlled model is some uncertain, and the parameters of N(s) and D(s) are variable in bounded region. So, the model is a group of transfer function denoted by {G(s)}. The control system is shown in figure 1.

(1)

(2)

and adjustment time Ts are considered as dynamic

%,*TS* (3)

**2.2 Performance robustness criterion based on Monte-Carlo method**

& Zhang, 2000).

Consider the SISO system as follows:

Fig. 1. Control system structure

The controller is PID controller:

controller group denoted by {PID}.

systems are obtained. The overshoot %

dynamic performance index is a collection, denoted by:

or

system.

e

when the parameters of model change in a certain interval, the dynamic performances of system are still in desired range.

This chapter discusses the new idea mentioned previous – Performance Robustness. Based on the famous Monte-Carlo method, the performance robustness criterion is proposed. The performance robustness criterion could give us a new view to study the important issue that how the PID controller performs while the parameters of model are uncertain. Not only the stability, but also the time-domain specifications such as overshoot and adjusting time, and the frequency-domain specifications such as gain margin and phase margin can be obviously clear on the specification figures.

The structure of this chapter is as follows. A brief history of Monte-Carlo method is given in section 2. The origin, development and latest research of Monte-Carlo method are introduced. The performance robustness criterion is discussed in detail. This section also contains several formulas to explain the proposed criterion. In section 3, the performance robustness criterion is applied on typical PID control systems comparison, the detailed comparisons between DDE method and IMC method, and between DDE method and GPM method. Finally, section 4 gives out a conclusion.

### **2. Monte-Carlo method in performance robustness criterion**

### **2.1 A brief history of Monte-Carlo method**

Monte-Carlo method is also called random sampling technology or statistical testing method. In 1946, a physicist named Von Neumann simulated neutron chain reaction on computer by random sampling method called Monte-Carlo method. This method is based on the probability statistics theory and the random sampling technology. With the further development of computer, the vast random sampling test became viable. So it was consciously, widely and systematic used in mathematical and physical problems. The Monte-Carlo method is also a new important branch of computational mathematics.

In the late 20th century, Monte-Carlo method is closely linked the computational physics, computational statistical probability, interface science of computer science and statistics, and other boundary discipline. In addition, the Monte-Carlo method also plays a role for the development of computer science. In order to show the new performance evaluation method of mainframe which has multi-program, variable word length, random access and time-shared system, the performance of developed computer was simulated and analysed on the other computer. The relationship could be clear via the study on different target.

Large numbers of practical problems on nuclear science, vacuum technology, geological science, medical statistics, stochastic service system, system simulation and reliability were solved by Monte-Carlo method, and the theory and application results have gained. It was used in simulation of continuous media heat transfer and flow (Cui et al., 2000), fluid theory and petroleum exploration and development (Lu & Li, 1999). Monte-Carlo method was combined with heat network method to solve the temperature field of spacecraft, and the steady-state temperature field of satellite platform thermal design was calculated and analysed (Sun et al., 2001). In chemical industry, Yuan calculated the stability of heat exchanger with Monte-Carlo method, and it was used in selection and design (Yuan, 1999). In power system, Monte-Carlo method was applied in reliability assessment of generation and transmission system, the software was design and the application was successful (Ding & Zhang, 2000).

#### **2.2 Performance robustness criterion based on Monte-Carlo method**

Consider the SISO system as follows:

$$\mathbf{G}(\mathbf{s}) = \frac{\mathbf{N}(\mathbf{s})}{D(\mathbf{s})} e^{-Ls} \tag{1}$$

In this system, N(s) and D(s) are coprime polynomials, and D(s)'s order is greater than or equal N(s)'s order, L is rational number greater than or equal to zero. The controlled model is some uncertain, and the parameters of N(s) and D(s) are variable in bounded region. So, the model is a group of transfer function denoted by {G(s)}. The control system is shown in figure 1.

Fig. 1. Control system structure

The controller is PID controller:

$$
\mu(s) = K\_p \left( 1 + \frac{1}{T\_{\bar{i}}s} + T\_d s \right) e(s) \tag{2}
$$

or

188 PID Controller Design Approaches – Theory, Tuning and Application to Frontier Areas

when the parameters of model change in a certain interval, the dynamic performances of

This chapter discusses the new idea mentioned previous – Performance Robustness. Based on the famous Monte-Carlo method, the performance robustness criterion is proposed. The performance robustness criterion could give us a new view to study the important issue that how the PID controller performs while the parameters of model are uncertain. Not only the stability, but also the time-domain specifications such as overshoot and adjusting time, and the frequency-domain specifications such as gain margin and phase margin can be