**4. Robust decentralized PID controller design in the frequency domain**

This section deals with an original frequency domain robust decentralized controller design methodology applicable for uncertain systems described by a set of transfer function matrices. The design methodology is based on the Equivalent Subsystems Method (ESM) - a frequency domain decentralized controller design technique to guarantee stability and specified performance of multivariable systems and is applicable for both continuous- and discrete-time controller designs (Kozáková et al., 2009). In contrast to the two stage robust decentralized controller design method based on the M-structure stability conditions (Kozáková & Veselý, 2009), the recent innovation (Kozáková et al., 2011) consists in that robust stability conditions are directly integrated into the ESM, thus providing a one-step (direct) robust decentralized controller design for robust stability and plant-wide performance.

#### **4.1 Preliminaries and problem formulation**

Consider a MIMO system described by a transfer function matrix ( ) *m m Gs R* and a controller ( ) *m m Rs R* in the standard feedback configuration according to Fig. 5,

Fig. 5. Standard feedback configuration

where w, u, y, e, d are respectively vectors of reference, control, output, control error and disturbance of compatible dimensions. Necessary and sufficient conditions for closed-loop stability are given by the Generalized Nyquist Stability Theorem applied to the closed-loop characteristic polynomial

$$\det F(\mathbf{s}) = \det[I + Q(\mathbf{s})] \tag{40}$$

where *Qs GsRs* () () () *m m R* is the open-loop transfer function matrix.

Characteristic functions of *Q s*( ) are the set of m algebraic functions ( ), 1,..., *i qs i m* defined as follows:

$$\det[q\_i(\mathbf{s})I\_m - \mathbb{Q}(\mathbf{s})] = 0 \quad \text{i} = 1, \ldots, m \tag{41}$$

Characteristic loci (CL) are the set of loci in the complex plane traced out by the characteristic functions of Q(s), *s j* .

*Theorem 4.1* (Generalized Nyquist Stability Theorem)

The closed-loop system in Fig. 1 is stable if and only if

Robust Decentralized PID Controller Design 151

*u*

Fig. 6. Standard feedback configuration with additive perturbation (left) recast into the

According to the general robust stability condition (Skogestad & Postlethwaite, 2009), if both

For individual uncertainty forms ( ) ( ), , *M k k s M s k ao* the corresponding matrices ( ) *Mk s* are given by (49) and (50), respectively (disregarding negative signs which do not affect resulting robustness condition). The nominal model 0 *G s*( ) is usually obtained as a

Consider an uncertain system that consists of m subsystems and is given as a set of N transfer function matrices obtained in N working points of plant operation. Let the uncertain system be described by a nominal model 0 *G s*( ) and any unstructured uncertainty

is to be designed to guarantee stability over the whole operating range of the plant specified by (46) or (47) (robust stability) and a specified plant-wide performance (nominal

 [ ( )] 1 , *M j*

( )*s* : max

1 0 ( ) ( ) ( )[ ( ) ( )] ( ) ( ) *M a a a s sRs I G sRs sM s* (49)

1 0 0 ( ) ( ) ( ) ( )[ ( ) ( )] ( ) ( ) *M o o o s sG sRs I G sRs sM s* (50)

0() () () *Gs Gs G s d m* (51)

0 *G s Gs Gs m d* () () () (53)

( ) { ( )} , det ( ) 0 *G s diag G s G s d i mm d* (52)

( ) { ( )} *R s diag R s i mm* det ( ) 0 *R s* (54)

() 1 if and only if

*u*

(48)

( )*s* are stable, the *M*

*M(s)*

*(s)*

*y*

*w e y*

a

*R(s) G0(s)*

system in Fig. 2 is stable for all perturbations

*y*

the nominal closed-loop system M(s) and the perturbations

form (46), (47). Consider the following splitting of 0 *G s*( ) **:**

max

*(s)*

structure (right)

model of mean parameter values.

**4.1.1 Problem formulation**

A decentralized controller

performance).

where

*M*

$$\text{a. } \det F(s) \neq 0 \qquad \forall s$$

$$\text{b. } N[0, \det F(\mathbf{s})] = \sum\_{i=1}^{m} N\{0, [1 + q\_i(\mathbf{s})] \} = n\_q$$

where *Fs I Qs* ( ) ( ( )) and nq is the number of unstable poles of Q(s). Let the uncertain plant be given as a set of N transfer function matrices

$$\text{If } \Pi = \{\mathbf{G}^k(\mathbf{s})\}, k = 1, 2, \dots, N \quad \text{where} \quad \mathbf{G}^k(\mathbf{s}) = \left\langle \mathbf{G}^k\_{ij}(\mathbf{s}) \right\rangle\_{m \times m} \tag{43}$$

The simplest uncertainty model is the unstructured perturbation. A set of unstructured perturbations DU is defined as

$$D\_{\rm II} \coloneqq \{ E(jo) \colon \sigma\_{\rm max}[E(jo)] \le \ell(o) \}, \quad \ell(o) = \max\_{k} \ \sigma\_{\rm max}[E(jo)] \} \tag{44}$$

where ( ) is a scalar weight on the norm-bounded perturbation *m m s R* , max [ ( )] 1 *j* over given frequency range, max ( ) is the maximum singular value of (.), hence

$$E(j\alpha) = \ell(\alpha)A(j\alpha) \tag{45}$$

Using unstructured perturbation, the set can be generated by either additive (Ea), multiplicative input (Ei) or multiplicative output (Eo) uncertainties, or their inverse counterparts (Skogestad & Postlethwaite, 2009) thus specifying pertinent uncertainty regions. In the sequel, just additive (a) and multiplicative output (o) perturbations will be considered; results for other uncertainty types can be obtained by analogy.

Denote *G s*( ) any member of a set of possible plants , , *k k ai* ; 0 *G s*( ) the nominal model used to design the controller, and ( ) *k* the scalar weight on a normalized perturbation. The sets *k* generated by the two considered uncertainty forms are: *Additive uncertainty:*

$$\begin{aligned} \Pi\_a &:= \{ \mathbf{G}(\mathbf{s}) : \mathbf{G}(\mathbf{s}) = \mathbf{G}\_0(\mathbf{s}) + E\_a(\mathbf{s}), E\_a(joo) \le \ell\_a(oo)A(joo) \} \\ \ell\_a(oo) &= \max\_k \sigma\_{\max} \{ \mathbf{G}^k(joo) - \mathbf{G}\_0(joo) \}, \ k = 1, 2, \dots, N \end{aligned} \tag{46}$$

*Multiplicative output uncertainty:*

$$\begin{aligned} \Pi\_o &:= \{ \mathbf{G}(s) : \mathbf{G}(s) = [I + \mathbf{E}\_o(s)] \mathbf{G}\_0(s), \ A(joo) \le \ell\_o(joo)A(joo) \} \\ \mathcal{L}\_o(oo) &= \max\_k \sigma\_{\max} \{ [\mathbf{G}^k(joo) - \mathbf{G}\_0(joo)] \mathbf{G}\_0^{-1}(joo) \}, \ k = 1, 2, \dots, N \end{aligned} \tag{47}$$

Standard feedback configuration with uncertain plant modelled using any unstructured uncertainty form can be recast into the *M* structure (for additive perturbation see Fig. 6) where M(s) is the nominal model and *s* is the norm-bounded complex perturbation.

a. det ( ) 0 *Fs s*

1 [0,det ( )] {0,[1 ( )]} *m*

*Gs k N* where () () *k k*

The simplest uncertainty model is the unstructured perturbation. A set of unstructured

max max *U* : { ( ) : [ ( )] ( ), ( ) max [ ( )]} *k*

*Ej j* ( ) ( )( )

multiplicative input (Ei) or multiplicative output (Eo) uncertainties, or their inverse counterparts (Skogestad & Postlethwaite, 2009) thus specifying pertinent uncertainty regions. In the sequel, just additive (a) and multiplicative output (o) perturbations will be

: { ( ) : ( ) ( ) ( ), ( ) ( ) ( )}

0 1 max 0 0 : { ( ) : ( ) [ ( )] ( ), ( ) ( ) ( )} ( ) max {[ ( ) ( )] ( )}, 1,2, ,

*Gs Gs I E s G s j j j*

Standard feedback configuration with uncertain plant modelled using any unstructured

*Gs Gs G s E s E j j*

( ) max [ ( ) ( )], 1,2, , *a aa a*

*Gj Gj k N*

(46)

*Gj Gj G j k N*

(47)

considered; results for other uncertainty types can be obtained by analogy.

0 max 0

*o oo*

Denote *G s*( ) any member of a set of possible plants , , *k*

*k a k*

*k o k*

*k* generated by the two considered uncertainty forms are:

is a scalar weight on the norm-bounded perturbation *m m*

*i N Fs N q s n*

where *Fs I Qs* ( ) ( ( )) and nq is the number of unstable poles of Q(s). Let the uncertain plant be given as a set of N transfer function matrices

{ ( )}, 1,2,..., *k*

*D Ej Ej*

*j* over given frequency range, max

*i q*

*ij m m Gs Gs* (43)

 *E j* (44)

( ) is the maximum singular value of (.),

can be generated by either additive (Ea),

the scalar weight on a normalized perturbation. The

structure (for additive perturbation see

*s* is the norm-bounded complex

(45)

*k ai* ; 0 *G s*( ) the nominal model

*s R* ,

(42)

b.

Using unstructured perturbation, the set

used to design the controller, and ( ) *k*

*Multiplicative output uncertainty:*

uncertainty form can be recast into the *M*

Fig. 6) where M(s) is the nominal model and

perturbations DU is defined as

where ( )

 [ ( )] 1

max

hence

sets

*Additive uncertainty:*

perturbation.

Fig. 6. Standard feedback configuration with additive perturbation (left) recast into the *M* structure (right)

According to the general robust stability condition (Skogestad & Postlethwaite, 2009), if both the nominal closed-loop system M(s) and the perturbations ( )*s* are stable, the *M* system in Fig. 2 is stable for all perturbations ( )*s* : max () 1 if and only if

$$
\sigma\_{\text{max}}[M(j\rho)] < 1 \quad \forall \, \rho \tag{48}
$$

For individual uncertainty forms ( ) ( ), , *M k k s M s k ao* the corresponding matrices ( ) *Mk s* are given by (49) and (50), respectively (disregarding negative signs which do not affect resulting robustness condition). The nominal model 0 *G s*( ) is usually obtained as a model of mean parameter values.

$$M(\mathbf{s}) = \ell\_a(\mathbf{s})R(\mathbf{s})[I + \mathbf{G}\_0(\mathbf{s})R(\mathbf{s})]^{-1} = \ell\_a(\mathbf{s})M\_a(\mathbf{s})\tag{49}$$

$$M\text{(s)} = \ell\_o \text{(s)} G\_0 \text{(s)} R \text{(s)} \text{[}I + G\_0 \text{(s)} R \text{(s)}\text{]}^{-1} = \ell\_o \text{(s)} M\_o \text{(s)}\tag{50}$$

#### **4.1.1 Problem formulation**

Consider an uncertain system that consists of m subsystems and is given as a set of N transfer function matrices obtained in N working points of plant operation. Let the uncertain system be described by a nominal model 0 *G s*( ) and any unstructured uncertainty form (46), (47). Consider the following splitting of 0 *G s*( ) **:**

$$\mathbf{G}\_0(\mathbf{s}) = \mathbf{G}\_d(\mathbf{s}) + \mathbf{G}\_m(\mathbf{s}) \tag{51}$$

where

$$G\_d(\mathbf{s}) = \operatorname{diag} \{ G\_i(\mathbf{s}) \}\_{m \times m'} \qquad \det G\_d(\mathbf{s}) \neq 0 \tag{52}$$

$$\mathbf{G}\_m(\mathbf{s}) = \mathbf{G}\_0(\mathbf{s}) - \mathbf{G}\_d(\mathbf{s}) \tag{53}$$

A decentralized controller

$$R(s) = \operatorname{diag} \{ R\_i(s) \}\_{m \times m} \quad \text{det} \, R(s) \neq 0 \tag{54}$$

is to be designed to guarantee stability over the whole operating range of the plant specified by (46) or (47) (robust stability) and a specified plant-wide performance (nominal performance).

Robust Decentralized PID Controller Design 153

characteristic functions of [ ( )] *G s m* (the set of characteristic functions are denoted

*m km k i*

det[ ( ) ( )] det{ [ ( ) ( )] ( )}

is a diagonal matrix of m equivalent subsystems generated as follows

*I GsRs I G s G s Rs*

1 det[ ( ) ( )] det[ ] [ ( ) ( )] 0, 1,2,... *m*

(57)

( ) 1 ( ) ( ) 1,2,..., *eq eq CLCP s R s G s i m i i i* (61)

(58)

(62)

*i Ps G s pI G g s g s k m*

With respect to stability, the interactions matrix *G s m*( ) can thus be replaced by [-P(s)]

det[ ( ) ( ) ( )]det ( ) det[ ( ) ( )]

( ) { ( )} *eq eq G s diag G s i m m* (59)

( ) ( ) ( ), 1,2, , *eq G s Gs g s i m ik i k* (60)

As all matrices are diagonal, on subsystems level (58) breaks down into m equivalent closed-

Considering (58)-(61), stability conditions stated in the Generalized Nyquist Stability

The closed-loop in Fig. 3 comprising the system (51) and the decentralized controller (54) is

1. det[ ( ) ] 0, *p sI G k m* for a fixed *k m* {1,..., }

2. all equivalent characteristic polynomials (61) have roots with Re{ } 0 *s* ;

3. [0,det ( )] *N Fs n q*

where N[0,g(s)] is number of anticlockwise encirclements of the complex plane origin by the

The decentralized controller design technique for nominal stability resulting from *Corollary 4.1* enables to independently design stabilizing local controllers for individual single inputsingle output equivalent subsystems using any standard frequency-domain design method, e.g. (Bucz et al., 2010; Drahos, 2000). In the originally developed ESM version (Kozáková et al., 2009) it was proved that local controllers tuned for a specified feasible degree-of-stability of equivalent subsystems constitute the decentralized controller guaranteeing the same

stable if and only if there exists a diagonal matrix () ()() *Ps p sIs k* such that

Nyquist plot of g(s); *nq* is number of open loop poles with Re{ } 0 *s* .

*R s G s Ps Rs I G sRs*

*d m eq d*

( ), 1,2,..., *i gsi m* ); thus

where

yielding the important relationship

loop characteristic polynomials (CLCP)

Theorem modify as follows:

*Corollary 4.1*

1

To solve this problem, a frequency domain robust decentralized controller design technique has been developed (Kozáková and Veselý, 2009; Kozáková et. al., 2011); the core of it is the Equivalent Subsystems Method (ESM).

### **4.2 Decentralized controller design for performance: Equivalent Subsystems Method**

The Equivalent Subsystems Method (ESM) is a Nyquist-based technique to design decentralized controller for stability and specified plant-wide performance. According to it, local controllers ( ), 1,..., *Rsi m i* are designed independently for so-called equivalent subsystems obtained from frequency responses of decoupled subsystems by shaping each of them using one of m characteristic loci of the interactions matrix Gm(s). If local controllers are independently tuned for specified degree-of-stability of equivalent subsystems, the resulting decentralized controller guarantees the same degree-of-stability plant-wide (Kozáková et al., 2009). Unlike standard robust approaches, the proposed technique considers full nominal model of mean parameter values, thus reducing conservatism of resulting robust stability conditions. In the context of robust decentralized controller design, the Equivalent Subsystems Method is directly applicable to design DC for the nominal model (Fig. 3).

Fig. 7. Standard feedback loop under decentralized controller

The key idea behind the method is factorization of the closed-loop characteristic polynomial (40) in terms of the nominal system (51) under the decentralized controller (54). Then

$$\det F(\mathbf{s}) = \det[\mathbf{R}^{-1}(\mathbf{s}) + \mathbf{G}\_d(\mathbf{s}) + \mathbf{G}\_m(\mathbf{s})] \det \mathbf{R}(\mathbf{s}) \tag{55}$$

Denote the sum of the diagonal matrices in (55) as

$$R^{-1}(\mathbf{s}) + G\_d(\mathbf{s}) = P(\mathbf{s})\tag{56}$$

where ( ) { ( )} *P s diag p s i mm* .

In order to "counterbalance" interactions *G s m*( ) , consider the closed-loop being at the limit of instability and choose the diagonal matrix () () *Ps p sI k* to have identical entries pk(s); then by similarity with (41) the bracketed term in (55) defines the k-th of the m characteristic functions of [ ( )] *G s m* (the set of characteristic functions are denoted ( ), 1,2,..., *i gsi m* ); thus

$$\det[P(\mathbf{s}) + \mathbf{G}\_m(\mathbf{s})] = \det[p\_k I + \mathbf{G}\_m] = \prod\_{i=1}^m [-\mathbf{g}\_k(\mathbf{s}) + \mathbf{g}\_i(\mathbf{s})] = \mathbf{0}, \ k = 1, 2, \dots \\ m \tag{57}$$

With respect to stability, the interactions matrix *G s m*( ) can thus be replaced by [-P(s)] yielding the important relationship

$$\begin{aligned} \det[I + G(s)R(s)] &= \det[I + [G\_d(s) + G\_m(s)]R(s)] = \\ &= \det[R^{-1}(s) + G\_d(s) - P(s)]\det R(s) = \det[I + G^{\circ \eta}(s)R(s)] \end{aligned} \tag{58}$$

where

152 Introduction to PID Controllers – Theory, Tuning and Application to Frontier Areas

To solve this problem, a frequency domain robust decentralized controller design technique has been developed (Kozáková and Veselý, 2009; Kozáková et. al., 2011); the core of it is the

**4.2 Decentralized controller design for performance: Equivalent Subsystems Method**  The Equivalent Subsystems Method (ESM) is a Nyquist-based technique to design decentralized controller for stability and specified plant-wide performance. According to it, local controllers ( ), 1,..., *Rsi m i* are designed independently for so-called equivalent subsystems obtained from frequency responses of decoupled subsystems by shaping each of them using one of m characteristic loci of the interactions matrix Gm(s). If local controllers are independently tuned for specified degree-of-stability of equivalent subsystems, the resulting decentralized controller guarantees the same degree-of-stability plant-wide (Kozáková et al., 2009). Unlike standard robust approaches, the proposed technique considers full nominal model of mean parameter values, thus reducing conservatism of resulting robust stability conditions. In the context of robust decentralized controller design, the Equivalent Subsystems Method is directly applicable to design DC for the nominal

*w***e** *u* **+ +** *y*

The key idea behind the method is factorization of the closed-loop characteristic polynomial

In order to "counterbalance" interactions *G s m*( ) , consider the closed-loop being at the limit of instability and choose the diagonal matrix () () *Ps p sI k* to have identical entries pk(s); then by similarity with (41) the bracketed term in (55) defines the k-th of the m

(40) in terms of the nominal system (51) under the decentralized controller (54). Then

1

*Gd(s)*

**G11 0 … 0 0 G22 … 0 ………………... 0 0 … Gmm**

1 det ( ) det[ ( ) ( ) ( )]det ( ) *Fs R s G s G s Rs d m* (55)

() () () *R s G s Ps d* (56)

*Gm(s)*

**0 G12 … G1m G21 0 … G2m ……………….… Gm1 Gm2 … 0**

*G0(s)*

Equivalent Subsystems Method (ESM).

**-**

*R(s)*

Fig. 7. Standard feedback loop under decentralized controller

Denote the sum of the diagonal matrices in (55) as

where ( ) { ( )} *P s diag p s i mm* .

**R1 0 … 0 0 R2 … 0 ……………….. 0 0 … Rm**

model (Fig. 3).

$$\mathbf{G}^{\alpha \dagger}(\mathbf{s}) = \operatorname{diag} \{ \mathbf{G}\_i^{\alpha \dagger}(\mathbf{s}) \}\_{m \times m} \tag{59}$$

is a diagonal matrix of m equivalent subsystems generated as follows

$$\mathbf{G}\_{lk}^{\prime \neq l}(\mathbf{s}) = \mathbf{G}\_l(\mathbf{s}) + \mathbf{g}\_k(\mathbf{s}), \qquad i = 1, 2, \dots, m \tag{60}$$

As all matrices are diagonal, on subsystems level (58) breaks down into m equivalent closedloop characteristic polynomials (CLCP)

$$\text{CLCP}\_{i}^{\text{eq}}\text{(s)} = 1 + R\_{i}(\text{s})\text{G}\_{i}^{\text{eq}}\text{(s)}\qquad i = 1, 2, \dots, m \tag{61}$$

Considering (58)-(61), stability conditions stated in the Generalized Nyquist Stability Theorem modify as follows:

#### *Corollary 4.1*

The closed-loop in Fig. 3 comprising the system (51) and the decentralized controller (54) is stable if and only if there exists a diagonal matrix () ()() *Ps p sIs k* such that

$$\mathbf{1.}\ \det[p\_k(\mathbf{s})\mathbf{I} + \mathbf{G}\_m] = \mathbf{0}, \quad \text{for a fixed} \quad k \in \{\mathbf{1}, \dots, m\}$$

2. all equivalent characteristic polynomials (61) have roots with Re{ } 0 *s* ; (62)

$$\text{3. } N[0, \det F(s)] = n\_q$$

where N[0,g(s)] is number of anticlockwise encirclements of the complex plane origin by the Nyquist plot of g(s); *nq* is number of open loop poles with Re{ } 0 *s* .

The decentralized controller design technique for nominal stability resulting from *Corollary 4.1* enables to independently design stabilizing local controllers for individual single inputsingle output equivalent subsystems using any standard frequency-domain design method, e.g. (Bucz et al., 2010; Drahos, 2000). In the originally developed ESM version (Kozáková et al., 2009) it was proved that local controllers tuned for a specified feasible degree-of-stability of equivalent subsystems constitute the decentralized controller guaranteeing the same

Robust Decentralized PID Controller Design 155

3. Specification of a minimum phase margin PM for equivalent subsystems using (63). 4. Local controller design for specified PM in equivalent subsystems using appropriate

**4.3 Decentralized controller design for robust stability using the Equivalent**

5. Verification of achieved performance by evaluating frequency domain performance

In the context of robust control approach, the ESM method in its original version is inherently appropriate to design decentralized controller guaranteeing stability and specified performance of the nominal model (nominal stability, nominal performance). If, in addition, the decentralized controller has to guarantee closed-loop stability over the whole operating range of the plant specified by the chosen uncertainty description (robust stability), the ESM can be used either within a two-stage design procedure or a direct design

1. Two stage robust decentralized controller design for robust stability and nominal

In the first stage, the decentralized controller for the nominal system is designed using ESM, afterwards, fulfilment of the M-stability condition (48) is examined; if satisfied, the design procedure stops, otherwise in the second stage the controller parameters are modified to satisfy robust stability conditions in the tightest possible way, or local controllers are

1

Expressions on the r.h.s. of (69) and (70) do not depend on a particular controller and can be

multiplicative output uncertainty (70)

*additive uncertainty* (69)

can be obtained using the singular value properties in manipulations of the M-condition (48) considering (49) or (50). The following bounds for the nominal complementary

00 0 *T s G sRs I G sRs* ( ) ( ) ( )[ ( ) ( )] (68)

redesigned using modified performance requirements (Kozáková & Veselý, 2009). 2. Direct decentralized controller design for robust stability and nominal performance By direct integration of robust stability condition (48) in the ESM, a "one-shot" design of local controllers for both nominal performance and robust stability can be carried out. In case of decentralized controller design for guaranteed maximum overshoot and specified settling time, the upper bound for the maximum peak of the nominal complementary

max , ts and MT using (66), (67).

1. Generating frequency responses of equivalent subsystems. 2. Specification of performance requirements in terms of

procedure for robust stability and nominal performance.

sensitivity over the given frequency range

sensitivity have been derived:

max 0

*T j*

max 0

1 [ ( )] ( ) ( ) *O o*

evaluated prior to designing the controller. In this way, if

*MT max{ [T ( j )]} max* 0

min 0

[ ( )] [ ( )] ( ) ( ) *A a G j T j L*

*L*

*Design procedure:*

**Subsystems Method**

performance

frequency domain method.

measure and via simulation.

degree-of-stability plant-wide. To design local controllers of equivalent subsystems, the general conditions in *Corollary 4.1* allow using any frequency domain performance measure that can appropriately be interpreted for the full system. In the next subsection, the plant wide performance is specified in terms of maximum overshoot which is closely related to phase margins of equivalent subsystems.

#### **4.2.1 Decentralized controller design for guaranteed maximum overshoot and specified settling time**

The ESM can be applied to design decentralized controller to guarantee specified maximum overshoot of output variables of the multivariable system. The design procedure evolves from the known relationship between the phase margin (PM) and the maximum peak of the complementary sensitivity (Skogestad & Postlethwaite, 2009)

$$PM \ge 2 \arcsin\left(\frac{1}{2M\_T}\right) \ge \frac{1}{M\_T} [rad] \tag{63}$$

where

$$M\_T = \sigma\_{\text{max}}[T(j\rho)]\tag{64}$$

is the maximum peak of the complementary sensitivity T(s) defined as

$$T(s) = G(s)R(s)[I + G(s)R(s)]^{-1} \tag{65}$$

Relation between the maximum overshoot max and MT is given by (Bucz et al., 2010)

$$
\eta\_{\text{max}} \le \frac{1.18M\_T - \left| T(0) \right|}{\left| T(0) \right|} 100 \left[ \% \right] \tag{66}
$$

According to the ESM philosophy, local controllers are designed using frequency domain methods; if PID controller is considered, the most appropriate ones are e.g. the Bode diagram design or the Neymark D-partition method. If using the Bode diagram design, in addition to max it is also possible to specify the required settling time ts related with the closed-loop bandwidth frequency0 defined as the gain crossover frequency. The following relations between ts and 0 are useful (Reinisch, 1974).

$$t\_s \approx \frac{3}{o\_0} \text{ for } M\_T \in \{1.3; 1.5\}$$

$$\frac{\pi}{t\_s} < o\_0 < \frac{4\pi}{t\_s} \tag{67}$$

In general, a larger bandwidth corresponds to a smaller rise time, since high frequency signals are more easily passed on to the outputs. If the bandwidth is small, the time response will generally be slow and the system will usually be more robust.

#### *Design procedure:*

154 Introduction to PID Controllers – Theory, Tuning and Application to Frontier Areas

degree-of-stability plant-wide. To design local controllers of equivalent subsystems, the general conditions in *Corollary 4.1* allow using any frequency domain performance measure that can appropriately be interpreted for the full system. In the next subsection, the plant wide performance is specified in terms of maximum overshoot which is closely related to

The ESM can be applied to design decentralized controller to guarantee specified maximum overshoot of output variables of the multivariable system. The design procedure evolves from the known relationship between the phase margin (PM) and the maximum peak of the

**4.2.1 Decentralized controller design for guaranteed maximum overshoot and**

1 1 2arcsin [ ] 2 *T T PM rad*

is the maximum peak of the complementary sensitivity T(s) defined as

max

0 are useful (Reinisch, 1974).

0 3 *st*

response will generally be slow and the system will usually be more robust.

*M M*

max[ ( )] *MT T j*

1.18 (0) 100[%] (0) *M T T T*

max it is also possible to specify the required settling time ts related with the

According to the ESM philosophy, local controllers are designed using frequency domain methods; if PID controller is considered, the most appropriate ones are e.g. the Bode diagram design or the Neymark D-partition method. If using the Bode diagram design, in

for (1.3; 1.5) *MT*

0 4

In general, a larger bandwidth corresponds to a smaller rise time, since high frequency signals are more easily passed on to the outputs. If the bandwidth is small, the time

*s s t t*

(63)

1 *Ts GsRs I GsRs* ( ) ( ) ( )[ ( ) ( )] (65)

(66)

0 defined as the gain crossover frequency. The following

(67)

max and MT is given by (Bucz et al., 2010)

(64)

complementary sensitivity (Skogestad & Postlethwaite, 2009)

phase margins of equivalent subsystems.

Relation between the maximum overshoot

**specified settling time**

where

addition to

relations between ts and

closed-loop bandwidth frequency

#### **4.3 Decentralized controller design for robust stability using the Equivalent Subsystems Method**

In the context of robust control approach, the ESM method in its original version is inherently appropriate to design decentralized controller guaranteeing stability and specified performance of the nominal model (nominal stability, nominal performance). If, in addition, the decentralized controller has to guarantee closed-loop stability over the whole operating range of the plant specified by the chosen uncertainty description (robust stability), the ESM can be used either within a two-stage design procedure or a direct design procedure for robust stability and nominal performance.

1. Two stage robust decentralized controller design for robust stability and nominal performance

In the first stage, the decentralized controller for the nominal system is designed using ESM, afterwards, fulfilment of the M-stability condition (48) is examined; if satisfied, the design procedure stops, otherwise in the second stage the controller parameters are modified to satisfy robust stability conditions in the tightest possible way, or local controllers are redesigned using modified performance requirements (Kozáková & Veselý, 2009).

2. Direct decentralized controller design for robust stability and nominal performance

By direct integration of robust stability condition (48) in the ESM, a "one-shot" design of local controllers for both nominal performance and robust stability can be carried out. In case of decentralized controller design for guaranteed maximum overshoot and specified settling time, the upper bound for the maximum peak of the nominal complementary sensitivity over the given frequency range

$$M\_T = \max\_{\mathbf{o}} \{ \sigma\_{\max} \{ T\_0(j \mathbf{o}) \} \} \qquad T\_0(\mathbf{s}) = \mathbf{G}\_0(\mathbf{s}) \mathbf{R}(\mathbf{s}) \mathbf{I} \mathbf{I} + \mathbf{G}\_0(\mathbf{s}) \mathbf{R}(\mathbf{s}) \}^{-1} \tag{68}$$

can be obtained using the singular value properties in manipulations of the M-condition (48) considering (49) or (50). The following bounds for the nominal complementary sensitivity have been derived:

$$\sigma\_{\text{max}}[T\_0(j\alpha)] < \frac{\sigma\_{\text{min}}[\mathcal{G}\_0(j\alpha)]}{\left|\ell\_d(o\rho)\right|} = L\_A(o) \qquad \forall \, o \quad additive \, uncertainty \tag{69}$$

$$\sigma\_{\text{max}}[T\_0(joo)] < \frac{1}{\left| \ell\_o(o) \right|} = \mathcal{L}\_0(o) \qquad \forall o \quad \text{multiplicative output uncertainty} \tag{70}$$

Expressions on the r.h.s. of (69) and (70) do not depend on a particular controller and can be evaluated prior to designing the controller. In this way, if

$$M\_T = \max\_{o \boldsymbol{o}} \{ \sigma\_{\max} [T\_0(jo)] \} \tag{71}$$

Robust Decentralized PID Controller Design 157

0 20 40 *s*

where s is sampling frequency, and 0 is control system bandwidth, i.e. the maximum frequency at which the system output still tracks and input sinusoid in a satisfactory manner (Lian et al., 2002). A proper choice of sampling period is crucial for achievable bandwidth and feasibility of the required phase margin. Given a discrete-time transfer function *G z*( ) , the frequency response can be studied by plotting Nyquist or Bode plots of

. The discrete-time robust controller design for maximum overshoot and settling

**4.5 Decentralized discrete-time PID Controller design for the Quadruple tank process**  In the frequency domain, the direct robust decentralized PID design procedure has been applied for the transfer function matrix (3) identified in three working points within the minimum and nonminimum phase regions (7) and (8), respectively. In both cases the

From three plant models (3) evaluated in working points taken from the minimum phase

62 1 (23 1)(62 1) ( ) 1.5667 3.1333

(30 1)(90 1) 90 1

*ss s*

All three transfer function matrices were discretized using the sampling period

0 50 100 150 200 250 300 350 400 450 500 0

time(s)

30 *T s S* chosen as approx. 1/10 of the settling time of plant step responses in Fig. 8.

2.4667 1.2333

*s ss*

uncertainty region as specified in (7), the resulting continuous-time nominal model is

( ) *j T z e G z*

time is described in the next Section.

*Minimum phase configuration*

nominal model is a mean value parameter model.

0

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Fig. 8. Step response of the quadruple tank process

Discrete-time transfer function matrix of the nominal plant is

y1,y2

*G s*

(72)

(73)

is used in the Design procedure, the resulting decentralized controller will simultaneously guarantee achieving the required maximum overshoot of all output variables (nominal performance) and stability over the whole operating range of the plant specified by selected working points (robust stability).

### **4.4 Discrete-time robust decentralized controller design using the Equivalent Subsystems Method**

Controllers for continuous-time plants are mostly implemented as discrete-time controllers. A common approach to discrete-time controller design is the continuous controller redesign i.e. conversion of the already designed continuous controller into its discrete counterpart. This approach, however, is only an approximate scheme; performance under these controllers deteriorates with increasing sampling period. This drawback may be improved by modifying the continuous controller design before it is discretized which can often allow significantly larger sampling periods (Lewis, 1992). Then, the ESM design methodology can be applied in a similar way as in the continuous-time case using discrete characteristic loci, discrete Nyquist plots and discrete Bode diagrams of equivalent subsystems. Local controllers designed as continuous-time ones are subsequently converted into their discretetime counterparts. Closed-loop performance under a discrete-time controller is verified using simulations and the discrete-time maximum singular value of the sensitivity *M*[ ( )] *S z* where

$$S(z) = \left[I + G(z)R(z)\right]^{-1}, \ z = e^{j\alpha T\_s} \tag{72}$$

The maximum singular value *j T max [S(e )] max* plotted as function of frequency should

be small at low frequencies where feedback is effective, and approach 1 at high frequencies, as the system is strictly proper, having a peak larger than 1 around the crossover frequency. The peak is unavoidable for real systems. Bandwidth frequency is

defined as frequency where [ ( )] *s j T M S e* crosses 0.7 from below (Skogestad & Postlethwaite, 2009). Similarly, a discretized version of robust stability conditions (69), (70) based on (46) and (47) is applied.

## **4.4.1 Design of continuous controllers for discretization**

The crucial step for the discrete controller design is proper choice of the sampling time T. Then, frequency response of the discretized system matches the one of the continuous time system up to a certain frequency / 2 *S* , and the discrete controller can be obtained by converting the continuous–time controller designed from the discrete frequency responses to its discrete-time counterpart.

The sampling period T is to be selected according to the Shannon-Kotelnikov sampling theorem, or using common rules of thumb, e.g. as ~ 1/10 of the settling time of the plant step response, or from control system bandwidth according to the relation

*MT* max{ [ ( )]} max 0 *T j*

is used in the Design procedure, the resulting decentralized controller will simultaneously guarantee achieving the required maximum overshoot of all output variables (nominal performance) and stability over the whole operating range of the plant specified by selected

Controllers for continuous-time plants are mostly implemented as discrete-time controllers. A common approach to discrete-time controller design is the continuous controller redesign i.e. conversion of the already designed continuous controller into its discrete counterpart. This approach, however, is only an approximate scheme; performance under these controllers deteriorates with increasing sampling period. This drawback may be improved by modifying the continuous controller design before it is discretized which can often allow significantly larger sampling periods (Lewis, 1992). Then, the ESM design methodology can be applied in a similar way as in the continuous-time case using discrete characteristic loci, discrete Nyquist plots and discrete Bode diagrams of equivalent subsystems. Local controllers designed as continuous-time ones are subsequently converted into their discretetime counterparts. Closed-loop performance under a discrete-time controller is verified

  (71)

*M*[ ( )] *S z*

**4.4 Discrete-time robust decentralized controller design using the Equivalent**

using simulations and the discrete-time maximum singular value of the sensitivity

1 ( ) [ ( ) ( )] , *s j T Sz I GzRz z e*

be small at low frequencies where feedback is effective, and approach 1 at high frequencies, as the system is strictly proper, having a peak larger than 1 around the crossover frequency. The peak is unavoidable for real systems. Bandwidth frequency is

Postlethwaite, 2009). Similarly, a discretized version of robust stability conditions (69),

The crucial step for the discrete controller design is proper choice of the sampling time T. Then, frequency response of the discretized system matches the one of the continuous time

converting the continuous–time controller designed from the discrete frequency responses

The sampling period T is to be selected according to the Shannon-Kotelnikov sampling theorem, or using common rules of thumb, e.g. as ~ 1/10 of the settling time of the plant

 (72)

plotted as function of frequency should

crosses 0.7 from below (Skogestad &

*S* , and the discrete controller can be obtained by

working points (robust stability).

The maximum singular value *j T max [S(e )] max*

defined as frequency where [ ( )] *s j T M S e*

(70) based on (46) and (47) is applied.

system up to a certain frequency / 2

to its discrete-time counterpart.

**4.4.1 Design of continuous controllers for discretization**

step response, or from control system bandwidth according to the relation

**Subsystems Method**

where

$$20 < \frac{\alpha\_s}{\alpha\_0} < 40\tag{72}$$

where s is sampling frequency, and 0 is control system bandwidth, i.e. the maximum frequency at which the system output still tracks and input sinusoid in a satisfactory manner (Lian et al., 2002). A proper choice of sampling period is crucial for achievable bandwidth and feasibility of the required phase margin. Given a discrete-time transfer function *G z*( ) , the frequency response can be studied by plotting Nyquist or Bode plots of ( ) *j T z e G z* . The discrete-time robust controller design for maximum overshoot and settling time is described in the next Section.

#### **4.5 Decentralized discrete-time PID Controller design for the Quadruple tank process**

In the frequency domain, the direct robust decentralized PID design procedure has been applied for the transfer function matrix (3) identified in three working points within the minimum and nonminimum phase regions (7) and (8), respectively. In both cases the nominal model is a mean value parameter model.

#### *Minimum phase configuration*

From three plant models (3) evaluated in working points taken from the minimum phase uncertainty region as specified in (7), the resulting continuous-time nominal model is

$$\mathbf{G}\_{0}(\mathbf{s}) = \begin{bmatrix} \frac{2.4667}{62s+1} & \frac{1.2333}{(23s+1)(62s+1)}\\ \frac{1.5667}{(30s+1)(90s+1)} & \frac{3.1333}{90s+1} \end{bmatrix} \tag{73}$$

All three transfer function matrices were discretized using the sampling period 30 *T s S* chosen as approx. 1/10 of the settling time of plant step responses in Fig. 8.

Fig. 8. Step response of the quadruple tank process

Discrete-time transfer function matrix of the nominal plant is

Robust Decentralized PID Controller Design 159

Relevant parameters read form discrete Bode plots of uncompensated equivalent

subsystem PM Crossover

12( ) *eq G z* 53.90 0.048 rad/ s-1 22( ) *eq G z* 58.350 0.0448 rad/s-1

For both equivalent subsystems the required settling time and maximum overshoot were chosen with respect to plant dynamics: 600 , 1.05 *s T t sM* corresponding to max

Related values of other design parameters obtained from (63) and (67) respectively are:

*PM PM req* min was chosen 0 65 *PMreq* . To design local controllers, Bode design procedure (Kuo, 2003) has been applied independently for each equivalent subsystem to achieve the

is designed. If 0 ( ) *PM PM*

designed. The resulting PID controller is obtained in the series form

. Achieved design results are summarized in Tab. 4.

subsyst. Ri(s) Ri(z) PMachieved achieved

Table 4. Design results and achieved frequency domain performance measures (minimum

1 1 0.199 0.082 ( ) 1 *z R z*

2 1 0.221 0.119 ( ) 1 *z R z*

*z*

*z*

is found on the magnitude Bode plot; if 0 ( ) *PM PM*

1

1

58.350 0.0122

65.70 0.0121

frequency

*req* , a PD controller

, and subsequently a PI controller is

0.0131 . The required phase margin

5% .

*req* , a

rad/s-1

rad/s-1

subsystems in Fig. 10 are summarized in Tab. 3.

0

required phase margin: 0 *PM*( )

PI controller ( ) *I*

( ) ( )(1 ) *I PID P D K G s K Ks s*

Eq.

12( ) *eq G z* 1

22( ) *eq G z* 2

phase configuration)

*PI P K Gs K*

0.0039 *R s*( ) 0.1988

0.0034 *R s*( ) 0.2212

min *PM* 56.87 and required crossover frequency 0

*s*

*s*

*s*

() 1 *G s Ks PD D* is designed first, to provide 0 ( ) *PMreq*

Equivalent

Table 3. Relevant parameters of equivalent subsystems generated by g2(z)

$$G(z) = \begin{bmatrix} \frac{0.9462z^{-1}}{1 - 0.6164z^{-1}} & \frac{0.2221z^{-1} + 0.1226z^{-2}}{1 - 0.8877z^{-1} + 0.1673z^{-2}}\\ \frac{0.1710z^{-1} + 0.1097z^{-2}}{1 - 1.0840z^{-1} + 0.2636z^{-2}} & \frac{0.8882z^{-1}}{1 - 0.7165z^{-1}} \end{bmatrix} \tag{74}$$

From the discretized transfer function matrices and the nominal model (74), upper bounds for max 0 [ ( )] *T j*were evaluated according to (69) and (70).

Fig. 9. Upper bounds for max 0 [ ( )] *T j*evaluated according to (69) and (70)

Inspection of Fig. 9 reveals, that \_ min 0.77 1 *M L TA A* is not feasible for the local controller design (closed-loop design magnitude less than 1 does not guarantee proper setpoint tracking, even at =0); hence \_ min 1.22 *MM L T TO O* has been considered in the sequel.

Characteristic loci g1(z), g2(z) of Gm(z) were calculated; 2 *g z*( ) was selected to generate the equivalent subsystems according to (60). Bode plots of resulting equivalent subsystems are shown in Fig. 10.

Fig. 10. Discrete Bode plots of equivalent subsystems generated by g2(z): 12 ( ) *eq G z* (left), 22( ) *eq G z* (right) (min. phase case)

1 0.6164 1 0.8877 0.1673 ( ) 0.1710 0.1097 0.8882 1 1.0840 0.2636 1 0.7165

*z zz G z*

1 2 1 1 2 1

10-2

has been considered in the sequel.

0 10 20

Phase [deg]

Fig. 10. Discrete Bode plots of equivalent subsystems generated by g2(z): 12 ( ) *eq G z* (left),

magnitude [dB]

evaluated according to (69) and (70)

omega [rad/s]

design (closed-loop design magnitude less than 1 does not guarantee proper setpoint tracking,

Characteristic loci g1(z), g2(z) of Gm(z) were calculated; 2 *g z*( ) was selected to generate the equivalent subsystems according to (60). Bode plots of resulting equivalent subsystems are

10-1

is not feasible for the local controller

10-4 10-3 10-2 10-1 100 -10

10-4 10-3 10-2 10-1 100 -200

omega [rad/s]

omega [rad/s]

*zz z zz z*

From the discretized transfer function matrices and the nominal model (74), upper bounds

LA LO

were evaluated according to (69) and (70).

10-3

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

even at =0); hence \_ min 1.22 *MM L T TO O*

10-4 10-3 10-2 10-1 100 -5

omega [rad/s]

10-4 10-3 10-2 10-1 100 -200

22( ) *eq G z* (right) (min. phase case)

omega [rad/s]

Inspection of Fig. 9 reveals, that \_ min 0.77 1 *M L TA A*

 [ ( )] *T j*

Fig. 9. Upper bounds for max 0

shown in Fig. 10.

Phase [deg]

magnitude [dB]

for max 0

 [ ( )] *T j*

0.9462 0.2221 0.1226

1 1 2 1 1 2

(74)

*z zz*

Relevant parameters read form discrete Bode plots of uncompensated equivalent subsystems in Fig. 10 are summarized in Tab. 3.

Table 3. Relevant parameters of equivalent subsystems generated by g2(z)

For both equivalent subsystems the required settling time and maximum overshoot were chosen with respect to plant dynamics: 600 , 1.05 *s T t sM* corresponding to max 5% . Related values of other design parameters obtained from (63) and (67) respectively are: 0 min *PM* 56.87 and required crossover frequency 0 0.0131 . The required phase margin *PM PM req* min was chosen 0 65 *PMreq* . To design local controllers, Bode design procedure (Kuo, 2003) has been applied independently for each equivalent subsystem to achieve the required phase margin: 0 *PM*( ) is found on the magnitude Bode plot; if 0 ( ) *PM PM req* , a

PI controller ( ) *I PI P K Gs K s* is designed. If 0 ( ) *PM PM req* , a PD controller () 1 *G s Ks PD D* is designed first, to provide 0 ( ) *PMreq* , and subsequently a PI controller is designed. The resulting PID controller is obtained in the series form ( ) ( )(1 ) *I PID P D K G s K Ks s* . Achieved design results are summarized in Tab. 4.

Table 4. Design results and achieved frequency domain performance measures (minimum phase configuration)

Robust Decentralized PID Controller Design 161

yq g

5 0 dB

Achieved nominal performance was verified via plotting sensitivity magnitude plot in Fig.

10-4 10-3 10-2 10-1 100 0

w [rad/sec]

Fulfilment of robust stability condition (70) is examined in Fig. 15. The closed-loop system is

Real Axis

around the crossover frequency proves good

6 dB 4 dB

2 dB

14. Sensitivity peak max{ [ ( )]} 2 *M S j*

0.2

stable over the whole minimum phase region (7).

0.4

0.6

sigmaMax(S)

0.8

1

1.2

1.4

closed-loop performance.

Fig. 14. [ ( )] *M j T z e S z*

Fig. 13. Stability test using the Nyquist plot of det[ ( ) ( )] *I GzRz*

Imaginary Axis

Design results in Tab. 4 along with Bode plots of compensated equivalent subsystems in Fig.11 prove achieving required design parameters. Closed-loop step responses are in Fig. 12.

Fig. 11. Discrete Bode plots of equivalent subsystems under designed PI controllers: 12( ) *eq G z* (left), 22( ) *eq G z* (right)

Fig. 12. Nominal closed-loop step responses of the quadruple tank process (reference steps 0.1m occurred at t=0s at the input of the 1st subsystem, and at t=300s and t=10s, respectively, at the input of the 2nd subsystem). Maximum overshoot and settling time (600s) were kept in both cases.

Nominal closed-loop stability was verified both by calculating closed-loop poles and using the Generalized Nyquist encirclement criterion (Fig. 13).

Roots\_of\_CLCP { 0.7019 0.2572i,0.8313, 0.7167, 0.7165, 0.6164, 0.3720, 0.2637

Design results in Tab. 4 along with Bode plots of compensated equivalent subsystems in Fig.11 prove achieving required design parameters. Closed-loop step responses are in

0

Fig. 12. Nominal closed-loop step responses of the quadruple tank process (reference steps 0.1m occurred at t=0s at the input of the 1st subsystem, and at t=300s and t=10s, respectively, at the input of the 2nd subsystem). Maximum overshoot and settling time (600s) were kept in

Nominal closed-loop stability was verified both by calculating closed-loop poles and using

Roots\_of\_CLCP { 0.7019 0.2572i,0.8313, 0.7167, 0.7165, 0.6164, 0.3720, 0.2637

y1,y2

Phase

Fig. 11. Discrete Bode plots of equivalent subsystems under designed PI controllers:

y1 y2 Magnitude

50

10-4 10-3 10-2 10-1 100 -50

10-4 10-3 10-2 10-1 100 -200

0 100 200 300 400 500 600 700 800 900 1000 20

t

omega

y

y1 y2

Fig. 12.

0

both cases.

y1,y2 [cm]

Phase

Magnitude

50

10-4 10-3 10-2 10-1 100 -50

10-4 10-3 10-2 10-1 100 -200

0 100 200 300 400 500 600 700 800 900 100 20

t [s]

the Generalized Nyquist encirclement criterion (Fig. 13).

12( ) *eq G z* (left), 22( ) *eq G z* (right)

omega

Fig. 13. Stability test using the Nyquist plot of det[ ( ) ( )] *I GzRz*

Achieved nominal performance was verified via plotting sensitivity magnitude plot in Fig. 14. Sensitivity peak max{ [ ( )]} 2 *M S j* around the crossover frequency proves good closed-loop performance.

Fig. 14. [ ( )] *M j T z e S z* - versus –frequency plot

Fulfilment of robust stability condition (70) is examined in Fig. 15. The closed-loop system is stable over the whole minimum phase region (7).

Robust Decentralized PID Controller Design 163

Obviously, proper setpoint tracking can be guaranteed for both uncertainty types, just on a limited frequency range. Hence, 1.05 *MT* and multiplicative output uncertainty will be

Bode plots of equivalent subsystems generated using 2 *g z*( ) are shown in Fig. 17, and their

Phase [deg]

Fig. 17. Discrete Bode plots of equivalent subsystems generated by g2(z): 12 ( ) *eq G z* (left),

subsystem PM Crossover

22( ) *eq G z* 44.040 0.0344

12 ( ) *eq G z* 43.810 0.040rad/s-1

For both equivalent subsystems the required settling time and maximum overshoot were chosen the same as in the minimum phase case: 600 , 1.05 *s T t sM* corresponding to

0 60 *PMreq* . Achieved design results are summarized in Tab. 6 and Bode plots of compensated equivalent subsystems in Fig.18 prove achieving required design

frequency

rad/s-1

0.0131 . The required phase margin *PM PM req* min was chosen

Equivalent

Table 5. Relevant parameters of equivalent subsystems generated by g2(z).

5% . Related values of other design parameters are: 0

magnitude [dB]

10-4 10-3 10-2 10-1 100 -20

10-4 10-3 10-2 10-1 100 -200

omega [rad/s]

min *PM* 56.87 and required

omega [rad/s]

considered in the sequel.

max

parameters.

crossover frequency 0

Phase [deg]

magnitude [dB]

relevant parameters are summarized in Tab. 5.

10-4 10-3 10-2 10-1 100 -20

10-4 10-3 10-2 10-1 100 -200

22( ) *eq G z* (right) (non-minimum phase case)

omega [rad/s]

omega [rad/s]

Fig. 15. Verification of the robust stability condition max 0 1 [ ( )] ( ) ( ) *O o T j L*

#### *Non-minimum phase configuration*

To design robust decentralized PI controller for the non-minimum phase configuration, the continuous-time nominal model was evaluated for 1 2 , taken from the non-minimum phase uncertainty region (8) and interchanged columns of the transfer function matrix (due to opposite pairing as suggested in Section 2):

$$\mathbf{G}\_{0}(\mathbf{s}) = \begin{bmatrix} \frac{3.0830}{(23s+1)(62s+1)} & \frac{0.6167}{62s+1} \\ \frac{0.7833}{90s+1} & \frac{3.9170}{(30s+1)(90s+1)} \end{bmatrix} \tag{73}$$

Discrete-time transfer function matrix of the nominal plant obtained for 30 *T s S* is

$$\mathbf{G}(z) = \begin{bmatrix} \frac{0.5554z^{-1} + 0.3065z^{-2}}{1 - 0.8877z^{-1} + 0.1673z^{-2}} & \frac{0.2366z^{-1}}{1 - 0.6164z^{-1}}\\ \frac{0.2220z^{-1}}{1 - 0.7165z^{-1}} & \frac{0.4275z^{-1} + 0.2743z^{-2}}{1 - 1.0840z^{-1} + 0.2636z^{-2}} \end{bmatrix} \tag{74}$$

Upper bounds for max 0 [ ( )] *T j*evaluated according to (69) and (70) are in Fig. 16.

Fig. 16. Upper bounds for max 0 [ ( )] *T j*evaluated according to (69) and (70)

LO sigma max (T)

1 [ ( )] ( ) ( ) *O o*

taken from the non-minimum

 *L*

(73)

(74)

10-4 10-3 10-2 10-1 100 0

w [rad/sec]

To design robust decentralized PI controller for the non-minimum phase configuration, the

phase uncertainty region (8) and interchanged columns of the transfer function matrix (due

(23 1)(62 1) 62 1 ( ) 0.7833 3.9170

0.5554 0.3065 0.2366

Discrete-time transfer function matrix of the nominal plant obtained for 30 *T s S* is

1 0.8877 0.1673 1 0.6164 ( )

*zz z G z*

3.0830 0.6167

*ss s*

90 1 (30 1)(90 1)

*s ss*

1 2 1 1 2 1

*zz z*

10-4 10-3 10-2 10-1 100 0.5

omega [rad/s]

0.2220 0.4275 0.2743 1 0.7165 1 1.0840 0.2636

1 1 2 1 1 2

evaluated according to (69) and (70) are in Fig. 16.

evaluated according to (69) and (70)

LA LO

*z zz z zz*

*T j*

> ,

0.5

*Non-minimum phase configuration*

Upper bounds for max 0

Fig. 16. Upper bounds for max 0

 [ ( )] *T j*

L

,LO

A

1 1.5 2 2.5 3 3.5 4 4.5

 [ ( )] *T j*

Fig. 15. Verification of the robust stability condition max 0

continuous-time nominal model was evaluated for 1 2

to opposite pairing as suggested in Section 2):

0

*G s*

1

1.5

2

2.5

Obviously, proper setpoint tracking can be guaranteed for both uncertainty types, just on a limited frequency range. Hence, 1.05 *MT* and multiplicative output uncertainty will be considered in the sequel.

Bode plots of equivalent subsystems generated using 2 *g z*( ) are shown in Fig. 17, and their relevant parameters are summarized in Tab. 5.

Fig. 17. Discrete Bode plots of equivalent subsystems generated by g2(z): 12 ( ) *eq G z* (left), 22( ) *eq G z* (right) (non-minimum phase case)

Table 5. Relevant parameters of equivalent subsystems generated by g2(z).

For both equivalent subsystems the required settling time and maximum overshoot were chosen the same as in the minimum phase case: 600 , 1.05 *s T t sM* corresponding to max 5% . Related values of other design parameters are: 0 min *PM* 56.87 and required crossover frequency 0 0.0131 . The required phase margin *PM PM req* min was chosen 0 60 *PMreq* . Achieved design results are summarized in Tab. 6 and Bode plots of compensated equivalent subsystems in Fig.18 prove achieving required design parameters.

Table 6. Design results and achieved frequency domain performance measures for the nonminimum phase case

Fig. 18. Discrete Bode plots of equivalent subsystems under designed PI controllers: 12 ( ) *eq G z* (left), 22( ) *eq G z* (right)

y

Robust Decentralized PID Controller Design 165

10-4 10-3 10-2 10-1 100 0

w [rad/sec]

Fulfilment of robust stability condition (70) is examined in Fig. 21. The closed-loop system is

10-4 10-3 10-2 10-1 100 0

w [rad/sec]

The robust decentralized PID controller design procedures have been developed both in frequency and time domains. The proposed controller design schemes are based on different principles, with the same control aim: to achieve robust stability and specified performance. The comparative study of both approaches is presented on robust decentralized discrete-

*T j*

LO sigma max (T)

1 [ ( )] ( ) ( ) *O o*

 *L*

 around the

Roots\_of\_CLCP { 0.6768 0.2761i, 0.7335 0.2262i, 0. 7165, 0.6164, 0.5876, 0.3313

The sensitivity magnitude plot in Fig. 20 with the peak max{ [ ( )]} 2 *M S j*

crossover frequency proves good closed-loop nominal performance.

0.5

stable over the whole non-minimum phase region (8).

0.5 1 1.5 2 2.5 3 3.5 4 4.5

Fig. 21. Verification of the robust stability condition max 0

sigmaMax(S)

Fig. 20. [ ( )] *M j T z e S z*

**5. Conclusion**

1

1.5

Fig. 19. Nominal closed-loop step responses of the quadruple tank system in non-minimum phase configuration (reference steps 0.1m occurred at t=0s at the input of the 1st subsystem, and at t=300s and t=10s, respectively, at the input of the 2nd subsystem). Maximum overshoot and settling time (600s) were kept in both cases.

Nominal closed-loop poles verify nominal stability.

subsyst. Ri(s) Ri(z) PMachieved achieved

Table 6. Design results and achieved frequency domain performance measures for the non-

1 1 0.2083 0.0923 ( ) 1 *z R z*

2 1 0.2376 0.1832 ( ) 1 *z R z*

0

Phase

Fig. 18. Discrete Bode plots of equivalent subsystems under designed PI controllers:

y1 y2

y1,y2 [cm]

Fig. 19. Nominal closed-loop step responses of the quadruple tank system in non-minimum phase configuration (reference steps 0.1m occurred at t=0s at the input of the 1st subsystem,

and at t=300s and t=10s, respectively, at the input of the 2nd subsystem). Maximum

Magnitude

50

*z*

*z*

1

1

54.170 0.0122

56.89 0.0120

10-4 10-3 10-2 10-1 100 -50

10-4 10-3 10-2 10-1 100 -200

omega

y

y1 y2

0 100 200 300 400 500 600 700 800 900 1000 20

t[s]

rad/s-1

rad/s-1

Eq.

12 ( ) *eq G z* 1

22( ) *eq G z* 2

0

y1,y2

Phase

Magnitude

50

minimum phase case

0.0039 *R s*( ) 0.2083

0.0030 *R s*( ) 0.2376

10-4 10-3 10-2 10-1 100 -50

10-4 10-3 10-2 10-1 100 -200

12 ( ) *eq G z* (left), 22( ) *eq G z* (right)

omega

0 100 200 300 400 500 600 700 800 900 100 20

t

Nominal closed-loop poles verify nominal stability.

overshoot and settling time (600s) were kept in both cases.

*s*

*s*

Roots\_of\_CLCP { 0.6768 0.2761i, 0.7335 0.2262i, 0. 7165, 0.6164, 0.5876, 0.3313

The sensitivity magnitude plot in Fig. 20 with the peak max{ [ ( )]} 2 *M S j* around the crossover frequency proves good closed-loop nominal performance.

Fig. 20. [ ( )] *M j T z e S z* - versus –frequency plot

Fulfilment of robust stability condition (70) is examined in Fig. 21. The closed-loop system is stable over the whole non-minimum phase region (8).

Fig. 21. Verification of the robust stability condition max 0 1 [ ( )] ( ) ( ) *O o T j L*

## **5. Conclusion**

The robust decentralized PID controller design procedures have been developed both in frequency and time domains. The proposed controller design schemes are based on different principles, with the same control aim: to achieve robust stability and specified performance. The comparative study of both approaches is presented on robust decentralized discrete-

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**2. Methodology and types of analysis employed in feeding and ontogenetic dietary shifts studies**

The majority of researchers have conducted feeding studies in feral fish populations based on diet descriptions of the stomach contents, using occurrence, numerical, gravimetric and volumetric methods. The main disadvantage of feeding studies is that fish are systematically killed in order to study their stomach contents. However, due to the decline of many natural fish populations, the studies that use non-lethal methods are now more frequent. Different techniques have been used to collect stomach contents without harming the fish such as gastric lavage, emetics or forceps [5-7]. The effectiveness of the gastric lavage is not related to the size of the trouts, but rather to the prey's own morphological characteristics, the degree of repletion of the stomach and the extent of digestion of the food [8]. The effectiveness of this method is inversely related to the degree of repletion [8].

#### **2.1. Prey selection analysis**

prey accessibility, prey size, energetic selection criteria and prey preference. In this context, the trophic niche for brown trout is very flexible and is usually broader in adults than juveniles.

272 New Advances and Contributions to Fish Biology

**Figure 1.** Graphical example of the trophic base and niche for brown trout.

A knowledge of the foraging ecology of fishes is fundamental to understanding the processes that function at the individual, population and community levels since the factors that influence the acquisition and assimilation of food can have significant consequences for the condition, growth, survival and recruitment of fishes [4]. In this context, the development of effective conservation programmes requires a clear understanding of fish ecological require‐ ments, so the knowledge of its feeding habits is essential to achieve this objective. For example, the knowledge on how food is shared among individuals of the same population is critical for understanding its functioning. Hence, conclusions of field studies on feeding could help wildlife managers to take measures to preserve fish populations, especially for threatened and exploited species. In this chapter we will briefly discuss the variables that are involved in the feeding behaviour of brown trout as an example of a predatory freshwater fish species.

Prey selection is an important part of fish feeding ecology. In order to study prey selection of fishes, several indices have been employed, such as the Savage index [9] and Ivlev's selectivity index [10]. The Savage index varies from zero (maximum negative selection) to infinity (maximum positive selection), whereas possible values of Ivlev's selectivity index range from –1 to +1, with negative values indicating rejection or inaccessibility of the prey, zero indicating random feeding, and positive values indicating active selection. Moreover, several researchers have demonstrated that studies based on food selection provide insight into factors involved in prey choice of brown trout [e.g. 11-13].

#### **2.2. Stomach content analysis**

In the early 80s Hyslop reviewed the methods used to study the feeding behaviour of fishes and their application to stomach content analyses [14]. Hyslop pointed out the difficulties in the application of these methods and, where appropriate, proposed alternative approaches. Food overlap between age classes can be assessed with Schoener's overlap index [15]. The overlap index has a minimum of 0 (no prey overlap), and a maximum of 1 (all prey items in equal proportions), and diet overlap is usually considered significant when value of the index exceeds 60% [16]. A chi-square (χ2) test can be used to test for significant differences in the diet composition between age classes [e.g. 17].

#### **2.3. Graphical methods**

Graphical methods proposed by Costello [18] and Tokeshi [19] were used to illustrate the relative importance of prey species and to assess the feeding strategy of fish species. Amundsen and collaborators designed an alternative method of Costello graphical method, by plotting prey abundance (Ai) (y - axis) against the frequency of occurrence in diet (Fi) (x - axis) for each prey species. Information on prey importance, feeding strategy and niche breadth can be obtained by examining the distribution of points along the diagonals and axes of the graph [20] (Figure 5 and Section 4.1).

in freshwater fish species. As shown in sections 4 and 5, it is an important tool to disentangle

Ontogenetic Dietary Shifts in a Predatory Freshwater Fish Species: The Brown Trout as an Example of a Dynamic...

http://dx.doi.org/10.5772/54133

275

In brown trout populations there is strong evidence for a critical period with high mortality in the first few weeks after fry emergence [3]. Furthermore, the most critical stage for popula‐ tion regulation in the whole life cycle is the density-dependent mortality of young trout in the first few weeks of the life cycle soon after the young fish start to feed [3 and references therein]. Thus, first feeding of newly emerged fry is very important for brown trout survival in this phase of the life cycle, and in newborns of brown trout first feeding can occur even prior to emergence [13,31,32]. In this sense, the feeding behaviour of newly emerged brown trout fry has been studied in both laboratory conditions and in natural spawning areas. Results of those studies show that feeding in recently emerged fry can be initiated before complete yolk exhaustion [13,31,32]. Zimmerman and Mosegaard observed that alevins of brown trout began feeding in experimental conditions when yolk constituted approximately 40% of the total alevin dry weight [31]. Other researchers have indicated that brown trout fry under natural conditions start feeding when having almost 30% of yolk sac remaining compared to the presumed original size of the yolk sac at hatching [32], while in a recent study no food particles

the food resource partitioning among both sympatric age classes and fish species.

have been found in the stomachs of fry having >10% of the yolk sac remaining [13].

spite of their abundance in the benthos [13].

The optimal foraging theory (OFT) explains adaptation via natural selection through quanti‐ tative models, which led to a better understanding of foraging behaviour. Hence, OFT predicts that predators should select prey that maximise the energetic gains available in relation to the energetic costs of capturing, ingesting and digesting the prey [33,34]. In this context, Many researchers have found that chironomid larvae and baetid nymphs seem to be the most important food items for newborns in different geographical areas [e.g. 13,32,35]. These are probably the most accessible invertebrates living in the gravel interstices on nesting grounds at the moment of emergence, providing over 80% of the energetic input [13]. However, although chironomid larvae and baetid nymphs seem to be the most important food items for newborns, newly emerged brown trout fry can show differences in the selection of these prey items. Although Baetidae is abundant in the benthos, this taxon is negatively selected accord‐ ing to Ivlev's selectivity index, whereas Chironomidae remains positively selected (Figure 2), demonstrating that abundance of prey items in the benthos is not the only factor explaining the complex mechanism that operates in the food selection during this phase of the ontogeny. Thus, prey size may affect the prey ingestion in early fish larvae, and much literature focuses on the relationship between prey size and mouth size as the primary factor of prey selection [e.g. 36]; but in general, other factors apart from size, such as locomotor skills of fish or accessibility and antipredator behaviour of prey items play an important role in feeding behaviour. These hypotheses that could explain the absence of some items in the stomachs in

**3. Diet composition of newly emerged brown trout fry**

#### **2.4. Niche breadth indexes**

Marshall and Elliott compared univariate and multivariate numerical and graphical techni‐ ques for determining inter- and intraspecific feeding relationships in estuarine fish [21] and on the basis of this study, different indices have been employed by ichthyologists to study niche breadth and diet specialisation. Generally, the Shannon diversity index was combined with the Levin's index to assess niche breadth [21] and the evenness index was used to evaluate diet specialisation, these being indices employed to study feeding habits in brown trout populations [22,23]. However, stable isotope analysis is a potentially powerful method of measuring trophic niche width, particularly when combined with conventional approaches [24]. For this reason, over the past two decades this methodology has been employed to study the trophic interactions and dietary niche in different fish species, and it has been recently used to study ontogeny and dietary specialization in brown trout [25,26].

#### **2.5. Multivariate approaches**

Recently prey trait analysis has been proposed as a functional approach to understand mechanisms involved in predator–prey relationships [27,28]. Despite the disadvantages of this methodology [29 and references therein], it has been used in order to get a deeper insight into the mechanisms that regulate diet composition and feeding habits of fishes, providing extremely valuable ecological information and complementing traditional diet analysis [23,29,30]. For the application of prey trait analysis, researchers have to use the same trait database and trait analyses as de Crespin de Billy [27]. To evaluate the potential vulnerability of invertebrates to fish predation, de Crespin de Billy and Usseglio-Polatera created a total of 71 different categories for 17 invertebrate traits [(1) macrohabitat, (2) current velocity, (3) substratum, (4) flow exposure, (5) mobility/attachment to substratum, (6) tendency to drift in the water column, (7) tendency to drift at the water surface, (8) trajectory on the bottom substratum or in the drift, (9) movement frequency, (10) diel drift behaviour, (11) agility, (12) aggregation tendency, (13) potential size, (14) concealment, (15) body shape (including cases/ tubes), (16) body flexibility (including cases/tubes) and (17) morphological defences] [28]. The information of this trait database is structured using a 'fuzzy coding' procedure; thus, a score is assigned to each taxon describing its affinity for each category of each trait, with '0' indicating 'no affinity' to '5' indicating 'high affinity'. The taxonomic resolution (order, family and genus) use in the classification process corresponds to the lowest possible level of determination of taxa in fish gut contents. When identification to genus is not possible or in the case of missing information for a certain genus, the value assign for a trait is that of the family level, using the average profile of all other genus of the same family. Additionally, all the taxa and their assigned scores for each category can be found in previous works [27,28]. Prey trait analysis should be carried out with the software R (version 2.11.1), its ADE4 library for the analysis in R is free and downloadable at http://cran.es.r-projet.org/. Finally, the analysis of prey traits has provided ichthyologists with important clues for understanding the ontogenetic dietary shifts in freshwater fish species. As shown in sections 4 and 5, it is an important tool to disentangle the food resource partitioning among both sympatric age classes and fish species.


4.1.4 Effect of superconductor temperature on GTM structure

In this subsection, we study the influence of superconductor's temperature on transmission spectrum of 1D hybrid GTM structure for different incidence levels. Thus, we evaluate the characteristics of multichannel. Indeed,

## Figure 5.

Variation of factor quality Q of the GTM quasiperiodic multilayered stack containing a superconducting material versus frequency f (THz) at the frequency range between 170 and 171 THz.

Photonic Quasicrystals for Filtering Application DOI: http://dx.doi.org/10.5772/intechopen.81572

## Figure 6.

A schematic view of transmittance spectra through the one dimensional photonic quasicrystals arranged according to GTM sequence for n = 2 and p = 7.

Figure 7.

A schematic view of distributed transmission of hybrid GTM photonic heterostructure versus frequency and incident angle for T = 20, 40, 60 and 80 K.

Figure 7 shows that GTM multilayer stack exhibits a specific zone with zero transmission (the yellow area) for different incident angles. In the corresponding band, the propagation wave is prohibited and reached the maximum recovers for θ = 1.5 rad.

Moreover, the spectrum presented a stack of band gaps and separated by sharp transmission peaks (the blue areas) allows the propagation of wave in this specific region of frequencies. The size of propagate zone within all PBG is sensitive to temperature T of YBa2Cu3O7. The width of transmission peak within the channels increases progressively with the increase of T. A large zero of reflection bands is also noticed for T = 80 K, it covers all optical telecommunication frequency range and it constitutes perfect reflectors in these region.