= *Kx*(*t*) + *ψ* (*x*,*e*,L, *t*) (4.7)

= F*e*(*t*) + L(*x*,*e*, *t*)*e*(*t*) (4.8)

= *AK* + *B*F. Note that if asymptotical stability of

F *BT*XF + QF = 0 (4.10)

P1 + P2 + ··· + PN + P*kBBT*P*k*

= *A*F *e*(*t*) + DΔ(*t*)E*x*(*t*) + *B*L(*x*,*e*, *t*)*e*(*t*) (4.9)

= P1 + P2 + ··· + PN + P*kBBT*P*k*(*k* = 1, ··· , N ) *satisfy the relation*

= *x*(*t*) − *x*(*t*). Beside, using the

matrix *AK*

input.

because *e*(*t*)

next subsection.

�

*Suppose that the matrices* S*k*

*γ* (*k*)

+ N −1 ∑ *j*=1

�

= *A* + *BK* is stable.

matrix. From (4.1), (4.5), (4.7) and (4.8), we have

*d*

In (4.9), *A*F ∈ **R***n*×*n* is a matrix given by *A*F

unique solution of the algebraic Riccati equation *AT*

�

P1 + P2 + ··· + PN + P*kBBT*P*k*

† i.e. S*k* ∈ **R***n*×*n* are symmetric positive definite matrices.

the parameter uncertainty, we introduce the error vector *e*(*t*)

*u*(*t*) �

*ψ* (*x*,*e*, L,*t*)

and nonlinear perturbations, and it is supposed to have the following structure.

�

non-quadratic Lyapunov functions have been used to improve the estimate of robust stability and to design robust stabilizing controllers(7; 30; 34). We have also proposed variable gain controllers and adaptive gain controllers based on Piecewise Lyapunov functions (PLFs) for a class of uncertain linear systems(23; 24). However, the resulting variable gain robust controllers may occur the chattering phenomenon. In this section, we propose a variable gain robust state feedback controller avoiding chattering phenomenon for a class of uncertain linear systems via PLFs and show that sufficient conditions for the existence of the proposed variable gain robust state feedback controller.

### **4.1 Problem formulation**

Consider a class of linear systems with non-linear perturbations represented by the following state equation (see **Remark 2**).

$$\frac{d}{dt}\mathbf{x}(t) = \left(A + \mathcal{D}\Delta(t)\mathcal{E}\right)\mathbf{x}(t) + \mathcal{B}u(t) \tag{4.1}$$

where *x*(*t*) ∈ **R***n* and *u*(*t*) ∈ **R***m* are the vectors of the state (assumed to be available for feedback) and the control input, respectively. In (4.1), the matrices *A* and *B* denote the nominal values of the system, and the matrix *B* has full column rank. The matrices D and E which have appropriate dimensions represent the structure of uncertainties. The matrix Δ(*t*) ∈ **R***p*×*q* represents unknown time-varying parameters and satisfies the relation Δ(*t*) ≤ 1. Note that the uncertain term DΔ(*t*)E consists of matched part and unmatched one. Additionally, introducing the integer N ∈ **Z**+ defined as

$$\mathcal{N} \stackrel{\triangle}{=} \arg\min\_{\mathcal{Z} \in \mathbb{Z}^+} \left\{ \mathcal{Z} \mid (\mathcal{Z}m - n) \ge 0 \right\} \tag{4.2}$$

we assume that there exist symmetric positive definite matrices S*k* ∈ **R***n*×*n* (*k* = 1, ··· , N ) which satisfies the following relation(23; 24).

$$\bigcap\_{k=1}^{N} \Omega\_{\mathcal{S}\_k} = \{0\} \tag{4.3}$$

where ΩS*k* represents a subspace defined as

$$\Omega\_{\mathcal{S}\_k} \stackrel{\triangle}{=} \left\{ \mathbf{x} \in \mathbb{R}^n \mid \boldsymbol{B}^T \mathcal{S}\_k \mathbf{x} = \mathbf{0} \right\} \tag{4.4}$$

The nominal system, ignoring the unknown parameter in (4.1), is given by

$$\frac{d}{dt}\overline{\mathbf{x}}(t) = A\overline{\mathbf{x}}(t) + B\overline{u}(t) \tag{4.5}$$

where *x*(*t*) ∈ **R***n* and *u*(*t*) ∈ **R***m* are the vectors of the state and the control input, respectively. First of all, we adopt the standard linear quadratic LQ control theory for the nominal system (4.5) in order to generate the desirable transient response for the plant systematically, i.e. the control input is given by *u*(*t*) = *Kx*(*t*). Note that some other design method so as to generate the desired response for the controlled system can also be used (e.g. pole assignment). Thus the feedback gain matrix *K* ∈ **R***m*×*n* is derived as *K* = −R−1*BT*P where P ∈ **R***n*×*n* is unique solution of the algebraic Riccati equation

$$A^T \mathcal{P} + \mathcal{P}A - \mathcal{P}B\mathcal{R}^{-1}B^T \mathcal{P} + \mathcal{Q} = 0\tag{4.6}$$

18 Will-be-set-by-IN-TECH

non-quadratic Lyapunov functions have been used to improve the estimate of robust stability and to design robust stabilizing controllers(7; 30; 34). We have also proposed variable gain controllers and adaptive gain controllers based on Piecewise Lyapunov functions (PLFs) for a class of uncertain linear systems(23; 24). However, the resulting variable gain robust controllers may occur the chattering phenomenon. In this section, we propose a variable gain robust state feedback controller avoiding chattering phenomenon for a class of uncertain linear systems via PLFs and show that sufficient conditions for the existence of the proposed

Consider a class of linear systems with non-linear perturbations represented by the following

where *x*(*t*) ∈ **R***n* and *u*(*t*) ∈ **R***m* are the vectors of the state (assumed to be available for feedback) and the control input, respectively. In (4.1), the matrices *A* and *B* denote the nominal values of the system, and the matrix *B* has full column rank. The matrices D and E which have appropriate dimensions represent the structure of uncertainties. The matrix Δ(*t*) ∈ **R***p*×*q*

that the uncertain term DΔ(*t*)E consists of matched part and unmatched one. Additionally,

we assume that there exist symmetric positive definite matrices S*k* ∈ **R***n*×*n* (*k* = 1, ··· , N )

*x* ∈ **R***n* | *BT*S*kx* = 0

where *x*(*t*) ∈ **R***n* and *u*(*t*) ∈ **R***m* are the vectors of the state and the control input, respectively. First of all, we adopt the standard linear quadratic LQ control theory for the nominal system (4.5) in order to generate the desirable transient response for the plant systematically, i.e. the control input is given by *u*(*t*) = *Kx*(*t*). Note that some other design method so as to generate the desired response for the controlled system can also be used (e.g. pole assignment). Thus the feedback gain matrix *K* ∈ **R***m*×*n* is derived as *K* = −R−1*BT*P where P ∈ **R***n*×*n* is unique

represents unknown time-varying parameters and satisfies the relation

= arg min

 N

*k*=1

*dt x*(*t*) = (*A* + DΔ(*t*)E) *x*(*t*) + *Bu*(*t*) (4.1)

Z∈**Z**+ {Z |(Z*m* − *n*) ≥ 0} (4.2)

ΩS*k* = {0} (4.3)

*dt x*(*t*) = *Ax*(*t*) + *Bu*(*t*) (4.5)

*AT*P + P *A* − P*B*R−1*BT*P + Q = 0 (4.6)

Δ(*t*)

≤ 1. Note

(4.4)

variable gain robust state feedback controller.

introducing the integer N ∈ **Z**+ defined as

which satisfies the following relation(23; 24).

where ΩS*k* represents a subspace defined as

solution of the algebraic Riccati equation

*d*

N �

ΩS*k* � =

The nominal system, ignoring the unknown parameter in (4.1), is given by *d*

**4.1 Problem formulation**

state equation (see **Remark 2**).

In (4.6), the matrices Q ∈ **R***n*×*n* and R ∈ **R***m*×*m* are design parameters and Q is selected such that the pair (*A*, C) is detectable, where C is any matrix satisfying Q = CC*T*, and then the matrix *AK* � = *A* + *BK* is stable.

Now on the basis of the works of Oya et al.(21; 22), in order to obtain on-line information on the parameter uncertainty, we introduce the error vector *e*(*t*) � = *x*(*t*) − *x*(*t*). Beside, using the optimal gain matrix *K* ∈ **R***m*×*n* for the nominal system (4.5), we consider the following control input.

$$
\mu(t) \stackrel{\triangle}{=} \text{Kx}(t) + \psi\left(\mathbf{x}, e, \mathcal{L}, t\right) \tag{4.7}
$$

where *ψ* (*x*,*e*,L, *t*) ∈ **R***m* is a compensation input so as to reduce the effect of uncertainties and nonlinear perturbations, and it is supposed to have the following structure.

$$
\psi\left(\mathbf{x},\mathbf{e},\mathcal{L},t\right) \stackrel{\triangle}{=} \mathcal{F}e(t) + \mathcal{L}(\mathbf{x},\mathbf{e},t)e(t) \tag{4.8}
$$

where F ∈ **R***m*×*n* is a fixed gain matrix and L(*x*,*e*, *t*) ∈ **R***m*×*n* is an adjustable time-varying matrix. From (4.1), (4.5), (4.7) and (4.8), we have

$$\frac{d}{dt}e(t) = \left(A + \mathcal{D}\Delta(t)\mathcal{E}\right)\mathbf{x}(t) + B\left\{\mathbf{K}\mathbf{x}(t) + \boldsymbol{\psi}\left(\mathbf{x}, \mathbf{e}, \mathcal{L}, t\right)\right\}$$

$$= A\_{\mathcal{F}}e(t) + \mathcal{D}\Delta(t)\mathcal{E}\mathbf{x}(t) + B\mathcal{L}(\mathbf{x}, \mathbf{e}, t)e(t) \tag{4.9}$$

In (4.9), *A*F ∈ **R***n*×*n* is a matrix given by *A*F � = *AK* + *B*F. Note that if asymptotical stability of the uncertain error system (4.9) is ensured, then the uncertain system (4.1) is robustly stable, because *e*(*t*) � = *x*(*t*) − *x*(*t*). Here, the fixed gain matrix F ∈ **R***m*×*n* is determined by using LQ control theory for the nominal error system. Namely F = −RF *BT*XF and XF ∈ **R***n*×*n* is unique solution of the algebraic Riccati equation

$$A\_K^T \mathcal{X}\_{\mathcal{F}} + \mathcal{X}\_{\mathcal{F}} A\_K - \mathcal{X}\_{\mathcal{F}} B \mathcal{R}\_{\mathcal{F}}^{-1} B^T \mathcal{X}\_{\mathcal{F}} + \mathcal{Q}\_{\mathcal{F}} = 0 \tag{4.10}$$

where QF ∈ **R***n*×*n* and RF ∈ **R***m*×*m* are design parameters and symmetric positive definite matrices. A decision method of the time-varying matrix L(*x*,*e*, *t*) ∈ **R***m*×*n* will be stated in the next subsection.

From the above discussion, our control objective in this section is to design the robust stabilizing controller for the uncertain error system (4.9). That is to design the variable gain matrix L(*x*,*e*, *t*) ∈ **R***m*×*n* that the error system with uncertainties (4.9) is asymptotically stable.

#### **4.2 Synthesis of variable gain robust state feedback controllers via PLFs**

The following theorem gives sufficient conditions for the existence of the proposed controller.

**Theorem 4.** *Consider the uncertain error system (4.9) and the control input (4.7) and (4.8).*

*Suppose that the matrices* S*k* � = P1 + P2 + ··· + PN + P*kBBT*P*k*(*k* = 1, ··· , N ) *satisfy the relation (4.3), where* P*k* ∈ **R***n*×*n are symmetric positive definite matrices*† *satisfying the matrix inequalities*

$$\begin{aligned} \left(\mathcal{P}\_1 + \mathcal{P}\_2 + \dots + \mathcal{P}\_N + \mathcal{P}\_k \mathcal{B} \mathcal{B}^T \mathcal{P}\_k\right) A\_{\mathcal{F}} + A\_{\mathcal{F}}^T \left(\mathcal{P}\_1 + \mathcal{P}\_2 + \dots + \mathcal{P}\_N + \mathcal{P}\_k \mathcal{B} \mathcal{B}^T \mathcal{P}\_k\right) \\ + \sum\_{j=1}^{N-1} \gamma\_j^{(k)} \mathcal{P}\_k \mathcal{B} \mathcal{B}^T \mathcal{P}\_k + \mathcal{Q}\_k < 0 \quad (k = 1, \dots, N) \end{aligned} \tag{4.11}$$

† i.e. S*k* ∈ **R***n*×*n* are symmetric positive definite matrices.

for a Class of Uncertain Dynamical Systems 21

Synthesis of Variable Gain Robust Controllers for a Class of Uncertain Dynamical Systems 331

D*T*S*ke*(*t*)

D*T*S*ke*(*t*)

*eT*(*t*)P*kBBT*P*ke*(*t*)

*e*(*t*) < 0 for *k* = arg max*k*

*j* ≥ 0 (*j* = 1, ··· , N − 1, *k* = 1, ··· , N ) satisfying

*j* PN *BBT*PN − *γ*(N )

S*kA*F + *AT*

*k* {*λ*min {Q*k*}}, we obtain

*j* P1*BBT*P1 − *γ*(1)

By using **Lemma 4** (S-procedure), the inequality (4.19) is satisfied if and only if there exist

Noting that since the condition (4.11) is a sufficient condition for the matrix inequalities (4.20), if the inequalities (4.11) are satisfied, then the condition (4.20) is also satisfied. Therefore, we

> F S*k*

2 + *σ*(*t*) for *k* = arg max*k*

are given by

On the other hand, from the definition of the piecewise quadratic function, there always exist

≤ V (*e*, *t*) ≤ *η*+

D*T*S*ke*(*t*)

 E*x*(*t*)

 E*x*(*t*) 2

*e*(*t*) + *σ*(*t*) for *k* = arg max*k*

1 P2*BBT*P2 −···− *γ*(1)

1 P2*BBT*P2 −···− *γ*(N )

*e*(*t*)

 E*x*(*t*)

*BT*S*ke*(*t*)

*eT*(*t*)P*kBBT*P*ke*(*t*)

N −1PN *BBT*PN < 0

*e*(*t*) < −*eT*(*t*)Q*ke*(*t*) (4.21)

*eT*(*t*)P*kBBT*P*ke*(*t*)

  2 *BT*S*k*

 *e*(*t*)

*eT*(*t*)P*kBBT*P*ke*(*t*)

N −1PN −1*BBT*PN −1 < 0

(4.23)

2 (4.24)

(4.18)

(4.19)

(4.20)

(4.22)

*e*(*t*) + 2

function V(*e*, *t*).

*dt* V(*e*, *t*) ≤ *eT*(*t*)

*eT*(*t*)

F S1 +

F SN +

have the following relation.

*dt* V(*e*, *t*) ≤ −*ζk*

*e*(*t*)

Besides, by letting *ζk*

*d*

where *η*−

S*k* > 0 and *γ*(*k*)

S1*A*F + *AT*

SN *A*F + *AT*

+ 2*eT*(*t*)S*kB*

≤ *eT*(*t*)

for *k* = arg max*k*

Now we consider the following inequality.

S*kA*F + *AT*

N −1 ∑ *j*=1

*γ*(1)

*γ*(N )

*eT*(*t*)

. . .

N −1 ∑ *j*=1

> � = min

> > *e*(*t*)

and *η*+

two positive constants *δ*min and *δ*max such that for any *t* ≥ *t*0, *η*− *e*(*t*)

> *e*(*t*)

> > *η*− *e*(*t*) � = *δ*min *e*(*t*) 2

*η*+ *e*(*t*) � = *δ*max *e*(*t*)

S*kA*F + *AT*

 −

S*kA*F + *AT*

F S*k*  F S

 *σ*(*t*) +

F S

*d*

*In (4.11), γ*(*k*) *j* (*k* = 1, ··· , N , *j* = 1, ··· , N − 1) *are positive scalars and* Q*k* ∈ **R***n*×*n are symmetric positive definite matrices.*

*By using the matrices* S*k* ∈ **R***n*×*n,* L(*x*,*e*, *t*) ∈ **R***m*×*n is determined as*

$$\mathcal{L}(\mathbf{x}, e, t) = -\frac{\left( \left| \left| \mathcal{D}^T \mathcal{S}\_k e(t) \right| \left| \left| \mathcal{E} \mathbf{x}(t) \right| \right| \right)^2}{\left( \sigma(t) + \left| \left| \mathcal{D}^T \mathcal{S}\_k e(t) \right| \right| \left| \left| \mathcal{E} \mathbf{x}(t) \right| \right| \right) \left| \left| B^T \mathcal{S}\_k e(t) \right| \right|^2} B^T \mathcal{S}\_k$$

$$\text{for} \quad k = \arg\max\_k \left\{ e^T(t) \mathcal{P}\_k B B^T \mathcal{P}\_k e(t) \right\} \tag{4.12}$$

*In (4.12), σ*(*t*) ∈ **R**1 *is any positive uniform continuous and bounded function which satisfies*

$$\int\_{t\_0}^{t} \sigma(\tau) d\tau \le \sigma^\* < \infty \tag{4.13}$$

*where σ*∗ *is any positive constant and t*0 *denotes an initial time. Then the uncertain error system (4.9) are bounded and*

$$\lim\_{t \to \infty} e(t; t\_0, e(t\_0)) = 0 \tag{4.14}$$

*Namely, asymptotical stability of the uncertain error system (4.9) is ensured.*

*Proof.* Using symmetric positive definite matrices P*k* ∈ **R***n*×*n* (*k* = 1, ··· , N ) which satisfy (4.11), we introduce the following piecewise quadratic function.

$$\mathcal{V}(e,t) = e^T(t)\mathcal{S}\_k e(t) \quad \text{for} \quad k = \arg\max\_k \left\{ e^T(t)\mathcal{P}\_k B B^T \mathcal{P}\_k e(t) \right\} \quad \text{and} \quad k = 1, \cdots, N$$

$$= \max\_k \left\{ e^T(t)\mathcal{S}\_k e(t) \right\} \tag{4.15}$$

Note that the piecewise quadratic function V(*e*, *t*) is continuous and its level set is closed. The time derivative of the piecewise quadratic function V(*e*, *t*) can be written as

$$\frac{d}{dt}\mathcal{V}(e,t) = e^T(t)\left(\mathcal{S}\_k A\_{\mathcal{F}} + A\_{\mathcal{F}}^T \mathcal{S}\_k\right)e(t) + 2e^T(t)\mathcal{S}\_k \mathcal{D}\Delta(t)\mathcal{E}\mathbf{x}(t) + 2e^T(t)\mathcal{S}\_k \mathcal{B}\mathcal{L}(\mathbf{x}, e, t)e(t)$$

$$\text{for} \quad k = \arg\max\_k \left\{ e^T(t)\mathcal{P}\_k \mathcal{B}\mathcal{B}^T \mathcal{P}\_k e(t) \right\} \tag{4.16}$$

Now, using **Lemma 1**, we can obtain

$$\frac{d}{dt}\mathcal{V}(e,t) \le e^T(t) \left(\mathcal{S}\_k A\_{\mathcal{F}} + A\_{\mathcal{F}}^T \mathcal{S}\_k\right) e(t) + 2 \left\|\mathcal{D}^T \mathcal{S}\_k e(t)\right\| \left\|\mathcal{E}x(t)\right\|$$

$$+ 2e^T(t) \mathcal{S}\_k B \mathcal{L}(x, e, t) e(t) \quad \text{for} \quad k = \arg\max\_k \left\{ e^T(t) \mathcal{P}\_k B B^T \mathcal{P}\_k e(t) \right\} \tag{4.17}$$

Also, using the time-varying gain matrix (4.12) and the relation (4.17) and some trivial manipulations give the following relation for the time derivative of the piecewise quadratic function V(*e*, *t*).

20 Will-be-set-by-IN-TECH

D*T*S*ke*(*t*)

D*T*S*ke*(*t*)

*In (4.12), σ*(*t*) ∈ **R**1 *is any positive uniform continuous and bounded function which satisfies*

*By using the matrices* S*k* ∈ **R***n*×*n,* L(*x*,*e*, *t*) ∈ **R***m*×*n is determined as*

 *σ*(*t*) + for *k* = arg max*k*

 *t t*0

lim

*Namely, asymptotical stability of the uncertain error system (4.9) is ensured.*

(4.11), we introduce the following piecewise quadratic function.

F S*k*

F S*k*

+ 2*eT*(*t*)S*kB*L(*x*,*e*, *t*)*e*(*t*) for *k* = arg max*k*

S*kA*F + *AT*

V(*e*, *t*) = *eT*(*t*)S*ke*(*t*) for *k* = arg max*k*

*eT*(*t*)S*ke*(*t*)

S*kA*F + *AT*

= max *k*

Now, using **Lemma 1**, we can obtain

*dt* V(*e*, *t*) ≤ *eT*(*t*)

for *k* = arg max*k*

*dt* V(*e*, *t*) = *eT*(*t*)

L(*x*,*e*, *t*) = −

*j* (*k* = 1, ··· , N , *j* = 1, ··· , N − 1) *are positive scalars and* Q*k* ∈ **R***n*×*n are symmetric*

 E*x*(*t*) 2

*eT*(*t*)P*kBBT*P*ke*(*t*)

*eT*(*t*)P*kBBT*P*ke*(*t*)

D*T*S*ke*(*t*)

 E*x*(*t*)

*eT*(*t*)P*kBBT*P*ke*(*t*)

*BT*S*ke*(*t*)

*σ*(*τ*)*dτ* ≤ *σ*∗ < ∞ (4.13)

*t*→∞ *e*(*t*; *t*0,*e*(*t*0)) = 0 (4.14)

*e*(*t*) + 2*eT*(*t*)S*k*DΔ(*t*)E*x*(*t*) + 2*eT*(*t*)S*kB*L(*x*,*e*, *t*)*e*(*t*)

and *k* = 1, ··· , N

2 *BT*S*k*

(4.12)

(4.15)

(4.16)

(4.17)

 E*x*(*t*)

*where σ*∗ *is any positive constant and t*0 *denotes an initial time. Then the uncertain error system (4.9)*

*Proof.* Using symmetric positive definite matrices P*k* ∈ **R***n*×*n* (*k* = 1, ··· , N ) which satisfy

Note that the piecewise quadratic function V(*e*, *t*) is continuous and its level set is closed.

*eT*(*t*)P*kBBT*P*ke*(*t*)

*e*(*t*) + 2

Also, using the time-varying gain matrix (4.12) and the relation (4.17) and some trivial manipulations give the following relation for the time derivative of the piecewise quadratic

The time derivative of the piecewise quadratic function V(*e*, *t*) can be written as

*In (4.11), γ*(*k*)

*are bounded and*

*d*

*d*

*positive definite matrices.*

$$\begin{split} \frac{d}{dt}\mathcal{V}(e,t) &\leq \,\,\,^{T}(t)\left(\mathcal{S}\_{k}A\_{\mathcal{F}}+A\_{\mathcal{F}}^{\top}\mathcal{S}\right)e(t)+2\left\|\mathcal{D}^{\top}\mathcal{S}\_{k}e(t)\right\|\,\,\left\|\mathcal{E}x(t)\right\| \\ &\quad +2\,^{T}(t)\mathcal{S}\_{k}B\left\{-\frac{\left(\left\|\mathcal{D}^{\top}\mathcal{S}\_{k}e(t)\right\|\left\|\mathcal{E}x(t)\right\|\right)^{2}}{\left(\sigma(t)+\left\|\mathcal{D}^{\top}\mathcal{S}\_{k}e(t)\right\|\left\|\mathcal{E}x(t)\right\|\right)\left\|\mathcal{B}^{\top}\mathcal{S}\_{k}e(t)\right\|^{2}}B^{\top}\mathcal{S}\_{k}\right\}e(t) \\ &\quad\quad\,\,\,^{\text{for}}\,\,\,\,k=\operatorname\*{arg\,max}\_{k}\left\{e^{T}(t)\mathcal{P}\_{k}BB^{\top}\mathcal{P}\_{k}e(t)\right\} \\ &\leq \,\,e^{T}(t)\left(\mathcal{S}\_{k}A\_{\mathcal{F}}+A\_{\mathcal{F}}^{\top}\mathcal{S}\right)e(t)+\sigma(t) \quad\,\,\,\,\,k=\operatorname\*{arg\,max}\_{k}\left\{e^{T}(t)\mathcal{P}\_{k}BB^{\top}\mathcal{P}\_{k}e(t)\right\} \end{split} \tag{4.18}$$

Now we consider the following inequality.

$$e^{T}(t)\left(\mathcal{S}\_{k}A\_{\mathcal{F}} + A\_{\mathcal{F}}^{T}\mathcal{S}\_{k}\right)e(t) < 0 \quad \text{for} \quad k = \arg\max\_{k}\left\{e^{T}(t)\mathcal{P}\_{k}BB^{T}\mathcal{P}\_{k}e(t)\right\} \tag{4.19}$$

By using **Lemma 4** (S-procedure), the inequality (4.19) is satisfied if and only if there exist S*k* > 0 and *γ*(*k*) *j* ≥ 0 (*j* = 1, ··· , N − 1, *k* = 1, ··· , N ) satisfying

$$\begin{aligned} \mathcal{S}\_1 A\_{\mathcal{F}} + A\_{\mathcal{F}}^T \mathcal{S}\_1 + \sum\_{j=1}^{N-1} \gamma\_j^{(1)} \mathcal{P}\_1 \mathcal{B} \mathcal{B}^T \mathcal{P}\_1 - \gamma\_1^{(1)} \mathcal{P}\_2 \mathcal{B} \mathcal{B}^T \mathcal{P}\_2 - \dots - \gamma\_{N-1}^{(1)} \mathcal{P}\_N \mathcal{B} \mathcal{B}^T \mathcal{P}\_N &< 0 \\ \vdots \\ \mathcal{S}\_N A\_{\mathcal{F}} + A\_{\mathcal{F}}^T \mathcal{S}\_N + \sum\_{j=1}^{N-1} \gamma\_j^{(N)} \mathcal{P}\_N \mathcal{B} \mathcal{B}^T \mathcal{P}\_N - \gamma\_1^{(N)} \mathcal{P}\_2 \mathcal{B} \mathcal{B}^T \mathcal{P}\_2 - \dots - \gamma\_{N-1}^{(N)} \mathcal{P}\_{N-1} \mathcal{B} \mathcal{B}^T \mathcal{P}\_{N-1} &< 0 \end{aligned} \tag{4.20}$$

Noting that since the condition (4.11) is a sufficient condition for the matrix inequalities (4.20), if the inequalities (4.11) are satisfied, then the condition (4.20) is also satisfied. Therefore, we have the following relation.

$$\left(\boldsymbol{\varepsilon}^{T}(t)\left(\mathcal{S}\_{k}\boldsymbol{A}\_{\mathcal{F}} + \boldsymbol{A}\_{\mathcal{F}}^{T}\boldsymbol{\mathcal{S}}\_{k}\right)\boldsymbol{\varepsilon}(t) < -\boldsymbol{\varepsilon}^{T}(t)\,\mathcal{Q}\_{k}\boldsymbol{\varepsilon}(t)\tag{4.21}$$

Besides, by letting *ζk* � = min *k* {*λ*min {Q*k*}}, we obtain

$$\frac{d}{dt}\mathcal{V}(e,t) \le -\zeta\_k \left\| e(t) \right\|^2 + \sigma(t) \quad \text{for} \quad k = \arg\max\_k \left\{ e^T(t) \mathcal{P}\_k B B^T \mathcal{P}\_k e(t) \right\} \tag{4.22}$$

On the other hand, from the definition of the piecewise quadratic function, there always exist two positive constants *δ*min and *δ*max such that for any *t* ≥ *t*0,

$$
\eta^-\left(||e(t)||\right) \le \mathcal{V}\left(e, t\right) \le \eta^+\left(||e(t)||\right)\tag{4.23}
$$

where *η*− *e*(*t*) and *η*+ *e*(*t*) are given by

$$\begin{array}{l} \eta^{-} \left( ||e(t)|| \right) \stackrel{\triangle}{=} \delta\_{\text{min}} \left||e(t)|| \right|^{2} \\ \eta^{+} \left( ||e(t)|| \right) \stackrel{\triangle}{=} \delta\_{\text{max}} \left||e(t)|| \right|^{2} \end{array} \tag{4.24}$$

for a Class of Uncertain Dynamical Systems 23

Synthesis of Variable Gain Robust Controllers for a Class of Uncertain Dynamical Systems 333

**Remark 3.** *In order to get the proposed controller, symmetric positive definite matrices* S*k* ∈ **R***n*×*n* (*k* = 1, ··· , N ) *satisfying the assumption (4.3) are required. The condition (4.3) is reduced*

S1*B* S2*B* ··· SN *B*

*However there is not a globally effective method to obtain matrices satisfying the conditions (4.32). In*

**Remark 4.** *In this section, we introduce the compensation input (4.8). From (4.8) and (4.12), one can see that if e*(*t*) = 0*, then the relation ψ* (*x*,*e*,L, *t*) ≡ 0 *is satisfied. Beside, we find that the variable*

Now, we consider the condition (4.11) in **Theorem 4**. The condition (4.11) requires symmetric

section, on the basis of the works of Oya et al.(23; 24), we consider the following inequalities

1, ··· , N ) and using **Lemma 3** (Schur complement), we find that the condition (4.33)

*Ψ* (P1, ··· ,PN ) + Q*k* P*kB* P*kB* ··· P*kB*

(*k*)

. . . .

*BT*P*k* 0 00 −*ξ*

Note that if there exist symmetric positive definite matrices P*k* ∈ **R***n*×*n* and positive scalars

From the above discussion, one can see that in order to get the proposed robust controller,

satisfy the LMIs (4.34) and the assumption (4.3) are needed. Therefore firstly, we solve the

*j* ∈ **R**1 which satisfy the matrix inequalities (4.34), then the matrix inequality condition

�*T*�

F (P1 + P2 + ··· + PN )

*j* P*kBBT*P*k* + Q*k* < 0 (*k* = 1, ··· , N ) (4.33)

(*k*) *j* � = � *γ*(*k*) *j* �−1

1 *Im* 0 ··· 0

2 *Im* ··· 0

. ... .

*j* ∈ **R**1 and the symmetric positive definite matrices P*k* ∈ **R***n*×*n* which

. . ⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ < 0,

F (P1 + P2 + ··· + PN ) (4.35)

(*k*) N −1*Im*

(*k*)

*j* > 0 (*j* = 1, ··· , N − 1, *k* = 1, ··· , N )

= *n* (4.32)

*j* ∈ **R**1 for stability. In this

(*j* = 1, ··· , N − 1, *k* =

(4.34)

rank ��

*future work, we will examine the assumption (4.3) and the condition (4.32).*

*gain matrix* L(*x*,*e*, *t*) ∈ **R***m*×*n can be calculated except for e*(*t*) = 0 *(see (24)).*

(P1 + P2 + ··· + PN ) *A*F + *AT*

*BT*P*k* −*ξ*

*BT*P*k* 0 −*ξ*

. .

positive definite matrices P*k* ∈ **R***n*×*n* and positive scalars *γ*(*k*)

*γ*(*k*)

In addition, introducing complementary variables *ξ*

. .

(*k*)

LMIs (4.34) and next, we check the rank condition (4.32).

where *Ψ* (P1, ··· ,PN ) in (1, 1)-block of the LMIs (4.34) is given by *Ψ* (P1, ··· ,PN ) = (P1 + P2 + ··· + PN ) *A*F + *AT*

P*k* > 0 and *ξ*

*to the following rank condition.*

instead of (4.11).

+ N −1 ∑ *j*=1

equivalent to the following LMIs. ⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

(4.11) is also satisfied (23; 24).

the positive scalars *γ*(*k*)

*γ*(*k*)

It is obvious that any solution *e*(*t*; *t*0,*e*(*t*0)) of the error system is continuous. In addition, it follows from (4.22) and (4.23), that for any *t* ≥ *t*0, the following relation holds.

$$\begin{split} 0 \le \eta^- \left( \| \| e(t) \| \right) \le \mathcal{V} \left( e, t \right) = \mathcal{V} \left( e, t\_0 \right) + \int\_{t\_0}^t \frac{d}{dt} \mathcal{V} (e, \tau) d\tau \\ \mathcal{V} \left( e, t\_0 \right) + \int\_{t\_0}^t \frac{d}{dt} \mathcal{V} (e, \tau) d\tau \le \eta^+ \left( \| \| e(t\_0) \| \right) - \int\_{t\_0}^t \eta^\* \left( \| e(\tau) \| \right) d\tau + \int\_{t\_0}^t \sigma(\tau) d\tau \end{split} \tag{4.25}$$

In (4.25), *η*∗ *e*(*t*) is defined as

$$\eta^\*\left(\left||e(t)\right||\right) \stackrel{\triangle}{=} \zeta\_k\left||e(t)\right||^2\tag{4.26}$$

Therefore, from (4.25) we can obtain the following two results. Firstly, taking the limit as *t* approaches infinity on both sides of the inequality (4.25), we have

$$0 \le \eta^+\left(||e(t\_0)||\right) - \lim\_{t \to \infty} \int\_{t\_0}^t \eta^\*\left(||e(\tau)||\right)d\tau + \lim\_{t \to \infty} \int\_{t\_0}^t \sigma(\tau)d\tau\tag{4.27}$$

Thus one can see from (4.13) and (4.27) that

$$\lim\_{t \to \infty} \int\_{t\_0}^t \eta^\* \left( ||\boldsymbol{\varepsilon}(\tau)|| \right) d\tau \le \eta^+ \left( ||\boldsymbol{\varepsilon}(t\_0)|| \right) + \sigma^\* \tag{4.28}$$

On the other hand, from (4.25), we obtain

$$0 \le \eta^- \left( \| \| e(t) \| \right) \le \eta^+ \left( \| e(t\_0) \| \right) + \int\_{t\_0}^t \sigma(\tau) d\tau \tag{4.29}$$

It follows from (4.13) and (4.29) that

$$0 \le \eta^- \left( \| \| e(t) \| \right) \le \eta^+ \left( \| e(t\_0) \| \right) + \sigma^\* \tag{4.30}$$

The relation (4.30) implies that *e*(*t*) is uniformly bounded. Since *e*(*t*) has been shown to be continuous, it follows that *e*(*t*) is uniformly continuous. Therefore, one can see from the definition that *η*∗ *e*(*t*) is also uniformly continuous. Applying the **Lemma 3** ( Barbalat's lemma ) to (4.28) yields

$$\lim\_{t \to \infty} \eta^\* \left( \left||e(t)\right|| \right) = \lim\_{t \to \infty} \mathbb{Z}\_k \left||e(t)\right|| = 0 \tag{4.31}$$

Namely, asymptotical stability of the uncertain error system (4.9) is ensured. Thus the uncertain linear system (4.1) is also stable. Thus the proof of **Theorem 4** is completed.

**Remark 2.** *In this section, we consider the uncertain dynamical system (4.1) which has uncertainties in the state matrix only. The proposed robust controller can also be applied to the case that the uncertainties are included in both the system matrix and the input one. By introducing additional actuator dynamics and constituting an augmented system, uncertainties in the input matrix are embedded in the system matrix of the augmented system(36). Therefore the same design procedure can be applied.*

22 Will-be-set-by-IN-TECH

It is obvious that any solution *e*(*t*; *t*0,*e*(*t*0)) of the error system is continuous. In addition, it

 *t t*0 *d*

Therefore, from (4.25) we can obtain the following two results. Firstly, taking the limit as *t*

*dτ* ≤ *η*+

*e*(*t*0) + *t t*0

≤ *η*+

The relation (4.30) implies that *e*(*t*) is uniformly bounded. Since *e*(*t*) has been shown to be continuous, it follows that *e*(*t*) is uniformly continuous. Therefore, one can see from the

> = lim *t*→∞ *ζk e*(*t*)

Namely, asymptotical stability of the uncertain error system (4.9) is ensured. Thus the

**Remark 2.** *In this section, we consider the uncertain dynamical system (4.1) which has uncertainties in the state matrix only. The proposed robust controller can also be applied to the case that the uncertainties are included in both the system matrix and the input one. By introducing additional actuator dynamics and constituting an augmented system, uncertainties in the input matrix are embedded in the system matrix of the augmented system(36). Therefore the same design procedure*

*e*(*t*0)

is also uniformly continuous. Applying the **Lemma 3** ( Barbalat's

*dt* V(*e*, *τ*)*dτ*

*dτ* + lim *t*→∞

*e*(*t*0)   *t t*0

*σ*(*τ*)*dτ*

*σ*(*τ*)*dτ* (4.27)

+ *σ*∗ (4.28)

*σ*(*τ*)*dτ* (4.29)

+ *σ*∗ (4.30)

= 0 (4.31)

2 (4.26)

(4.25)

follows from (4.22) and (4.23), that for any *t* ≥ *t*0, the following relation holds.

*η*∗ *e*(*t*) � = *ζk e*(*t*)

approaches infinity on both sides of the inequality (4.25), we have

*e*(*t*0) − *t t*0 *η*∗ *e*(*τ*) *dτ* + *t t*0

 *t t*0 *η*∗ *e*(*τ*)

≤ *η*+

≤ V (*e*, *t*) = V (*e*, *t*0) +

*dt* V(*e*, *τ*)*dτ* ≤ *η*+

is defined as

*e*(*t*0) − lim *t*→∞

lim *t*→∞  *t t*0 *η*∗ *e*(*τ*)

0 ≤ *η*−

*e*(*t*)

> *e*(*t*)

> *e*(*t*)

0 ≤ *η*−

lim *t*→∞ *η*∗

0 ≤ *η*−

V (*e*, *t*0) +

In (4.25), *η*∗

*e*(*t*)

*e*(*t*)

 *t t*0 *d*

0 ≤ *η*+

Thus one can see from (4.13) and (4.27) that

On the other hand, from (4.25), we obtain

*e*(*t*)

uncertain linear system (4.1) is also stable. Thus the proof of **Theorem 4** is completed.

It follows from (4.13) and (4.29) that

definition that *η*∗

*can be applied.*

lemma ) to (4.28) yields

**Remark 3.** *In order to get the proposed controller, symmetric positive definite matrices* S*k* ∈ **R***n*×*n* (*k* = 1, ··· , N ) *satisfying the assumption (4.3) are required. The condition (4.3) is reduced to the following rank condition.*

$$\text{rank}\left\{ \left( \mathcal{S}\_1 \mathcal{B} \; \mathcal{S}\_2 \mathcal{B} \; \cdots \; \mathcal{S}\_N \mathcal{B} \right)^T \right\} = n \tag{4.32}$$

*However there is not a globally effective method to obtain matrices satisfying the conditions (4.32). In future work, we will examine the assumption (4.3) and the condition (4.32).*

**Remark 4.** *In this section, we introduce the compensation input (4.8). From (4.8) and (4.12), one can see that if e*(*t*) = 0*, then the relation ψ* (*x*,*e*,L, *t*) ≡ 0 *is satisfied. Beside, we find that the variable gain matrix* L(*x*,*e*, *t*) ∈ **R***m*×*n can be calculated except for e*(*t*) = 0 *(see (24)).*

Now, we consider the condition (4.11) in **Theorem 4**. The condition (4.11) requires symmetric positive definite matrices P*k* ∈ **R***n*×*n* and positive scalars *γ*(*k*) *j* ∈ **R**1 for stability. In this section, on the basis of the works of Oya et al.(23; 24), we consider the following inequalities instead of (4.11).

$$\left(\mathcal{P}\_1 + \mathcal{P}\_2 + \dots + \mathcal{P}\_N\right) A\_{\mathcal{F}} + A\_{\mathcal{F}}^T \left(\mathcal{P}\_1 + \mathcal{P}\_2 + \dots + \mathcal{P}\_N\right)$$

$$+ \sum\_{j=1}^{N-1} \gamma\_j^{(k)} \mathcal{P}\_k B B^T \mathcal{P}\_k + \mathcal{Q}\_k < 0 \quad (k = 1, \dots, N) \tag{4.33}$$

In addition, introducing complementary variables *ξ* (*k*) *j* � = � *γ*(*k*) *j* �−1 (*j* = 1, ··· , N − 1, *k* = 1, ··· , N ) and using **Lemma 3** (Schur complement), we find that the condition (4.33) equivalent to the following LMIs.

$$\begin{pmatrix} \Psi \begin{pmatrix} \mathcal{P}\_{1} \dots \mathcal{P}\_{N} \end{pmatrix} + \mathcal{Q}\_{k} & \mathcal{P}\_{k}B & \mathcal{P}\_{k}B & \cdots & \mathcal{P}\_{k}B \\\\ B^{T}\mathcal{P}\_{k} & -\mathfrak{f}\_{1}^{(k)}I\_{m} & 0 & \cdots & 0 \\\\ B^{T}\mathcal{P}\_{k} & 0 & -\mathfrak{f}\_{2}^{(k)}I\_{m} & \cdots & 0 \\\\ \vdots & \vdots & \vdots & \ddots & \vdots \\\\ B^{T}\mathcal{P}\_{k} & 0 & 0 & 0 & -\mathfrak{f}\_{N-1}^{(k)}I\_{m} \end{pmatrix} < 0,\\\\ \mathcal{P}\_{k} > 0 \text{ and } \mathfrak{f}\_{j}^{(k)} > 0 \quad (j = 1, \dots, N-1, \ k = 1, \dots, N) \end{pmatrix} < 0. \tag{4.34}$$

where *Ψ* (P1, ··· ,PN ) in (1, 1)-block of the LMIs (4.34) is given by

$$\Psi\left(\mathcal{P}\_{1},\cdots,\mathcal{P}\_{N}\right) = \left(\mathcal{P}\_{1} + \mathcal{P}\_{2} + \cdots + \mathcal{P}\_{N}\right)A\_{\mathcal{F}} + A\_{\mathcal{F}}^{T}\left(\mathcal{P}\_{1} + \mathcal{P}\_{2} + \cdots + \mathcal{P}\_{N}\right)\tag{4.35}$$

Note that if there exist symmetric positive definite matrices P*k* ∈ **R***n*×*n* and positive scalars *γ*(*k*) *j* ∈ **R**1 which satisfy the matrix inequalities (4.34), then the matrix inequality condition (4.11) is also satisfied (23; 24).

From the above discussion, one can see that in order to get the proposed robust controller, the positive scalars *γ*(*k*) *j* ∈ **R**1 and the symmetric positive definite matrices P*k* ∈ **R***n*×*n* which satisfy the LMIs (4.34) and the assumption (4.3) are needed. Therefore firstly, we solve the LMIs (4.34) and next, we check the rank condition (4.32).

for a Class of Uncertain Dynamical Systems 25

Synthesis of Variable Gain Robust Controllers for a Class of Uncertain Dynamical Systems 335

Besides, for numerical simulations, the initial values for the uncertain linear system (4.36)

The results of the simulation of this example are depicted in Figures 8–10. In these Figures, "Case 1)" and "Case 2)" represent the time-histories of the state variables *x*1(*t*) and *x*2(*t*) and the control input *u*(*t*) for the proposed variable gain robust controller. "Desired" shows the desired time-response and the desired control input generated by the nominal system. From Figures 8–10, we find that the proposed robust controller stabilizes the uncertain system (4.36) in spite of uncertainties. one can see from Figure 10 the proposed controller can avoid

0 0.5 1 1.5 2 2.5 3

Case 1) Case 2) Desired

Time

0 0.5 1 1.5 2 2.5 3

Case 1) Case 2) Desired

Time

2.0 −1.0*T* (i.e. *e*(0) =

0.0 0.0*T*),

 .

−1.0 × 10−3*t*

0 − sin(3.0*πt*)

respectively, and we choose *σ*(*t*) ∈ **R**+ in (4.12) as *σ*(*t*) = 5.0 × 1012 × exp

serious chattering. Therefore the effectiveness of the proposed controller is shown.

and the nominal system are selected as *x*(0) = *x*(0) =

State

Fig. 9. Time histories of the state *x*2(*t*)

0

0

0.5

State

Fig. 8. Time histories of the state *x*1(*t*)

1

1.5

2

• Case 2) : Δ(*t*) = cos(3.0*πt*) 0

#### **4.3 Illustrative examples**

Consider the following uncertain linear system, i.e. Z = 2.

$$\frac{d}{dt}\mathbf{x}(t) = \begin{pmatrix} -4 \ 1 \\ 0 \ 2 \end{pmatrix} \mathbf{x}(t) + \begin{pmatrix} 5 \ -1 \\ 0 \ 1 \end{pmatrix} \Delta(t) \begin{pmatrix} 1 \ 1 \\ 0 \ 1 \end{pmatrix} \mathbf{x}(t) + \begin{pmatrix} 0 \\ 1 \end{pmatrix} \boldsymbol{\mu}(t) \tag{4.36}$$

By applying **Theorem 4**, we consider deriving the proposed robust controller. Now we select the weighting matrices Q ∈ **R**2×2 and R ∈ **R**1×1 such as Q = 1.0*I*2 and R = 4.0 for the quadratic cost function for the standard linear quadratic control problem for the nominal system, respectively. Then solving the algebraic Riccati equation (4.6), we obtain the optimal gain matrix

$$K = \begin{pmatrix} -5.15278 \times 10^{-3} \ -4.06405 \end{pmatrix} \tag{4.37}$$

In addition, setting the design parameters QF and RF such as QF = 10.0 × 106 *I*2 and RF = 1.0, respectively, we have the following fixed gain matrix.

$$\mathcal{F} = \begin{pmatrix} -1.23056 \ -9.99806 \end{pmatrix} \times 10^3 \tag{4.38}$$

Besides, selecting the matrices Q*k* (*k* = 1, 2) in (4.34) as

$$\mathcal{Q}\_1 = \begin{pmatrix} 20.0 \ 1.0 \\ \\ 1.0 \ 1.0 \end{pmatrix}, \quad \mathcal{Q}\_2 = \begin{pmatrix} 1.0 \ 0.0 \\ \\ 0.0 \ 20.0 \end{pmatrix} \tag{4.39}$$

and solving the LMI condition (4.34), we get

$$\begin{array}{l} \mathcal{P}\_1 = \begin{pmatrix} 7.59401 \times 10^1 & 6.82676 \times 10^{-4} \\ 6.82676 \times 10^{-4} & 2.00057 \times 10^{-3} \\ 7.59401 \times 10^1 & 5.96286 \times 10^{-4} \\ 5.96286 \times 10^{-4} & 5.76862 \times 10^{-2} \end{pmatrix} \\ \mathcal{P}\_2 = \begin{pmatrix} 7.59402 \times 10^{-4} & 5.76862 \times 10^{-2} \\ 7.13182 \times 10^{-3} & \gamma\_2 = 7.13182 \times 10^{-3} \end{pmatrix} \end{array} \tag{4.40}$$

From (4.36) and (4.40), ΩS*k* (*k* = 1, 2) can be written as

$$\begin{aligned} \Omega\_{\mathcal{S}\_1} &= \{ \mathbf{x} \in \mathbb{R}^2 \mid 1.28240 \mathbf{x}\_1 + 7.80246 \mathbf{x}\_2 = 0 \} \\ \Omega\_{\mathcal{S}\_2} &= \{ \mathbf{x} \in \mathbb{R}^2 \mid 1.28032 \mathbf{x}\_1 + 7.77319 \mathbf{x}\_2 = 0 \} \end{aligned} \tag{4.41}$$

and thus the assumption (4.3) is satisfied.

On the other hand for the uncertain linear system (4.36), the quadratic stabilizing controller based on a fixed quadratic Lyapunov function cannot be obtained, because the solution of the LMI of (A.1) does not exist.

In this example, we consider the following two cases for the unknown parameter Δ(*t*).

$$\begin{pmatrix} \bullet & \text{Case 1} \\ \end{pmatrix} : \Delta(t) = \begin{pmatrix} -4.07360 \ 8.06857 \\ -4.41379 \ 3.81654 \end{pmatrix} \times 10^{-1}$$

$$\begin{array}{rcl} \bullet & \text{Case 2} \end{array} : \Delta(t) = \begin{pmatrix} \cos(3.0\pi t) & 0\\ 0 & -\sin(3.0\pi t) \end{pmatrix}.$$

24 Will-be-set-by-IN-TECH

� Δ(*t*)

By applying **Theorem 4**, we consider deriving the proposed robust controller. Now we select the weighting matrices Q ∈ **R**2×2 and R ∈ **R**1×1 such as Q = 1.0*I*2 and R = 4.0 for the quadratic cost function for the standard linear quadratic control problem for the nominal system, respectively. Then solving the algebraic Riccati equation (4.6), we obtain the optimal

In addition, setting the design parameters QF and RF such as QF = 10.0 × 106 *I*2 and RF =

−1.23056 −9.99806�

⎟⎠ , Q2 =

� 7.59401 × 101 6.82676 × 10−4 6.82676 × 10−4 2.00057 × 10−3

� 7.59401 × 101 5.96286 × 10−4 5.96286 × 10−4 5.76862 × 10−2

*γ*1 = 7.13182 × 10−3, *γ*2 = 7.13182 × 10−3

On the other hand for the uncertain linear system (4.36), the quadratic stabilizing controller based on a fixed quadratic Lyapunov function cannot be obtained, because the solution of the

× 10−1

In this example, we consider the following two cases for the unknown parameter Δ(*t*).

�

*x* ∈ **R**2 | 1.28240*x*1 + 7.80246*x*2 = 0}

⎛

1.0 0.0

0.0 20.0

⎜⎝

⎞

� 1 1

0 1

�

*x*(*t*) +

−5.15278 × 10−3 −4.06405� (4.37)

⎞

�

�

*x* ∈ **R**2 | 1.28032*x*1 + 7.77319*x*2 = 0} (4.41)

� 0 1

�

× 103 (4.38)

⎟⎠ (4.39)

(4.40)

*u*(*t*) (4.36)

� 5 −1 0 1

**4.3 Illustrative examples**

gain matrix

*d dt x*(*t*) =

Consider the following uncertain linear system, i.e. Z = 2.

�

*K* = �

F = �

⎛

20.0 1.0

1.0 1.0

⎜⎝

1.0, respectively, we have the following fixed gain matrix.

Besides, selecting the matrices Q*k* (*k* = 1, 2) in (4.34) as

P1 =

P2 =

From (4.36) and (4.40), ΩS*k* (*k* = 1, 2) can be written as

ΩS1 = �

ΩS2 = �

� −4.07360 8.06857 4.41379 3.81654

and thus the assumption (4.3) is satisfied.

LMI of (A.1) does not exist.

• Case 1) : Δ(*t*) =

and solving the LMI condition (4.34), we get

Q1 =

*x*(*t*) +

0 2

� −4 1

Besides, for numerical simulations, the initial values for the uncertain linear system (4.36) and the nominal system are selected as *x*(0) = *x*(0) = 2.0 −1.0*T* (i.e. *e*(0) = 0.0 0.0*T*), respectively, and we choose *σ*(*t*) ∈ **R**+ in (4.12) as *σ*(*t*) = 5.0 × 1012 × exp −1.0 × 10−3*t* .

The results of the simulation of this example are depicted in Figures 8–10. In these Figures, "Case 1)" and "Case 2)" represent the time-histories of the state variables *x*1(*t*) and *x*2(*t*) and the control input *u*(*t*) for the proposed variable gain robust controller. "Desired" shows the desired time-response and the desired control input generated by the nominal system.

From Figures 8–10, we find that the proposed robust controller stabilizes the uncertain system (4.36) in spite of uncertainties. one can see from Figure 10 the proposed controller can avoid serious chattering. Therefore the effectiveness of the proposed controller is shown.

Fig. 8. Time histories of the state *x*1(*t*)

Fig. 9. Time histories of the state *x*2(*t*)

for a Class of Uncertain Dynamical Systems 27

Synthesis of Variable Gain Robust Controllers for a Class of Uncertain Dynamical Systems 337

The future research subjects are an extension of the variable gain robust state feedback controller via PLFs to output feedback control systems. Besides, the problem for the extension to such a broad class of systems as uncertain large-scale systems, uncertain time-delay systems and so on should be tackled. Furthermore in future work, we will examine the condition (3.2)

On the other hand, the design of feedback controllers is often complicated by presence of physical constraints : saturating actuators, temperatures, pressures within safety margins and so on. If the constraints are violated, serious consequences may ensue, for example, physical components may be damaged, or saturation may cause a loss of closed-loop stability. In particular, input saturation is a common feature of control systems and the stabilization problems of linear systems with control input saturation have been studied (e.g. (17; 32)). Furthermore, some researchers have investigated analysis of constrained linear systems and reference managing for linear systems subject to input and state constraints (e.g. (10; 15)). Therefore, the future research subjects are to address the constrained robust control problems

The following lemma provides a LMI-based design method of a robust controller via

*There exists the state feedback gain matrix H* ∈ **R***m*×*n such that the control law u*(*t*) = *Hx*(*t*) *is a*

*If the solution* X ,Y *and δ of the LMI (A.1) exists, then the gain matrix H* ∈ **R***m*×*n is obtained as*

*T* P

*T* P

Here we have used the well-known relation (3.12). Thus the uncrtain linear system (4.1) is

*T* P + *δ*PDD*T*P +

= P−1 and consider the change of variable Y �

EX −*δIq*

*x*(*t*) + 2*xT*(*t*)PDΔ(*t*)E*x*(*t*)

*x*(*t*) + *δxT*(*t*)PDD*T*P*x*(*t*) + 1

1 *δ*

1 *δ*

= *exT*(*t*)P*x*(*t*) as a Lyapunov function

< 0 (A.1)

*δ*

E *T*E < 0 (A.3)

X E *T*EX < 0 (A.4)

= *H*X . Then, by pre-

*xT*(*t*)E *T*E*x*(*t*)

(A.2)

*A*X + X *AT* + *B*Y + Y*TBT* + *δ*DD*T* X E *T*

**Lemma A.1.** *Consider the uncertain linear system (4.1) and the control law u*(*t*) = *Hx*(*t*)*.*

*quadratic stabilizing control, if there exist* X > 0, Y *and δ* > 0 *satisfying the LMI*

in section 3 and assumptions (4.3) and (4.32) in section 4.

reducing the effect of unknown parameters.

*Proof.* Introducing the quadratic function V(*x*, *t*) �

P (*A* + *BH*) + (*A* + *BH*)

P (*A* + *BH*) + (*A* + *BH*)

robustly stable provided that the following relation is satisfied.

P (*A* + *BH*) + (*A* + *BH*)

*A*X + X *AT* + *B*Y + Y*TBT* + *δ*DD*T* +

and post-multiplying (A.3) by X = P−1 , we have

**6. Appendix**

*H* = YX −1*.*

*d*

candidate, we have

*dt* V(*x*, *t*) = *xT*(*t*)

≤ *xT*(*t*)

We introduce the matrix X �

**6.1 Quadratic stabilization**

Lyapunov stability criterion.

Fig. 10. Time histories of the control input *u*(*t*)

### **4.4 Summary**

In this section, we have proposed a design method of a variable gain robust controller for a class of uncertain nonlinear systems. The uncertainties under consideration are composed of matched part and unmatched one, and by using the concept of piecewise Lyapunov functions, we have shown that the proposed robust controller can be obtained by solving LMIs (4.34) and cheking the rank condition (4.32). By numerical simulations, the effectiveness of the proposed controller has been presented.

### **5. Conclusions and future works**

In this chapter, we have presented that the variable gain robust controller for a class of uncertain linear systems and through the numerical illustrations, the effectiveness of the proposed vaiable gain robust controllers has been shown. The advantage of the proposed controller synthesis is as follows; the proposed variable gain robust controller in which the real effect of the uncertainties can be reflected as on-line information is more flexible and adaptive than the conventional robust controller with a fixed gain which is derived by the worst-case design for the parameter variations. Additionally the proposed control systems are constructed by renewing the parameter which represents the perturbation region of unknown parameters, and there is no need to solve any other equation for the stability.

In Section 2 for linear systems with matched uncertainties, a design problem of variable gain robust state feedback controllers in order to achieve satisfactory transient behavior as closely as possible to desirable one generated by the nominal system is considered. Section 3 extends the result for the variable gain robust state feedback controller given in Section 2 to variable gain robust output feedback controllers. In this Section, some assumptions for the structure of the system parameters are introduced and by using these assumptions, an LMI-based the variable gain robust output feedback controller synthesis has been presented. In Section 4, the design method of variable gain robust state feedback controller via piecewise Lyapunov functions has been suggested. One can see that the crucial difference between the existing results and the proposed variable gain controller based on PLFs is that for uncertain linear systems which cannot be statilizable via the conventional quadratic stabilizing controllers, the proposed design procedure can stabilize it. Besides, it is obvious that the proposed variable robust control scheme is more effective for linear systems with larger uncertainties.

The future research subjects are an extension of the variable gain robust state feedback controller via PLFs to output feedback control systems. Besides, the problem for the extension to such a broad class of systems as uncertain large-scale systems, uncertain time-delay systems and so on should be tackled. Furthermore in future work, we will examine the condition (3.2) in section 3 and assumptions (4.3) and (4.32) in section 4.

On the other hand, the design of feedback controllers is often complicated by presence of physical constraints : saturating actuators, temperatures, pressures within safety margins and so on. If the constraints are violated, serious consequences may ensue, for example, physical components may be damaged, or saturation may cause a loss of closed-loop stability. In particular, input saturation is a common feature of control systems and the stabilization problems of linear systems with control input saturation have been studied (e.g. (17; 32)). Furthermore, some researchers have investigated analysis of constrained linear systems and reference managing for linear systems subject to input and state constraints (e.g. (10; 15)). Therefore, the future research subjects are to address the constrained robust control problems reducing the effect of unknown parameters.

### **6. Appendix**

26 Will-be-set-by-IN-TECH

Case 1) Case 2) Desired

0 0.5 1 1.5 2 2.5 3

Time

In this section, we have proposed a design method of a variable gain robust controller for a class of uncertain nonlinear systems. The uncertainties under consideration are composed of matched part and unmatched one, and by using the concept of piecewise Lyapunov functions, we have shown that the proposed robust controller can be obtained by solving LMIs (4.34) and cheking the rank condition (4.32). By numerical simulations, the effectiveness of the proposed

In this chapter, we have presented that the variable gain robust controller for a class of uncertain linear systems and through the numerical illustrations, the effectiveness of the proposed vaiable gain robust controllers has been shown. The advantage of the proposed controller synthesis is as follows; the proposed variable gain robust controller in which the real effect of the uncertainties can be reflected as on-line information is more flexible and adaptive than the conventional robust controller with a fixed gain which is derived by the worst-case design for the parameter variations. Additionally the proposed control systems are constructed by renewing the parameter which represents the perturbation region of unknown

In Section 2 for linear systems with matched uncertainties, a design problem of variable gain robust state feedback controllers in order to achieve satisfactory transient behavior as closely as possible to desirable one generated by the nominal system is considered. Section 3 extends the result for the variable gain robust state feedback controller given in Section 2 to variable gain robust output feedback controllers. In this Section, some assumptions for the structure of the system parameters are introduced and by using these assumptions, an LMI-based the variable gain robust output feedback controller synthesis has been presented. In Section 4, the design method of variable gain robust state feedback controller via piecewise Lyapunov functions has been suggested. One can see that the crucial difference between the existing results and the proposed variable gain controller based on PLFs is that for uncertain linear systems which cannot be statilizable via the conventional quadratic stabilizing controllers, the proposed design procedure can stabilize it. Besides, it is obvious that the proposed variable

parameters, and there is no need to solve any other equation for the stability.

robust control scheme is more effective for linear systems with larger uncertainties.

0

1

2

Control input

Fig. 10. Time histories of the control input *u*(*t*)

**4.4 Summary**

controller has been presented.

**5. Conclusions and future works**

3

4

5

### **6.1 Quadratic stabilization**

The following lemma provides a LMI-based design method of a robust controller via Lyapunov stability criterion.

**Lemma A.1.** *Consider the uncertain linear system (4.1) and the control law u*(*t*) = *Hx*(*t*)*. There exists the state feedback gain matrix H* ∈ **R***m*×*n such that the control law u*(*t*) = *Hx*(*t*) *is a quadratic stabilizing control, if there exist* X > 0, Y *and δ* > 0 *satisfying the LMI*

$$
\begin{pmatrix}
\mathcal{A}\mathcal{X} + \mathcal{X}A^T + \mathcal{B}\mathcal{Y} + \mathcal{Y}^T\mathcal{B}^T + \delta\mathcal{D}\mathcal{D}^T & \mathcal{X}\mathcal{E}^T \\
\mathcal{E}\mathcal{X} & -\delta I\_\emptyset
\end{pmatrix} < 0\tag{A.1}
$$

*If the solution* X ,Y *and δ of the LMI (A.1) exists, then the gain matrix H* ∈ **R***m*×*n is obtained as H* = YX −1*.*

*Proof.* Introducing the quadratic function V(*x*, *t*) � = *exT*(*t*)P*x*(*t*) as a Lyapunov function candidate, we have

$$\begin{split} \frac{d}{dt} \mathcal{V}(\mathbf{x}, t) &= \mathbf{x}^T(t) \left\{ \mathcal{P} \left( A + BH \right) + \left( A + BH \right)^T \mathcal{P} \right\} \mathbf{x}(t) + 2\mathbf{x}^T(t) \mathcal{P} \mathcal{D} \Delta(t) \mathcal{E} \mathbf{x}(t) \\ &\leq \mathbf{x}^T(t) \left\{ \mathcal{P} \left( A + BH \right) + \left( A + BH \right)^T \mathcal{P} \right\} \mathbf{x}(t) + \delta \mathbf{x}^T(t) \mathcal{P} \mathcal{D} \mathcal{D}^T \mathcal{P} \mathbf{x}(t) + \frac{1}{\delta} \mathbf{x}^T(t) \mathcal{E}^T \mathcal{E} \mathbf{x}(t) \end{split} \tag{A.2}$$

Here we have used the well-known relation (3.12). Thus the uncrtain linear system (4.1) is robustly stable provided that the following relation is satisfied.

$$\mathcal{P}\left(A+\mathcal{B}H\right) + \left(A+\mathcal{B}H\right)^{T}\mathcal{P} + \delta\mathcal{P}\mathcal{D}\mathcal{D}^{T}\mathcal{P} + \frac{1}{\delta}\mathcal{E}^{T}\mathcal{E} < 0\tag{A.3}$$

We introduce the matrix X � = P−1 and consider the change of variable Y � = *H*X . Then, by preand post-multiplying (A.3) by X = P−1 , we have

$$A\mathcal{X} + \mathcal{X}A^T + \mathcal{B}\mathcal{Y} + \mathcal{Y}^T\mathcal{B}^T + \delta\mathcal{D}\mathcal{D}^T + \frac{1}{\delta}\mathcal{X}\mathcal{E}^T\mathcal{E}\mathcal{X} < 0\tag{A.4}$$

for a Class of Uncertain Dynamical Systems 29

Synthesis of Variable Gain Robust Controllers for a Class of Uncertain Dynamical Systems 339

[19] H. Oya and K. Hagino: "Robust Servo System with Adaptive Compensation Input for Linear Uncertain Systems", Proc. of the 4th Asian Contr. Conf., pp.972-977, SINGAPORE,

[20] H. Oya and K. Hagino : "Observer-based Robust Control Giving Consideration to Transient Behavior for Linear Systems with Structured Uncertainties", Int. J. Contr.,

[21] H. Oya and K. Hagino, "Robust Control with Adaptive Compensation Input for Linear Uncertain Systems", IEICE Trans. Fundamentals of Electronics, Communications and

[22] H. Oya and K. Hagino, "Adaptive Robust Control Scheme for Linear Systems with Structured Uncertainties", IEICE Trans. Fundamentals of Electronics, Communications

[23] H. Oya, K. Hagino, S. Kayo and M. Matsuoka, "Adaptive Robust Stabilization for a Class of Uncertain Linear Systems via Variable Gain Controllers", Proc. of the 45th IEEE Conf.

[24] H. Oya, K. Hagino and S. Kayo, "Adaptive Robust Control Based on Piecewise Lyapunov Functions for a Class of Uncertain Linear Systems", Proc. of the European Contr. Conf.

[25] H. Oya, K. Hagino and S. Kayo, "Synthesis of Adaptive Robust Output Feedback Controllers for a Class of Uncertain Linear Systems", Proc. of the 47th IEEE Conf.on

[26] H. Oya and K. Hagino, "A New Adaptive Robust Controller Avoiding Chattering Phenomenon for a Class of Uncertain Linear Systems", Proc. of the 28th IASTED Int.

[27] I. R. Petersen, "A Riccati Equation Approach to the Design of Stabilizing Controllers and Observers for a Class of Uncertain Linear Systems", IEEE Trans. Automat. Contr., Vol.30,

[28] I. R. Petersen and D. C. McFarlane, "Optimal Guaranteed Cost Control and Filtering for Uncertain Linear Systems", IEEE Trans. Automat. Contr., Vol.39, No.9, pp.1971-1977,

[29] I. R. Petersen and C. C. Hollot, "A Riccati Equation Approach to the Stabilization of

[30] E. S. Pyatnitskii, V. I. Skrodinskii, "Numerical Methods of Lyapunov Function Contruction and Their Application to the Absolute Stability Problem", Syst. & Contr.

[31] S. O. Reza Moheimani and I. R. Petersen : "Optimal Guaranteed Cost Control of Uncertain Systems via Static and Dynamic Output Feedback", Automatica, Vol.32, No.4,

[32] M. C. Turner, I. Postlethwaite and D. J. Walker, "Non-linear Tracking Control for Multivariable Constrained Input Linear Systems", Int. J. Contr., Vol.73, No.12,

[33] S. Ushida, S. Yamamoto and H. Kimura, "Quadratic Stabilization by H∞ state feedback controllers with Adjustable Parameters", Proc. of the 35th IEEE Conf. Decision and

[34] V. Veselý, "Design of Robust Output Affine Quadratic Controller", Proc. of 2002 IFAC

Conf. on Modeling, Identification and Contr., pp.236–241, Innsbruck, 2009.

Uncertain Linear Systems", Automatica, Vol.22, No.4, pp.397-411, 1986.

2002.

Vol.75, No.15, pp.1231-1240, 2002.

No.9, pp.904-907, 1985.

Lett., Vol.2, No.1, 1986.

pp.575-579, 1996.

pp.1160-1172, 2000.

Contr., pp.1003-1008, Kobe, JAPAN, 1996.

World Congress, pp.1-6, Barcelona, SPAIN, 2002.

1994.

Computer Sciences, Vol.E86-A, No.6, pp.1517-1524, 2003.

and Computer Sciences, Vol.E87-A, No.8, pp.2168-2173, 2004.

on Decision and Contr., pp.1183–1188, San Diego, USA, 2006.

Decision and Contr., pp.995-1000, Cancun, MEXICO, 2008.

2007 (ECC2007), pp.810–815, Kos, GREECE, 2007.

One can see from **Lemma 2** (Schur complement) that the inequaity (A.4) is equivalent to the LMI (A.1).

### **7. References**

28 Will-be-set-by-IN-TECH

One can see from **Lemma 2** (Schur complement) that the inequaity (A.4) is equivalent to the

[1] H. L. S. Almeida, A. Bhaya and D. M. Falcão, "A Team Algorithm for Robust Stability Analysis and Control Design of Uncertain Time-Varying Systems using Piecewise Quadratic Lyapunov Functions", Int. J. Robust and Nonlinear Contr., Vol.11, No.1,

[2] R. E. Benton, JR and D. Smith, "A Non-itarative LMI-based Algorithm for Robust Static-output-feedback Stabilization", Int. J. Contr., Vol.72, No.14, pp.1322-1330, 1999. [3] B. R. Bermish , M. Corless and G. Leitmann, "A New Class of Stabilizing Controllers for Uncertain Dynamical Systems", SIAM J. Contr. Optimiz. Vol.21, No.2, pp.246-255, 1983. [4] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System

[5] Y. K. Choi, M. J. Chung and Z. Bien, "An Adaptive Control Scheme for Robot

[6] J. C. Doyle, K. Glover, P. P. Khargonekar and B. A. Francis, "State-Space Solutions to Standarad H2 and H∞ Control Problems", IEEE Trans. Automat. Contr., Vol.34, No.8,

[7] P. Gahinet, P. Apkarian and M. Chilali, "Affine Parameter Dependent Lyapunov Functions and Real Parameter Uncertainty", IEEE Trans. Automat. Contr., Vol.41, No.3,

[8] F. R. Gantmacher, "The Theory of Matrices", Vol.1, Chelsea Publishing Company, New

[9] J. C. Geromel, C. C. De Souza and R. E. Skelton, "LMI Numerical Solution for Output Feedbadck Stabilization", Proc. of the 1994 American Contr. Conf., Baltimore, MD, USA,

[10] E. G. Gilbert and I. Kolmanovsky, "Nonlinear Tracking Controlin the Presence of State and Control Constraints : A Generalized Reference Govonor", Automatica, Vol.38,

[11] T. Iwasaki, R. E. Skelton and J. C. Geromel, "Linear Quadratic Suboptimal Control with Static Output Feedback", Syst. & Contr. Lett., Vol.23, No.6, pp.421-430, 1994. [12] F. Jabbari and W. E. Schmitendolf, "Effect of Using Observers on Stabilization of Uncertain Linear Systems", IEEE Trans. Automat. Contr., Vol.38, No.2, pp.266-271, 1993.

[14] P. P. Khargonekar and M. A. Rotea, "Mixed H2/H∞ Control , A Convex Optimization

[15] K. Kogiso and K. Hirata, "Reference Governor for Contrained Systems with Time-Varying References", J. Robotics and Autonomous Syst., Vol.57, Issue 3, pp.289-295,

[16] V. Kuˇcera and C. E. De Souza, "A Necessary and Sufficient Conditions for Output

[17] Z. Lin, M. Pachter and S. Band, "Toward Improvement of Tracking Performance – Nonlinear Feedback for Linear Systems", Int. J. Contr., Vol.70, No.1, pp.1-11 (1998) [18] M. Maki and K. Hagino, "Robust Control with Adaptation Mechanism for Improving

[13] H. K. Khalil, "Nonlinear Systems, Third Edition", Prentice Hall, 2002.

Approach", IEEE Trans. Automat. Contr., Vol.36, No.7, pp.824-837, 1991.

Feedback Stabilizability", Automatica , Vol.31, No.9, pp.1357-1359, 1995.

Transient Behavior", Int. J. Contr., Vol.72, No.13, pp.1218-1226, 1999.

and Control Theory, SIAM Studies in Applied Mathmatics, 1994.

Manipulators", Int. J. Contr., Vol.44, No.4, pp.1185-1191, 1986.

LMI (A.1).

**7. References**

pp.357-371, 2001.

pp.831-847, 1989.

pp.436-442, 1996.

York, 1960.

2009.

pp.40-44, 1994.

No.12, pp.2071-2077, 2002.

**15**

Ciprian Lupu

*Romania*

**Simplified Deployment of Robust**

**Real-Time Systems Using Multiple**

**Model and Process Characteristic**

**Architecture-Based Process Solutions**

*Department of Automatics and Computer Science, University "Politehnica" Bucharest*

A common industrial practice is to find some specific control structures for the nonlinear processes that reduce, as much as possible, the design techniques to classic control approaches. There are a lot of situations when the designing of robust controller leads to complex hardware and software requirements. In international literature there are some interesting solutions (Kuhnen & Janocha, 2001; Dai et al., 2003; Wang & Su, 2006) for solving

In following sections there will be presented, in the first part, some elements of classic robust design of RST control algorithm and on the second, two alternative solutions based

The robustness of the systems is reported mainly to model parameters change or the structural model estimation uncertainties (Landau et al., 1997). A simple frequency analysis shows that the critical Nyquist point (i.e. the point (-1, 0) in the complex plane) plays an important role in assessing the robustness of the system. In this plan, we can trace hodograf (Nyquist place) open-loop system, i.e. the frequency response. The distance from the hodograf critical point system (edge module), i.e. radius centered at the critical point and tangent to hodograf is a measure of the intrinsic robustness of the system. The distance is

**1. Introduction**

implementation reduction.

greater, the system is more robust.

Fig. 1. RST control algorithm structure

on multiple model and nonlinear compensators structures.

**2. Some elements about classic RST robust control design**

## **Simplified Deployment of Robust Real-Time Systems Using Multiple Model and Process Characteristic Architecture-Based Process Solutions**

Ciprian Lupu

*Department of Automatics and Computer Science, University "Politehnica" Bucharest Romania*

## **1. Introduction**

30 Will-be-set-by-IN-TECH

340 Recent Advances in Robust Control – Novel Approaches and Design Methods

[35] L. Xie, S. Shishkin and M. Fu, "Piecewise Lyapunov Functions for Robust Stability of

[36] K. Zhou and P. P. Khargonekar, "Robust Stabilization on Linear Systems with Norm Bounded Time-Varying Uncertainty", Syst. & Contr. Lett., Vol.10, No.1, pp.17-20, 1988.

[37] K. Zhou, "Essentials of Robust Control", Prentice Hall Inc., New Jersey, USA, 1998.

Linear Time-Varying Systems", Syst. & Contr. Letters, Vol.31, No.3, 1997.

A common industrial practice is to find some specific control structures for the nonlinear processes that reduce, as much as possible, the design techniques to classic control approaches. There are a lot of situations when the designing of robust controller leads to complex hardware and software requirements. In international literature there are some interesting solutions (Kuhnen & Janocha, 2001; Dai et al., 2003; Wang & Su, 2006) for solving implementation reduction.

In following sections there will be presented, in the first part, some elements of classic robust design of RST control algorithm and on the second, two alternative solutions based on multiple model and nonlinear compensators structures.

## **2. Some elements about classic RST robust control design**

The robustness of the systems is reported mainly to model parameters change or the structural model estimation uncertainties (Landau et al., 1997). A simple frequency analysis shows that the critical Nyquist point (i.e. the point (-1, 0) in the complex plane) plays an important role in assessing the robustness of the system. In this plan, we can trace hodograf (Nyquist place) open-loop system, i.e. the frequency response. The distance from the hodograf critical point system (edge module), i.e. radius centered at the critical point and tangent to hodograf is a measure of the intrinsic robustness of the system. The distance is greater, the system is more robust.

Fig. 1. RST control algorithm structure

Simplified Deployment of Robust Real-Time Systems Using

0

with the following initial conditions:

represents the measures vector.

are calculated using the next expression:

Fig. 2. Sensibility function graphic representation

The estimated ˆ

sensibility function.

module margin.

Multiple Model and Process Characteristic Architecture-Based Process Solutions 343

*k k Fk k k k N*

*T T*

*Fk k kFk Fk Fk k N*

*k yk k t k N*

1 *F I GI I* (0) ( ) ,0 1

This approach allows the users to verify, and if necessary, to calibrate the algorithm's robustness (Landau et al., 1997). Next expression and Fig. 2 present "disturbance-output"

( )( ) , ( )( ) ( ) ( )

In the same time, the negative maximum value of the sensibility function represents the

Based on this value, in an "input-output" representation (Landau et al., 1997), process nonlinearity can be bounded inside the "conic" sector, presented in Fig. 3, where *a1* and *a2*

> 1 2 1 1 1 1 *a a M M*

*Ae Se R*

max ( ) *j dB vy R dB M Se*

(11)

(12)

*j j jj j j*

*Ae Se Be Re*

ˆ ˆ ( 1) ( ) ( 1) ( ) ( 1),

()() ()() ( 1) ( ) , 1 ()()()

ˆ ( 1) ( 1) ( ) ( ),

() ()

*Se He*

*def j j vy vy*

  0

*T*

*kFk k*

( ) *k* represents the parameters of the polynomial plant model and ( ) *T*

(9)

(8)

(10)

*k*

For this study we use a RST algorithm. For robustification there are used pole placement procedures (Landau et al., 1997). Fig. 1 presents a RST algorithm. The R, S, T polynomials are:

$$\begin{aligned} R\left(q^{-1}\right) &= r\_0 + r\_1 q^{-1} + \dots + r\_{nr} q^{-nr} \\ S\left(q^{-1}\right) &= s\_0 + s\_1 q^{-1} + \dots + s\_{ns} q^{-ns} \\ T\left(q^{-1}\right) &= t\_0 + t\_1 q^{-1} + \dots + t\_{nt} q^{-nt} \end{aligned} \tag{1}$$

The RST control algorithm is:

$$(S(q^{-1})u(k) + R(q^{-1})y(k) = T(q^{-1})y^\*(k)\tag{2}$$

or:

$$\mu(k) = \frac{1}{s} \text{l} - \sum\_{i=1}^{n\_S} s\_i \mu(k - i) - \sum\_{i=0}^{n\_R} r\_i y(k - i) + \sum\_{i=0}^{n\_T} t\_i y^\*(k - i) \text{l} \tag{3}$$

where: u(k) - algorithm output, y(k) - process output, y\*(k) - trajectory or filtered set point. When necessary, an imposed trajectory can be generated using a trajectory model generator:

$$\text{s.t.}\begin{aligned} \text{s.t.} &(\text{s}^{-1})\\ &\frac{m}{A} \binom{\text{q}^{-1}}{\text{m}} r(k) \end{aligned} \tag{4}$$

with Am and Bm like:

$$\begin{aligned} \, ^0A\_m(q^{-1}) &= \, 1 + a\_{m1}q^{-1} + \dots + a\_{mm} \, ^0q \, ^{-m}Am \\ \, ^0A\_m \, ^{-1} &= \, ^0Am \, ^0q \, ^{-1} + \dots + b\_{mm} \, ^0q \, ^{-m} \end{aligned} \tag{5}$$

Algorithm pole placement design procedure is based on the identified process' model.

$$y(k) = \frac{q^{-d}B(q^{-1})}{A(q^{-1})}u(k)\tag{6}$$

where

$$\begin{aligned} \, \_B\left(q^{-1}\right) &= b\_1 q^{-1} + b\_2 q^{-2} + \dots + b\_{nb} q^{-nb} \\ \, \_A\left(q^{-1}\right) &= 1 + a\_1 q^{-1} + \dots + a\_{na} q^{-na} \end{aligned} \tag{7}$$

The identification (Landau & Karimi, 1997; Lainiotis & Magill, 1969; Foulloy et al., 2004) is made in a specific process operating point and can use recursive least square algorithm exemplified in next relations developed in (Landau et al., 1997):

$$\begin{aligned} \hat{\theta}(k+1) &= \hat{\theta}(k) + F(k+1)\phi(k)x^{0}(k+1), \forall k \in N\\ F(k+1) &= F(k) - \frac{F(k)\phi(k)\phi^{T}(k)F(k)}{1 + \phi^{T}(k)F(k)\phi(k)}, \forall k \in N\\ \varepsilon^{0}(k+1) &= y(k+1) - \hat{\theta}^{T}(k)\phi(t), \forall k \in N\end{aligned} \tag{8}$$

with the following initial conditions:

342 Recent Advances in Robust Control – Novel Approaches and Design Methods

For this study we use a RST algorithm. For robustification there are used pole placement

...

*nr nr*

*ns ns*

(1)

(3)

(5)

(7)

*nt nt*

1 1 1\* *S*( )() ( )() ( ) () *q uk R q y k T q y k* (2)

(4)

*n*

*n*

(6)

...

...

*R q r rq r q*

1 \* ( ) [ ( ) ( ) ( )] 10 0

1 ( ) \*( 1) ( ) 1 ( ) *B q y k r m k A q m*

*Aq aq a q Am m mn*

*Bq b bq b q Bm m m mn m Bm*

*m Am*

Algorithm pole placement design procedure is based on the identified process' model.

1 12 ... 1 2

*nb Bq bq bq b q nb*

*na Aq aq a q na*

The identification (Landau & Karimi, 1997; Lainiotis & Magill, 1969; Foulloy et al., 2004) is made in a specific process operating point and can use recursive least square algorithm

1 1 1 ... 1

1 1 ( ) 1 1

1 1 ( ) 0 1

exemplified in next relations developed in (Landau et al., 1997):

1 1 ( ) ( ) ( ) ( ) *d q Bq y k u k A q*

where: u(k) - algorithm output, y(k) - process output, y\*(k) - trajectory or filtered set point. When necessary, an imposed trajectory can be generated using a trajectory model generator:

*n n n S R T uk suk i ry k i ty k i ii i s ii i*

*S q s sq s q*

*T q t tq t q*

procedures (Landau et al., 1997). Fig. 1 presents a RST algorithm.

0

The R, S, T polynomials are:

The RST control algorithm is:

with Am and Bm like:

where

or:

$$F(0) = \frac{1}{\delta}I = (GI)I,\\ 0 < \delta < 1\tag{9}$$

The estimated ˆ ( ) *k* represents the parameters of the polynomial plant model and ( ) *T k* represents the measures vector.

This approach allows the users to verify, and if necessary, to calibrate the algorithm's robustness (Landau et al., 1997). Next expression and Fig. 2 present "disturbance-output" sensibility function.

$$\begin{split} S\_{vy} \{ e^{j\alpha} \} &= H\_{vy} \{ e^{j\alpha} \} = \\ &= \frac{A(e^{j\alpha}) S(e^{j\alpha})}{A(e^{j\alpha}) S(e^{j\alpha}) + B(e^{j\alpha}) R(e^{j\alpha})}, \quad \forall \alpha \in R \end{split} \tag{10}$$

In the same time, the negative maximum value of the sensibility function represents the module margin.

$$\left. \Delta M \right|\_{dB} = -\max\_{o \bowtie R} \left| S\_{vy} \left( e^{j o \nu} \right) \right|\_{dB} \tag{11}$$

Based on this value, in an "input-output" representation (Landau et al., 1997), process nonlinearity can be bounded inside the "conic" sector, presented in Fig. 3, where *a1* and *a2* are calculated using the next expression:

$$\frac{1}{1 - \Delta M} \ge a\_1 \ge a\_2 \ge \frac{1}{1 + \Delta M} \tag{12}$$

Fig. 2. Sensibility function graphic representation

Simplified Deployment of Robust Real-Time Systems Using

the classic algorithm with a sufficient robustness reserve. In Figure 4 and 5, the blocks and variables are as follows:

Command calculus – unit that computes the process control law;

Process – physical system to be controlled;

Classic Alg. – control algorithm (PID, RST);

 u – output of the Command calculus block; u alg. – output of the classic algorithm; u i.m. – output of the inverse model block; r – system's set point or reference trajectory;

aspects met on designing of the presented structure.

a. determination of the process' (static) characteristic,

nonlinearity compensator serialized with real process.

c. robust control law design of classic algorithm. The second structure imposes following these steps:

y – outp5t of the process;

**3.1 Control design procedure**

b. construction of command generator,

a. determination of process' characteristic, b. construction of nonlinearity compensator,

**3.2 Determination of process characteristic**

points is obtained using extrapolation procedure.

p – disturbances.

will be presented.

process.

Multiple Model and Process Characteristic Architecture-Based Process Solutions 345

The second solution (serial structure) has the inverse model command generator between the classic algorithm and the process. The inverse model command generator acts as a nonlinear compensator and depends on the command value. The (classic) algorithm generates a command that, filtered by the nonlinearity compensator, controls the real

The presented solutions propose treating the inverse model mismatches that "disturb" the classic command as some algorithm's model mismatches. This approach imposes designing

Related to classical control loops, both solutions need addressing some supplementary specific aspects: determination of static characteristic of the process, construction of inverse model, robust control law design. In next sections we will focus on the most important

c. designing the classic algorithm based on "composed process" which contains the

These steps are more or less similar for the two structures. For the (a) and (c) steps it is obvious; for (b) the command generator and nonlinearity compensator have different functions but the same design and functioning procedure. Essential aspects for these steps

This operation is based on several experiments of discrete step increasing and decreasing of the command *u(k)* and measuring the corresponding stabilized process output *y(k)* (figure 6 (a)). The command *u(k)* covers all (0 to 100%) possibilities. Because the noise is present, the static characteristics are not identical. The final static characteristic is obtained by meaning of all correspondent positions of these experiments. The graphic between two "mean"

For the first structure the specific aspects of the control design procedure are:

Fig. 3. Robust control design procedure

## **3. Nonlinear compensator control solution**

Various papers and researches target the inverse model control approach; a few of these can be mentioned: (Tao & Kokotovic, 1996; Yuan et al., 2007) etc.

In these researches there have been proposed several types of structures based on the inverse model. According to those results, this section comes up with two very simple and efficient structures presented in Figures 4 and 5. Here, the inverse model is reduced to the geometric inversed process (nonlinear) characteristic – reflection from the first leap of static characteristic of the process, as presented in Figure 6(b).

The first solution (parallel structure) considers the addition of two commands: the first "a feedforward command" generated by the inverse model command generator and the second, generated by a classic, simple algorithm (PID, RST ).

The first command, based on the static process characteristic, depends on the set point value and is designed to generate a corresponding value that drives the process' output close to the imposed set point. The second (classic) algorithm generates a command that corrects the difference caused by external disturbances and, according to the set point, by eventual bias errors caused by mismatches between calculated inverse process characteristic and the real process.

Fig. 4. Proposed scheme for "parallel" structure

Fig. 5. Proposed scheme for "serial" structure

The second solution (serial structure) has the inverse model command generator between the classic algorithm and the process. The inverse model command generator acts as a nonlinear compensator and depends on the command value. The (classic) algorithm generates a command that, filtered by the nonlinearity compensator, controls the real process.

The presented solutions propose treating the inverse model mismatches that "disturb" the classic command as some algorithm's model mismatches. This approach imposes designing the classic algorithm with a sufficient robustness reserve.

In Figure 4 and 5, the blocks and variables are as follows:

344 Recent Advances in Robust Control – Novel Approaches and Design Methods

Various papers and researches target the inverse model control approach; a few of these can

In these researches there have been proposed several types of structures based on the inverse model. According to those results, this section comes up with two very simple and efficient structures presented in Figures 4 and 5. Here, the inverse model is reduced to the geometric inversed process (nonlinear) characteristic – reflection from the first leap of static

The first solution (parallel structure) considers the addition of two commands: the first "a feedforward command" generated by the inverse model command generator and the

The first command, based on the static process characteristic, depends on the set point value and is designed to generate a corresponding value that drives the process' output close to the imposed set point. The second (classic) algorithm generates a command that corrects the difference caused by external disturbances and, according to the set point, by eventual bias errors caused by mismatches between calculated inverse process characteristic and the real

Fig. 3. Robust control design procedure

process.

**3. Nonlinear compensator control solution**

be mentioned: (Tao & Kokotovic, 1996; Yuan et al., 2007) etc.

characteristic of the process, as presented in Figure 6(b).

Fig. 4. Proposed scheme for "parallel" structure

Fig. 5. Proposed scheme for "serial" structure

second, generated by a classic, simple algorithm (PID, RST ).

Related to classical control loops, both solutions need addressing some supplementary specific aspects: determination of static characteristic of the process, construction of inverse model, robust control law design. In next sections we will focus on the most important aspects met on designing of the presented structure.

## **3.1 Control design procedure**

For the first structure the specific aspects of the control design procedure are:

The second structure imposes following these steps:

These steps are more or less similar for the two structures. For the (a) and (c) steps it is obvious; for (b) the command generator and nonlinearity compensator have different functions but the same design and functioning procedure. Essential aspects for these steps will be presented.

## **3.2 Determination of process characteristic**

This operation is based on several experiments of discrete step increasing and decreasing of the command *u(k)* and measuring the corresponding stabilized process output *y(k)* (figure 6 (a)). The command *u(k)* covers all (0 to 100%) possibilities. Because the noise is present, the static characteristics are not identical. The final static characteristic is obtained by meaning of all correspondent positions of these experiments. The graphic between two "mean" points is obtained using extrapolation procedure.

Simplified Deployment of Robust Real-Time Systems Using

Fig. 7. Parallel RST feedback-feedforward control structure

characteristic and for systems with different functioning regimes.

component is not influenced by the noise.

replace the inverse model command generator.

law control doesn't need important resources.

flexibility to control it.

can be another disadvantage.

Narendra & Balakrishnan, 1997).

supplementary specific aspects:

Control law switching.

can be highlighted:

Selection of the best algorithm;

Dimension of multiple-model configuration;

developments.

**4. Multiple model control solution**

Multiple Model and Process Characteristic Architecture-Based Process Solutions 347

The system is very stabile due to the global command that contains a "constant" component generated by an inverse static model command block, according to the set point value. This

A fuzzy logic bloc that can "contain" human experience about some nonlinear processes can

Being not very complex in terms of real time software and hardware implementation, the

This structure is very difficult to use for the system that doesn't have a bijective static

Another limitation is that this structure can only be used for stabile processes. In the situations where the process is "running", the global command is likely to not have enough

The increased number of experiments for the determination of a correct static characteristic

The essential function of a real-time control system is to preserve the closed-loop performances in case of non-linearity, structural disturbances or process uncertainties. A valuable way to solve these problems is the multiple-models or multicontroller structure. The first papers that mentioned the "multiple-models" structure/system have been reported in the 90s. Balakrishnan and Narendra are among the first authors addressing problems of stability, robustness, switching and designing this type of structures in their papers

Research refinement in this field has brought extensions to the multiple-model control concept. Parametric adaptation procedures – Closed-Loop Output Error (Landau & Karimi, 1997), use of Kalman filter representation (Lainiotis & Magill, 1969), the use of neural networks (Balakrishnan, 1996) or the fuzzy systems are some of the important

Related to classical control loops, multiple-model based systems need addressing some

From the multiple-models control systems viewpoint, two application oriented problems

Fig. 6. (a) - left - Determination of process characteristic. Continuous line represents the final characteristic. (b) - right - Construction of nonlinearity compensator

According to system identification theory, the dispersion of process trajectory can be found using next expression (Ljung & Soderstroom, 1983).

$$\sigma^2\left[n\right] \cong \frac{1}{n-1} \sum\_{i=1}^{n} y^2\left[i\right], \forall n \in N^\* \; \left\langle 1 \right\rangle \tag{13}$$

This can express a measure of superposing of noise onto process, process' nonlinearity etc. and it is very important for the control algorithm robust design.

### **3.3 Construction of nonlinearity compensator (generator)**

This step deals with the process's static characteristic "transposition" operation. Figure 6 (b) presents this construction. According to this, *u(k)* is dependent to *r(k)*. This characteristic is stored in a table; thus we can conclude that, for the nonlinearity compensator based controller, selecting a new set point *r(k)* will impose finding in this table the corresponding command *u(k)* that determines a process output *y(k)* close to the reference value.

### **3.4 Control law design**

The control algorithm's duty is to eliminate the disturbances and differences between the nonlinearity compensator computed command and the real process behavior. A large variety of control algorithms can be used: PID, RST, fuzzy etc., but the goal is to have a very simple one. For this study we use a RST algorithm. This is designed using the pole placement procedure (Landau et al., 1997). Figure 7 presents a RST base algorithm structure. Finally, if it is imposed that all nonlinear characteristics be (graphically) bounded by the two gains, or gain limit to be great or equal to the process static maximal distance characteristic ΔG≥*mg*, a controller that has sufficient robustness was designed.

### **3.5 Analysis and conclusions for proposed structure**

The main advantage consists in using a classic procedure for designing the control algorithm and determining the nonlinearity compensator command block, comparative to robust control design procedures. Well known procedures for identification and law control design are used. All procedures for the inverse characteristic model identification can be included in a real time software application.

Fig. 7. Parallel RST feedback-feedforward control structure

346 Recent Advances in Robust Control – Novel Approaches and Design Methods

(a) (b)

Fig. 6. (a) - left - Determination of process characteristic. Continuous line represents the final

According to system identification theory, the dispersion of process trajectory can be found

 2 2\* 1

This can express a measure of superposing of noise onto process, process' nonlinearity etc.

This step deals with the process's static characteristic "transposition" operation. Figure 6 (b) presents this construction. According to this, *u(k)* is dependent to *r(k)*. This characteristic is stored in a table; thus we can conclude that, for the nonlinearity compensator based controller, selecting a new set point *r(k)* will impose finding in this table the corresponding

The control algorithm's duty is to eliminate the disturbances and differences between the nonlinearity compensator computed command and the real process behavior. A large variety of control algorithms can be used: PID, RST, fuzzy etc., but the goal is to have a very simple one. For this study we use a RST algorithm. This is designed using the pole placement procedure (Landau et al., 1997). Figure 7 presents a RST base algorithm structure. Finally, if it is imposed that all nonlinear characteristics be (graphically) bounded by the two gains, or gain limit to be great or equal to the process static maximal distance

The main advantage consists in using a classic procedure for designing the control algorithm and determining the nonlinearity compensator command block, comparative to robust control design procedures. Well known procedures for identification and law control design are used. All procedures for the inverse characteristic model identification can be

*n i n yi N n*

command *u(k)* that determines a process output *y(k)* close to the reference value.

characteristic ΔG≥*mg*, a controller that has sufficient robustness was designed.

**3.5 Analysis and conclusions for proposed structure**

included in a real time software application.

1 , n \ 1 1

(13)

characteristic. (b) - right - Construction of nonlinearity compensator

and it is very important for the control algorithm robust design.

**3.3 Construction of nonlinearity compensator (generator)**

using next expression (Ljung & Soderstroom, 1983).

**3.4 Control law design**

The system is very stabile due to the global command that contains a "constant" component generated by an inverse static model command block, according to the set point value. This component is not influenced by the noise.

A fuzzy logic bloc that can "contain" human experience about some nonlinear processes can replace the inverse model command generator.

Being not very complex in terms of real time software and hardware implementation, the law control doesn't need important resources.

This structure is very difficult to use for the system that doesn't have a bijective static characteristic and for systems with different functioning regimes.

Another limitation is that this structure can only be used for stabile processes. In the situations where the process is "running", the global command is likely to not have enough flexibility to control it.

The increased number of experiments for the determination of a correct static characteristic can be another disadvantage.

## **4. Multiple model control solution**

The essential function of a real-time control system is to preserve the closed-loop performances in case of non-linearity, structural disturbances or process uncertainties. A valuable way to solve these problems is the multiple-models or multicontroller structure. The first papers that mentioned the "multiple-models" structure/system have been reported in the 90s. Balakrishnan and Narendra are among the first authors addressing problems of stability, robustness, switching and designing this type of structures in their papers Narendra & Balakrishnan, 1997).

Research refinement in this field has brought extensions to the multiple-model control concept. Parametric adaptation procedures – Closed-Loop Output Error (Landau & Karimi, 1997), use of Kalman filter representation (Lainiotis & Magill, 1969), the use of neural networks (Balakrishnan, 1996) or the fuzzy systems are some of the important developments.

Related to classical control loops, multiple-model based systems need addressing some supplementary specific aspects:

From the multiple-models control systems viewpoint, two application oriented problems can be highlighted:

Simplified Deployment of Robust Real-Time Systems Using

u – output generate by Command calculus block

 r –set point system or reference trajectory; p – disturbances of physical process.

algorithms, respectively;

on the "switching" problem.

To be (very) fast.

the algorithm coefficients.

**4.1.1 Classic solutions**

**4.1 Control algorithms switching**

most appropriate model for the system's current state;

algorithm, which involves at least performance degradation.

y and y1, y2, …, yN – output of the process and outputs of the N models;

Multiple Model and Process Characteristic Architecture-Based Process Solutions 349

SELECTOR – based on adequate criteria evaluations, provides information about the

u and u1, u2, uN – output of the Command calculus block and outputs of the N control

As noted above, depending on the process specifics and the approach used to solve the "control algorithms switching" and/or "the best model choice" problems, the scheme can be adapted on the situation by adding/eliminating some specific blocks. This section focuses

The logic operation of multiple model system structure implies that after finding the best algorithm for the current operating point of the, the next step consists in switching the control algorithm. Two essential conditions must be verified with respect to this operation: To be designed so that no bumps in the applications of the control law are encountered;

Shocks determined by the switching operation cause non-efficient and/or dangerous behaviors. Moreover, a switch determines a slow moving area of action of the control

These are the main problems to be solved in designing block switching algorithms. From structurally point of view, this block may contain all implementation algorithms or at least

Present solutions (Landau et al., 1997; Dumitrache, 2005) solve more or less this problem and they are based on maintaining in active state all the control algorithms, also called "warm state". This supposes that every algorithm receives information about the process output y(k) and set the point value (eventually filtered) r(k), but only the control law ui(k) is applied on the real process, the one chosen by the switching block. This solution does not impose supplementary logic function for the system architecture and, for this reason, the switching time between algorithms is short. The drawback of this approach is that when

These supplementary conditions demand the match of the control algorithm outputs in the neighborhood switching zones. The superposition of models identification zones accomplishes this aspect. That can be seen in Fig. 9. As a result of this superposition, the

Other approaches (Dussud et al., 2000; Pages et al., 2002) propose the mix of two or more algorithms outputs. The "weighting" of each control law depends on the distance from the current process operating point and the action zone of each algorithm. Based on this, the switching from an algorithm to another one is done using weighting functions with a continuous evolution in [0–1] intervals. This technique can be easily implemented using fuzzy approach, An example is presented in Fig. 10. This solution involves solving control

designing the multi-model structure several supplementary steps are necessary.

multi-model structure will have an increased number of models.

gain problems, determined by mixing algorithm outputs.

Fig. 8. General scheme for multiple model structure

As function of the process particularity, several multiple-models structures are proposed (Balakrishnan, 1996). One of the most general architectures is presented in Figure 8.

As noted above, depending on the process specifics and the approach used to solve the "control algorithms switching" and/or "the best model choice" problems, the scheme can be adapted on the situation by adding/eliminating some specific blocks. This section focuses on the "switching" problem.

## **4.1 Control algorithms switching**

The logic operation of multiple model system structure implies that after finding the best algorithm for the current operating point of the, the next step consists in switching the control algorithm. Two essential conditions must be verified with respect to this operation:

348 Recent Advances in Robust Control – Novel Approaches and Design Methods

Class of systems with nonlinear characteristic, which cannot be controlled by a single

 Class of systems with different operating regimes, where different functioning regimes don't allow the use of a unique algorithm or imposes using a very complex one with

As function of the process particularity, several multiple-models structures are proposed

 Status or position identification system – component that provide information about the model–control algorithm "best" matching for the current state of the system; Mod. 1, Mod. 2, …, Mod. N - previously identified models of different regimes or

(Balakrishnan, 1996). One of the most general architectures is presented in Figure 8.

Command calculus – unit that computes the process law control ;

Alg. 1, Alg. 2, …, Alg. N – control algorithms designed for the N models;

Fig. 8. General scheme for multiple model structure

special problems on implementation.

In Fig 8, the blocks and variables are as follows: Process – physical system to be controlled;

SWITCH – mix or switch between the control laws;

algorithm;

operating points;

Shocks determined by the switching operation cause non-efficient and/or dangerous behaviors. Moreover, a switch determines a slow moving area of action of the control algorithm, which involves at least performance degradation.

These are the main problems to be solved in designing block switching algorithms. From structurally point of view, this block may contain all implementation algorithms or at least the algorithm coefficients.

## **4.1.1 Classic solutions**

Present solutions (Landau et al., 1997; Dumitrache, 2005) solve more or less this problem and they are based on maintaining in active state all the control algorithms, also called "warm state". This supposes that every algorithm receives information about the process output y(k) and set the point value (eventually filtered) r(k), but only the control law ui(k) is applied on the real process, the one chosen by the switching block. This solution does not impose supplementary logic function for the system architecture and, for this reason, the switching time between algorithms is short. The drawback of this approach is that when designing the multi-model structure several supplementary steps are necessary.

These supplementary conditions demand the match of the control algorithm outputs in the neighborhood switching zones. The superposition of models identification zones accomplishes this aspect. That can be seen in Fig. 9. As a result of this superposition, the multi-model structure will have an increased number of models.

Other approaches (Dussud et al., 2000; Pages et al., 2002) propose the mix of two or more algorithms outputs. The "weighting" of each control law depends on the distance from the current process operating point and the action zone of each algorithm. Based on this, the switching from an algorithm to another one is done using weighting functions with a continuous evolution in [0–1] intervals. This technique can be easily implemented using fuzzy approach, An example is presented in Fig. 10. This solution involves solving control gain problems, determined by mixing algorithm outputs.

Simplified Deployment of Robust Real-Time Systems Using

Fig. 11. Proposed multiple model switching solution

performance solution to this is proposed in the next section.

commands for the "new" selected one.

presented.

**4.2 Manual – automatic bumpless transfer**

Multiple Model and Process Characteristic Architecture-Based Process Solutions 351

corresponds to the manual control for all the other N-1 inactive algorithms., as presented in Fig. 11. In the switching situation, when a "better" *Aj* algorithm is found, the actual *Ai* active algorithm is commuted in an inactive state, and *Aj* in active state, respectively. For a bumpless commutation, the manual–automatic transfer problems must be solved, and the

The system can be implemented in two variants – first - with all inactive algorithms holding on manual regime, or – second - just a single operating algorithm (the active one) and activation of the "new" one after the computation of the currently corresponding manual regime and switching on automatic regime. Both variants have advantages and disadvantages. Choosing one of them requires knowledge about the hardware performances

In all cases, it is considered that the active algorithm output values represent manual

The "key" of proposed multiple model switching solution performances is based on manual-to-automatc bumpless transfer, so in this section some important elements about are

The practice implementation highlights important problems like manual-to-automatic (MA)/automatic-to-manual (AM) regime commutations, respectively turning out/in from the control saturation states; (i.e. manual operation is the situation where the command is calculated and applied by human operator). Of course, these problems exist in

of the structure. After a general view, the first variant seems to be more reasonable.

Fig. 9. Superposition of identification zones for two neighbor-models and their corresponding control actions

Fig. 10. Algorithms weighting functions for a specified operating position

### **4.1.2 Proposed solution**

In this subsection, there is presented a solution that provides very good results for fast processes with nonlinear characteristics. The main idea is that, during the current functioning of multiple-models control systems with N model-algorithm pairs, it is supposed that just one single algorithm is to be maintained active, the good one, and all the other N-1 algorithms rest inactive. The active and inactive states represent automatic, respectively manual, regimes of a law control. The output value of the active algorithm 350 Recent Advances in Robust Control – Novel Approaches and Design Methods

Fig. 9. Superposition of identification zones for two neighbor-models and their

Fig. 10. Algorithms weighting functions for a specified operating position

In this subsection, there is presented a solution that provides very good results for fast processes with nonlinear characteristics. The main idea is that, during the current functioning of multiple-models control systems with N model-algorithm pairs, it is supposed that just one single algorithm is to be maintained active, the good one, and all the other N-1 algorithms rest inactive. The active and inactive states represent automatic, respectively manual, regimes of a law control. The output value of the active algorithm

corresponding control actions

**4.1.2 Proposed solution**

Fig. 11. Proposed multiple model switching solution

corresponds to the manual control for all the other N-1 inactive algorithms., as presented in Fig. 11. In the switching situation, when a "better" *Aj* algorithm is found, the actual *Ai* active algorithm is commuted in an inactive state, and *Aj* in active state, respectively. For a bumpless commutation, the manual–automatic transfer problems must be solved, and the performance solution to this is proposed in the next section.

The system can be implemented in two variants – first - with all inactive algorithms holding on manual regime, or – second - just a single operating algorithm (the active one) and activation of the "new" one after the computation of the currently corresponding manual regime and switching on automatic regime. Both variants have advantages and disadvantages. Choosing one of them requires knowledge about the hardware performances of the structure. After a general view, the first variant seems to be more reasonable.

In all cases, it is considered that the active algorithm output values represent manual commands for the "new" selected one.

## **4.2 Manual – automatic bumpless transfer**

The "key" of proposed multiple model switching solution performances is based on manual-to-automatc bumpless transfer, so in this section some important elements about are presented.

The practice implementation highlights important problems like manual-to-automatic (MA)/automatic-to-manual (AM) regime commutations, respectively turning out/in from the control saturation states; (i.e. manual operation is the situation where the command is calculated and applied by human operator). Of course, these problems exist in

Simplified Deployment of Robust Real-Time Systems Using

Multiple Model and Process Characteristic Architecture-Based Process Solutions 353

Fig. 12. Computation of the set point value for imposed manual command

M3 (52-100%). These will be the zones for corresponding algorithms.

We have evaluated the achieved performances of the multi-model control structure and nonlinear compensator control using a hardware and software experimental platform, developed on National Instruments LabWindows/CVI. In figure 13, one can see a positioning control system. The main goal is the vertical control of the ball position, placed inside the pipe; here, the actuator is air supply unit connected to cDAQ family data

The obtained results are comparead to very complex (degree = 8) RST robust algorithm. Total operations number for robust structure is 24 multiplies and 24 adding or subtraction. The nonlinear relation between the position Y (%) and actuator command U (%) is presented in Figure 14. One considers three operating points P1, P2, and P3 on the plant's nonlinear diagram (Figure 14). Three different models are identified like: M1 (0-21%), M2 (21-52%) and

According to the models-algorithms matching zones (Lupu et al., 2008), we have identified the models M1, M2 and M3, as being appropriated to the following intervals (0-25%), (15- 55%) (48-100%), respectively. For a sampling period Te=0.2 sec, the least-squares identification method from Adaptech/WinPIM platform (Landau et al., 1997) identifies the

> 1 1 2 0.35620 0.05973 1 0.454010 0.09607

2 1 2 1.23779 0.33982 1 0.98066 0.17887 *q M*

3 1 2 2.309530 0.089590 1 0.827430 0.006590 *q M*

*q M*

1

1

1

*q q*

*q q*

*q q*

**5. Experimental results**

acqusition module.

next models:

analogical systems and have specific counteracting procedures, which are not applicable on numerical systems.

In real functioning, MA transfer is preceded by "driving" the process in the nominal action zone. To avoid command switching "bumps", one must respect the following two conditions:

Neglecting these conditions leads to "bumps" in the transfer because the control algorithm output value is computed using the actual, but also the past, values of the command, process and set point, respectively.

At the same time, there are situations when the perfect "matching" between process output and set point value is very difficult to obtained and/or needs a very long time. Hence, the application of this procedure becomes impossible in the presence of important disturbances. In the following, these facts will be illustrated using an RST control algorithm (Foulloy et al.,

2004), Fig. 1.

In this context, for a inactive algorithm – possible candidate for next active one, since the algorithm output is the manual command set by operator (or active algorithm) and the process output depends on command, the set point remains the only "free" variable in the control algorithm computation. Therefore, the proposed solution consists in the modification of the set point value, according to the existent control algorithm, manual command and process output (Lupu et al., 2006).

Memory updating control algorithm is done similarly as in the automatic regime. For practical implementatio a supplementary memory location for the set point value is necessary. From Eq(3), results the expression for the set point value:

$$y^\*(k) = \frac{1}{t\_0} \left[ \sum\_{i=0}^{n\_s} s\_i \mu(k-i) + \sum\_{i=0}^{n\_R} r\_i y(k-i) - \sum\_{i=1}^{n\_T} t\_i y^\*(k-i) \right] \tag{14}$$

When the set point (trajectory) generator Eq(4) exists, keeping all the data in correct chronology must be with respect to the following relation:

$$r(k) = \frac{A\_m(q^{-1})}{B\_m(q^{-1})} y^\*(k) \tag{15}$$

System operation scheme is presented in Fig. 12.

Concluding, this solution proposes the computation of that set point value that determines, according to the algorithm history and process output, a control equal to the manual command applied by the operator (or active algorithm). At the instant time of the MA switching, there are no gaps in the control algorithm memory that could determine bumps. An eventually mismatching between the set point and process output is considered as a simple change of the set point value. Moreover, this solution can be successfully used in cases of command limitation.

The only inconvenient of this solution is represented by the necessary big computation power when approaching high order systems, which is not, however, a problem nowadays.

Fig. 12. Computation of the set point value for imposed manual command

## **5. Experimental results**

352 Recent Advances in Robust Control – Novel Approaches and Design Methods

analogical systems and have specific counteracting procedures, which are not applicable on

In real functioning, MA transfer is preceded by "driving" the process in the nominal action zone. To avoid command switching "bumps", one must respect the following two

 According to the algorithm complexity (function of the degrees of controller polynomials), the complete algorithm memory actualization must be waited for. Neglecting these conditions leads to "bumps" in the transfer because the control algorithm output value is computed using the actual, but also the past, values of the command,

At the same time, there are situations when the perfect "matching" between process output and set point value is very difficult to obtained and/or needs a very long time. Hence, the application of this procedure becomes impossible in the presence of important disturbances. In the following, these facts will be illustrated using an RST control algorithm (Foulloy et al.,

In this context, for a inactive algorithm – possible candidate for next active one, since the algorithm output is the manual command set by operator (or active algorithm) and the process output depends on command, the set point remains the only "free" variable in the control algorithm computation. Therefore, the proposed solution consists in the modification of the set point value, according to the existent control algorithm, manual

Memory updating control algorithm is done similarly as in the automatic regime. For practical implementatio a supplementary memory location for the set point value is

When the set point (trajectory) generator Eq(4) exists, keeping all the data in correct

1 ( ) \* ( ) ( ) 1 ( ) *A q rk y k m B q m*

Concluding, this solution proposes the computation of that set point value that determines, according to the algorithm history and process output, a control equal to the manual command applied by the operator (or active algorithm). At the instant time of the MA switching, there are no gaps in the control algorithm memory that could determine bumps. An eventually mismatching between the set point and process output is considered as a simple change of the set point value. Moreover, this solution can be successfully used in

The only inconvenient of this solution is represented by the necessary big computation power when approaching high order systems, which is not, however, a problem nowadays.

(14)

(15)

\* \* 001 0 1 () [ ( ) ( ) ( )] *nS n n R T ii i ii i y su k i ry k i ty k i t*

Process output must be perfectly matched with the set point value;

numerical systems.

process and set point, respectively.

command and process output (Lupu et al., 2006).

*k*

System operation scheme is presented in Fig. 12.

cases of command limitation.

chronology must be with respect to the following relation:

necessary. From Eq(3), results the expression for the set point value:

conditions:

2004), Fig. 1.

We have evaluated the achieved performances of the multi-model control structure and nonlinear compensator control using a hardware and software experimental platform, developed on National Instruments LabWindows/CVI. In figure 13, one can see a positioning control system. The main goal is the vertical control of the ball position, placed inside the pipe; here, the actuator is air supply unit connected to cDAQ family data acqusition module.

The obtained results are comparead to very complex (degree = 8) RST robust algorithm. Total operations number for robust structure is 24 multiplies and 24 adding or subtraction.

The nonlinear relation between the position Y (%) and actuator command U (%) is presented in Figure 14. One considers three operating points P1, P2, and P3 on the plant's nonlinear diagram (Figure 14). Three different models are identified like: M1 (0-21%), M2 (21-52%) and M3 (52-100%). These will be the zones for corresponding algorithms.

According to the models-algorithms matching zones (Lupu et al., 2008), we have identified the models M1, M2 and M3, as being appropriated to the following intervals (0-25%), (15- 55%) (48-100%), respectively. For a sampling period Te=0.2 sec, the least-squares identification method from Adaptech/WinPIM platform (Landau et al., 1997) identifies the next models:

$$\begin{aligned} M\_1 &= \frac{0.35620 - 0.05973q^{-1}}{1 - 0.454010q^{-1} - 0.09607q^{-2}} \\\\ M\_2 &= \frac{1.23779 - 0.33982q^{-1}}{1 - 0.98066q^{-1} - 0.17887q^{-2}} \\\\ M\_3 &= \frac{2.309530 - 0.089590q^{-1}}{1 - 0.827430q^{-1} - 0.006590q^{-2}} \end{aligned}$$

Simplified Deployment of Robust Real-Time Systems Using

Multiple Model and Process Characteristic Architecture-Based Process Solutions 355

In this case, we have computed three corresponding RST algorithms using a pole placement procedure from Adaptech/WinREG platform (Landau et al., 1997). The same nominal performances are imposed to all systems, through a second order system, defined by the dynamics 0 = 3.0, = 2.5 (tracking performances) and 0 = 7.5, = 0.8 (disturbance rejection

> 1 1 2 1 *R* ( ) 1.670380 -0.407140 -0.208017 *q q q*

1 1 2 1 *S* ( ) 1.000000 -1.129331 0.129331 *q qq*

1 1 2 1 *T q*( ) 3.373023 -3.333734q 1.015934q

1 1 2 2 *R* ( ) 0.434167 0.153665 -0.239444 *q qq*

1 1 2 2 *S* ( ) 1.000000 -0.545100 -0.454900 *q q q*

1 1 2 2 *T q*( ) 1.113623 -1.100651q 0.335417q

1 1 2 3 *R* ( ) 0.231527 -0.160386 -8.790E-04 *q qq*

1 1 2 3 *S* ( ) 1.000000 -0.988050 -0.011950 *q q q*

1 1 2 3 *T q*( ) 0.416820 -0.533847q 0.187289q

Fig. 15. Multi-model controller real-time software application

performances) respectively, keeping the same sampling period as for identification. All of these algorithms control the process in only their corresponding zones.

Fig. 13. Process experimental platform

Fig. 14. Nonlinear diagram of the process

354 Recent Advances in Robust Control – Novel Approaches and Design Methods

Fig. 13. Process experimental platform

Fig. 14. Nonlinear diagram of the process

In this case, we have computed three corresponding RST algorithms using a pole placement procedure from Adaptech/WinREG platform (Landau et al., 1997). The same nominal performances are imposed to all systems, through a second order system, defined by the dynamics 0 = 3.0, = 2.5 (tracking performances) and 0 = 7.5, = 0.8 (disturbance rejection performances) respectively, keeping the same sampling period as for identification.  Multi-model controller real-time software application

Simplified Deployment of Robust Real-Time Systems Using

effectuated to verify the structure. These tests are:

(0-100%) in 10 subinterval (0-10, 10-20 etc).

keeping the same sampling period as for identification.

7 multiplies and 7 adding or subtraction operations.

Fig. 18. a) Process static determination test; b) functioning test;

evolutions.

functioning points.

Multiple Model and Process Characteristic Architecture-Based Process Solutions 357

user interface is presented on Figure 17. This application implement the scheme proposed in Fig 7 and allows the user in a special window, to construct the nonlinear compensator. Using this application, that contains a simple second order RST algorithm, few tests were

a. Determination of inverse model characteristic. Figure 18(a) presents this evolutions and contains the corresponding *r(k)-u(k)* data pairs obtained by dividing the total domain

b. Testing structure stability on different functioning point. Figure 18(b) presents these

On (a) test one can see the nonlinear process model characteristics identification procedures. The second one, present that there are no shocks and the system is stable on different

For proposed control structure, presented in Figure 7 was identified a wery simple model:

*M*

1.541650 1 0.790910

*q*

In this case, we have computed the corresponding RST algorithms using a pole placement procedure from Adaptech/WinREG platform. The nominal performances are imposed, through a second order system, defined by the dynamics 0 = 2, = 0.95 (tracking performances) and 0 = 1.1, = 0.8 (disturbance rejection performances) respectively,

1 1 *R*( ) 0.083200 -0.056842 *q q*

1 1 *S q*( ) 1.000000 -1.000000*q*

1 1 2 *T q*( ) 0.648656 -1.078484q 0.456187q

To calculate corresponding command for a single controller presented before, there are used

For the second control structure, in addition to command calculus operation here is the calculus of direct command. This depends on software implementation. For PLC, particular

1

To verify the proposed switching algorithm, a multi-model controller real-time software application was designed and implemented, that can be connected to the process. The user interface is presented on Figure 15.

On the top of Figure 15, there are respectively the set point, the output and control values, manual-automatic general switch, general manual command and graphical system evolution display. On the bottom of Figure 15, one can see three graphical evolution displays corresponding to the three controllers (Ri, Si, Ti, i=1...3). The colors are as follows: yellow – set point value, red – command value, blue – process output value and green – filtered set point value.

Using this application, few tests were done to verify the switching between two algorithms. The switching procedure is determinate by the change of the set point value. These tests are:

In both tests, one can see that there are no shocks or that there are very small oscillations in the control evolution by applying this approach. Increasing the number of modelsalgorithms to 4 or 5 could eliminate the small oscillations.

To verify the nonlinear compensator control structure, a second real-time software application was designed and implemented, that can be connected with the process. The

Fig. 16. a) (left) switching test; b) (right) switching test

Fig. 17. Nonlinear compensator controller real-time software application

356 Recent Advances in Robust Control – Novel Approaches and Design Methods

To verify the proposed switching algorithm, a multi-model controller real-time software application was designed and implemented, that can be connected to the process. The user

On the top of Figure 15, there are respectively the set point, the output and control values, manual-automatic general switch, general manual command and graphical system evolution display. On the bottom of Figure 15, one can see three graphical evolution displays corresponding to the three controllers (Ri, Si, Ti, i=1...3). The colors are as follows: yellow – set point value, red – command value, blue – process output value and green –

Using this application, few tests were done to verify the switching between two algorithms. The switching procedure is determinate by the change of the set point value. These tests are: a. from 20% (where algorithm 1 is active) to 40% (where algorithm 2 is active). The effective switching operation is done when the filtered set point (and process output)

b. from 38% (where algorithm 2 is active) to 58% (where algorithm 3 is active). The effective switching operation is done when the filtered set point (and process output)

In both tests, one can see that there are no shocks or that there are very small oscillations in the control evolution by applying this approach. Increasing the number of models-

To verify the nonlinear compensator control structure, a second real-time software application was designed and implemented, that can be connected with the process. The

becomes greater than 21%. Figure 16(a) presents the evolutions.

becomes greater than 52%. Figure 16(b) presents the evolutions.

algorithms to 4 or 5 could eliminate the small oscillations.

Fig. 16. a) (left) switching test; b) (right) switching test

Fig. 17. Nonlinear compensator controller real-time software application

interface is presented on Figure 15.

filtered set point value.

user interface is presented on Figure 17. This application implement the scheme proposed in Fig 7 and allows the user in a special window, to construct the nonlinear compensator.

Using this application, that contains a simple second order RST algorithm, few tests were effectuated to verify the structure. These tests are:

On (a) test one can see the nonlinear process model characteristics identification procedures. The second one, present that there are no shocks and the system is stable on different functioning points.

For proposed control structure, presented in Figure 7 was identified a wery simple model:

$$M = \frac{1.541650}{1 - 0.790910q^{-1}}$$

In this case, we have computed the corresponding RST algorithms using a pole placement procedure from Adaptech/WinREG platform. The nominal performances are imposed, through a second order system, defined by the dynamics 0 = 2, = 0.95 (tracking performances) and 0 = 1.1, = 0.8 (disturbance rejection performances) respectively, keeping the same sampling period as for identification.

$$R(q^{-1}) = 0.083200 \quad \text{-} 0.056842q^{-1}$$

$$S(q^{-1}) = 1.0000000 \quad \text{-} 1.0000000q^{-1}$$

$$T(q^{-1}) = 0.648656 \quad \text{-} 1.078484 \mathbf{q}^{-1} \quad + \quad 0.456187 \mathbf{q}^{-2}$$

To calculate corresponding command for a single controller presented before, there are used 7 multiplies and 7 adding or subtraction operations.