**3. Robust decentralized PID controller design in the time domain**

In this section, robust decentralized controller in time domain is designed based on robust stability conditions formulated and solved as linear (or bilinear) matrix inequalities. To include performance evaluation, the quadratic performance index is used. Decentralized robust control problem is formulated in general framework for augmented system, including the model of controlled system as well as controller dynamics. The robust stability conditions from literature are recalled, using D-stability concept which enables unified formulation for continuous-time and discrete-time cases. Our modification of these results includes derivative term of PID controller as well as a term for guaranteed cost. Thus, the decentralized control design procedure is presented in the general form comprising both continuous and discrete-time system models.

Notation: for a symmetric square matrix X, *X >* 0 denotes positive definiteness; \* in matrices denotes the respective transposed term to make the matrix symmetric, 0 in matrices denotes zero block of the corresponding dimensions, In denotes identity matrix of dimensions nxn; dimension index is often omitted, when the dimension is clear from the context. Argument t denotes either continuous time for continuous-time, or sampled time for discrete-time system models; we intentionally use the same symbol t for both cases to underline that the formulation of developed results is general, applicable for both cases.

#### **3.1 Preliminaries and problem formulation**

#### **3.1.1 Decentralized control of uncertain system, guaranteed cost control**

Consider a linearized model of interconnected system, where subsystems with polytopic uncertainty are assumed, described by

$$\mathbf{S} \colon \delta \mathbf{x}\_i(t) = A\_i(\boldsymbol{\alpha}) \mathbf{x}\_i(t) + B\_i(\boldsymbol{\alpha}) \boldsymbol{\mu}\_i(t) + \sum\_{\substack{j=1 \\ j \neq i}}^N (A\_{ij}(\boldsymbol{\alpha}) \mathbf{x}\_j(t) + B\_{ij}(\boldsymbol{\alpha}) \boldsymbol{\mu}\_j(t))$$

$$y\_i(t) = \mathbb{C}\_i \mathbf{x}\_i(t); \ i = \mathbf{1}, \ldots, \mathbf{N} \tag{9}$$

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

γ2

1 WP1

0 1

WP3 WP2

γ1

a) minimum phase configuration b) nonminimum phase configuration

In this section, robust decentralized controller in time domain is designed based on robust stability conditions formulated and solved as linear (or bilinear) matrix inequalities. To include performance evaluation, the quadratic performance index is used. Decentralized robust control problem is formulated in general framework for augmented system, including the model of controlled system as well as controller dynamics. The robust stability conditions from literature are recalled, using D-stability concept which enables unified formulation for continuous-time and discrete-time cases. Our modification of these results includes derivative term of PID controller as well as a term for guaranteed cost. Thus, the decentralized control design procedure is presented in the general form comprising both

Notation: for a symmetric square matrix X, *X >* 0 denotes positive definiteness; \* in matrices denotes the respective transposed term to make the matrix symmetric, 0 in matrices denotes zero block of the corresponding dimensions, In denotes identity matrix of dimensions nxn; dimension index is often omitted, when the dimension is clear from the context. Argument t denotes either continuous time for continuous-time, or sampled time for discrete-time system models; we intentionally use the same symbol t for both cases to underline that the

Consider a linearized model of interconnected system, where subsystems with polytopic

*i i i ii ij j ij j j j i*

() () *i ii y t Cx t* ; i=1,...,N (9)

 *xt A xt B ut A xt B ut* 

1 ( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( )) *N*

   

**3. Robust decentralized PID controller design in the time domain** 

γ1

formulation of developed results is general, applicable for both cases.

**3.1.1 Decentralized control of uncertain system, guaranteed cost control** 

Fig. 2. Uncertainty domain specified by working points

WP2

WP3

0 1

γ2

1

WP1

continuous and discrete-time system models.

**3.1 Preliminaries and problem formulation** 

Si:

uncertainty are assumed, described by

where *xt xt* () () for continuous-time system model; *xt xt* ( ) ( 1) for discrete-time system model; ( ) *ni <sup>i</sup> xt R* , ( ) *mi ut R <sup>i</sup>* , ( ) *<sup>i</sup> <sup>p</sup> <sup>i</sup> <sup>y</sup> t R* are the subsystem state, control and output vectors respectively, 1 *N i i n n* , 1 *N i i m m* , 1 *N i i p p* ; *Ci* are matrices with corresponding dimensions. Uncertain model matrices ( ) *Ai* , ( ) *Bi* , ( ) *Aij* , ( ) *Bij* are from polytopic uncertainty domains

$$A\_i(\boldsymbol{\alpha}) \in \left\{ \sum\_{k=1}^K \alpha\_k \tilde{A}\_{ik}, \sum\_{k=1}^K \alpha\_k = 1, \alpha\_k \ge 0 \right\}, B\_i(\boldsymbol{\alpha}) \in \left\{ \sum\_{k=1}^K \alpha\_k \tilde{B}\_{ik}, \sum\_{k=1}^K \alpha\_k = 1, \alpha\_k \ge 0 \right\},$$

$$A\_{ij}(\boldsymbol{\alpha}) \in \left\{ \sum\_{k=1}^K \alpha\_k \tilde{A}\_{ijk}, \sum\_{k=1}^K \alpha\_k = 1, \alpha\_k \ge 0 \right\}, B\_{ij}(\boldsymbol{\alpha}) \in \left\{ \sum\_{k=1}^K \alpha\_k \tilde{B}\_{ijk}, \sum\_{k=1}^K \alpha\_k = 1, \alpha\_k \ge 0 \right\}.\tag{10}$$

The whole interconnected system model in the compact form is

$$\begin{aligned} \text{S: } \,\,\delta\mathbf{x}(t) &= A(\alpha)\mathbf{x}(t) + B(\alpha)\boldsymbol{\mu}(t) \\\\ \mathbf{y}(t) &= \mathbf{C}\_d \mathbf{x}(t) \end{aligned} \tag{11}$$

uncertain system matrix ( ) ( ) *AA A d m* , *BB B* ( ) *d m* ( ) and

$$A(\alpha) \in \left\{ \sum\_{k=1}^{K} \alpha\_k \tilde{A}\_{(k)}, \sum\_{k=1}^{K} \alpha\_k = 1, \alpha\_k \ge 0 \right\}, \text{ } B(\alpha) \in \left\{ \sum\_{k=1}^{K} \alpha\_k \tilde{B}\_{(k)}, \sum\_{k=1}^{K} \alpha\_k = 1, \alpha\_k \ge 0 \right\} \tag{12}$$

where *A*( ) *<sup>k</sup>* has diagonal blocks *Aik* and off-diagonal blocks *Aijk* , *<sup>B</sup>*( ) *<sup>k</sup>* has diagonal blocks *Bik* and off-diagonal blocks *Bijk* respective to (10); and

1 2 1 2 1 2 ( ) ( ... ), ( ) ( ... ), ( ) ( ... ) *TT T TT T TT T xt x x x ut u u u yt y y y NNN* are state, control and output vectors of the overall system S;

<sup>1</sup> ( ) { ( ),..., ( )}, *A diag A A d N* <sup>1</sup> ( ) { ( ),..., ( )}, *B diag B B d N* <sup>1</sup> { ,..., } *C diag C C d N* are overall system matrices of corresponding dimensions respective to the subsystems, matrices *Am*( ) , *Bm*( ) correspond to interconnections.

A closed loop system performance is assessed considering the guaranteed cost notion; the quadratic cost function known from LQ theory is used.

$$\begin{aligned} \mathbf{J}\_c &= \underset{\mathbf{0}}{\overset{\alpha}{\mathbf{I}}} \[\mathbf{x}(t)^T Q \mathbf{x}(t) + \boldsymbol{\mu}(t)^T R \mathbf{x}(t)\} dt \qquad \text{for a continuous-time and} \\\\ \mathbf{J}\_d &= \underset{\mathbf{k} = \mathbf{0}}{\overset{\alpha}{\mathbf{I}}} \[\mathbf{x}(t)^T Q \mathbf{x}(t) + \boldsymbol{\mu}(t)^T R \mathbf{x}(t)\} \text{ for a discrete-time systems} \end{aligned} \tag{13}$$

where , *nn mm QR RR* are symmetric positive semidefinite and positive definite block diagonal matrices respectively, with block dimensions respective to the subsystems. The concept of guaranteed cost control is used in a standard way: let there exist a control law *u t*( ) and a constant 0*J* such that

$$\|\cdot\|\_{0}\tag{14}$$

Robust Decentralized PID Controller Design 141

Then the closed-loop system (11) with PID controller (17) can be described by augmented

() 0 () ( ) <sup>0</sup> 0 0 0

or

0 () 0 () 0 ()

*I zC z z*

 

0 () 0 ( ) 0 , 00 0

*I C*

() 0 () 0

 

*C C*

*d d*

*d*

*d d*

(20)

(21)

0 ( ) *t*

, also y(t-1) is

*i y i* 

*d I B KC x A x B x*

() () *Md nC n*

*IB C*

*AB C*

 

( ) 00 0

A discrete-time PID (often denoted as PSD) controller is described by control algorithm

0 ( ) ( ) ( ) [ ( ) ( 1)] *t PI D i ut k et k ei k et et* 

0 ( ) ( ) ( ) [ ( ) ( 1)] *t PI D i ut k yt k yi k yt yt* 

of PID controller (21) requires two state variables, since besides

, then 2 1 *<sup>y</sup>*( 1) ( ) ( ) *t zt zt* . Rewriting (21) as

State space description of PID controller can be derived in the following way. The dynamics

needed. One possible choice of controller state variables is: 1 2 ( ) [ ( ) ( )] *T TT zt z t z t* ,

where *u t*( ) , *et yt wt* () () () , *w t*( ) are discrete time counterparts to the continuous time signals; , , *k kk PID* are controller parameter matrices to be designed. By analogy with

*C P I*

*A K K*

*d D*

*M K*

 

 

*A xB x B x*

*C z z z*

() 0 () 0 0

0 00 0 0

 

 

*xA xB x u zC z*

*d*

*d*

 

*n*

which in a compact form yields

argument t is omitted for brevity.

2 1

0 0 ( ) ( ), ( ) ( ) *t t*

*i i z t y izt y i* 

1 2

*D d*

continuous time case, for constant *w t*( ) we write

model

where

() () () () *ut K C xt KC zt K C xt Pd Id Dd* . (17)

   

*Pd Id*

*K C KC*

*xA x* (18)

(19)

*Pd Id D d*

*K C KC K C*

holds for the closed loop system (9). Then the respective control *u t*( ) is called the *guaranteed cost control* and the value 0*J* is the *guaranteed cost*.

#### *Decentralized Control Problem*

The control design aim is to find decentralized control law ( ( )) *uxt i i* , or ( ( )) *uyt i i* , i=1,…,N , i.e. the overall system is controlled using local control loops for subsystems, such that uncertain dynamic system (11) is robustly stable in uncertainty domain (12) with guaranteed cost.

Basically, control design problem will be transformed into the output feedback form: () () *u t Fy t i ii* , employing augmented system model to include controller dynamics, as it is using PID controller.

#### **3.1.2 Augmented system model for continuous and discrete-time PID controller**

The augmented system model including PID controller dynamics is developed in this section in general form appropriate both for continuous and discrete-time PID controllers. Firstly, recall PID control algorithms for both cases.

Control algorithm for continuous-time PID is

$$\ln(t) = K\_P e(t) + K\_I \int\_0^t e(t)dt + K\_D \dot{e}(t) \tag{15}$$

where *et yt wt* () () () is control error, *w k*( ) is reference value (negative feedback sign is included in matrices , , *K KK PID* ); , , *K KK PID* are controller parameter matrices (for SISO system they are scalars) to be designed.

Generally, different output variables can be considered for proportional, integral and derivative controller terms, for better readability we assume that all outputs enter all three controller terms. We further assume that the reference value is constant, *wk w* ( ) and that the system states in model (11) correspond to the deviations from working point (these assumptions correspond to step change of reference value). Then the control law (15) can be rewritten as

$$u(t) = K\_P y(t) + K\_I \int\_0^t y(t)dt + K\_D \dot{y}(t) \,. \tag{16}$$

Integral term can be included into the state vector in the common way defining the auxiliary state 0 ( ) *t z yt* , i.e. () () () *<sup>d</sup> zt yt C xt* and PID controller algorithm is

$$\mathbf{u}(t) = \mathbf{K}\_P \mathbf{C}\_d \mathbf{x}(t) + \mathbf{K}\_I \mathbf{C}\_d \mathbf{z}(t) + \mathbf{K}\_D \mathbf{C}\_d \dot{\mathbf{x}}(t) \,. \tag{17}$$

Then the closed-loop system (11) with PID controller (17) can be described by augmented model

 () 0 () 0 0 () 0 () ( ) <sup>0</sup> 0 0 0 *n d Pd Id D d d xA xB x u zC z A xB x B x K C KC K C C z z z* or 0 () 0 () 0 () 0 00 0 0 *D d Pd Id d I B KC x A x B x K C KC I zC z z* 

which in a compact form yields

$$M\_d(\boldsymbol{\alpha})\dot{\mathbf{x}}\_n = A\_\mathbb{C}(\boldsymbol{\alpha})\mathbf{x}\_n \tag{18}$$

where

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

where , *nn mm QR RR* are symmetric positive semidefinite and positive definite block diagonal matrices respectively, with block dimensions respective to the subsystems. The concept of guaranteed cost control is used in a standard way: let there exist a control law

holds for the closed loop system (9). Then the respective control *u t*( ) is called the *guaranteed* 

The control design aim is to find decentralized control law ( ( )) *uxt i i* , or ( ( )) *uyt i i* , i=1,…,N , i.e. the overall system is controlled using local control loops for subsystems, such that uncertain dynamic system (11) is robustly stable in uncertainty domain (12) with guaranteed

Basically, control design problem will be transformed into the output feedback form: () () *u t Fy t i ii* , employing augmented system model to include controller dynamics, as it is

> 0 () () () () *t*

where *et yt wt* () () () is control error, *w k*( ) is reference value (negative feedback sign is included in matrices , , *K KK PID* ); , , *K KK PID* are controller parameter matrices (for SISO

Generally, different output variables can be considered for proportional, integral and derivative controller terms, for better readability we assume that all outputs enter all three controller terms. We further assume that the reference value is constant, *wk w* ( ) and that the system states in model (11) correspond to the deviations from working point (these assumptions correspond to step change of reference value). Then the control law (15) can be

> 0 () () () () *t*

Integral term can be included into the state vector in the common way defining the auxiliary

*z yt* , i.e. () () () *<sup>d</sup> zt yt C xt* and PID controller algorithm is

*u t K e t K e t dt K e t PI D* (15)

*u t K y t K y t dt K y t PI D* . (16)

**3.1.2 Augmented system model for continuous and discrete-time PID controller**  The augmented system model including PID controller dynamics is developed in this section in general form appropriate both for continuous and discrete-time PID controllers.

<sup>0</sup> *J J* (14)

*u t*( ) and a constant 0*J* such that

*Decentralized Control Problem* 

using PID controller.

rewritten as

state

0 ( ) *t*

cost.

*cost control* and the value 0*J* is the *guaranteed cost*.

Firstly, recall PID control algorithms for both cases. Control algorithm for continuous-time PID is

system they are scalars) to be designed.

$$\begin{aligned} \boldsymbol{M}\_{d}(\boldsymbol{\alpha}) &= \begin{bmatrix} \begin{bmatrix} I & 0 \\ 0 & I \end{bmatrix} - \begin{bmatrix} B(\boldsymbol{\alpha}) \\ 0 \end{bmatrix} \begin{bmatrix} \boldsymbol{K}\_{D} & \boldsymbol{0} \end{bmatrix} \begin{bmatrix} \mathbf{C}\_{d} & \mathbf{0} \\ \mathbf{0} & \mathbf{C}\_{d} \end{bmatrix} \end{aligned} \tag{19}$$
 
$$\boldsymbol{A}\_{\boldsymbol{C}}(\boldsymbol{\alpha}) = \begin{bmatrix} \boldsymbol{A}(\boldsymbol{\alpha}) & \mathbf{0} \\ \mathbf{C}\_{d} & \mathbf{0} \end{bmatrix} + \begin{bmatrix} B(\boldsymbol{\alpha}) \\ \mathbf{0} \end{bmatrix} \begin{bmatrix} \boldsymbol{K}\_{P} & \boldsymbol{K}\_{I} \end{bmatrix} \begin{bmatrix} \mathbf{C}\_{d} & \mathbf{0} \\ \mathbf{0} & \mathbf{C}\_{d} \end{bmatrix}$$

argument t is omitted for brevity.

A discrete-time PID (often denoted as PSD) controller is described by control algorithm

$$u(t) = k\_P e(t) + k\_I \sum\_{i=0}^{l} e(i) + k\_D [e(t) - e(t-1)] \tag{20}$$

where *u t*( ) , *et yt wt* () () () , *w t*( ) are discrete time counterparts to the continuous time signals; , , *k kk PID* are controller parameter matrices to be designed. By analogy with continuous time case, for constant *w t*( ) we write

$$u(t) = k\_P y(t) + k\_I \sum\_{i=0}^{t} y(i) + k\_D [y(t) - y(t-1)] \tag{21}$$

State space description of PID controller can be derived in the following way. The dynamics of PID controller (21) requires two state variables, since besides 0 ( ) *t i y i* , also y(t-1) is needed. One possible choice of controller state variables is: 1 2 ( ) [ ( ) ( )] *T TT zt z t z t* , 2 1 1 2 0 0 ( ) ( ), ( ) ( ) *t t i i z t y izt y i* , then 2 1 *<sup>y</sup>*( 1) ( ) ( ) *t zt zt* . Rewriting (21) as

Robust Decentralized PID Controller Design 143

*Rd R*

In a decentralized PID controller design, controller gain matrices are restricted to block

The presented general closed loop augmented system polytopic model (26) is

In this section we recall several recent results on robust stability for linear uncertain systems with polytopic model. These results are formulated as robust stability conditions in LMI form. Let us start with basic notions concerning Lyapunov stability and D-stability concept (Peaucelle et al., 2000; Henrion et al., 2002), used to receive the robust stability conditions in

{ iscomplex number : 0}

(For simplicity, we use in Def. 3.1 scalar values of parameters rij, in general, the stability domain can be defined using matrix values of parameters rij with the respective dimensions.) The standard choice of rij is r11 = 0, r12 = 1, r22 = 0 for a continuous-time system; r11 = -1, r12 = 0, r22 = 1 for a discrete-time system, corresponding to open left half plane and

The D-stability concept enables to formulate robust stability condition for uncertain polytopic system in general way, (deOliveira et al., 1999; Peaucelle et al., 2000). The following robust stability condition is based on the existence of Lyapunov function

> 

Uncertain system (30) is *robustly D-stable* in the convex uncertainty domain (12) if and only if

\* <sup>12</sup> <sup>12</sup> 11 22 ()() ()() () ()()() 0 *T T rP A rA P rP rA P A*

  **,** 

*F FF F k k kF k k k* <sup>121</sup> ; , *PID DID* <sup>2</sup> for a discrete-time case. (28b)

*aug*

\* 11 12 \* 12 22 1 1

*r r*

*s s r r*

*xt A xt* () ( )() (30)

(31)

2 0

*p*

and ( ) *Md* 

for a continuous-time case; (28a)

*I* **.** (27b)

(29)

0 *d*

*C <sup>C</sup> <sup>I</sup>* 

*BC A* 

*A*

for a discrete-time PID: () 0 ( ) *aug*

diagonal structure respective to subsystem dimensions.

Consider the D-domain in the complex plain defined as

Linear system is D-stable if and only if all its poles lie in the D-domain.

for linear uncertain polytopic system

is from uncertainty domain (12).

  such that

 

*D s*

advantageously used in next developments.

PID controller parameters are:

**3.1.3 Robust stability** 

more general form. *Definition 3.1* (D-stability)

unit circle respectively.

*Definition 3.2* (Robust stability)

there exists a matrix () () 0 *<sup>T</sup> P P*

*Vt xtP xt* () () ( )() 

where *A*( ) 

*A*

*FFF K K* 1 2 *P I* and *KD* included in ( ) *Md*

$$\begin{split} u(t) &= k\_P y(t) + k\_I \sum\_{i=0}^{t-1} y(i) + k\_I y(t) + k\_D [y(t) - y(t-1)] \\ &= (k\_P + k\_I + k\_D) y(t) + k\_I z\_2(t) - k\_D (z\_2(t) - z\_1(t)) \end{split} \tag{21}$$

we obtain the respective description of the discrete-time PID controller in state space as

$$\begin{aligned} z(t+1) &= \begin{bmatrix} 0 & I \\ 0 & I \end{bmatrix} z(t) + \begin{bmatrix} 0 \\ I \end{bmatrix} y(t) = A\_R z(t) + B\_R y(t) \\ u(t) &= \begin{bmatrix} k\_D & k\_I - k\_D \end{bmatrix} z(t) + (k\_P + k\_I + k\_D) y(t) = \\ &= \mathcal{C}\_R z(t) + D\_R y(t) \end{aligned} \tag{22}$$

where z(t) is controller dynamics state vector, <sup>2</sup> ( ) *<sup>p</sup> zt R* **.** 

The respective augmented model for discrete-time version of system (11) with PID controller is

$$\mathbf{x}\_n(t+1) = \begin{bmatrix} \mathbf{x}(t+1) \\ \mathbf{z}(t+1) \end{bmatrix} = \begin{bmatrix} A(\alpha) & 0 \\ B\_R \mathbf{C}\_d & A\_R \end{bmatrix} \begin{bmatrix} \mathbf{x}(t) \\ \mathbf{z}(t) \end{bmatrix} + \begin{bmatrix} B(\alpha) \\ 0 \end{bmatrix} \begin{bmatrix} (D\_R \mathbf{C}\_d & \mathbf{C}\_R) \end{bmatrix} \begin{bmatrix} \mathbf{x}(t) \\ \mathbf{z}(t) \end{bmatrix} \tag{23}$$

where 2 2 <sup>0</sup> , <sup>0</sup> *p p R R I AR A I* **,** <sup>2</sup> <sup>0</sup> , *p p BR B R R <sup>I</sup>* **,**  <sup>2</sup> , *m p C R C k kk R R DID* **,** 

*D k kk RPID* **.** 

Analogically as in continuous time case, the augmented system (23) can be rewritten in a compact form as

$$\mathbf{x}\_n(t+1) = A\_\mathbb{C}(\alpha)\mathbf{x}\_n(t) \tag{24}$$

$$\begin{array}{ll}\text{where} & A\_{\mathbb{C}}(a) = \begin{bmatrix} A(a) & 0 \\ B\_{R}\mathbb{C}\_{d} & A\_{R} \end{bmatrix} + \begin{bmatrix} B(a) \\ 0 \end{bmatrix} \begin{bmatrix} D\_{R} & \mathbb{C}\_{R} \end{bmatrix} \begin{bmatrix} \mathbb{C}\_{d} & 0 \\ 0 & I\_{2p} \end{bmatrix} .\end{array} \tag{25}$$

Summarizing the augmented closed loop system models (18), (19) and (24), (25) for continuous and discrete-time PID controllers respectively, we can finally, using denotation *x t*( ) **,** introduced in (9), rewrite both of them in general form

$$M\_d(a)\delta \mathbf{x}\_n(t) = A\_\mathbb{C}(a)\mathbf{x}\_n(t) \tag{26}$$

where ( ) *Md* is assumed to be invertible,

$$A\_C(\alpha) = A\_{\text{aug}}(\alpha) + \begin{bmatrix} B(\alpha) \\ 0 \end{bmatrix} \begin{bmatrix} F\_1 & F\_2 \end{bmatrix} \mathbf{C}\_{\text{aug}} = A\_{\text{aug}}(\alpha) + B\_{\text{aug}}(\alpha) \mathbf{F} \mathbf{C}\_{\text{aug}} \quad \text{and} \quad \mathbf{C}\_{\text{aug}}$$

for a continuous PID: () 0 ( ) <sup>0</sup> *aug <sup>d</sup> A A C* **,** <sup>0</sup> 0 *d aug d C C C* and

$$M\_d(\mathbf{a}) = \begin{pmatrix} \begin{bmatrix} I & 0 \\ 0 & I \end{bmatrix} - \begin{bmatrix} B(\mathbf{a})K\_D \mathbf{C}\_d & 0 \\ 0 & 0 \end{bmatrix} \end{pmatrix};\tag{27a}$$

$$\text{for a discrete-time PID: } A\_{\text{aug}}(\alpha) = \begin{bmatrix} A(\alpha) & 0 \\ B\_R \mathbb{C}\_d & A\_R \end{bmatrix}, \ \mathbb{C}\_{\text{aug}} = \begin{bmatrix} \mathbb{C}\_d & 0 \\ 0 & I\_{2p} \end{bmatrix} \text{ and } \ M\_d(\alpha) = I. \text{ (27b)}$$

PID controller parameters are:

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

( ) ( ) ( ) ( ) [ ( ) ( 1)]

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

*k k k yt kz t k z t z t*

2 21

*R R*

(21)

(22)

**,**  <sup>2</sup> , *m p C R C k kk R R DID* **,** 

(24)

**.** (25)

**;** (27a)

2

*xt A xt* (26)

and

and

*d*

*C*  *d*

(23)

1

*t*

*i*

*R R*

*I*

*I*

where

*x t*( ) **,** introduced in (9), rewrite both of them in general form

is assumed to be invertible,

 1 2 ( ) () () () () <sup>0</sup> *C aug aug aug aug aug*

*A A F F C A B FC*

*d*

*M*

where z(t) is controller dynamics state vector, <sup>2</sup> ( ) *<sup>p</sup> zt R* **.** 

controller is

where 2 2 <sup>0</sup> , <sup>0</sup> *p p R R*

*D k kk RPID* **.** 

compact form as

where ( ) *Md*

  *B*

for a continuous PID: () 0 ( ) <sup>0</sup> *aug <sup>d</sup>*

*A*

*AR A*

*C zt D yt*

() ()

*I*

0

0 0

 

*PI ID*

*ut k yt k yi kyt k yt yt*

*PID I D*

we obtain the respective description of the discrete-time PID controller in state space as

( 1) ( ) ( ) ( ) ( ) <sup>0</sup>

*z t zt yt A zt B yt I I*

( ) () ( )()

*ut k k k zt k k k yt*

*DID PID*

The respective augmented model for discrete-time version of system (11) with PID

( 1) ( ) 0 ( ) ( ) ( ) ( 1) ( ( 1) () 0 ( ) *<sup>n</sup> Rd R*

**,** <sup>2</sup> <sup>0</sup> , *p p BR B R R <sup>I</sup>* 

Analogically as in continuous time case, the augmented system (23) can be rewritten in a

( 1) ( ) ( ) *n Cn xt A xt* 

*A B C*

Summarizing the augmented closed loop system models (18), (19) and (24), (25) for continuous and discrete-time PID controllers respectively, we can finally, using denotation

( ) () ( ) () *Md n Cn*

 

( ) <sup>0</sup> <sup>0</sup>

*C R R*

*A D C*

*A*

*C* 

() 0 () 0

*Rd R p*

 

> 

**,** <sup>0</sup> 0 *d*

*C*

*D d*

*aug*

*C*

*I B KC*

0 () 0 ( ) 0 00

*I*

*BC A I*

 

*xt A xt B x t*

 

*zt B C A zt z t* 

*Rd R*

*x t DC C*

$$F = \begin{bmatrix} F\_1 & F\_2 \end{bmatrix} = \begin{bmatrix} K\_P & K\_I \end{bmatrix} \text{ and } \begin{aligned} K\_D \text{ included in } \ M\_d(a) \text{ for a continuous-time case:}\\ \end{aligned} \tag{28a}$$

$$F = \begin{bmatrix} F\_1 & F\_2 \end{bmatrix}; \ F\_1 = k\_P + k\_I + k\_{D \ \prime}, \ F\_2 = \begin{bmatrix} k\_D & k\_I - k\_D \end{bmatrix} \text{ for a discrete-time case.} \tag{28b}$$

In a decentralized PID controller design, controller gain matrices are restricted to block diagonal structure respective to subsystem dimensions.

The presented general closed loop augmented system polytopic model (26) is advantageously used in next developments.

#### **3.1.3 Robust stability**

In this section we recall several recent results on robust stability for linear uncertain systems with polytopic model. These results are formulated as robust stability conditions in LMI form. Let us start with basic notions concerning Lyapunov stability and D-stability concept (Peaucelle et al., 2000; Henrion et al., 2002), used to receive the robust stability conditions in more general form.

*Definition 3.1* (D-stability)

Consider the D-domain in the complex plain defined as

$$D = \{ \text{s is complex number}: \begin{bmatrix} 1 \\ s \end{bmatrix} \begin{bmatrix} r\_{11} & r\_{12} \\ r\_{12} & r\_{22} \end{bmatrix} \begin{bmatrix} 1 \\ s \end{bmatrix} < 0 \}\tag{29}$$

Linear system is D-stable if and only if all its poles lie in the D-domain.

(For simplicity, we use in Def. 3.1 scalar values of parameters rij, in general, the stability domain can be defined using matrix values of parameters rij with the respective dimensions.) The standard choice of rij is r11 = 0, r12 = 1, r22 = 0 for a continuous-time system; r11 = -1, r12 = 0, r22 = 1 for a discrete-time system, corresponding to open left half plane and unit circle respectively.

The D-stability concept enables to formulate robust stability condition for uncertain polytopic system in general way, (deOliveira et al., 1999; Peaucelle et al., 2000). The following robust stability condition is based on the existence of Lyapunov function *Vt xtP xt* () () ( )() for linear uncertain polytopic system

$$
\delta \mathbf{x}(t) = A(\alpha)\mathbf{x}(t) \tag{30}
$$

where *A*( ) is from uncertainty domain (12).

#### *Definition 3.2* (Robust stability)

Uncertain system (30) is *robustly D-stable* in the convex uncertainty domain (12) if and only if there exists a matrix () () 0 *<sup>T</sup> P P* such that

$$r\_{12}P(a)A(a) + r\_{12}^\* A^T(a)P(a) + r\_{11}P(a) + r\_{22}A^T(a)P(a)A(a) < 0\tag{31}$$

For one Lyapunov function for the whole uncertainty domain, i.e. *P P* () 0 , the *quadratic D-stability* is guaranteed by (31). Generally, robust stability condition (31) with parameter dependent matrix *P*( ) is less conservative (provides bigger stability domain for *A*( ) than quadratic stability one), however stability is guaranteed only for relatively slow changes of system parameters within uncertainty domain (12) (in comparison with system dynamics). On the other hand, quadratic stability guards against arbitrary quick changes of system parameters within uncertainty domain (12) at the expense of sufficient, relatively strong, stability condition; which can be overly conservative for the case of slow parameter changes.

We consider the parameter dependent Lyapunov function (PDLF) defined as

$$V(t) = \mathbf{x}(t)P(\alpha)\mathbf{x}(t)\tag{32}$$

Robust Decentralized PID Controller Design 145

*r P A H HA Q C F RFC r P HM A G r P M H GA r P M G GM*

 

*Proof.* The proof is analogical to the one presented in (Rosinová & Veselý, 2007) for the continuous-time PID. Firstly, we formulate the sufficient stability condition for uncertain system (26) using the respective Lyapunov function. The assumption that ( ) *Md*

system. Then the sufficient D-stability condition (31) can be rewritten in the following form

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

 

To prove Theorem 3.1, it is sufficient to prove that (35) implies (36). This can be shown

It is important to note that robust stability condition (35) is linear with respect to

matrix inequality (35) is equivalent to the set of matrix inequalities respective to the

Uncertain linear system (26) with cost function (13) is robustly *D*-stable with parameter

*TT T T T k dk Ck k dk dk*

*A A B FC Ck aug k aug k aug* , and *Aaug k* , *Baug k* correspond to the k-th vertex of uncertainty

dependent Lyapunov function (32), (33) and guaranteed cost 0 (0) ( ) (0) *<sup>T</sup> JJ x P x*

*T T T T T k Ck Ck k dk Ck*

*r P M H GA r P M G GM* 

*A A B FC*

 

*r P A H HA Q C F RFC r P HM A G*

*I A M left handside of M A*

<sup>1</sup> () () (35) 0

**.** Therefore, for convex polytopic uncertainty domain (12) and PDLF (33),

 

() () () ()() 0

*d d d d*

1\*1 12 12 11

*<sup>T</sup> <sup>T</sup> T T*

*r A M P M A Q C F RFC*

*rP M A rA M P rP*

*d d*

 

1 1

 

*<sup>T</sup> <sup>T</sup>*

 

then the system (26) is robustly *D*-stable with guaranteed cost: 0 (0) ( ) (0) *<sup>T</sup> JJ x P x*

() () () () () () <sup>0</sup> () () () () ()

*T T T T T C C d C*

> *TT T T T d C kd d*

 

> 

*xt M A xt* and use parameter

 

() ()

 

*d C*

() () () , 1, 0

*C aug aug aug k Ck k k*

  *I*

*Vt Vt Vt* ( ) ( 1) ( ) for a discrete-time

*V t*( ) to consider the guaranteed cost.

 

> .

(35)

is

(36)

(37)

, k=1,...,K (38)

0

 *A*

1 1

*K K*

*k k*

 

,

if the

11 12

 

12 22

invertible, enables us to rewrite (26) as <sup>1</sup> () ( ) ( )() *<sup>d</sup>*

dependent Lyapunov function (32) to write robust stability condition.

*Vt Vt* for a continuous-time system,

(known from LQ theory, for details see e.g. Rosinová et al., 2003)

 

 

1

 

11 12

12 22

*<sup>T</sup> <sup>T</sup> C d*

22

applying congruence transformation on (35):

polytope vertices, as summarized in Corollary 3.1.

which immediately yields (36).

following matrix inequalities hold

\*

domain of the overall system (10), (12);

where

parameter

*Corollary 3.1* 

where the term *T T Q C F RFC d d* has been appended to

\*

Denote ( ) ( ) 

$$P(\boldsymbol{\alpha}) = \sum\_{k=1}^{K} \alpha\_k P\_k \quad \text{where} \ P\_k = P\_k^T > 0 \tag{33}$$

PDLF given by (32), (33) enables to transform robust stability condition (31) for uncertain linear polytopic system (9), (10) into the set of N Linear Matrix Inequalities (LMIs). Several respective sufficient robust stability conditions have been developed in the literature, e.g. (deOliveira et al., 1999; Peaucelle et al., 2000; Henrion et al., 2002). Recall the sufficient robust *D*-stability condition proposed in (Peaucelle et al., 2000), which to the authors best knowledge belongs to the least conservative (Grman et al., 2005).

*Lemma 3.1* 

If there exist matrices , *nxn nxn HR GR* and K symmetric positive definite matrices *nxn P R <sup>k</sup>* such that for all k = 1,…, K:

$$\begin{bmatrix} r\_{11}P\_k + \tilde{A}\_{\{k\}}^T H^T + H \tilde{A}\_{\{k\}} & r\_{12}P\_k - H + \tilde{A}\_{\{k\}}^T G \\\ r\_{12}^\* P\_k - H^T + G^T \tilde{A}\_{\{k\}} & r\_{22}P\_k - \{G + G^T\} \end{bmatrix} < 0 \tag{34}$$

then uncertain system (30) is robustly D-stable in uncertainty domain (12).

Note that matrices H and G are not restricted to any special form; they were included to relax the conservatism of the sufficient condition. Robust stability condition for more general dynamic system model (26), including also the term for guaranteed cost will be presented in the next section.
