**3.2 Robust decentralized PID controller design**

In this section, the robust decentralized PID controller is designed, based on robust stability condition developed in our recent papers, (Rosinová & Veselý, 2007; Veselý & Rosinová, 2011). Robust stability condition with guaranteed cost for closed loop uncertain system (26) is provided in the next theorem.

#### *Theorem 3.1*

Consider uncertain linear system (26) with cost function (13). If there exist symmetric matrix *P*() 0 and matrices H, G and F of the respective dimensions such that

*quadratic D-stability* is guaranteed by (31). Generally, robust stability condition (31) with

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

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

( ) where 0

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

If there exist matrices , *nxn nxn HR GR* and K symmetric positive definite matrices

*T T T kk k k k T T T*

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

In this section, the robust decentralized PID controller is designed, based on robust stability condition developed in our recent papers, (Rosinová & Veselý, 2007; Veselý & Rosinová, 2011). Robust stability condition with guaranteed cost for closed loop uncertain system (26)

Consider uncertain linear system (26) with cost function (13). If there exist symmetric matrix

11 ( ) ( ) 12 ( )

*r P A H HA r P H A G r P H GA r P G G* 

12 ( ) 22

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

and matrices H, G and F of the respective dimensions such that

*k kk*

*P P PP*

*kk k k*

*T*

(33)

0

(34)

( )

than quadratic stability one), however stability is guaranteed only for relatively

(32)

is less conservative (provides bigger stability domain

, the

For one Lyapunov function for the whole uncertainty domain, i.e. *P P* () 0

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

1

 

*K*

*k*

knowledge belongs to the least conservative (Grman et al., 2005).

parameter dependent matrix *P*( )

*nxn P R <sup>k</sup>* such that for all k = 1,…, K:

presented in the next section.

is provided in the next theorem.

*Theorem 3.1* 

*P*() 0 

\*

**3.2 Robust decentralized PID controller design** 

for *A*( ) 

*Lemma 3.1* 

parameter changes.

$$\begin{bmatrix} r\_{11}P(a) + A\_{\mathbb{C}}(a)^{\top}H^{T} + HA\_{\mathbb{C}}(a) + Q + \mathbf{G}^{T}F^{T}RFC & r\_{12}P(a) - HM\_{d}(a) + A\_{\mathbb{C}}(a)^{\top}G \\\ r\_{12}^{\ast}P(a) - M\_{d}(a)^{\top}H^{T} + \mathbf{G}^{T}A\_{\mathbb{C}}(a) & r\_{22}P\_{k} - M\_{d}(a)\mathbf{G} - \mathbf{G}^{T}M\_{d}(a)^{\top} \end{bmatrix} < 0 \tag{35}$$

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

*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* is invertible, enables us to rewrite (26) as <sup>1</sup> () ( ) ( )() *<sup>d</sup> xt M A xt* and use parameter dependent Lyapunov function (32) to write robust stability condition.

Denote ( ) ( ) *Vt Vt* for a continuous-time system, *Vt Vt Vt* ( ) ( 1) ( ) for a discrete-time system. Then the sufficient D-stability condition (31) can be rewritten in the following form (known from LQ theory, for details see e.g. Rosinová et al., 2003)

$$\begin{aligned} r\_{12}P(a)M\_d^{-1}(a)A(a) + r\_{12}^\star A^T(a) \left(M\_d^{-1}(a)\right)^T P(a) + r\_{11}P(a) + \\ r\_{22}A^T(a) \left(M\_d^{-1}(a)\right)^T P(a)M\_d^{-1}(a)A(a) + Q + C\_d^T F^T RFC\_d < 0 \end{aligned} \tag{36}$$

where the term *T T Q C F RFC d d* has been appended to *V t*( ) to consider the guaranteed cost. To prove Theorem 3.1, it is sufficient to prove that (35) implies (36). This can be shown applying congruence transformation on (35):

$$\left[I \quad A\_{\mathbb{C}}^{\mathrm{T}}(a) \Big(M\_{d}^{-1}(a)\Big)^{\mathrm{T}}\right] \Big| \{\mathrm{left}{h}\mathrm{and}\,\mathrm{side}\,\mathrm{of}\,\mathrm{(35)}\} \Big| \begin{matrix} I \\ \left(M\_{d}^{-1}(a)\right)A\_{\mathbb{C}}(a) \end{matrix} \Big| < 0 \tag{37}$$

which immediately yields (36).

It is important to note that robust stability condition (35) is linear with respect to parameter **.** Therefore, for convex polytopic uncertainty domain (12) and PDLF (33), matrix inequality (35) is equivalent to the set of matrix inequalities respective to the polytope vertices, as summarized in Corollary 3.1.

#### *Corollary 3.1*

Uncertain linear system (26) with cost function (13) is robustly *D*-stable with parameter dependent Lyapunov function (32), (33) and guaranteed cost 0 (0) ( ) (0) *<sup>T</sup> JJ x P x* if the following matrix inequalities hold

$$\begin{bmatrix} r\_{11}P\_k + A\_{\text{CK}}^T H^T + H A\_{\text{CK}} + Q + \mathbf{C}^T F^T R \mathbf{F} \mathbf{C} & r\_{12}P\_k - H M\_{\text{dk}} + A\_{\text{CK}}^T \mathbf{G} \\\ r\_{12}^\* P\_k - M\_{\text{dk}}^T H^T + \mathbf{G}^T A\_{\text{CK}} & r\_{22}P\_k - M\_{\text{dk}} \mathbf{G} - \mathbf{G}^T M\_{\text{dk}}^T \end{bmatrix} < 0, \text{ } k = 1, \dots, \mathbf{K} \tag{38}$$

$$\begin{array}{rcl} \text{where} & A\_{\mathbb{C}}(\boldsymbol{\alpha}) = A\_{\text{aug}}(\boldsymbol{\alpha}) + B\_{\text{aug}}(\boldsymbol{\alpha}) \, \mathrm{FC}\_{\text{aug}} \in \left\{ \sum\_{k=1}^{K} \alpha\_{k} A\_{\text{Cl}}, \sum\_{k=1}^{K} \alpha\_{k} = 1, \alpha\_{k} \ge 0 \right\}, \\\ & \cdot \text{ } \cdot \text{ } \cdot \text{ } \cdot \text{ } \cdot \text{ } \cdot \text{ } \cdot \text{ } \end{array}$$

*A A B FC Ck aug k aug k aug* , and *Aaug k* , *Baug k* correspond to the k-th vertex of uncertainty domain of the overall system (10), (12);

Robust Decentralized PID Controller Design 147

2 4 *A A cm* 35 [ ];

1

*x*

4

*x*

2 4 *a a cm* 0.0785 [ ] ;

2nd subsyst. controller

6.45 2.578 1

1

1

*z*

*z*

1

 

1

*z*

 

*z*

1.3833 1.1361 1

2 1 3 *A A cm* 30[ ]; <sup>2</sup>

2 1 3 *a a cm* 0.0977 [ ]; <sup>2</sup>

10 20 <sup>30</sup> <sup>40</sup> *h h cm h cm h cm* 20 [ ]; 2.75 [ ]; 2.22 [ ] ;

<sup>2</sup> *g cm s* 981 [ / ]; 1 2 *k k* 1.790; 1.827 .

3 3 2 1 2 2 2 2

> 1 2 2 3

Subsystems are indicated via the splitting dashed lines. Polytope vertices respective to working points (7) or (8) for minimum phase or nonminimum phase configurations respectively determine the corresponding uncertainty domains indicated in Fig.2. State space model has been discretized with sampling period 5[ ] *T s <sup>s</sup>* (sampling period was

In the minimum phase case, robust decentralized controller is designed for chosen pairing 1 12 2 *v yv y* , (see Section 2.1) using alternatively solution of LMI (39) or BMI (38) for decentralized discrete-time PI controller design. The resulting controller parameters are in Tab.1, the respective simulation results are illustrated and compared on step responses in

1

*z*

1

*z*

1

1

 

 

*z*

*z*

1.3002 1.0351 1

controller

Table 1. Decentralized PID controller parameters – minimum phase case

5.862 2.602 1

*y x y x*

1000 0010

*x xu x x u*

0 0.0435 0 0 0 0.0595(1 ) 0 0 0.0111 0.0333 0 0.0522 0 0 0 0.0333 0.052(1 ) 0

0.0161 0.0435 0 0 0.0596 0

**3.3 Decentralized PID controller design for the Quadruple tank process** 

We consider quadruple tank linearized model (2) with parameters:

1 1 1

*x x*

 

*x x*

chosen with respect to the process dynamics).

one tested point from uncertainty domain, in Fig. 3.

LMI (39)

BMI (38) Q=0.01\*I, R=5\*I

Design approach 1st subsyst.

*Minimum phase configuration* 

4 4 1

$$M\_d(a) \in \left\{ \sum\_{k=1}^K a\_k M\_{dk} \; \sum\_{k=1}^K a\_k = 1, a\_k \ge 0 \right\}, \; M\_{dk} \text{ is for PID controller given by (27a) or (27b), and}$$
  $B(a) \text{ is given by (12).}$ 

Robust stability condition (38) is LMI for stability analysis, for controller synthesis it is in the BMI form. Therefore, (38) can be used for robust controller design either directly – using appropriate BMI solver (Henrion et al., 2005) or using some convexifying approach, (for discrete-time case see e.g. (Crusius **&** Trofino, 1999; deOliveira et al., 1999)). We have relatively good experience with the following simple convexified LMI procedure for static output feedback discrete-time controller design, which is directly applicable for discretetime PID controller design problem formulated by (26), (27b), (28b).

The controller gain block diagonal matrix F is obtained by solving LMIs (39) for unknown matrices F, M, G and Pk of appropriate dimensions, the Pk being block diagonal symmetric, and M, G block diagonal with block dimensions conforming to subsystem dimensions. This convexifying approach does not allow including a term corresponding to performance index, therefore the resulting control guarantees only robust stability within considered uncertainty domain.

$$\begin{bmatrix} -\mathbf{P}\_{\mathbf{k}} & \mathbf{A}\_{\text{aug}\cdot\mathbf{k}}\mathbf{G} + \mathbf{B}\_{\text{aug}\cdot\mathbf{k}}\mathbf{K}\mathbf{C}\_{\text{aug}}\\ \mathbf{G}^{T}\mathbf{A}\_{\text{aug}\cdot\mathbf{k}}^{T} + \mathbf{C}^{T}\mathbf{K}^{T}\mathbf{B}\_{\text{aug}\cdot\mathbf{k}}^{T} & -\mathbf{G} - \mathbf{G}^{T} + \mathbf{P}\_{\text{k}} \end{bmatrix} < \mathbf{0}, \quad \text{,} \quad \mathbf{k} = \mathbf{1}, \dots, \mathbf{K}$$

$$\mathbf{M}\mathbf{C}\_{\text{aug}} = \mathbf{C}\_{\text{aug}}\mathbf{G} \tag{39}$$

$$F = \mathbf{K}\mathbf{M}^{-1}$$

F is the corresponding output feedback gain matrix.

The main advantage of the use of LMI (39) for controller design is its simplicity. The major drawbacks are, that the performance index cannot be considered, and that due to convexifying constraint ( *MC CG aug aug* ), it need not provide a solution even in a case when

feasible solution is received through BMI (38). (This is the case in our example in Section 3.3, in nonminimum phase configuration.)

To conclude this section we summarize the described decentralized PID controller design procedure, assuming that the state space model is in the form of (9) with polytopic uncertainty domain given by (10), where columns of control input matrix B are arranged respectively to chosen pairing.

#### *Design procedure for decentralized PID design in time domain*

Step 1. Formulate the augmented state space model (26) for given system and chosen type of PID controller.

Step 2. Compute decentralized PID controller parameters using one of design alternatives:


Robust stability condition (38) is LMI for stability analysis, for controller synthesis it is in the BMI form. Therefore, (38) can be used for robust controller design either directly – using appropriate BMI solver (Henrion et al., 2005) or using some convexifying approach, (for discrete-time case see e.g. (Crusius **&** Trofino, 1999; deOliveira et al., 1999)). We have relatively good experience with the following simple convexified LMI procedure for static output feedback discrete-time controller design, which is directly applicable for discrete-

The controller gain block diagonal matrix F is obtained by solving LMIs (39) for unknown matrices F, M, G and Pk of appropriate dimensions, the Pk being block diagonal symmetric, and M, G block diagonal with block dimensions conforming to subsystem dimensions. This convexifying approach does not allow including a term corresponding to performance index, therefore the resulting control guarantees only robust stability within considered

0, *<sup>k</sup> aug k aug k aug*

<sup>1</sup> *F KM*

The main advantage of the use of LMI (39) for controller design is its simplicity. The major drawbacks are, that the performance index cannot be considered, and that due to convexifying constraint ( *MC CG aug aug* ), it need not provide a solution even in a case when feasible solution is received through BMI (38). (This is the case in our example in Section 3.3,

To conclude this section we summarize the described decentralized PID controller design procedure, assuming that the state space model is in the form of (9) with polytopic uncertainty domain given by (10), where columns of control input matrix B are arranged

Step 1. Formulate the augmented state space model (26) for given system and chosen type of

 BMI alternative – guarantees robust stability and guaranteed cost for quadratic performance index (13): solve BMI (38) for unknown block diagonal matrices F, Pk>0 and matrices G, H, of appropriate dimensions, PID controller parameters are given by F

and *Mdk* respectively to (28) and (27), *Mdk* is for PID controller given by (27).

Step 2. Compute decentralized PID controller parameters using one of design alternatives: LMI alternative for discrete-time case – guarantees robust stability: solve LMI (39) for unknown block diagonal matrices F, M, G and Pk>0, of appropriate dimensions; PID

, k=1,...,K

*MC CG aug aug* (39)

*P A G B KC*

 

, *Mdk* is for PID controller given by (27a) or (27b), and

1 1 ( ) , 1, 0 *K K d k dk k k k k*

 

time PID controller design problem formulated by (26), (27b), (28b).

*T T T TT T aug k aug k k*

F is the corresponding output feedback gain matrix.

*Design procedure for decentralized PID design in time domain* 

controller parameters are given by F respectively to (28b).

in nonminimum phase configuration.)

respectively to chosen pairing.

PID controller.

*GA CKB G G P*

 

*M M* 

uncertainty domain.

*B*( ) 

 

is given by (12).

## **3.3 Decentralized PID controller design for the Quadruple tank process**

We consider quadruple tank linearized model (2) with parameters:

2 1 3 *A A cm* 30[ ]; <sup>2</sup> 2 4 *A A cm* 35 [ ]; 2 1 3 *a a cm* 0.0977 [ ]; <sup>2</sup> 2 4 *a a cm* 0.0785 [ ] ; 10 20 <sup>30</sup> <sup>40</sup> *h h cm h cm h cm* 20 [ ]; 2.75 [ ]; 2.22 [ ] ; <sup>2</sup> *g cm s* 981 [ / ]; 1 2 *k k* 1.790; 1.827 . 1 1 1 3 3 2 1 2 2 2 2 4 4 1 0.0161 0.0435 0 0 0.0596 0 0 0.0435 0 0 0 0.0595(1 ) 0 0 0.0111 0.0333 0 0.0522 0 0 0 0.0333 0.052(1 ) 0 *x x x xu x x u x x* 1 1 2 2 3 4 1000 0010 *x y x y x x* 

Subsystems are indicated via the splitting dashed lines. Polytope vertices respective to working points (7) or (8) for minimum phase or nonminimum phase configurations respectively determine the corresponding uncertainty domains indicated in Fig.2. State space model has been discretized with sampling period 5[ ] *T s <sup>s</sup>* (sampling period was chosen with respect to the process dynamics).

#### *Minimum phase configuration*

In the minimum phase case, robust decentralized controller is designed for chosen pairing 1 12 2 *v yv y* , (see Section 2.1) using alternatively solution of LMI (39) or BMI (38) for decentralized discrete-time PI controller design. The resulting controller parameters are in Tab.1, the respective simulation results are illustrated and compared on step responses in one tested point from uncertainty domain, in Fig. 3.


Table 1. Decentralized PID controller parameters – minimum phase case

Robust Decentralized PID Controller Design 149

Comparison of simulation results for minimum and nonminimum phase cases shows the

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

Consider a MIMO system described by a transfer function matrix ( ) *m m Gs R* and a

*d*

det ( ) det[ ( )] *Fs I Qs* (40)

det[ ( ) ( )] 0 1,..., *q sI Qs i m i m* (41)

*R(s) G(s)*

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

Characteristic functions of *Q s*( ) are the set of m algebraic functions ( ), 1,..., *<sup>i</sup> qs i m* defined

Characteristic loci (CL) are the set of loci in the complex plane traced out by the

*w e u y*

controller ( ) *m m Rs R* in the standard feedback configuration according to Fig. 5,

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

.

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

deteriorating influence of nonminimum phase on settling time.

performance.

**4.1 Preliminaries and problem formulation** 

Fig. 5. Standard feedback configuration

characteristic functions of Q(s), *s j*

*Theorem 4.1* (Generalized Nyquist Stability Theorem) The closed-loop system in Fig. 1 is stable if and only if

characteristic polynomial

as follows:

Fig. 3. Step response of y1 and y2 to setpoint step changes: w1 in 400s and w2 in 800s; comparison of LMI and BMI design results from Tab.1

Obviously, the results for the BMI solution including performance index outperform the ones obtained using simpler LMI approach.

#### *Nonminimum phase configuration*

In the nonminimum phase case, robust decentralized controller is designed for chosen pairing 1 22 1 *v yv y* , (see Section 2.1) using a solution of BMI (38) for decentralized discrete-time PI controller design, (in this case LMI procedure (39) does not provide a feasible solution). The resulting controller parameters are in Tab.2, the respective simulation results are illustrated on step responses in one tested point from uncertainty domain, in Fig. 4.


Table 2. Decentralized PID controller parameters – nonminimum phase case

Fig. 4. Step response of y1 and y2 to setpoint step changes: w1 (for y2) in 1000s and w2 (for y1) in 2000s

y2 [cm]

1

*z*

1

0 500 1000 1500 2000 2500 3000

Fig. 4. Step response of y1 and y2 to setpoint step changes: w1 (for y2) in 1000s and w2

**step response in the considered working point**

 

*z*

<sup>0</sup> <sup>500</sup> <sup>1000</sup> <sup>1500</sup> <sup>0</sup>

**step response in the considered working point**

2nd subsyst. controller

0.7221 0.6941 1

t [s]

output y1 output y2 1

*z*

1

 

*z*

t [s]

LMI design BMI design

<sup>0</sup> <sup>500</sup> <sup>1000</sup> <sup>1500</sup> <sup>0</sup>

ones obtained using simpler LMI approach.

BMI (38) Q=0.01\*I, R=5\*I

0

(for y1) in 2000s

5

10

15

20

25

30

*Nonminimum phase configuration* 

comparison of LMI and BMI design results from Tab.1

**step response in the considered working point**

t [s]

step responses in one tested point from uncertainty domain, in Fig. 4.

Design approach 1st subsyst.

Fig. 3. Step response of y1 and y2 to setpoint step changes: w1 in 400s and w2 in 800s;

controller

Table 2. Decentralized PID controller parameters – nonminimum phase case

0.5371 0.5099 1

Obviously, the results for the BMI solution including performance index outperform the

In the nonminimum phase case, robust decentralized controller is designed for chosen pairing 1 22 1 *v yv y* , (see Section 2.1) using a solution of BMI (38) for decentralized discrete-time PI controller design, (in this case LMI procedure (39) does not provide a feasible solution). The resulting controller parameters are in Tab.2, the respective simulation results are illustrated on

LMI design BMI design

y1 [cm] Comparison of simulation results for minimum and nonminimum phase cases shows the deteriorating influence of nonminimum phase on settling time.
