Robust controller for uncertain parameters systems

ISA Transactions 51 (2012) 632–640

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Robust controller for uncertain parameters systems Maher Ben Hariz a,c,n, Faouzi Bouani b,c, Mekki Ksouri b,c a b c

Tunis El Manar University, Faculty of Sciences of Tunis, Tunisia Tunis El Manar University, National Engineering School of Tunis, Tunisia Analysis, Conception and Control of Systems Laboratory, Tunisia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 13 January 2012 Received in revised form 19 March 2012 Accepted 22 April 2012 Available online 29 June 2012

In this paper, we present the synthesis of a robust controller for Linear Time Invariant (LTI) uncertain systems. A linear parametric uncertainties model is used to describe the system dynamic behavior. The main purpose of this controller is to guarantee some step response performances such as the settling time and the overshoot. The controller synthesis is formulated as a min–max optimization problem which takes in account the desired closed-loop performances and the uncertainties on the model parameters. Then the controller parameters represent the best solution for the worst case of all possible models. In order to emphasize its performances and its efficiency, a real time implementation of the proposed controller on a laboratory pilot plant has been presented. & 2012 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Robust controller Uncertain systems Step response performances Non-convex optimization problem

1. Introduction The study of robust control has attached the attention of many researchers. The model accuracy plays a crucial role in control theory. But, in real applications, there is always a difference between the plant and the model. This difference represents the uncertainty. Previously, the uncertainties are modeled by additive uncertainties presented under the shape of a Gaussian noise and usually represent the measurement errors on the output signal. Moreover, this type of uncertainty is not suited to characterize the uncertain behavior of the physical system. Nevertheless, parametric uncertainties allow to describe the uncertain dynamic behavior of the real system. By adopting this type of uncertainty, each model parameter can be represented by an uncertain bounded variable [1]. The robust control consists in taking into account uncertainties during the development of the control law. Hence, the control law which is obtained by solving a min–max optimization problem (worst case strategy) constitutes the best solution for the worst case of all possible models [2]. The basic idea of min–max optimization method is to choose the best result of the worst cases. This method which represents an effective mean to resolve multi-objective optimization problem has been used with different robust controllers.

n Corresponding author at: Tunis El Manar University, Faculty of Sciences of Tunis, Tunisia. Tel.: þ216 983 864 33. E-mail addresses: [email protected] (M. Ben Hariz), [email protected] (F. Bouani), [email protected] (M. Ksouri).

Zafiriou [3] and Zheng and Morari [4] have used the min–max optimization in Model Predictive Control (MPC) with finite impulse response models. In [5], Lee and Cooley have used the min–max optimization in MPC based on state space models with time-varying parametric uncertainty in the control matrix. In Gruber et al. [6], the Min–Max MPC (MMMPC) based on an upper bound of the worst case cost with guaranteed stability is presented. Indeed, MMMPC provides the opportunity to take into account disturbances and model uncertainties in the prediction model. The optimal control law, in MMMPC policies, is calculated by the minimization of the worst case cost which can be computed by the maximization of the cost function compared with all possible instances of uncertainties and disturbances. Casavola et al. [7] have presented a robust predictive control algorithm for uncertain norm-bounded linear systems. The idea is to minimize, at each instant, a semi-definite convex optimization problem subject to Linear Matrix Inequality (LMI) constraints. In Kothare et al. [8] a robust constrained MPC using LMI is proposed. The objective is to design a control law that minimizes a worstcase infinite horizon objective function by taking into account the control input and system output. Ding et al. [9] have proposed an approach for output feedback robust MPC for systems with polytopic uncertainty and bounded disturbance. By solving LMI optimization problems, they calculated off-line a sequence of output feedback laws based on the state estimators. But, on-line, they chose an appropriate output feedback law from this sequence at each sampling time. In [10], authors treated the problem of designing an output/state feedback robust MPC for linear polytopic systems with input constraints. They proposed a robust control algorithm which guarantees quadratic stability and

0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2012.04.007

M. Ben Hariz et al. / ISA Transactions 51 (2012) 632–640

guaranteed cost under input constraints in the uncertainty domain for the closed-loop system. Kheriji et al. [11] solved a min–max predictive controller for class of constrained linear Multi-Input Multi Output (MIMO) systems in presence of parametric uncertainty in the state space model. In Bouzouita et al. [12], the Robust Predictive Control (RPC) based on Controlled Auto Regressive Integrated Moving Average (CARIMA) model using parametric uncertainties is studied. Lovaas et al. [13], have presented a robust output-feedback MPC for discrete-time systems with constraints and unstructured uncertainty. In Mayne et al. [14], a robust model predictive controller for constrained linear discrete-time systems with bounded disturbances is proposed. In [15], Lavaei and Aghdam are interested on the high-performance robust control of discrete-time Linear Time Invariant (LTI) uncertain systems. In this paper, we propose an extension of the work given in [16] which is interested in the synthesis of a fixed low-order controller for LTI systems with some step response specifications such as the settling time and the overshoot. Our work deals with parametric uncertainties systems. The controller synthesis is formulated as a min–max optimization problem which takes in account the desired closed-loop performances and the uncertainties on the model parameters. The desired closed loop characteristic equation is fixed by the user as given in [17]. The controller parameters are found in minimizing the worst case objective function. The introduction of parametric model uncertainties leads to a non-convex min–max optimization problem. The paper is organized as follows. In Section 2 the problem statement is presented. The synthesis of the controller is developed in Section 3. In order to illustrate the effectiveness of the proposed controller some simulation results are given in Section 4. To highlight the obtained results, we propose the practical implementation of the controller on a laboratory pilot plant in Section 5 and a comparison between different types of controllers has been exposed. Finally, conclusion and prospect are given in Section 6.

633

The controller is given by: CðsÞ ¼

BðsÞ AðsÞ

ð4Þ

AðsÞ ¼ st þat1 st1 þ    þ a1 s þa0

ð5Þ

BðsÞ ¼ br sr þbr1 sr1 þ    þ b1 s þ b0

In the case of low-order controller, we have: r rt ol  1. In order to obtain a closed loop transfer function with a unit static gain, we introduce a polynomial F(s): FðsÞ ¼ f q sq þ f q1 sq1 þ    þ f 1 s þf 0

ð6Þ

By using (1) and (4) the closed-loop transfer function will be given by: HðsÞ ¼

FðsÞBðsÞNðs,nÞ AðsÞDðs,dÞ þ BðsÞNðs,nÞ

ð7Þ

The closed-loop characteristic equation is given by the following relation:

dðsÞ ¼ AðsÞDðs,dÞ þ BðsÞNðs,nÞ ¼ dn sn þ dn1 sn1 þ    þ d1 s þ d0

ð8Þ

where n¼ lþt The uncertain model and its structure being fixed, the objective, now, is to determine a controller which ensures some step response performances such as the overshoot and the settling time and takes in account the model uncertainties. Then, the desired closed-loop characteristic equation must be fixed by using the polynomial characteristics ratios [17] which will be explained in the following section. 2.1. Choice of the K-polynomial d(s) Consider the following polynomial:

dðsÞ ¼ dn sn þ    þ d1 s þ d0 ¼

n X

di si

ð9Þ

i¼0

The characteristic ratios ai (i¼1,y,n  1) and the generalized time constant t are defined in [17,18] as:

2. Problem statement A Single Input Single Output (SISO) system can be represented by a transfer function G(s). The basic structure for uncertain systems can be regarded as the association of a nominal model G0(s) and an unknown bounded uncertainty. Then the uncertainty can be represented by DG(s) which is added to the nominal model G0(s). Thus the transfer function of the uncertain system is given by: GðsÞ ¼ Gðs,n,dÞ ¼

Nðs,nÞ Dðs,dÞ ð1Þ

þ where ni and dj are bounded by intervals ½n i ,ni  and respectively. So ni and dj can be given by: þ n i r ni rni

dj A R;

dj r dj rdj

t¼

d2i , i ¼ 1,. . .,n1 di1 di þ 1

 þ ½dj ,dj ,

for

i ¼ 0,1,. . .,m

ð2Þ

for

j ¼ 0,1,. . .,l

ð3Þ

ð10Þ

d1 d0

ð11Þ

Then the coefficients of d(s) can be represented in terms of ai and t as follows:

d1 ¼ d0 t, d2 ¼

nm sm þ nm1 m1 sm1 þ    þn0 ¼ dl sl þdl1 sl1 þ    þ d0

ni A R;

ai ¼

d0 t2

a1

,. . .,

dn ¼

tn

a

d0 2 n1 n2

a

   an1 1

ð12Þ

The K-polynomial is defined as the polynomial having the characteristic ratio of: ( a1 4 2 ð13Þ þ sinðp=nÞ ak ¼ sinðk2p=nÞ a1 for k ¼ 2,. . .,n1 sinðkp=nÞ So, the K-polynomial is determined by only a1 for a given t and



þ

Let consider an all pole transfer function Hj(s) whose denominator is a K-polynomial d(s):

The closed-loop system is depicted in Fig. 1. R(s)

E(s) F(s)

U(s) C(s)

+

d0 by using (12) and (13).

Y(s) G(s)

-

Fig. 1. A feedback control system with cascade configuration.

Hj ðsÞ ¼

d0 d0 ¼ dðsÞ dn sn þ    þ d1 s þ d0

ð14Þ

We can control the speed of the step response of a linear all pole system by adjusting the value of t only if its characteristic ratios are kept the same, while maintaining the exact shape of the response.

634

M. Ben Hariz et al. / ISA Transactions 51 (2012) 632–640

transfer function H1 (s):

Table 1 ak of Hj(s) resulting in no overshoot when t ¼ 1. 

a ¼ a1

n



3



4



5



6



7



8

a2

a3

a4

a5

a6

a7

H1 ðsÞ ¼



2:836 2:259

2:646

 

2:538

2:053

2:053

2:538

2:464

1:943

1:848

1:943

2:464

2:411

1:874

1:742

1:742

1:874

2:411

2:370

1:830

1:680

1:640

1:680

1:830

  2:37 0



t2 ¼ 

Consider two all pole transfer functions of the same degree H1(s) and H2(s), of which the generalized time constants are t1 and t2, respectively. Let yi(t) be the step response of Hi(s) for i¼1,2. Then   t1 t ð15Þ y1 ðtÞ ¼ y2

t2

if and only if both H1(s) and H2(s) have the same characteristic ratios. For the transfer function Hj(s), the damping can be successfully controlled by adjusting the single parameter a1. Indeed, by increasing a1 the damping increases, subsequently the overshoot decreases. It results that we can easily determine the a1 value corresponding to no overshoot. Then we use a method for substituting a target transfer function by using K-polynomial. This transfer function satisfies the desired overshoot and settling time as mentioned in [16]. When n¼3,4,y,8, these ak are given in Table 1. The task is achieved in two steps: Step 1: (i) We set t1 ¼ 1 and we choose a value for a1 higher or equals to the recommended value in Table 1. (ii) We calculate the characteristic ratios ak, for k¼2,y,n 1 by using (13). (iii) We calculate the target polynomial coefficients dk, for k ¼1,y,n with d0 ¼ 1 by using (12). (iv) We determine the target step response and its settling time ts1.

(i) We fix the desired settling time ts2 and we calculate the corresponding generalized time constant t2 t s2 t1 t s1

t s2 0:5 ¼ 0:2174 t1 ¼ 2:3 t s1

Then a new K-polynomial d(s) is computed by using a ¼   

a1 a2 a3 ¼ 2:7 2:3046 2:7 and t2 ¼0.2174. That is

H2 ðsÞ ¼

d0 1 ¼ dðsÞ ð7:913Þ106 s4 þ 0:0006115s3 þ 0:0175s2 þ 0:2174s þ 1 ð18Þ

The step response of H2(s) is also given in Fig. 2. The desired overshoot is obtained by adjusting the parameter a1. 3. Synthesis of the controller Let the controller parameter vector be:  T x ¼ b0    br a0    at1

ð19Þ

We define the coefficients vectors of the closed-loop characteristic polynomial d(s) and the closed-loop desired polynomial d ðsÞ respectively as follows: h iT dðsÞ ¼ d0 d1    dn1 dn ð20Þ h

d ðsÞ ¼ d0

d1



dn1

dn

iT

ð21Þ

3.1. Controller for fixed parameters systems In this section, we suppose that the process is described by the nominal model. In this case, the closed-loop characteristic polynomial is given by the following equation:

dðsÞ ¼ AðsÞD0 ðsÞ þ BðsÞN 0 ðsÞ ¼ dn sn þ dn1 sn1    þ d1 s þ d0

Step 2:

t2 ¼

ð17Þ

Now, we determine the settling time from the step response of H1(s) shown in Fig. 2. It is ts1 ¼2.3 s. Next step is to compute a new K-polynomial d2(s) that achieves a desired settling time ts2 ¼0.5 s. Applying (16), we have



2:836 2:646

d0 1 ¼ dðsÞ 0:003543s4 þ 0:05952s3 þ0:3704s2 þ s þ1

ð22Þ

where N0(s) and D0(s) represent, respectively, the numerator and the denominator of the nominal model where the uncertainties are not taken into account.

ð16Þ Step Responses

(ii) In taking into account the new value of the generalized time constant t2 we compute the coefficients dk by using (12). (iii) Finally, we find the desired step response.

Let us obtain an all pole transfer function of degree 4 that meets no overshoot and 2% settling time of 0.5 s. From Table 1, we note that for polynomials of order n ¼4 we have a1 ¼ 2.646. According to the algorithm, we must choose a value of a1 higher or equals to the value mentioned in Table 1. So we can choose a1 ¼2.7. Hence for this value of a1, and with d0 ¼1 and t1 ¼1, we obtain a polynomial having a step response with no overshoot.     It means that a ¼ a1 a2 a3 ¼ 2:7 2:3046 2:7 by (13). Using (12), we obtain the characteristic polynomial d(s) and the

0.8 Amplitude

2.2. Example

1

0.6 0.4 0.2 H1 H2

0

0

0.5

1

1.5

2

2.5

3

3.5

Time (sec) Fig. 2. Settling time adjustment with t.

4

4.5

5

M. Ben Hariz et al. / ISA Transactions 51 (2012) 632–640

The coefficient vector of the closed-loop characteristic polynomial d(s) can be expressed as a function of x.

dðsÞ ¼ Px þ q where 2 n0 6 n1 6 6 6 ^ 6 6n 6 m P¼6 6 ^ 6 6 0 6 6 4 ^

0



0

d0

0



n0



0

d1

d0



^

d1 ^

 

^ nm1



nmr

d2 ^

^

dl

dl1



nm

^

^

^

0

0





0

0

0



^ 0 ^

0 h q¼ 0

ð23Þ

0 

0

d0



dlt



dl

0

3

7 7 7 0 7 7 ^ 7 7 7 dlt þ 1 7 7 ^ 7 7 7 dl 5 0

2

Dn0 6 Dn1 6 6 6 ^ 6 6 Dn m DP ¼ 6 6 6 ^ 6 6 0 6 6 4 ^

0



0

Dn0



0



 

Dd0 Dd1



Dnmr

^

^

^

Ddl

Dnm

^

^

^

0

0



0

0

0



^ 0 0



d0

h

iT



^ 0

dlt



Dd0

Dq ¼ 0    0

0

0

^

Dd0 Dd1 Dd2

^

Dnm1

0 h q¼ 0

635





Ddlt

dl



Ddl1   

0

3

7 7 7 7 0 7 7 ^ 7 7 Ddlt þ 1 7 7 7 ^ 7 7 Ddl 5 0

0

iT



Ddl

iT

P A Rðn þ 1Þðr þ t þ 1Þ ,x A Rr þ t þ 1 , qA Rn þ 1 We note by O the set of uncertainties on DP and Dq where O is given by:

P A Rðn þ 1Þðr þ t þ 1Þ ,x A Rr þ t þ 1 , q A Rn þ 1 The controller parameters are computed such that the difference between d(s) and d ðsÞ is minimum. This can be formulated by the following weighted cost function:

O ¼ fOij =Oij A DP,wi =wi A Dqg

f 0 ðxÞ ¼ ½dðsÞd ðsÞT W½dðsÞd ðsÞ

f ðx, OÞ ¼ ½dðxÞd ðsÞT W½dðxÞd ðsÞ ¼ xT F 1 x þ 2F 2 x þ F 3

ð24Þ

where W is a weighting matrix. It was indicated in [19] that the coefficients of lower powers of s in the transfer function are the most related to the step response. For example, d0, d1 and d2 have much more influence on the step response of the rest of di [17]. So, the weighting matrix may be chosen so that the weights for the low powers of s have greater values than those for higher powers. The Diophantine Eq. (22) cannot be resolved as any low order controller is employed. So, we use the Partial Model Matching (PMM) approach to obtain an approximate solution. The PMM method can be regarded as a Least Square Estimation (LSE) problem in the sense of minimizing f0(x) with regards to x. The controller can be obtained by resolving the following problem:

ð29Þ

The weighted cost function becomes: ð30Þ

where F 1 ¼ ½ðP 7 DPÞT WðP 7 DPÞ F 2 ¼ ½ððq þ DqÞd ðsÞÞT WðP 7 DPÞ F 3 ¼ ½ððq 7 DqÞd ðsÞÞT Wððq 7 DqÞd ðsÞÞ The introduction of the uncertainties leads to a min–max optimization problem: min max f ðx, OÞ x

ð31Þ

O

where f0(x) is obtained by using relations (23) and (24) as follows:

The min–max optimization problem is bi-level. It gives the solution of the best design in terms of the controller parameters  þ and the worst case referred to as ni A ½ni  ,ni þ  and dj A ½dj ,dj  (i ¼0,y,m and j ¼0,y,l). The first optimization problem consists to find the maximum of the objective function f(x,O) compared upon uncertainties O. In this case the unknown vector is given by:

f 0 ðxÞ ¼ xT ½PT WPx þ2½ðqd ðsÞÞT WPx þ ½ðqd ðsÞÞT Wðqd ðsÞÞ

x¼

min f 0 ðxÞ

ð25Þ

x

ð26Þ

h

b0



br

a

0



at1

Dn0



Dnm

Dd0



In presence of uncertainties on the model parameters, the closed-loop characteristic polynomial d(s) becomes:

dðsÞ ¼ AðsÞDðs,dÞ þ BðsÞNðs,nÞ ¼ dn sn þ dn1 sn1 þ    þ d1 s þ d0

ð27Þ

The coefficient vector of the closed-loop characteristic polynomial d(s) can be expressed as a function of x by:

dðsÞ ¼ ðP 7 DPÞx þ ðq7 DqÞ

0

ð28Þ

0



0

d0

0



n0 ^



0 ^

d1 d2

d0 d1

 

nm1



nmr

^

^

^

dl

dl1

^ 0



^ 0





nm

^

^

^

0

0



0

0

0



0

0

The obtained solution is used in the second optimization problem which consists in minimizing f(x,O) respect to the controller parameters. The optimization problem can be resolved by using a standard optimization technique. In the present work, we have exploited the defined function of MATLAB ‘‘fmincon’’ which is used to solve nonlinear convex optimization problems. The min–max optimization criterion is used to find the best controller parameters for the worst case of the all possible models. This strategy will lead to only one controller which will be used to control the uncertain dynamic plant.

3

7 7 7 7 7 ^ 7 7 7 dlt þ 1 7 7 ^ 7 7 7 dl 5 0 0

iT

ð32Þ

3.2. Controller for uncertain parameters systems

where 2 n0 6 n1 6 6 6 ^ 6 6n 6 m P¼6 6 ^ 6 6 0 6 6 4 ^

Ddl

4. Simulation results In this section two simulation examples are presented to illustrate the effectiveness of the synthesis method represented in Section 3.We consider the following system: G0 ðsÞ ¼

12:615 s5 þ4:3s4 þ14:21s3 þ23:135s2 þ 20:59s þ 12:615

ð33Þ

636

M. Ben Hariz et al. / ISA Transactions 51 (2012) 632–640

In this simulation, we have supposed that each parameter of the nominal model is uncertain and the uncertainties are assumed to be bounded. So, it can be represented by means of a median value and absolute value of the maximum deviations Dni and Ddj with respect to their median values that are: ( dj ¼ dj þ ei Ddj ni ¼ ni þ ei Dni with 9ei9 r1 Let consider the following fixed third order controller: ( AðsÞ ¼ s3 þ a2 s2 þ a1 s þa0 BðsÞ ¼ b1 s þb0

ð34Þ

Table 2 Coefficients of the target model and the characteristic polynomials in first simulation. Index (i)

di

di0

di1

di2

0 1 2 3 4 5 6 7 8

17.593 87.965 199.920 268.011 230.963 130.862 47.663 10.240 1

16.300 81.550 189.100 255.300 220.000 118.300 41.120 9.130 1

17.590 87.970 200.200 267.300 232.000 130.200 46.960 10.130 1

15.650 78.270 180.500 249.100 216.500 117.900 41.7900 9.330 1

Assume that our objective is to synthesize a controller with the following specifications: (i) Overshoot r2%. (ii) 2% Settling time r10 s.

1.2

The K-polynomial is computed by using the parameters a1 ¼ 2.2 and t ¼5 which satisfy the specifications of synthesis [16]. In this case, the wij elements of the weighting matrix W are given by: 8 for i,j ¼ 0,. . .,6 and i ¼ j, > < 0:3 wij ¼ 0:025 for i,j ¼ 7,8 and i ¼ j, > :0 for i aj

0.9

Step Responses

1.1 1

Amplitude

0.8 0.7 0.6 0.5 0.4

The resolution of the min–max optimization problem leads to the following controller:

0.3

0:202s þ 1:22 C 1 ðsÞ ¼ 3 s þ 4:83s2 þ 6:145s þ 0:0721

0.1

ð35Þ

13:615 s5 þ 5:3s4 þ 15:21s3 þ 24:135s2 þ 21:59s þ 13:615

ð36Þ

12:115 s5 þ 4:5s4 þ 13:91s3 þ 23s2 þ 19:59s þ 12:11

ð37Þ

and G2 ðsÞ ¼

The coefficients dij (j¼ 0,1,2) of the closed-loop characteristic polynomials and the target model are given in Table 2. The closed loop responses and the control signals obtained with the found controller are plotted respectively in Figs. 3 and 4. We note that the proposed controller leads the process output to meet the desired specifications (an overshoot about 2% and a settling time equals to 10 s) despite the change of the process dynamic. Our purpose, now, is to synthesize a controller with the following specifications: (i) Overshoot r5%. (ii) 2% Settling time r5 s.

3

1

2

3

4

5

6

7 8 9 10 11 12 13 14 15 Time (sec)

Fig. 3. Step response of the close loop system for the first example simulation.

To test the performance of this controller we considered the same transfer functions G0(s), G1(s) and G2(s) of the first simulation. The coefficients dij (j¼0,1,2) of the closed-loop characteristic polynomials and the target model are given in Table 3. Fig. 5 shows the output and the set point. The control signals are given in Fig. 6. It is clear from Fig. 5 that the specification performances (overshoot about 5% and a 2% settling time equals to 5 s) are reached despite the presence of uncertainties.

5. Practical results In order to illustrate the performances of the presented approach, we make a practical implementation of the proposed controller on a liquid level control system. The system is constituted by one tank. The outlet flow and the inlet flow are controlled respectively by a digital valve and a proportional valve. The objective is to control the liquid level by manipulating the inlet flow with the proportional valve.

2

AðsÞ ¼ s þ a2 s þ a1 s þa0 BðsÞ ¼ b1 s þb0

By applying the same procedure, we obtain the following controller: C 2 ðsÞ ¼

0

5.1. Identification

Let the conceived controller be: (

G0(s) G1(s) G2(s)

0

In order to test the performance of this controller, we have considered three situations. In the first situation, we have supposed that the plant is described by the nominal model. In the second and the third situations, we have considered uncertainties on the nominal model parameters and the behavior of the plant will be described by the following transfer functions: G1 ðsÞ ¼

0.2

1:9290s7:7361 s3 þ 6:3879s2 þ15:8563s þ 27:6693

ð38Þ

The identification consists in applying signals to the system input and in analyzing the system output in order to obtain a mathematical model [20–22]. To model the physical system, we have applied to the proportional valve the input sequences given in Fig. 7. The obtained open loop system outputs are plotted in Fig. 8. We note that for the identification procedure the sampling period is Ts ¼1 s.

M. Ben Hariz et al. / ISA Transactions 51 (2012) 632–640

Control signals

Control signals 1.1

1.2

1

1.1 1

0.9

0.9 0.8

0.7

Amplitude

Amplitude

0.8

0.6 0.5 0.4

0.7 0.6 0.5 0.4

0.3

0.3

0.2

0.2

u0 u1 u2

0.1 0

637

0

0

1

2

3

4

5

6

7

8

u0 u1 u2

0.1

9 10 11 12 13 14 15

0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

Tim e (sec)

Time (sec)

Fig. 6. Control signals for the second example simulation.

Fig. 4. Control signals for the first example simulation.

Table 3 Coefficients of the target model and the characteristic polynomials in second simulation.

Input signals 4

Index (i)

di

di0

di1

di2

0 1 2 3 4 5 6 7 8

271.390 787.000 1097.300 954.500 564.500 232.200 64.900 11.300 1

251.500 745.400 1047.000 904.200 512.700 209.800 57.530 10.690 1

271.400 787.000 1097.000 955.100 563.600 233.000 64.920 11.690 1

241.400 710.700 1024.000 886.800 511.600 210.900 58.510 10.890 1

u1 u2

3.5

Tension (v)

3

2.5

2

Step Responses

1.5

1.2 1.1

1

1

0

0.9

Iteration

0.8 Amplitude

1000 2000 3000 4000 5000 6000 7000 8000 9000

Fig. 7. Input signals.

0.7 0.6 0.5 0.4

Output signals

0.7

y1 y2

0.3 0.2

0

1

2

3

4

5

6

7

8

Time (sec) Fig. 5. Step response of the closed loop system for the second example simulation.

By using the toolbox ident of Matlab, we were able to determine the system transfer functions which are given by: G1 ðsÞ ¼

0:002853s2 þ 0:01116s þ 0:024 s3 þ 1:482s2 þ 10:42s þ0:009175

0.4 0.3 0.2 0.1

ð39Þ 0

and 2

G2 ðsÞ ¼

0.5

9 10 11 12 13 14 15 Level (m)

0

0.6

G0(s) G1(s) G2(s)

0.1

0:004317s þ 0:0169s þ0:0363 s3 þ 1:465s2 þ 10:41s þ0:009138

0

1000 2000 3000 4000 5000 6000 7000 8000 9000 Iteration

ð40Þ Fig. 8. Output signals.

638

M. Ben Hariz et al. / ISA Transactions 51 (2012) 632–640

Table 4 Parameters of the PI controller by the first method of Ziegler–Nichols. Kp

Ti ¼Kp/Ki

PI

0.9T/L

L/0.3

9 8 7 Tension (v)

Controller

Set point and output

0.5

Control signal

10

6 5 4

0.45

Level (m)

3 0.4

2

0.35

1 0

0.3

0

0.25

100

200

300 Iteration

400

500

600

Fig. 10. Control signal obtained with the PI controller.

0.2 Output Set point

0

100

200

300 Iteration

400

500

600

Fig. 9. Set point and output obtained with the PI controller.

The model validation consists in carrying out several tests of checking. For this step, we have used other measures different from those used during the identification stage and we have obtained a small error between the system and the models outputs. 5.2. Synthesis of controllers In order to compare the closed-loop performances, we take an interest in synthesis of three types of digital controllers namely: – A Proportional Integral (PI) controller. – A controller for fixed parameters system. – A controller for uncertain parameters system.

From these figures, we remark that the output fluctuates around the set point and the control signal presented many variations as well as peaks. We can calculate the control variance by the following relation: 1X M¼ uðkÞ2 ð43Þ N In the case of PI controller, we find: M1 ¼15.435. 5.2.2. Controller for fixed parameters system To synthesize controller, we use the system transfer function G1(s) given by (39). Consider the following fixed second order controller: ( AðsÞ ¼ s2 þa1 s þ a0 BðsÞ ¼ b1 s þ b0 Let our objective is to synthesize a controller with the following specifications: (i) Overshoot r2%. (ii) 2% Settling time r 700 s.

For the synthesis of different types of controllers, the sampling period is Ts ¼ 10 s. 5.2.1. PI controller In order to determine the PI controller parameters, we have used the first method of Ziegler–Nichols which takes in account the step response of the open loop system [23]. In this case, the controller is given by the following expression:

These performances can be obtained by choosing the characteristic ratio a1 ¼ 2.7 and the generalized time constant t ¼1. Thus, we can, determine the target model. The minimization of the objective function allows us to obtain the following controller:

ð42Þ

11760 ð44Þ s2 þ 3s þ 5029 The evolutions of the output, the set point and the control signals are shown in Figs. 11 and 12 respectively. It is noted that, in spite of the variation of the set point, the controller synthesized meets the specified performances which are a response with an overshoot less than or equals to 2% and a settling time of 700 s. According to Fig. 12, we notice that the control signal shows many peaks at the beginning and in the change of the set point as well as fluctuations. This signal presents fewer variations than in the case of a PI controller. In this case, the control variance is M2 ¼8.427.

The evolutions of the set point, the output signal and the control signal, obtained with the PI controller are represented in Figs. 9 and 10.

5.2.3. Controller for uncertain parameters system Before proceeding to the synthesis of this controller, it is necessary to take into account the variation range of

CðsÞ ¼ K p þ K i

1 s

ð41Þ

where Kp and Ki parameters are defined in Table 4. The parameters T and L are determined from the response curve of Ziegler–Nichols first method. By applying this method on the found model G1(s) given by (39), the PI controller will be defined by: C 1 ðsÞ ¼

22:5s þ 0:135 s

C 2 ðsÞ ¼

M. Ben Hariz et al. / ISA Transactions 51 (2012) 632–640

Set point and output

Set point and output

0.5

0.45

0.45

0.4

0.4 Level (m)

Level (m)

0.5

0.35 0.3

639

0.35 0.3 0.25

0.25

0.2

0.2

Output Set point

Output Setpoint

0

50

100

150

200

250

300

350

400

450

0

50

100

150

Iteration Fig. 11. Set point and output signals obtained with the fixed parameters system.

400

450

9

8

8

7

7

6

Tension (v)

Tension (v)

350

Control signal

10

9

5 4

6 5 4

3

3

2

2

1

1

0

300

Fig. 13. Set point and output signals for the uncertain parameters system.

Control signal

10

200 250 Iteration

0

0

50

100

150

200

250

300

350

400

450

0

50

100

150

200

250

300

350

400

450

Iteration

Iteration Fig. 12. Control signal obtained with the fixed parameters system.

Fig. 14. Control signal for the uncertain parameters system.

uncertainties. From the relations (39) and (40), we determined the average transfer function of the system which is given by:

With an aim of achieving these specifications and to determine the characteristic polynomial d ðsÞ, we chose a1 ¼ 2.7 and t ¼1.6. The determination of the desired controller parameters amounts to solving the following min–max optimization problem

GðsÞ ¼

0:0036s2 þ 0:0140s þ0:0302 s3 þ1:4735s2 þ 10:4150s þ 0:0091565

ð45Þ

We consider the following uncertainties O which are determined by taking into account the transfer functions given by the expressions (39), (40) and (45).   DO ¼ 7 0:002, 7 0:004, 7 0:007, 7 0:00002, 7 0:01, 7 0:01 By considering the system transfer function given by the relation (45) and by adopting the uncertainties DO, we proceed the synthesis of the controller by exploiting the approach exposed previously. The controller is described by the following polynomials: ( AðsÞ ¼ s2 þ a1 s þa0 BðsÞ ¼ b1 s þb0 Our purpose, now, is to synthesize a controller with the following specifications: (i) Overshoot r2%. (ii) 2% Settling time r700 s.

min max f ðx, OÞ x

O

The resolution of this problem leads to the following controller: C 3 ðsÞ ¼

1337:3 s2 þ 2:1s þ709:2

ð46Þ

By using this controller with the real system, we obtain the output signal which is plotted in Fig. 13, and the control signal shown in Fig. 14. From Fig. 13, we note that the output system meets the desired requirements, which are a response having an overshoot about 2% and a settling time less than 700 s. We notice also that the control signal obtained with this last controller provides a small variation compared to the control signals obtained with a PI controller or a controller for fixed parameters system. This control sequence presents the least control variance which equals to 6.907.

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M. Ben Hariz et al. / ISA Transactions 51 (2012) 632–640

6. Conclusion and prospect The research conducted as part of this paper focused first on the synthesis of a controller for fixed parameters systems then in the case of uncertain parameters systems. For fixed parameters system, we determined the controller parameters by minimizing an optimization criterion. In the case of uncertain parameters systems, and assuming that the uncertainties are bounded, the determination of the controller parameters amounts to minimizing the maximum of these uncertainties with regard to the controller parameters. Therefore, the controller is obtained by solving a min–max optimization problem which represents the best solution for the worst case of all possible models. The study of the physical reality of the systems must take into account the uncertainties affecting the nominal model parameters. By adopting this assumption, and by following methodology carried out on a laboratory pilot plant, we could exploit the obtained theoretical results. In this work, the criterion to be optimized is non-convex what implies the use of a global optimization method in order to ensure good performances in closed loop. To look further into research in this context and to highlight the effectiveness of this controller, it is desirable to proceed to a comparison with other optimization methods. References [1] Adrot O. Diagnostic a base de mode les incertains utilisant lanalyse par intervalles: lapproche bornante. PhD thesis. Institut National Polytechnique de Lorraine; 2000. [2] Ramirez DR, Arahal MR, Camacho EF. Min–max predictive control of a heat exchanger using a neural network solver. IEEE Transactions on Control Systems Technology 2004;12:776–86. [3] Zafiriou E. Robust model predictive control of processes with hard constraints. Computers and Chemical Engineering 1990;14:359–71. [4] Zheng ZQ, Morari M. Robust stability of constrained model predictive control. In: Proceedings of the 1993 American control conference; 1993. p. 379–83. [5] Lee JH, Cooley BL. Min–max predictive control techniques for a linear statespace system with a bounded set of input matrices. Automatica 2000; 36:463–73.

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