Proceedings of the 2001 IEEE International Conference on Control Applications September 5-7,2001 Mexico Ciiy, Mexico

EDF Benchmark Control Problem : - a Multiobjective Robust Design Approach

Jean-Pierre Folcher*, Pascale Bendotti**, ClStment-Marc Falinower**

* Laboratoire I3S, CNRS-UNSA, * * Direction des Etudes et des Recherches, Les Algorithmes / Batiment Euclide B, 2000 route des Lucioles BP 121, F-06903 Sophia-Antipolis, France e-mail : folcher@ibs.unice.fr. e-mail: pascale.bendotti.clement-marc.falinoverQder.edfgdf.fr.

Electricit6 de France, 6 quai Vatier - BP49, 78401 Chatou Cedex, France,

Abstract- A solution to the EDF benchmark problem is presented in this paper. To reflect all the control features, a multiobjective synthesis approach is proposed. The method consists in ensuring simultaneously specifications as robust stability and time-domain bounds (output and command in- put peak). Synthesis conditions are Linear Matrix Inequal- ities (LMI) associated with a set of non-convex conditions. They are checked using an efficient linearization algorithm. The output feedback controller is explicitly derived from the numerical solution. The achieved performances are eval- uated using the parameter-dependent simulator in accor- dance with the benchmark experiment planning.

I. INTRODUCTION

This paper gives a solution to the EDF benchmark prob- lem presented in [ 2 ] . This benchmark considers the wa- ter level control of a Steam Generator (SG). The plant is a time-varying system with dynamics that change slowly as the internal power variations. Both the control input and the outputs are constrained. To reflect control objec- tives an LMI-based multiobjective synthesis method close to those presented in 161, 131 is proposed. The output feed- back control of LTI systems subject to “structured” dissipa- tive perturbations is studied. The key idea is that a number of design specifications, such as robust input peak bound and robust output peak bound can be translated into LMI constraints associated with a nonconvex constraint of the form ST = I , where S and T are two (block-diagonal) ma- trix variables. The synthesis problem can be solved using LMI optimization associated with an efficient cone com- plementarity linearization aZgorithm described in 151. This makes the design problem amenable to an efficient numer- ical solution. Futhermore, explicit formulas are given to compute the output feedback controller parameters.

We applied the multiobjective approach to the design of the water level control of the steam generator. In accor- dance with the benchmark experiment planning the per- formances achieved by the multiobjective output feedback controller in closed loop with the plant are validated using a parameter-dependent simulator.

The paper is organized as follows: the next section de- scribes the main features of the control problem. The mul- tiobjective design framework is exposed in section 111. The water level controller design is performed in the following section. Numerical experiments are presented in section V

and some concluding remarks end the paper.

11. EDF BENCHMARK CONTROL PROBLEM A detailled discussion of the EDF benchmark control

problem is done in [ 2 ] . The control objective is to adjust the feed-water flow-rate Qe to maintain the steam genera- tor water level within allowable limits, in the face of chang- ing steam demand Qu, considered as a disturbance. The water level (due to the two-phase nature of the steam water mixture) is not a well defined quantity. Two measurements estimate the water level: the narrow range level Nge and the wide range level Ngl. In addition, steam and feed-water flow-rate measurements are available. A simplified model of the complex physical system is given in 121. This model described by the following transfert function based diagram shown in figure 1 exhibits the main characteristics of the plant behavior. Both & and & represent the integrator

describe the nonminimum phase behavior on Qe and Qv respectively. The dynamics of the SG change with the op- erating power, this is reflected in the model by considering parameters values r*, a*, ,& to be power dependent.

effect of the SG. Transfert functions s z ~ ~ ~ ~ a , and +,s+l B

Fig. 1. Benchmark simplified model.

The following control objectives specifies bounds on the command input and the regulated output. (i) the error magnitude on the level N g , should lie prefer- ably between f 5 and strictly between f 2 0 ,

0-7803-6733-21011$10.00 0 2001 IEEE 866

(ii) the error magnitude on the level Nge should die off in less than 100 seconds, then the error should be less than f0.5, (iii) the saturation on the control is [urnin, uma2] = [4,120].

111. MULTIOBJECTIVE ROBUST SYNTHESIS

In this section, main results about the multiobjective de- sign approach based on Lyapunov approach combined with LMI (in the spirit of 111) are presented. For a more detailled presentation and complete proofs we refer the reader to [3].

A . Problem Statement

We consider the uncertain system defined as follows

where x E R" is the state, U E R"- the command in- put, w E R"- the exogeneous input, t E R"= is the reg- ulated output, y E R"y is the measured output. We as- sume that Dqw, Dzp, D,, are zero. We consider that Ai is {Vi, V,, Wi}-dissipative, i.e. , for every p i , qi E RnAi with pi = Ai ( q i ) we have

(2) where Vi _< 0, x, Wi 2 0 are given real matrices. Note that the composite operator A defined as p = A(q) = [A, (ql)T, ..., A,(qm)T]T is also {SU, SV, SW}-dissipative (see 171) with

U = diag ( U I , . . . ,Um) , V = diag(V1, ..., V,), W = diag(Wl, ..., Wm), S = diag ( X i I n n a l , ..., XmInnPm), X i > 0, i = 1 ,..., m.

(3)

B. Control Objectives

The purpose is to find a strictly proper LTI controller in closed loop with the system (1) which achieves simul- taneously a number of desired properties. The closed-loop system is represented in figure 2.

Fig. 2. Standard Closed-Loop System.

S.1 Robust Stability. For every A {U, V, W}-dissipative, the closed-loop system with w 0 is stable.

S.2 Robust output peak bound. For every A { U , V, W}- dissipative, the responses zz of the closed-loop system to zero initial conditions, and a unit-impulse in the i-th coor- dinate of w , the other inputs being zero, must satisfy

where z,,, is a given positive number.

S . 3 Robust input peak bound. For every A {U,V,W}- dissipative, for the trajectories defined in S . 2 , the result- ing command input satisfies, for some prespecified number

11u(t)II 5 uma, for every t 2 0.

urnax

C. Synthesis Conditions

The synthesis conditions presented in this paragraph are a special case of results in 131.

Theorem 1 If there exist matrices P,Q E RnX", Y E R"uxn, Z E RnXny, and structured matrices S,T and scalars T, v that satisfy constraints

(5) 1 U S + SVTD,,

BFP + DcpZT +DT SV +SVTC, + DLSWC, 4p

+D,,SWD,P

867

BpT+QCTV 1 QAT + AQ QCTW +BUY + Y T B T +YTDruW +YTDTuV

ST = I . (6) < ,

> 0, (7) 1 DFwZT 0 -B,TPB, - B,TzD,, -D;,z~B, + U I

ZDyw P I

1 0

1 [ T F I 0 ; ;] Q I > O , TU=^. (9)

then there exists a strictly proper full order output feedback controller with the state space matrices

= M-’ P A + ATQ-l + PBuYQ-’ + ZDyUYQ-’ ( +ZC, + CTWS (C, + D,,YQ-l)

- (PB, + ZD,, + CFVS + CFSWD,, ) D&VS + VTSD,, + DT,SWD,, + SU

l3TQ-l + V S (C, + D,,YQ-l)

an closed-loop with the system which ensures S.l-S.3 con- trol Objectives.

D. Checking the synthesis conditions The conditions of theorem 1 are not convex, due to the

coupling conditions ST = I and TU = 1. However, they are simple LMI conditions for fixed S and T . We may use the cone complementarity linearization algorithm proposed in [5]. Consider the LMIs

1. Find P, Q, Y, 2, So, TO, TO, uo that satisfy constraints (5), (6) , (7) , (8) and (9), where the non-convex equalities ST = I , TU = 1 are replaced by the LMIs (10). If the problem is infeasible, stop. Otherwise, set k = 1.

2. Find sk,Tk ?-k , vk that solve the LMI problem of mini-

LMI constraints of step 1.

reached a stationary point, stop. Otherwise, set k = IC + 1 and go to step 2.

mizing TI’(Tk-1Si-Sk-1T) +Vk-1T+Tk-1U7 subject to the

3. If the objective ~ ( T k - l S + S k - i T ) + U ~ - i T + T k - i U has

IV. WATER LEVEL CONTROLLER DESIGN

The control design reflects benchmarck specifications given in section I1 through the synthesis objectives S.1- S.3. “Embedding” the plant in the uncertain system (1) is the first step for the design. If S.l holds then the plant in closed-loop with the controller will be stable despite oper- ating point variations.

A crude way to do will consist to construct an uncertain system with the seven parameters variations (T,, a*, p* of the benchmark simplified model shown in figure 1). Note that parameters change slowly with the operating power while the synthesis method considers arbitrary fast param- eter variations. This approach might be conservative, thus leading to a controller with poor performances. To cope with this potential drawback we choose to consider uncer- tainty only for the critical parameter T~ (the time constant of the actuation). That is, we pose T, = T,“ + 67, where ST, is the parameter uncertainty. The opportunity of this choice will be confirmed by the numerical experimentation. This leads to the choice of signals q = Qe and p = 67,Qe, see figure 2.

Bounds on the command input U and on the regulated output z = Nge are also specified in section 11, in partic- ular for step input disturbance Qv. Synthesis objectives S.2-S.3 strongly reflect these specifications by adding an integrator on the disturbance input Qv with input w (to modeled the step disturbance input of size y). The associ- ated closed-loop synthesis system is represented in figure 3.

l y U The following algorithm is guaranteed to converge to a sta-

tionary point of the objective (see [ 5 ] ) . Fig. 3. Benchmarck Closed-Loop System.

a6a

The state-space matrices of the nominal part of the un- certain system represented by G in figure 2 is

1 - 7 0 0 0 0

p . 0 0 0 0 0 1 0 -a1 -a0 0 0

1 O a T C

0 I o 0 1 0 0 0 0

0 0 0 0 - 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

7 9

o + o 0 0 0 0

o + o 0 0 0 0 0 1 P1 Po 1 0 0

1 0 0 0 0 0 0 L o 0 0 0 0

0

0 0 0 0

Y 0 0 0 0 0 1

- - -

0 0

According to results presented in the companion papers of [2] in IFAC’99 World Congress, no fixed linear controller is able to ensure performance over the entire operating range. The approach we proposed consists of two con- trollers designs: one for the low power range and another for the high power range. The first controller KLP operates between 10%Pn-20%P, (P, is the nominal power of the steam generator). The parameters values T:,(YO, ..., T~ of the state-space data (11) are given in table I. The size of the variation of re, denoted by ISrel was maximized. Synthe- sis parameters are choosen as ( z rna2 ,urnas ,~ ) = (10,6,5). We proceed in the same way to design the controller KRP which covers the high power range 20%Pn-80%Pn. The state-space data parameter values r;,ao, ..., T~ for (11) are given in table I. The synthesis parameters are fixed to ( z ~ ~ ~ , u ~ ~ ~ , Y ) = (10,6,5). The numeric solutions of the synthesis problems were found using the heuristic presented in section 111-D, the LMI optimization code SP [8] and its matlab interface LMITOOL [4].

95

90-

d 2

85

80

-

I I

I I - I

\ ’

- TABLE I PARAMETERS FOR THE TWO NOMINAL MODELS.

r e &I PO LP 2.505 0.074 0.276 -0.045 ‘UP 4.006 0.467 0.683 -0.138

V. NUMERICAL EXPERIMENTS

The final controller switches between the two designed controllers for the intermediate operating power 20%P,.

PI P g

0.079 1.053 0.098 1.220

In order to evaluate performance, the final controller is validated on the parameter-dependent simulator described in 121.

The following experiment planning is considered (Vari- able ip represents the initial value of the operating power.).

1. A +lo% step change on disturbance Qv at t o = 50 s with ip = 85, see figures 4,5.

2. A +5% step change on disturbance Q,, at t o = 50 s with ip = 10, see figures 6,7.

3. A -15%/min ramp corresponding to a 70% change of Q,, with ip = 100, see figures 8,9.

4. A +5% step change on disturbance Qv at t o = 50 s with zp = 5, see figures 10, l l .

5. A -5% step change on disturbance Q,, at t o = 50 s with ip = 10, see figures 12,13.

The plots conventions are the following. For each exper- iment, the first figure represents the time response of the steam flow Q,, in solid line, the feedwater flow Qe in dashed line and the command input U in thin line. On the second figure, the narrow rang level Nge is plot in solid line and the time performance template in dashed line. The following comments can be done: . in all experiments, the f20 specification on Nge is satis- fied ;

the f 5 specification is satisfied on experiments 2, 3, 4 and overstepped for experiments 2, 5 ;

the f 0 . 5 specification is ensured on experiments 1, 2, 4 and overstepped for experiments 3, 5.

In view of these experiments, the controller can be judged as giving satisfying results.

100

50 100 150 200 t c3)

75 ’ 0

Fig. 4. Steam demand Q,,, feed-water flow Qe for +lo% step change on disturbance Q,, with ip = 85.

869

8

6 -

4 -

2 -

25 0

-2

-4

-6

-8

Fig. 5. Narrow water level measurement Nge for +lo% step change on disturbance Qu with i p = 85.

- I

- I I

I I . . - _ _ _ - - - _ _ _ _ - - _ I

I I

-

50 100 150 200

' - 1

9 50 1 0 0 150 200 250

I

Fig. 6. Steam demand Qu, feed-water flow Qe for +5% step change on disturbance Qu with i p = 10.

50 1 0 0 150 200 -8

' t (=)

Fig. 7. Narrow water level measurement Nge for +5% step change on disturbance Qu with zp = 10.

I I

I 100 200 300 400 500 600

1 (3)

Fig. 8. Steam demand Qu, feed-water flow Qe for -15%/min ramp corresponding to a 70% change of Qu with i p = 100.

- 1

t 100 200 300 400 500

1 (S)

-8 ' 0

Fig. 9. Narrow water level measurement Nge for -15%/min ramp corresponding to a 70% change of Qu with i p = 100.

4 50 100 150 200 250 300 350

I (D)

0

Fig. 10. Steam demand Q w , feed-water flow Qe for +5% step change on disturbance Qu with ip = 5.

870

8

1

t 1 50 100 150 2 W 250 3 W 350 400

-8

t (=)

Fig. 11. Narrow water level measurement N g e for +5% step change on disturbance Qv with ip = 5.

11 1

I 0 50 100 150 200 250 3 W 350 400

I W

Fig. 12. Steam demand Q v , feed-water flow Qe for -5% step change on disturbance Qv with zp = 10.

i I

I I -1 I I I I I

I 50 100 150 2 W 250 300 350 400

-8

(U

VI. CONCLUSION In this paper, a solution to the EDF benchmark prob-

lem has been obtained using a multiobjective synthesis ap- proach. The method ensured simultaneously specifications as robust stability and time-domain bounds (output and command input peak). The synthesis problem was casted as a nonconvex optimization problem under Linear Matrix Inequatilies contraints. This problem was solved numeri- cally by an LMI-based linearization algorithm and the out- put feedback controller parameters were derived explicitly. In our opinion, benchmark control specifications are tigth. Up to now, no synthesis method has provided a single con- troller which ensures the whole benchmark control specifi- cations. Most of the solutions presented in the companion papers of [2] are linear parameter-varying controllers. We proposed in this paper a controller which switches between two linear time invariant controllers for the intermediate operating power. The achieved performances obtained with the final controller in closed loop with the plant have been evaluated using a parameter-dependent simulator of the plant : benchmark specifications are almost satisfied.

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[3]

Fig. 13. Narrow water level measurement N,, for -5% step change on disturbance Qv with zp = 10.

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