MIMO adaptive constrained predictive control case study: An environmental test chamber

Automatica, Vol.27, No. 4, pp. 611-626, 1991

0005-1098/91 $3.00+ 0.00 PergamonPressplc (~) 1991InternationalFederationof AutomaticControl

Printedin Great Britain.

MIMO Adaptive Constrained Predictive Control Case Study: An Environmental Test Chamber* J. M. DION,t$ L. D U G A R D , t A. FRANCO,t NGUYEN MINH TRIt and D. REYt

Control of temperature and humidity in thermal plants suggests the design and installation of algorithms able to cope with nonlinearities, couplings, saturations and time-varying dynamics. Key Words--Multivariable systems; adaptive control; predictive control; saturation; thermal plants.

An interesting problem concerns the possible saturations affecting the process inputs and/or outputs. It is well known that it is very difficult to master the adaptation transients which often lead, in practice, to undesirable and unrealistic computed control inputs. These adaptation transients may happen during set-point changes and more generally when the process dynamics varies. They lead to control inputs, characterized by important and fast variations, and which cannot be actually applied due to the physical limitations on the actuators. Numerous papers deal with constrained control inputs. Most of them are based on the "Dynamic Matrix Control" approach (DMC) developed by Cutler and Ramaker (1980), using predicted output errors in the presence of physical constraints. Let us mention, for instance, Little and Edgar (1985), Arulalan and Deshpande (1986) and references listed therein. Notice that constrained optimal control has been first developed by using linear programming methods, but solution of the constrained problem was obtained using quadratic programming. In the literature, few papers deal with adaptive control in presence of saturating inputs: examples are Ohkawa and Yonezawa (1982), Payne (1986), Abramovitch et al. (1986) and Zhang and Evans (1987) who use saturated inputs instead of computed inputs. In the last two papers, a stability study has been carried out for open-loop stable processes. Another approach is also possible (see Ortega et al., 1984) where the reference model is recomputed whenever the computed control input is out of the permitted input range. In this approach, the problem is to

A b s t r a c t - - I n this paper, we develop a multivariable control

algorithm that presents some interesting features. First, it is based on the Generalized Predictive Control approach developed by Clarke (1987). This allows one to take into account explicitly constraints on inputs, outputs, linear combinations of them and also on their derivatives or increments, this over some receding horizon. It is certainly the main feature of the algorithm. Secondly, it takes into account couplings between outputs, using a multivariable model. Thirdly, some nonlinearities are taken into account using a piecewise linear model. Lastly, changes in process model parameters are coped with, using an additional loop that consists of on-line updating of the parameters of the estimated process model using a parameter adaptation algorithm. The proposed control algorithm is experimentally evaluated on an industrial test chamber. We emphasize the influence and usefulness of the various "ingredients" involved in the proposed control algorithm.

1. INTRODUCTION

Much WORK has been done in the development of highly sophisticated adaptive control algorithms. Robustness issues are now dealt with (w.r.t. bounded disturbances, modeling errors, time-varying parameters e t c . . . ) . Some specific issues are unfortunately not often addressed, or at least neither well understood nor solved. * Received 18 January 1990; revised 9 October 1990; received in final form 12 December 1990. The original version of this paper was presented at the IFAC Symposium on Adaptive Systems in Control and Signal Processing which was held in Glasgow, Scotland, U.K. during April 1989. The published proceedings of this IFAC Meeting may be ordered from: Pergamon Press plc, Headington Hill Hall, Oxford OX3 0BW, U.K. This paper was recommended for publication in revised form by Associate Editor P. J. Gawthrop under the direction of Editor P. C. Parks. t Laboratoire d'Automatique de Grenoble, CNRSENSIEG-INPG, B.P. 46, 38402 Saint Martin d'Heres, France. The authors are also with the GRECO CNRS "Automatique'. Author to whom all correspondence should be addressed. 611

612

J . M . DION et al.

show the boundedness of the reference model output. Another approach (Noguchi et al., 1987), is to filter the computed control input in order to meet the constraints. Toivonen (1983) deals with saturated input variances in the stochastic context. The above mentioned approaches are characterized by the fact that constraints on the future control inputs are not taken into account. For stable processes, Kahan Foigel (1980) propose an adaptive version of the Model Algorithm Control (MAC) developed by Richalet et al. (1978). In this paper, we develop a multivariable adaptive control algorithm that takes into account constraints on the process inputs, outputs and their derivatives. Furthermore, this algorithm is able to handle processes that cannot be decoupled by linear state-variable feedback. In that case, decoupling can be perfectly achieved whenever the process model is well identified and the control input is within the saturation domain. This work follows preliminary studies made by Nguyen Minh Tri et al. (1986) and Dion et al. (1987) where the monovariable case with input constraints was addressed, and Dion et al. (1989) where experimental results concerning the control of an environmental test chamber were given. The multivariable adaptive control algorithm developed in this paper is based on previous work. Clarke et al. (1987) proposed a new approach, the "Generalized Predictive Control" and Bornard and Gauthier (1977, 1983) took into account inputs and states constraints, in the known parameters case. In order to cope with nondecouplable systems, a precompensator can be added. The adaptive constrained control problem is then solved iteratively, using classical "quadratic programming with constraints" methods (Fletcher, 1981). McDermott (1987), independently, presented some simulation results for a similar algorithm based on long range predictive control that takes into account input constraints (present and future) and uses the co-ordinate search method (one of the simplest descent methods). Tsang and Clarke (1988) likewise presented a control algorithm whose ideas are similar to Dion et al. (1987). This algorithm has successfully been applied to simulated examples. In this paper we deal with approaches where the control input is computed on a receding horizon, such as the Generalized Predictive Control. The constraints on the inputs are not only applied to the present control input but also on the future control inputs. All these control inputs are not actually applied since, at each sampling time, only the present computed

constrained input is applied. However, it may be noticed, in the proposed algorithm, that the present control input has been computed, under the hypothesis that the future computed control inputs and/or their derivatives also satisfy the constraints. This fact is very important in practice, and this leads to a better control since it takes advantage of some knowledge about the future control inputs (which are also constrained). The proposed algorithm is used to control an environmental test chamber. Usually, original control devices, associated with these test chambers are very simple (independent SISO proportional loops) and not suitable when high performances for following precise temperature and humidity profiles are desired. This relatively bad behaviour can be explained by several salient features of environmental test chambers. Indeed these processes are highly nonlinear (the response is very different for positive or negative temperature steps as well as for positive or negative humidity steps). The actuators have physical limitations, leading to constraints on the inputs and their derivatives. These processes have time-varying parameters models, depending on the setpoint and on the load under test. Last but not least, couplings between temperature and humidity are very important. As it will be seen later, the proposed algorithm is able to cope successfully with the above mentioned characteristics of the process. The paper is organized as follows. In Section 2, the problem is formulated. The process model and the control objectives are given. Section 3 is devoted to the development of the algorithm: multivariable adaptive constrained predictive control. In Section 4, the environmental test chamber is described and analysed in detail. Some open loop responses are given as well as closed loop responses with the original controller and simple PID type controllers. In Section 5, experimental control results are presented that emphasize the different characteristics of the proposed algorithm. Some concluding remarks given in Section 6 end this paper. 2. STATEMENT OF THE PROBLEM We develop our control algorithm on the basis of a linear model. Let us consider the following multivariable process model: A(q-l)y(t)

= B(q-~)u(t)

where A(q-1)=I+A~q

~+. • • + A ~ q - "

B(q -1)-- B l q -~ + • • • + B . q - "

(1)

Adaptive control of a thermal plant are (p x p ) and (p x m) polynomial matrices respectively, n is an upper bound on the observability index of the process model. I is the identity matrix of appropriate dimension; u(t) and y(t) are, respectively, the input and output vectors. The transfer matrix is assumed to be of full row rank. Some Bi matrices may be null or singular. The process is not assumed to be either stable or minimum phase.

Remark. The case m > p is very interesting when constraints are taken into account. The m - p degrees of freedom can be used to meet the control objectives, while satisfying the constraints. The control objectives are the following: Let us consider the following finite horizon quadratic criterion N

J(t) = ~_~ (y(t + j) -- yM(t + j))rR(y(t + j) /=1

- yM(t + j)) N

+ ~ Aur(t +j-

1)Q1 Au(t + j -

1)

1=1 N

+ ~ ur(t + j - 1 ) Q 2 u ( t + j - 1 )

(2)

j=l

where N is the prediction horizon, {yM(t)} is the output reference vector sequence, A is the difference operator (1 - q-1)l and R, Q1 and Q2 are some positive definite matrices. Notice that the weight Q1 on the input increments incorporates some integral action and that the weight Q: on the input allows one to save control energy when we have degrees of freedom on the inputs. We want to compute at each time t, control inputs that minimize the quadratic criterion J(t) and satisfy constraints on the inputs and outputs of the following type:

C Aut >- ~p,( O, dp(t), t) O(O)y, >- Vy(O, dp(t), t)

(3)

613

C, D(O), Wu(.), ~Py(.) depend on the constraints and will be specified later. The inequalities given by (3) can express different types and combinations of constraints on u(.) and y(.), for instance constraints on the magnitude of u and y, the magnitude of the derivative of u and y or on a linear combination of u and y. In the next section, we derive a multivariable control algorithm based on the minimization of the quadratic criterion (2), while satisfying the constraints (3). 3. D E V E L O P M E N T O F T H E A L G O R I T H M

3.1. Multivariable control with constraints (known parameters) When the Ai and Bi parameter matrices are known, the main idea of the proposed algorithm is to transform the initial problem [minimization of (2) while satisfying (3)] into a classical one: quadratic minimization with constraints. The control law is then computed, using quadratic programming methods with constraints. Without constraints, the basic structure of the control law is that developed by Clarke et al. (1987). Some improvements have been added in order to cope with the MIMO case. The proposed algorithm is an extension of that given in Nguyen Minh Tri et al. (1986) and Dion et al. (1987). Let us first build j-step ahead predictors with the following Bezout identities: I = Ej(q-1)A(q -~) A + Fj(q-1)q -j, j=I,...,N where Ei(q -1) matrices

and

Fj(q -1)

are

polynomial

E/(q -1) = I + EJlq -1 +" • " + E~_lq -j+l ~(q-1)

= Flo + . . .

+ FJnq-n.

y(t + j) = Ej(q-1)B(q -1) Au(t + j) + Fj(q-1)y(t).

(7)

A u r = [Aur(t) . . . . . A u r ( t + N - 1)], (1 x N m ) yrt= [yr(t + 1) . . . . . y r ( t +N)], (1 × Np)

C and D(O) matrices;

An, B1 . . . . . are

dpr(t) = [--yr(t) . . . . .

some

AUTO 27:4-B

are

Bn], (p × (np + nm)). well

dimensioned

some

u r ( t - - n + 1)] function

Defining GJ(q -~) as Ej(q-~)B(q -1)

GJ(q -1) = GJoq-~ + ' "

+ G~+j_~q -j-n

(8)

we have the following set of predictors

y(t + 1) = G~ Au(t) + (qG~(q -1) - G 1) Au(t) + Fl(q-1)y(t) y(t + 2) = (G 2+ G2q -1) Au(t + 1) + (qG2(q -1) - G 2 _ G21q-~)

--yr(t -- n + 1),

ur(t) . . . . . ~pu(.), apy(.)

(6)

These polynomial matrices are uniquely defined by A ( q - l ) , A, and the prediction horizon j. Using equations (1) and (5), we get

where

0 = [.41. . . . .

(5)

(4)

vectors.

× Au(t + 1) + F2(q-1)y(t) (9)

614

J.M.

DION et al. (1 X N m ) u , may be rewritten as:

N-1

y(t+

(17)

u, = ISIN AU, + (t,_ 1

((q-l)

where

u~"GiNq-i )

+ qG N

i=0

× A u ( t + X - 1) + F s ( q - 1 ) y ( t ) .

It is easily shown (Clarke et al., 1987) that G O = G01 = G 2 . . . . .

O 0N - 1 = G 0N

o, = o~ = o 3 .....

o7

tTIN =

I

0o)

GN-1 = GJe-1.

Using equation becomes: J(t)

The set of equations (9) can be rewritten in a condensed form: y, = GN Au, +f,

(

criterion

(2)

AUt

+ ft -- YMt) -Jr- AUTt O I •U t

+ auIIZlr~O_2l:lu Au, +2

fT(t)]

)

~(t) = qaJ(q -1) - ~ G{q -~ Au(t + j - 1)

- TN Q-2 U-, _ , + t t t r - , O 2 / ~ , _ l . A u , TH

dJ(t) d Au,

(13)

Go

(18)

If there is no constraint on Aut, the solution Au, minimizing (18) can be explicitly found, using

i=0

+ Fj(q-1)y(t)

the

= (ON Au, + f , - yta,)rR(GN Au,

(11)

y T ( t + N)]

,'

(11),

+ ft -- Y~t,) + Aur, Q., Au, + uTQ.2 u,

txuT= [Aur(t),..., Aur(t + N - 1)] (12) ft = [far(t) . . . . .

I

= (ON Aut + ft --YMt)TR(GN

where y f = [ y r ( t + 1) . . . . .

El I

(Nm x Nm).

0

leading to:

G1

RUt = ((~T/~(~N "1- 0 1 + / ~ T 0 2 / t ~ r N ) - I

x [0T/~(ya~,--f,) -/-tT02/~t_I].

~=

G1

GN-1

Go (Np × Nm).

(14)

The term GN Au, contains present and future inputs. The term f, contains present and past outputs (up to time t), and past inputs (up to time t - 1). The quadratic criterion (2) can then be rewritten as J(t) = (Yt - YM,)T[¢(Y, -- Yta,)

Remark. In order to reduce computation complexity, it is always advisable to take into account the knowledge on the plant model (possible known delays in the process). It is also possible to reduce the computational burden by imposing a constant control input vector after a certain horizon N, which is called the control horizon ( A u ( t + j -- 1) = 0 for j > N.). In that case, equation (11) becomes y,

Au T = [Aur(t) .....

l~Ult -li-ft

(20)

A u r ( t + Nu - 1)l

m

Go

where Q1 = diag ( Q , . . . . .

= GN, N,'

with

+ Au,rOi Au, + urQ.2u, (15) h = diag (R . . . . .

(19)

G1 R) Q1)

utr = [ u r ( / ) , . . , , u r ( t + N Defining fi,r__1 = [ u r ( t - 1) . . . . .

Go

GN, N. =

02 = diag (Q2 . . . . . Q2) yrMt = [ y ~ ( t + 1) . . . . , y r ( t + N)]

1)](lxNm). u r ( t - 1)],

(16)

G1

Gu-,

GN--N,,.

(Np X N~m). (21)

Adaptive control of a thermal plant following input-output constraints:

The corresponding control law becomes -T

- -

SHU->u(t)

AUlt = [GN, N RGN, N~ + Q1,N. -T

-

-

- HN.Q2,NUL,_,]

>-SBU

SDU -> Au(t) - - S D U

+ IZlfv.O_.2,NflN.]-'[G~v.N.R(yM,-ft)

SHY->y(t)

(22)

where

(25)

->SBY

SDY - Ay(t) -> -SDY.

i] , mx um, U- lr, t - - 1

615

=[ur(t-1),.

.

. uT(t--1)],

0~,N~ = diag (Q~ . . . . .

Q~)

Q2,N. = diag (Q2 . . . . .

Q2).

(lxNum)

The inequalities should be read as: each component of the different vectors is bounded by the corresponding component of the constraints vectors. SHU, SBU, SDU, SHY, SBY, SDY are respectively the vectors upper threshold, lower threshold and derivative threshold of the control inputs and outputs. After some computations (see Appendix A) one obtains the following result:

The effectively implemented control law input at time t is u(t) = u(t - 1) + Au(/) (23) where Au(t) is given by (19) or (22). When dealing with constraints (3), it is no longer possible to express the solution in the explicit form (19) or (22). Thanks to expression (18), the problem given by (1), (2) and (3) is then expressed as:

Cu,(O) =

c L - c L cL

-cL cL - c W ~uy(O, #p(t), t) = [-~p~, _~pr, ~p~(t - 1 ) , -l/,~(t - 1), ~p4r(t), - ~psr(t), ~ ( t ) , - ~p~(t)]r with C 1 -~-/~

(26)

(Nm x Nm 1

min { A u r A 1 Au, - 2Blr Au,} AUt

with

°1

C mut > - lpu(O , dp(t), t)

• ..

D(O)y, > -- ~Oy(O, dp(t), t)

where

-

-

0

aL,Q2HN.

Go

using equation (11), the constraints can be written Cur(O) Au, >- ~Ouy(O, #p(t), t)

I

C3 = (~N

A, = Or/~(~N + Q, +/4NrQ2HN Brl = (YM,-- f ) T R G N

I

(Nm × Nm)

0 N GN m a -'

C4 ~ aN--

(24)

where Cur(O) is a matrix and ~Puy( ) is some function vector. Cur( ) and ~Puy( ) depend on the possible constraints on the inputs and outputs. At each sampling time, the problem is solved by iteratively computing the control input vector Au,, or Aul, using standard quadratic programming algorithms with constraints [for instance VEO2A of the HARWELL library, the method is that described by Fletcher (1981)] and applying u(t) = u(t - 1) + Au(t). The computation time varies with the number of iterations required, but are at least comparable with the time required to solve a set of linear equations in n variables and might typically take 10 times this amount. We will now explicit Cuy( ) and l # u y ( ) for the

_aN_

2

....

Go

O_

(27) and ~plr = [ S D U r , . . . , SDU r] ~p2r(t- 1) = [ ( S B U - u ( t - 1 ) ) r , . . . , (SBU

-

u(t - 1)) r]

~p~(t- 1) = [ ( S H U - u ( t - 1 ) ) r , . . . , (SHU - u(t - 1)) r] ~p4r(t) = [ ( S B Y - f l ( t ) ) r , . . . , (SBY--fN(t)) r] Wsr(t) = [ ( S H Y - f ~ ( t ) ) r , . . . , (SHY--fN(t)) r] W~'(t) = [ ( - S D Y - fx(t) + y(/))r, ( - S D Y - Af2(t))r, . . . , ( - S D Y - AfN(t)) r] ~,((t) = [ ( S D Y - f~(t) + y(t)) r, (SDY - A f 2 ( t ) ) r , . . . ,

(SDY - AfN(t)) r] (28)

J . M . DION et al.

616

where the f ( t ) have been previously defined in equation (12).

3.2. Adaptive multivariable control with constraints (unknown parameters) In order to derive an adaptive version of the control laws described previously, it is necessary to add a parameter estimation procedure. Hence, at each sampling time, the process model parameters (the A~ and B i matrices appearing in 0) are estimated using standard recursive identification algorithms. The design of the control law is made analogously to the known parameters case, using the estimated instead of the true ones. The estimated parameters can be obtained for instance via the following parameter estimation algorithm:

O(t) = O(t - 1) -t

e(t)cpr(t- 1 ) F ( t - 1)

I

1)

(29)

/

0---

or

J>[-VIm with

[5~,] AUt-L_w,

CI=/.

(m4)

- S D Y T Af2(t)

/

- S D Y - AfN(t)

/

(A15)

or lp7 >" C 4 A/At >" 1//6

(A16)

[ 5C4] Au, :> [ _ ~ 7 ]

(AI7)

or

The constraints on u(t), SHU->u(t)-> SBU, lead to the following relations SHU -> u(t) = Au(t) + u(t - 1) -> SBU

where Ca= G N - - G ~ defined in (27). With (A4), (A8), (A13), (A17) one gets

SHU -> u(t + 1) = Au(t + 1) + An(t) + u(t - 1) -> SBU

:

(A5)

SHU->u(t + N -

1)= Au(t + N -

1) + - • • + Au(t)

" Cil

"-~1"

-C

-~Pl

+ u(t - 1) -->SBU C2 I

which are equivalent to

-C2

lP2

A u , >-

-Vd3

(AI8)

SHU - u(t - 1) -> Au(t) -> SBU - u(t - 1) S H U - u ( t - 1)-> Au(t + N -

1) + - • • + Au(t)

(A6)

-> SBU - u(t - 1)

C31 - C3 C4

V'°4 - ~P5 1/'6

-- C 4

- ap7.

or

or ~P3>- C2 A u t > ~P2

'-

L-~P3J

(A7)

Cuy( O) Au, >- ~V,y( O, q~(t), t).

(A19)

Notice that only C 3 and C a depend on the process parameter 0.