Dev. Chem Eng. Mineral Process., 9(1L2),pp.49-56,2001. Robust Stability Conditions for SISO Dynamic Matrix Control L.K. Dai" and Z.Q. Cheng National Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou 310027, I? R. China This paper presents the robust stability conditions for single-inputhingle-output (SISO) dynamic matrix control (DMCJ). Model uncertainty is described by using a range of possible plant impulse responses, and a new form of DMC algorithm based on afinite impulse response (FIR) model 13discussed. Then robust stability conditions for the new DMC algorithm are derived by applying Jury's dominant coeficient lemma. These results provide theoretical foundations for analyzing and designing predictive control systems. Introduction Model predictive control (MPC) has been widely used in industrial processes for its good performance and strong robustness [l]. At the same time, many scholars researched the relationship between the performance of MPC systems and control parameters in depth. Garcia and Morari [I] firstly analyzed the closed loop stability of one antitype MPC algorithm under the fiame of internal model control [2]. Badgwell [3] derived the robust stability of the antitype MPC algorithm using the Jury's dominant coefficient lemma [4]. However, the results cannot be applied to the DMC algorithm for introducing the punishment on the change of control input. This paper extends the result of Badgwell [3] to DMC algorithm. We used the families of finite impulse response (FIR) models to describe the uncertainty of the controlled system and to research the DMC algorithm based on FIR model. Then we derived the robust stability condtions of closed loop systems. These results provide a theoretical foundation for analyzing and desigmng predictive control systems. Process Model and Uncertainty Description A linear open-loop stable SISO plant can be described by either a finte impulse response (FIR) h, c N y ( k )= h(i)u(k- i ) + d ( k ) ...(1) i=l or by a finite step response a: N-1 y ( k ) = c a ( i ) A u ( k- i ) + a(N)u(k - N ) + d ( k ) ...(2) k l both with N coefficients. *Authorfor correspondence (7kdai@mail.hz.zj.cn). 49 L. K. Dai and ZQ.Cheng The plant impulse h and step response a are related by: i ...(3) a(i) = C h ( j ) , h(i) = a(i)- a(i - 1) j=l + The controller uses an FIR model to predict future values of the output j’j(k j ) N .. j ( k + j ) = x f i ( i ) u ( k+ j - i) + 2(k + j ) i=l The nominal model &i) is Merent, in general, from the plant h. Here assume the duturbance &k) is step disturbancefor its inscrutability as follows: N 1 ...(5) &k+j)= &k)=y(k)-xh(i)u(k-i) i=l Here the y(k) is plant output at time k. Based on Equation (4), The j ( k + j ) can be split into three parts as follows: ...(6) j ( k + j ) = j f ( k + j ) + j p( k + j )+ i ( k ) j N 1 , j f ( k + j ) = x h ( i ) u ( k + j - i ) , j p ( k + j ) = x h ( i ) u ( k + j -i) i= j+l ill where j p( k + j ) and if(k + j ) are the contributions of past and fume input. The plant gain a(N) and model gain 2 ( N ) are assumed to be nonzero: i=l i=l Now we can define a family of plants for the uncertainty description. Definition 1. The family of mismatch plants Q is defined as : R = { h I h,,,,,,(i) ...(7) V i = 1,N) Ih(i) I h,(i), The family of mismatch plants includes a class of nominal SISO mismatch plants. All possible responses of each plant lying between the maximum{h-(i),i = 1,N) and minimum {h-(i),i = 1,N) curves. Definition 2. The total mismatch of one specific process M is defined as: N M = 1Ih(i)- &ill ...(8) i=l For each process in Q, Ih(i) - i ( i ) (I &(i) - hGn( i )= Ah, (i) ,therefore: N M sCu,(i) = M,, VI E R i=l where Mn is the maximum total mismatch of R. 50 ...(9) Robust Stability Conditionsfor SISO Dynamic Matrix Control DMC Algorithm Based on FIR Model DMC algorithm is a computer control algorithm, whch is presented firstly by Cutler et al. [ 5 ] . Garcia et al. [6] presented an unproved algorithm QDMC. It offered an objective, efficient method to deal with an inpuvoutput constrains system using quadratic programming. The DMC algorithm searches the future control change sequence (Au(k),.-.,Au(k+ m -1)) to minimize the objective: c P @(k) = m qi( r(k)- j ( k + j))' + j=1 qAu2(k+ i - 1) ...(10) i=l where m is control horizon, p is predictive horizon (in general p m ) , rjk) is the setpoint of process output, qj and ri are weights of output errors and input changes. The objective function @(k) makes the future output to its setpoint while minimizing the changes of input. Assume that the input keep steadiness after m steps. It can be described as follows: u(k + m + j ) = u(k + m - l),j 1 0 , or Au(k + m + j ) = 0 , j 2 0 . This paper describes the model mismatch with FIR, so we described the DMC algorithm based on FIR model firstly. We define the sequence of predictive error: ...(11) e ( k + j )= r ( k ) - ~ ( k + j ) = i ( k + j ) - ~ ~ ( k + j ) where &k +j ) = r(k)- j P ( k + j ) - i ( k ) ...(12) i-1 because u(k+m+i) =u(k+m-l),Vi>O and i / ( k + i ) = C h C i - i ) u ( k + i ) . i=O When j 2 m , i-1 .L where c ( j ) = x h ( j - i ) = c j-m+l h(i)=i(j-m+l), Vjj2m. Let H = ...( 14) L K.Dai and Z Q. Cheng then j f =Hu, e=i-Hu From A u ( k + z - l ) = u ( k + i - l ) - u ( k + i - 2 ) Mk) Let A u = [ Au(k +i m - 1) 1, b=[ y], ...(15) Vi2l. 1 0 -1 1 G=[ then AU = G u - h ( k - 1 ) Let Q = diag(q, q p ) , R = diag(r, rewritten as: 0 ... .-. . 0 *. 0 -1 1 ...(16) --- rm), then the objective function can be @(k) =(2-Hu)' Q(2 -Hu) +(Gu-b4k -l))'R(GU- bdk -1)) ...( 17) The lease square solution is: u = (H'QH + GrRG)-'(H5Qi + G'Rb u(k - 1)) ...(18) Because DMC adopts scroll optimization, only the first item of output is used by controlled system: ...(19) u(k) = k e i + k,u(k - 1) where k, = b'(H'QH +GrRG)-'HrQ,k, = br(HTQH+ GrRG)-'G'Rb or P ...(20) u(k) = x k e j i ( k + j ) + k , u ( k - l ) j =1 Robust Stability Condition for DMC Algorithm For N 1 N 1 q k + j ) = r(k)- x h ( i ) u ( k +j -i) - y(k) + x h ( i ) u ( k-i) i=l i=j+l N y ( k )= C h ( i ) u ( k - i ) + d ( k ) i=l N ...(21) :. u(k) = k , ( r ( k ) - d ( k ) ) + R , u ( k - l ) + C k i u ( k - i ) i=l where the disturbance d(k)is immeasurable, but the robust stability of the closed loop system does not depend on it. P k, = C k e j j=l P ki =Ckej[I;(i)-h(i)-I;(i+j)] i = l , ..., N j=1 The characteristic polynomial of the closed loop system is: 52 Robust Stabiliry Conditionsfor SISO Dynamic Matrix Control f( z ) = zN - k,zN-' + N 1kizN-i ...(22) i=l The sufficient and necessary condition for the stability of the closed loop system is that each eigenvalue is in the unit cycle. The following theorem offers a simple test for robust stability of DMC algorithm. Theorem I (robust stability conltion for DMC). To a family R of uncertainly plants, the sufficient condition of an DMC controller which is robustly stabilizing for all plants in the family is: ..(23) Proof: From Equation (9), Equation (23) means: ..(24) Based on the defintion of M in Equation (8) and triangle inequality, we can get: . .. kej(i(l + j ) + h(1) - i(1)) - k, IC + x k e j ( i ( i+ j ) + h(i) - i ( i ) )< 1. i=2 j=l 53 L.K. Dai and Z Q.Cheng then the controller can stabilize all plants in R . Where Ci(j)(j=l,...,N) is step response of the controller's model. Corollary 1 (Maximum tolerable mismatch of DMC controller) For a family $2 of uncertainly plants and one of it whose model is described as i ( j ) , j= l;.., N , if the maximum total mismatch M, of family R is smaller than the absolute value of model gain h(N), i.e. M, < lwq ...(26) Then there exists a DMC controller based on model i ( j ) , j = l,..-,N can stabilize all plants. This shows that, to uncertainty object family R, we should select the plant with large gain to design our controller if we want to enhance the robust stability of DMC controller. In addition, once the plant family R and model i(j ) ,j = l,...,N are given, if the constrain of plant performance changing satisfy Equation (21), then we can select suitable control parameters such as m, p, I;: ( i = l,.-.,m) and q ( j = 1,. ,p ) to stabilize the closed loop system robustly. - Simulation Consider an uncertain SISO controlled system described by KPe-" . ..(27) Gp(s)= (T,s+l)(T,s+l) where K, E [l.O 2.51, T E [8 lo], T, E [5 lO]min, T2 E [20 25]min. The impulse responses for Merent plant parameters are shown in Figure 1, where the sampling interval T, = 2 min. Figure l(a) shows the impulse responses of plants with following parameters: K = {l.O, 1.75, 2.5}, = (8, 9, lo}, T I = ( 5 , 7.5, -. l o } , T = (20, 22.5, 25). The h , h m Figure l(b) are bounds of the impulse responses. h,, h2, h3,and h., are unit impulse responses of the following models: e Model 1': G;(S)= 2.5 eaS (5s + 1)(20s + 1) Model 2': G&(s) = 1.75e4' (7.5s + 1)(225s+ 1) Figure 1. Impulse responses of an uncertain controlled system. 54 Robust Stability Conditionsfor SISO Dynamic Matrix Control a . _ _ _ I 50 45 10 85 I0 Model 3': G i (s)= Model 4': G$ (s)= 1.o edS (5s + 1x20s+ 1) 1.Oe-''' (10s + 1)(25s + 1) We can get the maximum mismatch M = 2.357 of the plant. From Corollary 1 we know that the controller based on the model can stab&e the closed loop piant by selecting suitable control parameters. Here the control parameters are selected to be N = 100, rn = 1, p = 100, q j = 1, V j E [l, p ] ,and I;: = 0, Vi E [l,m]. Figures 2 to 5 show the closed loop responses of DMC controllers based on 55 L K. Dai and Z Q. Cheng models 1'4'. The DMC controllers based on models 2'4' can stabilize the plant of R although they do not satisfy the condition of Corollary 1. Thus it can been seen that condition (21) is only the sufficiency condition for robust stability of DMC control system. But the degree of robust stability will decrease when the degree of departure to condition (21) is increasing. In addition, the relation between the closed loop performance of system and the process dynamic parameters: , TI, T2i s weak when the predictive horizon length of DMC controller is sufficiently large. Then the performance mainly depends on the plant gain. Conclusions Based on finite impulse response (FIR) models we have described the form of dynamic matrix control (DMC). Then the robust stability conditions for the new DMC algorithm have been derived by applying Jury's dominant coefficient lemma. Through the analysis of an uncertain SISO DMC controlled system, the following conclusion can be drawn: the performance mainly depends on the plant gain. To assume the robust stability, we should select the model with large gain to design DMC controller. It is appropriate for DMC applications. References 1. 2. 3. 4. 5. 6. Garcia, C.E.; h t t , D.M.,and Morari, M. 1989. Model predictive control: theory and practice - a survey. Automatica, 25(3), 335-348. Garcia, C.E., and Morari, M. 1982. Internal model control. 1. A unifylng review and some new results. Ind. Eng. Chem Roc. Des. Dev., 21,308-323. Badgwell, T.A. 1997. Robust stability conditions for SISO model predictive control algorithms. Automatica, 33(7), 1357-1361. Jury, E.I. 1964. Theory and application of the Z-transformmethod. Wiley, New York. Cutler, C.R., and Ramaker, B.L.1980. Dynamic mahix control - a computer conhol algorithm. Roc. Joint American Control Conf., San Francisco, WPS-B. Garcia, C.E., and Monhedi, A.M. 1986. Quadratic programming solution of dynamic rmmx control (QDMC). Chem. Eng. Commun.,46,73-87. Received: 13 July 1999;Accepted ajier revision: 15 June 2000. 56

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