Architectural issues in the control of tall multiple-input multiple-output plants

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1 Preprints of the 19th World Congress The International Federation of Autoatic Control Architectural issues in the control of tall ultiple-input ultiple-output plants Pedro Albertos Mario E. Salgado Departent of Syste Engineering and Autoatica, Polytechnic University of Valencia, Spain, (e-ail: Departent of Electronic Engineering, Universidad Técnica Federico Santa María, Valparaíso, Chile (e-ail: ) Abstract: This paper deals with the proble of controlling linear discrete-tie tall ultipleinput ultiple-output plants, using a cascade philosophy approach. The ain idea is to use a cascade architecture to regulate a subset of the plant outputs while keeping the rest of the bounded and, additionally, aiing at achieving a satisfactory disturbance copensation and a appreciable degree of odelling error robustness. These ideas assue that the two subsets of outputs have been assigned different iportance for the control designer. 1. INTRODUCTION Most, if not all, industrial plants can be odelled as ultiple-input ultiple-output (MIMO) systes. When the nuber of anipulable input signals is at least equal to the nuber of plant variables to be controlled, there are several control design approaches which can be successfully used (Maciejowski [1989, Goodwin et al. [2001, Albertos and Sala [2004, Skogestad and Postlethwaite [1996). That is not the case for tall plants. Tall MIMO plants are those systes where the nuber of anipulable inputs is saller than the nuber of plant variables of interest. This feature poses a fundaental liitation in the control of these plants: it is not possible to drive all those plant outputs to track arbitrary references and/or to fully copensate arbitrary disturbances. Fro a different perspective, we could expect this liitation, since it is ipossible to build an inverse for the plant odel, which is a paradig in control design Goodwin et al. [2001 Soe exaples of tall systes include distillation coluns Treiber [1984, das Litrico [2002, agnetic bearing systes Morse et al. [1998, cheical reactors Munro [1990, etc. Tall systes also arise when distributed systes are approxiated by finite diensional ones (see, e.g., Moheiani et al. [2003). It is well known that, regarding step references, perfect tracking is only possible if the reference vector lies in the space generated by the coluns of the atrix odelling the plant dc gain. Several authors have researched this central aspect in tall MIMO control (see e.g. Chen et al. [2002 and Freudenberg and Middleton [1998). One possible way to deal with this structural constraint is described in García [2011 where stationary control errors in all channels are accepted. In this paper, our contribution follows a different approach: we consider the control of a plant with p outputs and inputs (p > ), and we focus on a twofold objective, naely This work was supported by CONICYT grants and ACT53, and by MECESUP FSM1204 Regulating outputs, say y 1 [k R, such that they can be driven to prescribed set-points with zero steady state errors Maintaining the reaining p outputs, [k within acceptable values, even when the transfer function atrix fro the inputs to these outputs is unstable. A fundaental observation is that, by prioritizing the regulation of y 1 [k, it becoes useless to specify a reference for [k, given that the ain objective will force a particular control signal u[k, which in turn will define [k, at least in steady state. On the other hand, these constraints do not preclude the iproveent in the transient behaviour of [k, at the expense of the transient perforance of y 1 [k. This difference in perforance specifications akes sense when the control designer assigns different hierarchies to the two plant output subsets. In this work, we address the above fundaental issues related to the proposed architecture. The controller design itself is only briefly addressed without going into the intricacies of that subject. 2. BASIC SETTING We consider a discrete-tie linear MIMO tall plant with a strictly proper transfer function atrix G[z C p, p >, where Y [z = with G[z = Y1 [z = G[zU[z; (1) Y 2 [z G1 [z = G 2 [z Gb [z G G c [z a [z (2) where Y 1 [z C, Y 2 [z C p, U[z C, G 1 [z C, G 2 [z C (p), G a [z C, G b [z C, G c [z C (p) To attain the goals defined above we propose to use the cascade architecture shown in Figure 1. In that structure, Copyright 2014 IFAC 4704

2 we assue that we want to drive the vector output y 1 [k to track step references, as specified by r 1 [k; also, we ai to keep the vector output [k bounded. It is straightforward to see that these two goals are achieved if the whole loop is internally stable with C 1 [z having integral action in all channels. Although no disturbance has been explicitly considered, it is known that the presence of integration in C 1 [z suffices to copletely copensate the ipact of disturbances on y 1 [k, when those disturbances tend to be constant as k. Bearing in ind that Youla paraetrization of all stabilizing controllers will be used in the exaples to carry out the control synthesis, we define the following quantities and relationships Q 2 [z = (I C 2 [zg 2 [z) 1 C 2 [z (3) = C 2 [z(i G 2 [zc 2 [z) 1 (4) C 2 [z = Q 2 [z(i G 2 [zc 2 [z) 1 (5) G e [z = G 1 [z(i Q 2 [zg 2 [z) (6) Q 1 [z = (I C 1 [zg e [z) 1 C 1 [z (7) C 1 [z = Q 1 [z(i G e [zq 1 [z) 1 (8) leading to Ũ[z = Q 1 [zr 1 [z (9) U[z = (I Q 2 [zg 2 [z)ũ[z (10) = (I Q 2 [zg 2 [z)q 1 [zr 1 [z (11) Y 2 [z = (I G 2 [zq 2 [z)g 2 [zũ[z (12) Y 1 [z = G e [zq 1 [zr 1 [z (13) = G 1 [z(i Q 2 [zg 2 [z)q 1 [zr 1 [z (14) The above relations are specially useful for a stable MIMO plant. In this case, to achieve internal stability it is necessary and sufficient to choose the Youla paraeters Q 1 [z and Q 2 [z stable. The unstable plant case will be dealt with in a later section. In the expressions above the secondary or inner loop has the ain function of keeping the output [k bounded. Thus, a first logical design step is to choose Q 2 [z, which allows to copute the equivalent plant G e [z, to choose Q 1 [z. Since we are aiing to achieve zero steady state errors for the regulation of y 1 [k when the references are step signals we paraetrize Q 1 [z as Q 1 [z = (1 z 1 ) Q 1 [z (G e [1) 1 (15) This construction ensures perfect inversion at frequency zero. These ideas are next illustrated with an exaple. Exaple 1. (A 3 2 plant). Assue a 3 2 (p = 3, = 2 plant with [ z(z 0.8) (z 0.5)(z 0.8) 0.5(z 0.5)(z 0.8) 0.2z(z 0.5) 0.5z(z 0.8) 0.1z(z 0.5) G[z = (16) It is also assued that the outputs to be controlled are the first two, that is y[k = [y 1 [k [k T, where y 1 [k R 2 and [k R, with y 1 [k = [y 11 [k y 12 [k T (17) With the above choice we have that [ z(z 0.8) (z 0.5)(z 0.8) 0.5(z 0.5)(z 0.8) 0.2z(z 0.5) G 1 [z = ; (18) G 2 [z = Say we choose Q 2 [z = With this choice, [0.5z(z 0.8) 0.1z(z 0.5) [ z 0.5 2(z 0.1) and the equivalent plant is I G 2 [zq 2 [z = z 0.6 z 0.1 z 0.6 G e [z = (z 0.5)(z 0.1) 0.5(z 0.6) z(z 0.1) (19) T 2.5(z 0.8) (20) z 0.1 z 0.6 z(z 0.1) 0.2(z 0.6) (z 0.8)(z 0.1) (21) (22) Therefore, the Youla paraeter which guarantees integral action is Q 1 [z = (1 z 1 ) Q 1 [z G e [1 1 (23) For siplicity, we choose Q 1 [z = I, and a reference given by r[k = [µ[k 1 2µ[k 25 T (24) where µ[k k o is a unit step at tie k = k o The results for that reference are shown in Figure 2. We appreciate that the tracking error for y 1 [k tends to zero, leading to a steady state value for [k which depends, not on r 2 [k, but on the reference r 1 [k and the d.c. gain of G IMPACT OF DISTURBANCES Given that G 2 [z is a stable transfer atrix, one ight think that it would be acceptable to just focus on the control design to regulate y 1 [k, leaving G 2 [z in open loop; however, there are several reasons why the proposed architecture exhibits potentially better perforance in a ore realistic setting. In this section we consider the presence of an output disturbance d 2 [k R p, as shown in Figure 3. We can quantify the ipact of this disturbance on [k, after soe analytic work we obtain Y 2 [z d2 = (I G 2 [z(i G e [zq 1 [z)q 2 [z) D 2 [z (25) fro where it can be appreciated that, through a sensible choice of the design paraeters Q 1 [z and Q 2 [z, the ipact of the disturbance on [k can be daped. 4705

3 r 1 e 1 C 1 [z ũ C 2 [z u G a [z G c [z p (p ) (p ) G b [z y 1 Fig. 1. Undisturbed cascade architecture, with indication of the diensions of every transfer function y 11 y Fig. 2. Plant outputs under control (Exaple 1) ũ C 2 [z u G a[z G c[z d2 Q 1 [z = 4z 3(z 0.6) (29) We next siulate the control loop, fro where we obtain the results shown in Figure 4. Fig. 3. Section of the loop, showing a disturbance at the output of G 2 [z. To appreciate this feature, we consider a siple singleinput two-ouput tall plant. Say that G[z = G1 [z G 2 [z G 1 [z = 0.3 z 0.8 ; G 2[z = 0.5 z 0.5 (26) To isolate the perforance regarding disturbance copensation, we assue that r 1 [k = 0 k 0 and that the disturbance d 2 [k is a sequence of two steps, naely Disturbance and [k Tie [s d 2 d 2 [k = µ[k 1 2µ[k 20 (27) If G 2 were in open loop, we can anticipate that [k = d 2 [k for all k 0. On the other hand, if G 2 is in closed loop as in Figure 3, we know that [k will tend to track the disturbance steps (since the control input u[k will tend to zero, given that C 1 has integration). However, a sensible choice of the controllers ay yield an iproved transient behaviour in [k, as illustrated in this exaple. To synthesize the controllers, we again use the Youla paraetrization. We first choose Q 2 [z = 1.6 z 0.5 z = G e [z = 0.3 z (28) Fro where we choose a Q 1 [z satisfying Q 1 [1 = G e [1 1 = 10/3, to force integration in C 1. For instance Fig. 4. Copensation of an output disturbance in [k In this figure we firstly note two unavoidable features of the disturbance response: The plant output [k reproduces the initial steps in the disturbance (at k = 1 and at k = 20). This behaviour is due to the fact that we are dealing with an output disturbance. The ouput [k tends to the steps stationary values; this behaviour can be observed in the intervals k (10, 20) and k > 30. This is due to the fact that the reference for the output y 1 [k is zero, driving the stationary value of control signal u[k to zero. In spite of these two features, we see that the control loop provides an appreciable daping of the disturbance. 4706

4 4. ROBUTSNESS ISSUES As in the previous section one ight wonder whether it would be sipler to leave (a stable) G 2 [z in open loop, even if G 2 includes a significant odelling error. To provide an answer, we show next a way to quantify the ipact of that error in the configuration shown in Figure 1. Assue first that a calibration odel for the second part of the plant is G 2T [z, such that G 2T [z = (I G l [z)g 2 [z (30) where G l [z is the (left) ultiplicative odelling error (Goodwin et al. [2001). Then the achievable transfer function fro Ũ[z to U[z, originally given by (10), is now given by U[z = (I Q 2 [zg 2 [z)(i G l [zg 2 [zq 2 [z) 1 Ũ[z (31) It then becoes evident that the ipact of the odelling error is negligible if Q 2 [z is chosen in such a way that the product G l [zg 2 [zq 2 [z is sall for all ω. More specifically, it will suffice to ipose the constraint. σ { G l [e jω G 2 [e jω Q 2 [e jω } << 1 ω [π, π (32) where σ { } denotes the axiu singular value of the atrix. Since usually the ultiplicative odelling error is sall at high frequencies, that requireent can be satisfied by iposing an upper bound for the secondary loop bandwidth. The satisfaction of constraint (32) will also ake G e, given in (6), fairly insensitive to the considered odelling error. 5. UNSTABLE G 2 A second fundaental reason to have G 2 [z in closed loop appears when G 2 [z is unstable. Then, the synthesis of the controller C 2 [z, using the Youla paraetrization follows a sequence of steps which can be suarized as follows (Goodwin et al. [2001). (1) Express G 2 [z using a right atrix fraction description (RMFD), i.e. G 2 [z = G 2N [zg 2D [z 1 (33) (2) Find any stabilizing controller C o [z and express it in RMFD as C o [z = C on [zc od [z 1 (34) (3) Then, all stabilizing controllers C 2 [z, can be expressed in RMFD as C 2 [z = (C on [zg 2D [zω[z)(c od [zg 2N [zω[z) 1 (35) In fact, the stability of the atrix Ω[z is a necessary and sufficient condition for C 2 [z, given by (35), to stabilize G 2 [z. With this controller, the equivalent plant G e [z is stable, deanding then, the synthesis of an stable Youla paraeter Q 1 [z. Naturally, a siilar approach should be followed if G 1 [z is also unstable (leading to an unstable G e [z. Exaple 2. Consider the sae plant as in Exaple 1, except for the fact that now, G 2 [z is unstable, and given by G 2 [z = [ 0.5 z z 0.8 (36) with, for exaple, the RMFD given by 0.5(z 0.8) 0.1(z 0.5) G 2N [z = (37) z 2 z 2 (z 0.5)(z 0.8) G 2D [z = z 2 (38) We can verify that C o [z = [3 2 T stabilizes G 2 [z with a RMFD given by C on [z = [3 2 T ; C od [z = 1 (39) Then, using (35), all stabilizing controllers C 2 [z can be expressed as a bilinear function of a stable and proper Ω[z C 2. This paraeter, as well as Q 1 [z, should be chosen to satisfy the particular specifications of a given proble. 6. CONCLUSION A cascade based architecture has been proposed to deal with the control of tall MIMO plants. A key assuption is that the plant outputs can be organized in two subsets, one of the to be regulated, and the second one to be kept within reasonable boundaries. In essence, a sequential design is called for. A fundaental fact is that to achieve a good transient behaviour in the second subset, one ust sacrifice the perforance in the control of the first subset. As a copensation, we have achieved zero steady state errors in the regulation of that first subsets of plant outputs. Future work should include the foral proposal of a design strategy, as well as the usage of an interaction easure to siplify the procedures arising fro that strategy. REFERENCES P. Albertos and A. Sala. Multivariable Control Systes: An Engineering Approach. Springer, G. Chen, J. Chen, and R. Middleton. Optial tracking perforance for SIMO systes. IEEE Transactions on Autoatic Control, 47(10): , J. Freudenberg and R. Middleton. Properties of single input, two output feedback systes. Proceedings of the 1998 Aerican Control Conference, 4: , June Philadelphia, PA. M. García. Líites de desepeño y diseño de controladores para sisteas ultivariables estables con s salidas que entradas. Master s thesis, Departaento de Electrónica, Universidad Técnica Federico Santa María, G. Goodwin, S. Graebe, and M. Salgado. Control Syste Design. Prentice Hall,

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