w u P z K Figure 1: Model of the closed loop control sstem Model In this section the model for the control problems will be given. It is static, multi

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1 Optimal Linear Static Control with Moment and Yield Objectives Anders Hansson, Stephen Bod, Lieven Vandenberghe and Miguel Sousa Lobo Information Sstems Laborator Stanford Universit, Stanford, CA 94305{4055, USA and Abstract This report describes how to optimize linear static control problems involving rst and second moments and ield objectives for Gaussian distributions. These problems can be cast as second-order cone programs, which is a class of convex optimization problems that can be solved ver ecientl. 1 Introduction Yield is of importance in man manufacturing processes. The prot can be directl proportional to the ield, or the ield can be related to qualit levels in a nonlinear wa, i.e. onl as long as a certain percentage of the production meets the desired qualit, the customer is paing full price for the product. In the former case it is desirable to maximize the ield, in the latter case it is most likel desirable to minimize the production cost subject to constraints on the ield. Both these problems will be addressed in this paper. In Section the model to be controlled will be dened. For simplicit onl a static model will be considered, but the results presented can easil be extended to the dnamic case using ideas like in [BB91]. Control of static sstems in the H1 framework has been described in [SP96], and static performance robustness of thermal processes has been described in [KKB94]. In Section 3 the control problems will be dened. Then in Section 4 the solution procedure will be outlined. It will be seen that the problems can be cast as Second-order Cone Programs (SOCP):s, which is a class of convex optimization problems that can be solved ver ecientl, see [LVB97]. In sections 5 and 6 some extensions and examples will be investigated. Finall, in Section 7, a few concluding remarks are given. 1

2 w u P z K Figure 1: Model of the closed loop control sstem Model In this section the model for the control problems will be given. It is static, multi-variable, and depicted in Figure 1. The equations relating the dierent variables are z = P u = K u w where w R l is the exogenous input, z R q is the output to be controlled, R p is the sensed output, u R m is the actuator input, and where K and P, Pzu P zw P w P u are matrices of compatible dimensions. The matrix P is given and describes the plant, whereas the matrix K, called the feedback gain, describes the controller; it is to be determined to optimize some objective. For K such that I, P u K is invertible the control problem is said to be well-posed, and simple algebra shows that z = Hw, where H, P zw P zu K(I, P u K),1 P w

3 Most objectives are convex in H, but unfortunatel not in K, which is the optimization-variable. However, this problem can be circumvented b the following linear-fractional transformation: which ields Q, K(I, P u K),1 H(Q) =P zw P zu QP w where H is now ane in Q. This transformation can also be extended to the dnamic case, see e.g. [BB91]. In case I QP u is invertible K can be computed as K =(IQP u ),1 Q The case when I QP u is not invertible is discussed in Appendix A. For the purpose of this report it will be assumed that w is a Gaussian random vector with mean w and covariances. The covariance ma be degenerate, i.e. need not be positive denite, onl positive semidenite. Notice that z is also Gaussian with mean and covariance m(q) (Q), E fz(q)g = H(Q)w, E n [z(q),m(q)] [z(q), m(q)] To = H(Q)H T (Q) respectivel. 3 Control Problems In this section control problems involving ield and rst and second moments will be dened. First it will be noticed that the second moment ofzis given b M(Q), E z(q)z T (Q) = H(Q), ww T H T (Q) Furthermore for an entr z i (Q), e T i z(q) ofz(q) it holds that the corresponding means, variances, and second moments are given b m i (Q) = e T i m(q) i (Q) = e T i (Q)e i M i (Q) = e T i M(Q)e i Notice that the are linear and quadratic in Q, respectivel. Finall the ield of the signal z i with respect to the level z i R is dened as Y i (Q), Prob fz i (Q) z i g 3

4 It can be expressed in the standard Gaussian distribution function () as zi,m i (Q) Y i (Q)= i (Q) Furthermore it holds that Y i (Q) i if and onl if z i, m i (Q) i (Q),1 ( i ) This follows from the fact that () is an increasing function.,1 ( i ) > 0 if and onl if i > 1=. It is now clear that the sets Notice that f(q; i ):m i (Q) i g; (Q; i ): i (Q)i (Q; i ):M i (Q)i ; fq:y i (Q) i >1=g are convex. Hence natural optimization problems to formulate are: minimize m 1 (Q); minimize M 1 (Q); minimize 1(Q) maximize Y 1 (Q) respectivel, subject to m i (Q) i ; i I m i (Q) i ; i I M i (Q) i ; i I M Y i (Q) i > 1=; i I Y where I m, I, I M, and I Y are subsets of f1; ;:::;qg. Notice that more general problems can be obtained b considering linear combinations of m i :s and positive linear combinations of i :s and M i :s, respectivel. For notational simplicit the details are not given. 4 Solution In this section the control problems of the previous section will be recast as SOCP problems. It is fairl straight forward to see that the constraints common to all optimization problems can be expressed as p H T 1 (Q) 0 i, H i (Q)w; i I m i ; i I 4

5 p ww T Hi T (Q),1 ( i ) p H T i (Q) i ; ii M z i,h i (Q)w; i I Y The additional expressions for the dierent optimization problems are Problem 1: minimize t s:t: 0 t, H 1 (Q)w Problem : minimize t s:t: p H T 1 (Q) t Problem 3: Problem 4: minimize t s:t: maximize t s:t: t p ww T H1(Q) T t p H T 1 (Q) z 1, H 1 (Q)w B the results in Appendix B problems 1{3 are all SOCP problems in Q and t, since,1 ( i ) > 0 for i > 1=. Furthermore Problem 4 is an SOCP feasibilit problem in Q for an xed value of t>0. Hence it can be solved b bisectioning over t>0. Notice that there exists a Q satisfing the inequalit for t>0ifand onl if there exist a Q such that Y 1 (Q) > 1=. w u P z u 0 K Figure : Model of the closed loop control sstem with an ane controller. 5

6 5 Ane Controllers In this section it will be shown how the model of Section can be used to accommodate not onl linear feedback but also ane feedback, see Figure. The equations for this are: z = P u w u = K u 0 Dene K and via u =[K u 0 ] 1, K and Pzw, Pu, Pw, and w via z = 4 P zu P zw 0 P u P w u w 1 3 5, Pzu Pzw u P u Pw w from which it follows that the ane controller can be cast in the linear framework b augmenting and w appropriatel. Notice how this formulation also covers the case u = u 0 as a special case. The dierent strategies will in the sequel be called u = u 0 : feed-forward u = K: feedback u = K u 0 : feedback/feed-forward Trade-O between Yield and Production Cost This example is designed to demonstrate the trade-o between ield and production cost when the latter is proportional to the energ of the control variable. The plant is depicted in Figure 3, and it can be described with the following set of equations: x = Gu v = x e where G R pm and where v and e are independent random vectors of appropriate dimensions with means v and e and covariances v and e, respectivel. The objective to minimize is given b M = mx i=1 M i ; M i =E u i ; i=1;;:::;m 6

7 v u G x e u 0 K Figure 3: Model of the closed loop control sstem for trade-o between ield and production cost. and the constraints are given b Y mi = Prob fx i z i g > 1=; i = 1; ;:::;p. This can be cast in the framework of sections and 3 b dening P zu = I ; P G zw = P u = G; P w =[I I] 0 0 I 0 and w T =[v T e T ], and = diag( v ; e ). The matrix G has been taken to have uniforml distributed random entries on [0; 1]. In the same wa the mean v has uniforml distributed random entries on [0; 1], and the covariances v and e are given b v =D T Dand e = E T E, where D and E are p p matrices with uniforml distributed p random entries on [0; 1]. The mean of e is given b e =0,and z i = v i e T i ve i. The number of variables in this example is m for the feed-forward case, mp for the feedback case, and m(p 1) for the feedback/feed-forward case. The number of constraints are p 1. The optimal values of M for m = 10, p = 0, =1:0, and values of 1 =0:51; 0:5;:::;0:98 are shown in Figure 4. It is seen that the pure feed-forward design is the most costl design and that the combined feedback/feed-forward design is the cheapest one. Notice that the trade-o curves not necessaril are concave, which, however, is the case in this example. The computational time was about 8 hours on a reasonabl fast computer. 6 Worst Case Exogenous Inputs In this section it will be shown how to consider worst case exogenous inputs. To this end let w k dene a set of Gaussian random vectors with means w k 7

8 alpha M Figure 4: Maximal ield versus cost for feed-forward solid line, feedback dashed line, and feedback/feed-forward dotted line and covariances k ; k I P, where I P is a nite index set. Problem 1 is then generalized to minimize Q maximize k m 1 (Q; k) subject to m i (Q; k) i ; i I m ;k I P i(q; k) i ; i I ;k I P M i (Q; k) i ; i I M ;k I P Y i (Q; k) i > 1=; i I Y ;k I P which can be rewritten as minimize t s:t: 0 t, H 1 (Q)w k ; k I P and 0 i, H i (Q)w k ; ii m ;k I P 8

9 q p k H T 1 (Q) p k Hi T (Q) k w k w T k HT i (Q),1 ( i ) i ; i I ;k I P i ; i I M ;k I P z i,h i (Q)w k ; ii Y ;k I P The other problems can be generalized in a similar fashion. T o o e o H K o v u 0 u G T r e r K r r Figure 5: Model of the closed loop control sstem for the heating example. Heating of a Room Consider the model describing heating of a room depicted in Figure 5, which can be described with the equations: T r = Gu HT ok v r = T r e r o = T o e o 9

10 where T r R is the room temperature, T ok R; k I P denes a set of outdoor temperatures ft ok : k I P g, v R is a random variable with zero mean and variance v, r R is the measurement of the room temperature, e r R is a random variable with zero mean and variance e r, o R is the measurement of the out-door temperature, and where e o R is a random variable with zero mean and variance e o. It is assumed that the random variables are independent of one another. The control signal u R is given b the feedback/feed-forward structure u = K r r K o o u 0 The objective istochoose K r, K o and u 0 as to minimize the maximal control 40 Power in W To in deg C P{19 < Tr < } To in deg C E{Tr} To in deg C Figure 6: Objectives and expected value of in-door temperature as functions of the out-door temperature. energ with respect to the set of out-door temperatures. The control energ is proportional to m 1 (k) =Efu(k)gas long as heating and not cooling is being performed. It would be desirable to perform the optimization subject to the constraint Y (k) = Prob T r T r (k) Tr > 0. This is however not possible, but a sucient condition is that Y (k) = Prob ft r T r (k)g 1 (1) 10

11 and Y 3 (k) = Prob T r (k) Tr 1 ( 1), which are the constraints that will be used. This problem can be cast in the framework of the previous sections b dening P zu = P u = 4 1 3,G5 ; Pzw = ,H,1 0 0 G H G H ; P 0 w = K = [ K r K o ]; w T k =[T ok v e r e o ] 3 5 and z = [,T r Tr ], and where the mean and covariance of w k are given b w k T =[T ok ] and k = diag(0;v ; e r ;e o ), respectivel. The solution for G =1 o C=W,H =1,T r =19 o C, Tr = o C, =0:80, v =3 o C, er =1 o C, eo = o C, and the set of out-door temperatures f,10; 0; 10; 0g o C is given b u =,1:1 r, 0:76 o 66 The closed loop behavior as a function of the out-door temperature is shown in Figure 6. 7 Conclusions In this paper is at been described how linear static control problems involving rst and second moments and ield objectives can be solved with ecient solvers for SOCP problems. Both feed-forward, feedback and feedback/feed-forward control has been addressed. It has been seen that about 00 variables are possible to optimize with respect to about 0 constraints. The computational time is a few minutes on a standard computer. Also it has been demonstrated how the optimization can be made robust with respect to dierent distributions of the exogenous inputs. Acknowledgements This work has been partl sponsored b the Swedish Research Council for Engineering Sciences under contract No. 95{838. References [BB91] S. Bod and C. H. Barratt. Linear Controller Design Limits of Performance. Prentice Hall, Englewood Clis, [KKB94] M. G. Kabuli, R. L. Kosut, and S. Bod. Improving static performance robustness of thermal processes. In Proceedings of the 33rd IEEE 11

12 Conference on Decision and Control, pages 6{66, Orlando, Florida, [LVB97] [SP96] M. S. Lobo, L. Vandenberghe, and S. Bod. Second-order cone programming. draft, Information Sstems Laborator, Stanford Universit, Stanford, California, R. Smith and A. Packard. Optimal control of perturbed linear static sstems. IEEE Transactions on Automatic Control, 41(4):579{584, Appendix A The control variable u can be computed in two dierent was. In case (I QP u )K = Q can be solved for K it is given b u = K. Notice that for an > 0 there is a Q such that jjh( Q), H(Q)jj and such that, I QPu K = Q has a solution. This means that it is possible to get arbitraril close to an H(Q) using the feedback structure u = K. However, the matrix K ma be large in a certain subspace. There is a wa toavoid this. Compute ~ as ~, P w w = 0, P u u 0 for some arbitrar u 0, where 0 is the corresponding measured output. Then compute u as u = Q~ = Q( 0, P u u 0 ) How this relates to digital control is now going to be discussed. Assume that the optimal control signal designed for the static sstem is used in a digital control application using zero order hold, and that the sample interval is much larger than the computational time so that the control signal can be written to the D/A converters almost immediatel after the measurement signal has been read from the A/D converters. Then the following set of equations are appropriate for describing the closed loop behavior: z(k) = P zu u(k)p zw w(k) (k) = P u u(k, 1) P w w(k) u(k) = Q((k), P u u(k, 1)) where k is time index, not to be confused with the k used to index the dierent probabilit measures in the previous section. Notice that the control signal can be written as u(k)qp u u(k, 1) = Q(k) 1

13 It is seen that the case when I QP u is not invertible corresponds to integral control, and hence it is not surprising that it cannot be implemented using just a static feedback controller u = K. B writing ^(k) = P zu u(k, 1) it holds that u(k) = Q~(k), where ~(k) = (k),^(k) can be interpreted as the estimation error when estimating (k) with ^(k). Notice that the observer is of dead-beat tpe, i.e. the estimation error is minimized as fast as possible. This follows from the fact that ~(k) =P w w(k). It is now clear that this problem setup is a special case of the general dnamic case described in [BB91]. Also notice that all the results in the main bod of the paper can be extended to the dnamic case including plants and controllers with poles on the stabilit boundar. Finall it is concluded that the closed loop sstem is stable. This follows from the fact that z(k) =H(Q)w(k), u(k) =QP w w(k) and that (k) = P u QP zw w(k, 1) P zw w(k). Appendix B In this appendix the SOCP:s will be recast in their standard form subject to To this end write min f T x x jja i x b i jj c T i x d i ; Q = where Q i R m1 and dene Notice that Q T P T zu e i = 8 >: i I L 8 : Q1 Q Q p 9 ; q = Q T 1 P T zue i Q T P T zu e i. Q T p Pzue T i 8 >: 9 Q 1 Q. Q p 9 >; 8 >; = >: e T i P zuq 1 e T i P zuq 9. >; = S iq e T i P zuq p where S i = diag(e T i P zu) with p blocks. So with Ai (D) =DP T ws j and b j (D) = DP T zwe i it holds that Now write DH T i (Q) =DP T zw e i DP T w QT P T zu e i = Ai (D)q b i (D) P w w = 13 8 >: t 1 t. t p 9 >;

14 Then it holds that where T = QP w w = Tq 8 : t1 I m t I m t p I m 9 ; So with c T i = e T i P zut and d i = e T i P zw w it holds that H i (Q)w=e T i P zw w e T i P zuqp w w =c T i q d i For Problem 4 dene x = q and for the other problems dene x T = 8 9 : t q T ;. The rest of the boring book-keeping is left as an exercise to the reader. Notice that the extension to linear combinations discussed at the end of Section 3 is obtained b stacking more rows to A i and b i. Furthermore the extension to worst case distributions discussed in Section 6 is obtained b having one constraint for each distribution. 14

Chapter 7 Interconnected Systems and Feedback: Well-Posedness, Stability, and Performance 7. Introduction Feedback control is a powerful approach to o

Chapter 7 Interconnected Systems and Feedback: Well-Posedness, Stability, and Performance 7. Introduction Feedback control is a powerful approach to o Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology c Chapter 7 Interconnected