Parametrization of All Stabilizing Controllers Subject to Any Structural Constraint

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1 49th IEEE Conference on Decision and Control December 15-17, 2010 Hilton Atlanta Hotel, Atlanta, GA, USA Parametrization of All Stabilizing Controllers Subject to Any Structural Constraint Michael C Rotkowitz Abstract We consider the roblem of designing otimal stabilizing decentralized controllers subject to an arbitrary structural constraint, where each art of the controller has access to some measurements but not others We develo a arametrization of the stabilizing structured controllers such that the objective is a convex function of the arameter, with the arameter subject to a single quadratic equality constraint I INTRODUCTION This aer addresses the roblem of otimal structured control, where we have multile controllers, each of which may have access to some sensor measurements but not others Most conventional controls analysis breaks down when such decentralization is enforced Finding otimal controllers when different controllers can access different measurements is notoriously difficult even for the simlest such roblem [1], and there are results roving comutational intractability for the more general case [2], [3] When a conditional called quadratic invariance holds, which relates these information constraints on the controller to the system being controlled, then the otimal decentralized control roblem may be recast as a convex otimization roblem [4] For a articular Youla arametrization, which converts the closed-loo erformance objective into a convex function of the new arameter, the information constraint becomes an affine constraint on the arameter, and thus the resulting roblem is still convex The roblem of finding the best block diagonal controller however, which reresents the case where each subsystem controller may only access measurements from its own subsystem, is never quadratically invariant excet for the case where the lant is block diagonal as well; that is, when subsystems do not affect one another The arametrization and otimization of stabilizing block diagonal controllers was addressed in a similar fashion using Youla arametrization in [5], focusing on the 2-channel (or 2-block, 2-subsystem, etc) case The arametrization similarly converts the objective into a convex function, but the block diagonal constraint on the controller then becomes a quadratic equality constraint on the otherwise free arameter It is further suggested that the trick for achieving this can be imlemented n 1 times for n-channel control, resulting in n 1 quadratic equality constraints While this constraint causes the resulting roblem to be nonconvex, it still converts a generally intractable roblem MC Rotkowitz is with the Deartment of Electrical and Electronic Engineering, The University of Melbourne, Parkville VIC 3010 Australia, mcrotk@unimelbeduau into one where the only difficulty is a well-understood tye of constraint Solving this resulting constrained roblem is then further exlored in [6], and many other methods exist for addressing quadratic equality constraints In this aer, we address arbitrary structural constraints, which are not generally block diagonal nor quadratically invariant We first in Section IV show how to convert this general roblem to a block diagonal synthesis roblem In Section V, we then adat the main insight of [5] and show that this can then be similarly converted into a roblem on a stable Youla arameter with a convex objective, but subject to a single quadratic equality constraint, regardless of the number of blocks II PRELIMINARIES We suose that we have a generalized lant P R artitioned as [ ] P11 P P 12 G P 21 We define the closed-loo ma by f(p,k) P 11 +P 12 K(I GK) 1 P 21 The ma f(p,k) is also called the (lower) linear fractional transformation (LFT) of P and K Note that we abbreviate G P 22, since we will refer to that block frequently, and so that we may refer to its subdivisions without ambiguity This interconnection is shown in Figure 1 Fig 1 z y P 11 P 12 P 21 G K u w Linear fractional interconnection of P and K We suose that there are n y sensor measurements and n u control actions, and thus artition the sensor measurements and control actions as y [ y T 1 y T n y ] T u [ u T 1 u T n u ] T with artition sizes y i L i 2 i 1,,n y u j L mj 2 j 1,,n u /10/$ IEEE 108

2 and total sizes n y y L 2, i ; u L m 2, i1 and then further artition G and K as G 11 G 1nu G K G ny1 G nynu with block sizes and total sizes K 11 K nu1 G ij R i mj s, K ji R mj i G R m s K R m n u j1 m j m K 1ny K nuny i,j This will tyically reresent n subsystems, each with its own controller, in which case we will have n n y n u, but this does not have to be the case a) Transfer functions: We use the following standard notation Denote the imaginary axis by jr z C R(z) 0 and the closed right half of the comlex lane by C + z C R(z) 0 We define transfer functions for continuous-time systems therefore determined on jr, but we could also define transfer functions for discrete-time systems determined on the unit circle A rational function G : jr C is called realrational if the coefficients of its numerator and denominator olynomials are real Similarly, a matrix-valued function G : jr C m n is called real-rational if G ij is real-rational for all i,j It is called roer if lim G(jω) exists and is finite, ω and it is called strictly roer if Denote by R m n roer transfer matrices R m n and let R m n s lim G(jω) 0 ω the set of matrix-valued real-rational G : jr C m n G roer, real-rational be R m n s G R m n G strictly roer Also let RH be the set of real-rational roer stable transfer matrices RH m n G R m n G has no oles in C + It can be shown that functions in RH are determined by their values on jr, and thus we can regard RH as a subsace of R If A R n n we say A is invertible if lim ω A(jω) is an invertible matrix and A(jω) is invertible for almost all ω R Note that this is different from the definition of invertibility for the associated multilication oerator on L 2 If A is invertible we write B A 1 if B(jω) A(jω) 1 for almost all ω R When the dimensions are imlied by context, we omit the suerscrits of R m n,rs m n,rh m n Let I n reresent the n n identity A Stabilization Fig 2 z y P 11 P 12 P 21 G K u w Linear fractional interconnection of P and K We say that K stabilizes P if in Figure 1 the nine transfer matrices from w,v 1,v 2 to z,u,y belong to RH We say that K stabilizes G if in the figure the four transfer matrices from v 1,v 2 to u,y belong to RH P is called stabilizable if there exists K R m such that K stabilizes P The following standard result relates stabilization of P with stabilization of G Theorem 1: Suose G R m s and P R (nz+) (nw+m), and suose P is stabilizable Then K stabilizes P if and only if K stabilizes G Proof: See, for examle, Chater 4 of [7] For a given system P, all controllers that stabilize the system may be arameterized using the well-known Youla arametrization [8], stated below Theorem 2: Suose that we have a doubly corime factorization of G over RH, that is, M l,n l,x l,y l,m r,n r,x r,y r RH such that G N r Mr 1 M 1 l N l and [ ][ ] [ ] Xl Y l Mr Y r Im 0 (1) N l M l N r X r 0 I Then the set of all stabilizing controllers is given by K R K stabilizes G (Y r M r Q)(X r N r Q) 1 X r N r Q is invertible,q RH (X l QN l ) 1 (Y l QM l ) X l QN l is invertible,q RH Furthermore, the set of all closed-loo mas achievable with stabilizing controllers is f(p,k) K R, K stabilizes P T 1 T 2 QT 3 Xr N r Q is invertible, Q RH, (2) 109

3 where T 1,T 2,T 3 RH are given by T 1 P 11 +P 12 Y r M l P 21 T 2 P 12 M r (3) T 3 M l P 21 Proof: See, for examle, Chater 4 of [7] The arameter Q is usually referred to as the Youla arameter We now show that the two arametrizations above give the same change of variables Lemma 3: Suose G R s Then the set of all stabilizing controllers is given by K R K stabilizes P (Y r M r Q)(X r N r Q) 1 (X l QN l ) 1 (Y l QM l ) Q RH Proof: The equivalence of stabilizing G and the generalized lant P follows from Theorem 1 It follows from G strictly roer that N l,n r are strictly roer, and thus the invertibility conditions of Theorem 2 are always satisfied It just remains to show that the two arametrizations give the same maing from arameter to controller This follows very similarly from [5]: (Y r M r Q r )(X r N r Q r ) 1 (X l Q l N l )(Y r M r Q r ) (X l Q l N l ) 1 (Y l Q l M l ) (Y l Q l M l )(X r N r Q r ) r l N r l M r (X l Y r Y l X r ) +Q l (M l X r N l Y r ) 0 I (X l M r Y l N r ) Q +Q l (M N ) I m 0 Q l Q r Remark 4: Even ifgis not strictly roer, the invertibility conditions still hold for almost all arameters Q [9, 111] III PROBLEM FORMULATION We develo the otimization roblem that we need to solve for otimal synthesis subject to an arbitrary structural constraint A Structural Constraints Structural constraints, which secify that each controller may access certain sensor measurements but not others, manifest themselves as sarsity constraints on the controller to be designed We here introduce some notation for reresenting this tye of constraint Q r Let B 0,1 reresent the set of binary numbers Suose A bin B m n is a binary matrix We define the subsace Sarse(A bin ) B R B ij (jω) 0 for all i,j such that A bin ij 0 for almost all ω R giving all of the roer transfer function matrices which satisfy the given saristy constraint We then reresent the constraints on the overall controller with a binary matrix K bin B nu ny where 1, if control inut k Kkl bin may access sensor measurement l 0, if not The subsace of admissible controllers is then given as S Sarse(K bin ) We can now set u our main roblem of finding the best such controller B Problem Setu Given a generalized lant P and a subsace of admissible controllers S, we would then like to solve the following roblem: minimize f(p,k) subject to K stabilizes P (4) K S Here is any norm on the closed-loo ma chosen to encasulate the control erformance objectives The subsace of admissible controllers, S, has been defined above to encasulate the constraints on which controllers can access which sensor measurements We call the subsace S the information constraint Many decentralized control roblems may be exressed in the form of roblem (4) In this aer, we focus on the case where S is defined by structural constraints as discussed above This roblem is made substantially more difficult in general by the constraint that K lie in the subsace S Without this constraint, the roblem may be solved with many standard techniques Note that the cost function f(p,k) is in general a non-convex function of K If the information constraint is quadratically invariant [4] with resect to the lant, then roblem may be recast as a convex otimization roblem, but no comutationally tractable aroach is known for solving this roblem for arbitrary P and S IV DIAGONALIZATION In this section, we show how the roblem of finding the otimal structured controller K S can be converted to a roblem of finding an otimal block diagonal controller Similar equivalences of decentralization constraints were considered for a different urose in [10], and we use consistent notation where ossible 110

4 The concets in this section are fairly straightforward, but covering the general case rigorously is unfortunately of a tedious nature requiring multile subscrits The reader wishing to skim this section need only take away that the lant may be altered such that sensor measurements are reeated and then controller inuts reconstituted in such a way that the active arts of the controller are shuffled around to make a block diagonal controller, and such that the closed-loo ma remains the same Examle 5 illustrates the shuffling for a articular controller structure First, given structural constraints roscribed by a binary matrix K bin, define the total number of active blocks in an admissible controller as n y n u a Kkl bin k1 l1 For each active block, that is, for each air (k,l) such that Kkl bin 1, assign a unique α 1,,a, and let κ α k andλ α l To avoid the use of trile subscrits later, we also index the block sizes of our diagonal controller as b α m κα and c α λα If we construct a block diagonal controller with the active blocks along the diagonal, it will then have a total outut size and inut size, resectively, of a a a u b α, a y c α α1 α1 We then define the subsace of block diagonal controllers S R au ay as S diag( K 11,, K aa ) Kαα R b α c α α We now wish to show that we can construct a roblem with the objective of finding the otimal stabilizing block-diagonal K S which is equivalent to our original roblem of finding the otimal stabilizing structured K S Let s first consider the sensor measurements that such a controller would have access to and call this ỹ An active block K kl is able to see y l, and so the corresonding art of ỹ α has to be a coy of y l ỹ thus becomes an extended version of the original vector of sensor measurements y, with measurements reeated as necessary for each active block of the controller which has access to it We can then define ỹ L ay 2 as follows ỹ [ ] ỹ1 T ỹa T T where ỹ α y λα We can ma from y to ỹ by defining the matrix U R ay as I l, if λ α l U αl 0, otherwise Since each sensor measurement is accessed by at least one controller block (otherwise it wouldn t be a sensor measurement), U has full column rank, and we can define a left inverse U R ay as U (U T U) 1 U T We then have ỹ Uy and y U ỹ We next turn our attention to the outut of the block diagonal controller, ũ Kỹ This will comrise the outut of each active controller block, from which we will have to reconstitute the original controller inut u They are related as follows u k n y K kl y l l1 α:κ αk α:κ αk K α ỹ α and so we can ma from ũ to u by defining the matrix V R m au as I mk, if κ α k V kα 0, otherwise Since each controller inut can see at least one sensor measurement, V has full row rank, and we can define a right inverse V R au m as ũ α V V T (VV T ) 1 We then have u Vũ, but do not necessarily have ũ V u We can now define a new generalized lant P with the following comonents R (nz+ay) (nw+au) P 11 P 11 P12 P 12 V (5) P 21 UP 21 G UGV which mas (w,ũ) (z,ỹ) Examle 5: Suose we are trying to find the best controller K S where S Sarse(K bin ) and where the admissible controller structure is given by K bin that is, we need to find the best 3 4 controller where only 5 articular arts of the controller may be active We then have a 5, and assign (κ 1,λ 1 ) (1,1), (κ 2,λ 2 ) (3,1), (κ 3,λ 3 ) (2,2), (κ 4,λ 4 ) (1,3), and (κ 5,λ 5 ) (3,4) We then get U I I I I I V I 0 0 I I I 0 0 I such that ỹ Uy [ ] y1 T y1 T y2 T y3 T y4 T T and such that u Vũ [ (ũ 1 +ũ 4 ) T ũ T 3 (ũ 2 +ũ 5 ) T] T The matrix U reeats the first sensor measurement since the first two active arts of the controller, and thus the first two arts of the block diagonal controller, both need to access it, and then the matrix V takes the 5 signals from the block diagonal controller (ũ) and reconstitutes the 3 controller inuts (u) 111

5 We may then relace the given generalized lant P with P as in (5), and any admissible controller K S by the block diagonal controller K diag(kκ1λ1,,k κ5λ 5 ) diag(k 11,K 31,K 22,K 13,K 34 ) such that K V KU, and this maintains the same closed-loo ma Having motivated these transformations, we now need to show that they indeed yield a diagonal synthesis roblem which is equivalent to our original roblem We first give a lemma which verifies that we have roerly set u our bijection from admissible structured controllers K S to block diagonal controllers K S Lemma 6: Given K S, we can let K V KU, and then K S Given K S, we can let K kl, if α β,k κ α,l λ α K αβ (6) 0, otherwise, and then K S and K V KU Proof: First, given K S, let K V KU Then, a a K kl V kα Kαβ U βl α1β1 a V kα Kαα U αl α1 since K diagonal Kαα, if α such that (k,l) (κ α,λ α ) 0, otherwise Kαα, if Kkl bin 1, (k,l) (κ α,λ α ) 0, if Kkl bin 0 and so K S Now, given K S, we can define K as in (6) K S clearly as it is defined to be block diagonal, and then K V KU is verified by the above equalities Remark 7: Note that given K S, V KU is generally not in S If K V KU, then f(p,k) P 11 +P 12 (V KU)(I GV KU) 1 P 21 P 11 +(P 12 V) K(I UGV K) 1 UP 21 P 11 + P 12 K(I G K) 1 P21 f( P, K) where we used the ush-through identity in the second ste, and thus the closed-loo mas are identical With reeated use of the ush-through identity we also find [ ] 1 [ I K V 0 G I 0 U ][ I K G I ] 1 [ ] V 0 0 U Thus if K stabilizes G, then K stabilizes G The converse does not generally follow from this relation, but if an unstable mode is suressed by V,U,V, or U, we just need a stabilizing K which yields V K(I G K) 1 U K(I GK) 1 to achieve the same closed-loo ma Generalizing the technical conditions needed for this, or understanding in what manner enforcing internal stability on the diagonalized roblem reresents a stronger notion of stability enforced on the orignal roblem, is ongoing work V PARAMETRIZATION In this section, we assume that our roblem has been converted to one of finding an otimal block diagonal controller, and address the roblem of arametrizing all of the stabilizing controllers and all of the achievable closedloo mas, to further transform the otimization roblem to one over a stable arameter with a convex objective We show that this can be achieved, with the diagonalization manifesting itself as a quadratic equality constraint on the arameter The key insight in this section, that a block diagonal constraint can be exressed as in (7), which then becomes a quadratic constraint in the Youla arameter, is largely derived from Manousiouthakis [5] There this idea was introduced for 2-channel control (block diagonal with 2 blocks), and it was suggested that the same technique could be used n 1 times, resulting in n 1 quadratic equality constraints on the Youla arameter, to enforce a block diagonal constraint with n blocks Here, in addition to having first started with an arbitrary structural constraint, we now show how this arameterization can be achieved with just one quadratic equality constraint, regardless of the number of blocks Define L l R au au as and define L r R ay ay as L l diag(i b1,2i b2,,ai ba ) L r diag(i c1,2i c2,,ai ca ) and note that the two matrices are identical if all controller blocks are square or scalar We now show how these matrices can be used to enforce a block diagonal constraint Lemma 8: Given K R au ay, Proof: L l K KLr K S (7) L l K KLr a a (L l ) ik Kkj K ik (L r ) kj k1 k1 (L l ) ii Kij K ij (L r ) jj i K ij j K ij i,j K ij 0 i j K S i,j i,j since L l,l r diag The following theorem is the main result of this section It shows that all of the stabilizing block diagonal controllers can be arametrized by a stable Youla arameter, subject to a single quadratic equality constraint It further shows that the set of all achievable closed-loo mas may then be exressed 112

6 an an affine function of this Youla arameter, subject to the same quadratic equality constraint Theorem 9: Suose that P is stabilizable, and that we have a doubly corime factorization of G as in (1) Then K R K stabilizes P, K S where (Y r M r Q)(X r N r Q) 1 Q RH, q(q) 0 and where [ ] W1 W 2 W 3 W 4 Further, q(q) [ I au Q ][ ][ ] W 1 W 2 Iay W 3 W 4 Q (8) [ ][ ][ ] Xl Y l Ll 0 Yr M r N l M l 0 L r X r N r (9) f( P, K) K stabilizes P, K S T 1 T 2 QT 3 Q RH, q(q) 0 (10) where T i are given as in (3) Proof: With this change of variables, the equivalence of Q RH with K stabilizing P, as well as f( P, K) T 1 T 2 QT 3, follow from Theorem 2 It just remains to show that the diagonal constraint on the controller is equivalent to the quadratic constraint on the Youla arameter Note that we utilize Lemma 3 and the equivalence of the left and right arametrizations in the second ste: K S L l K KLr L l (Y r M r Q)(X r N r Q) 1 (X l QN l )L l (Y r M r Q) (X l QN l ) 1 (Y l QM l )L r (Y l QM l )L r (X r N r Q) (X l L l Y r Y l L r X r ) +(Y l L r N r X l L l M r ) Q W 1 W 2 +Q(M l L r X r N l L l Y r ) +Q(N l L l M r M l L r N r ) Q 0 W 3 W 4 q(q) 0 We may thus solve the following equivalent roblem minimize subject to T 1 T 2 QT 3 Q RH q(q) 0 (11) to find the otimal Q, recover the otimal diagonal controller as K (Y r M r Q )(X r N r Q ) 1, and then recover the otimal structured controller for our original roblem (4) as K V K U Remark 10: Note that the calculation of the Youla arameters (1) and the closed-loo arameters (3) in Theorem 9 must be based on G and P Remark 11: If the quadratic term, W 4, is 0, then the constraint if affine, and the resulting otimization roblem is convex One can show that this occurs if and only if G is block diagonal, which corresonds to a secial case of quadratic invariance ACKNOWLEDGMENTS The author would like to thank Vasilios Manousiouthakis, Michael Cantoni, and Laurent Lessard for helful discussions VI CONCLUSIONS We have considered the roblem of synthesizing otimal stabilizing controllers subject to an arbitrary structural constraint We first showed how to recast this as a block diagonal synthesis roblem, and then how to recast that as a roblem over a stable Youla arameter with a convex objective The general roblem we are addressing is known to be intractable, and so it is not surrising that the resulting otimization roblem is not convex in general However, we have taken a broad class of imortant intractable roblems, shown how to handle them in a unified manner, and shown how the inherent difficulty of the roblem can be concentrated into a single quadratic equality constraint The synthesis of otimal (decentralized) control via Youla arametrization can now be summarized as follows Without structural constraints, finding the otimal stabilizing controller can be cast as otimizing a convex function of the Youla arameter, where the arameter is free and stable If the controller is instead subject to a quadratically invariant structural constraint, the arameter is subject to an affine equality constraint If the controller is subject to a structural constraint which is not quadratically invariant, the arameter is subject to a quadratic equality constraint REFERENCES [1] H S Witsenhausen, A counterexamle in stochastic otimum control, SIAM Journal of Control, vol 6, no 1, , 1968 [2] C H Paadimitriou and J N Tsitsiklis, Intractable roblems in control theory, SIAM Journal on Control and Otimization, vol 24, , 1986 [3] V D Blondel and J N Tsitsiklis, A survey of comutational comlexity results in systems and control, Automatica, vol 36, no 9, , 2000 [4] M Rotkowitz and S Lall, A characterization of convex roblems in decentralized control, IEEE Transactions on Automatic Control, vol 51, no 2, , February 2006 [5] V Manousiouthakis, On the arametrization of all stabilizing decentralized controllers, Systems and Control Letters, vol 21, no 5, , 1993 [6] D S Manousiouthakis, Best achievable decentralized erformance, IEEE Transactions on Automatic Control, vol 40, no 11, , 1995 [7] B Francis, A course in H control theory, in Lecture notes in control and information sciences, vol 88 Sringer-Verlag, 1987 [8] D Youla, H Jabr, and J B Jr, Modern Wiener-Hof design of otimal controllers: art II, IEEE Transactions on Automatic Control, vol 21, no 3, , 1976 [9] M Vidyasagar, Control System Synthesis: A Factorization Aroach The MIT Press, 1985 [10] L Lessard and S Lall, Reduction of decentralized control roblems to tractable reresentations, in Proc IEEE Conference on Decision and Control, 2009,