A Singular Value Decomposition Based Closed Loop Stability Preserving Controller Reduction Method

Transcription

1 A Singular Value Decomposition Based Closed Loop Stability Preserving Controller Reduction Method Kin Cheong Sou and Anders Rantzer Abstract In this paper a controller reduction method which preserves closed loop stability is described. A Lyapunov inequality based sufficient condition is proposed in the search of the reduced controller. The reduced controller leads to a stable closed loop system with guaranteed approximation quality. Furthermore, the proposed problem can be formulated as a matrix approximation problem which can be solved efficiently using singular value decomposition. Numerical application examples are shown in the end to evaluate the generality of the proposed reduction method. I. INTRODUCTION In this paper the controller reduction problem of a plant/controller feedback setup in Fig. 1 is considered. A reduced controller is sought to balance the tradeoff between reducing the complexity of the controller and preserving closed loop performances e.g. closed loop stability and closed loop transfer function proximity. On one hand, the controller reduction problem has been an important research topic in its own right. On the other, controller reduction can be treated as a special case for structured model reduction in which individual subsystems are simplified in a concerted manner so that the interconnected reduced subsystems is similar to the original interconnected system e.g. 1,. Most of the available methods for solving the controller reduction problem are based on frequency weighted balanced truncation/hankel reduction 3, 4, 5, 6, 7. On the other hand, the controller reduction problem can also be solved via the reduced controller design procedure in 8, 9, 1 in which a reduced controller is designed so that some norm of the closed loop transfer function for some derived plant is minimized. Similar to 8, 9, 1, the proposed controller reduction method also certifies the stability and the approximation quality of the closed loop system with the reduced controller using a Lyapunov function argument. The usual difficulty with this approach is that the problem becomes nonconvex. To overcome the above intractability the proposed method chooses to utilize the state covariance matrix of the closed loop system with the original full order controller to construct the Lyapunov function. In particular, in Sec. III an important inequality i.e. 6b will be proposed specifying a sufficient condition guaranteeing both closed loop stability and approximation quality. However, the proposed inequality requires that the closed loop state covariance matrix with the reduced controller have the same dimension as the full order one. This restriction contradicts the standard treatment of order reduction by using system KC. Sou and A. Rantzer are with the Department of Automatic Control, Lund University, Lund, Sweden. {kcsou, rantzer}@control.lth.se matrices of smaller sizes. As a result, a new way of reduced controller complexity measure in terms of matrix rank will be defined in Sec. III, and accordingly order reduction will take the form of matrix rank minimization. Then in Sec. IV the issues with solving the rank minimization problem proposed in Sec. III will be investigated. In particular, it will be shown that the rank minimization problem can be reduced to a minimum rank generalized matrix approximation problem, to which an efficient solution procedure based on singular value decomposition SVD will be proposed. Finally in Sec. V application examples will be demonstrated to provide the numerical substantiation of the proposed method. II. PROBLEM STATEMENT A. Controller reduction problem In this paper, all the systems involved are in discretetime. Consider a standard feedback setting of a plant and a controller in Fig. 1. The plant is denoted as P, and w z u P K Fig. 1. Feedback setup of a plant P and a controller K. The problem of controller reduction is to find a lower order controller ˆK to replace K, so that closed loop transfer function is stable and well approximated. the dimensions of the signals w,u,z,y are n w,n u,n z,n y respectively. Following the traditional technical assumption in the H control community, it is assumed that the plant feed-through from u to y is zero, otherwise it should be incorporated as an additional positive feedback to the controller. The controller is denoted as Ak B K = k 1 C k where A k R n k n k and D k R nu ny. The closed loop plant from w to z is denoted as P K, and it is assumed to be stable. The task of the controller reduction problem is to find a reduced order controller Âk ˆBk ˆK = Ĉ k ˆDk y D k of order ˆn k, so that the following four criteria are satisfied. 1he order of the reduced controller ˆn k should be small to achieve low model complexity. he closed loop system P ˆK should be stable. 3he difference between P K and P ˆK should be small to achieve good accuracy. 4 The algorithm to obtain ˆK should be efficient.

2 B. Controller reduction as reduced controller design Instead of trying to find a reduced controller ˆK approximating K in Fig. 1, the setup in Fig. will be investigated. In w P u P K P ˆK Fig.. Feedback setup in difference form. Everything other than the reduced controller ˆK is collected as the plant P, which is the plant that the proposed method uses to design ˆK. A reduced controller design problem is to find a low order ˆK so that the closed loop system is stable and the closed loop transfer function from w to e is small. Fig., the plant i.e. everything except the reduced controller ˆK is denoted as P as follows. A B 1 B P = C 1 D 11 D 1 3 C D 1 where A R n n, D 1 R nz nu, D 1 R ny nw. By definition, P ˆK = P K P ˆK. The problems in Fig. 1 and Fig. are related in the following ways. Firstly, the D term in P in 3 is zero because of the zero feed-through assumption in Fig. 1. Secondly, to use the reduced controller ˆK in Fig. as a solution to Fig. 1 the closed loop system P ˆK should be stable and its transfer function should be small. For convenience, the following notations will be used. A B1 B Ā, B1, B, ˆnk C C Iˆnk, D1 D1 y e Iˆnk 4 Also, the controllers can be characterized by the matrices Dk C L k R n k+n u n k +n y 5a B k A k ˆDk Ĉ ˆL k R ˆB ˆn k+n u ˆn k +n y 5b k  k From now on, the K and L will be used interchangeably to denote the full controller, and the same principle applies to ˆK and ˆL. Certain expressions can be simplified with 4, 5a, 5b. For example, the A and B matrices of P K can be expressed as Ā+ B L C and B 1 + B L D 1 respectively, and a similar expression can be derived for ˆL. III. PROPOSED PROBLEM FORMULATION In this paper the proposed problem to solve is different, but related to the reduced controller design problem in Fig.. In Subsec. III-A the proposed problem will be described, and then in Subsec. III-B the connection to the problem in Fig. will be made clear when the properties of the proposed problem are explained in detail. A. Description of the proposed problem The data of the proposed problem are the bar quantities of the plant in 4 and L in 5a specifying the full order controller. In addition, there are two algorithm parameters: a α,1, and b V S n+n k ++ i.e. V is symmetric positive definite, n + n k by n + n k. The effect of these two parameters will be discussed in the Subsec. III-C. The decision variable is a matrix ˆL with the same dimension as L i.e. ˆn k = n k. The proposed problem is defined as minimize ˆL X rankˆl subject to 6a X Ā + B ˆL C B1 + B ˆL D1 B1 + B 6b ˆL D1 Ā + B ˆL C + α Inu ˆL = Dk C k where X can be computed from the data by solving X = Ā + B L C X Ā + B L C + B1 + B L D 1 B1 + B L D 1 + V 6c 6d Once an optimal solution ˆL is found solving 6a, 6b, 6c, controller ˆK in can be defined, and a minimal realization can be obtained as the proposed reduced controller of the original controller reduction problem in Fig. 1. Finally, it is emphasized that while the rank minimization in 6a might seem to be intractable, the problem in 6a, 6b, 6c can in fact be solved efficiently using SVD. The details will be discussed in Sec. IV. B. Properties of the proposed problem First of all, the feasibility of 6a, 6b, 6c is guaranteed since L is always a feasible solution. L satisfies 6c. In addition, replace ˆL in 6b with L and then apply 6d, then 6b becomes 1 α B1 + B L D 1 B1 + B L D 1 + V which is always true because α,1 and V. Hence, the proposed problem is always feasible. In addition, with an appropriate choice of α and V, the solution ˆL satisfying 6a, 6b, 6c should lead to a controller ˆK which satisfies the four requirements in Subsec. II-A and the equivalence in Subsec. II-B. Model complexity is addressed by the rank minimization in 6a. Even though ˆL has the same dimension as L as ˆn k = n k, the minimal realization resulted from ˆL can have an order much smaller than n k because of the rank minimization in 6a. The following statement provides some justification. Theorem 1: Let ˆK be the reduced controller defined in, and let ˆL be the associated matrix as defined in 5b. Then it holds that order ˆK rankˆl. Proof: See Appendix.

3 Theorem 1 states that if the minimum rank in 6a is small, then the order of the corresponding reduced model ˆK should be small as well. On the other hand, it is also possible that rankˆl order ˆK. In this case, the rank based model complexity measure can lead to restriction. Stability is addressed by 6b and 6d. In essence, 6b enables the use of the state covariance matrix X defined in 6d of the assumed stable closed loop system P K to construct a Lyapunov function to certify the stability of the reduced closed loop system P ˆK. The stability result is organized in the following statement. Theorem : Assume in Fig. that the full order closed loop system P K is stable. Assume also that α > and V in 6b and 6d. If ˆL satisfies 6b but not necessarily 6a and ˆK is the associated reduced controller, then the closed loop system P ˆK is stable. Proof: See Appendix. Note that Theorem implies that the closed loop system is stable with any minimal realization of ˆK because stability is independent of state space realization. The accuracy of the proposed reduced controller is affected by the parameters α and V as follows. Theorem 3: Let X v be defined as the solution of X v = Ā + B L C Xv Ā + B L C + V 7 In addition, denote the full order closed loop C matrix as C cl = C 1 + D 1 D k C D 1 C k. If ˆL satisfies 6b and 6c and ˆK is the corresponding reduced controller as related by 5b, then the H norm of the closed loop system P ˆK in Fig. satisfies P ˆK < α 1 Tr C cl X v Ccl T 8 Proof: See Appendix. C. Discussions on the parameters α and V In Theorem 3 8 states that increasing α should result in better approximation quality. On the other hand, 6b suggests that it would be more difficult to find a solution ˆL with low rank if α is increased, thus creating a tradeoff for α. Regarding V, 8 implies that V should be chosen so that Tr C cl X v Ccl T is small relative to, for instance, P K. However, V cannot be made arbitrarily small e.g. V = because L will no longer be a feasible solution to 6b if V =. Moreover, if V is increased, then X in 6d will also be increased with the same proportion. The effect is an indirect decrease of α in 6d in comparison. Another effect of increasing V can be understood as follows. From 6d a V with a larger smallest eigenvalue leads to an X with a larger smallest eigenvalue. Then 6b would allow more options for ˆL including the ones with lower ranks. This leads to the tradeoff for the choice of V. While in principle V can be chosen to optimize the tradeoff, in practice V is chosen as V = βi where β > can be treated as another tuning parameter. Finally, note that even if α i.e. closed loop H norm performance is ignored, only those stabilizing controllers sharing the same closed loop Lyapunov function defined by X in 6d with the full order controller L are feasible. This manifests the proposed reduction method s fundamental limit the tractability to be explained in the next section comes at a price of a restricted feasible set. IV. SOLVING THE PROPOSED REDUCTION PROBLEM The result of this section originates from 14, and it states that the proposed problem can be solved efficiently via SVD. First, constraints 6b and 6c, if feasible, can be shown to be equivalent to a simplified form to be defined in 16. Then proposed problem can be shown to be equivalent to a minimum rank generalized matrix approximation problem in 19, solvable by an extension of the theorem by Eckart- Young-Mirsky using SVD 15. A. Reformulation of 6b and 6c to a simplified matrix approximation constraint in 16 First note that 6b can be reorganized into Ā X B1 + B ˆL C D1 Ā X B1 + B ˆL 9 C D1 αi nw Then enforce constraint 6c such that Dk C k ˆL = ˆL with ˆL R n k n k +n y being the only independent decision variable. In addition, from 6d it can be seen that X because V and L is assumed to be a stabilizing controller. Therefore, multiplying both sides by X 1, 9 can be simplified to T F + GˆL H F + GˆL H I 1 σf + GˆL H < 1 where σ denotes the maximum singular value of a matrix and F, G and H can be defined from the problem data as Ā F X 1 B Dk B1 + C k nk n u C X 1 D1 αinw 11 G X 1 n nk I nk H X 1 C D1 αinw mf nf and it is denoted that F R and ˆL R ml nl. It is worthwhile to understand why 1 is not straightforward to handle. From 11 it can be seen that G is a full rank thin matrix with more rows than columns. In addition, in this paper an assumption is made on the dimensions of C and D 1 in 3 so that H in 11 is a full rank fat matrix. The assumed dimensions of G and H lead to the most difficult and typical situation. In this case, it would be inappropriate to simplify 1 with a substitution such as L = GˆLH because ˆL might not be found even if L is known

4 but the substitution would work if G and H were square and invertible. In fact, 1, when F, G and H were not defined by 11 could be infeasible. Nevertheless, the feasibility of 1 is guaranteed in here, as ˆL = B k A k is always a feasible solution. Before the main result can be presented, the following notations need to be defined first. Denote the rectangular, or economy size SVD of G and H as G = U G S G VG T s.t. U G R mf ml, U T G U G = I ml S G R ml ml, diagonal V G R ml ml, V T G V G = I ml1 T H = U H S H V H s.t. U H R nl nl, U T H U H = I nl S H R nl nl, diagonal V H R nf nl, V T H V H = I nl 13 In addition, define the basis matrices N G and N H for the kernels of U G in 1 and V H in 13 as, respectively N G R mf mf ml, N T G N G = I, U T G N G = N H R nf nf nl, N T H N H = I, V T H N H = 14 Then 14 states that if 1 is feasible, which is the case in this paper, then the following two matrices can be defined. 1 G I nf F T N G N T G F 1 15 I mf FN H N T H F T H In addition, 1 is equivalent to the following inequality. σ ˇF + Ľ < 1 16 where ˇF U G T H U G 1 U G T H FV H VH T G V H 1 Ľ P 1 1 ˆL P 1 P 1 V G S G 1 U G T H U G 1 P V T 1 H G V H S H 1 T U H 17 with data matrices computed in 1, 13, 14 and 15. To summarize, constraints 6b and 6c combined is equivalent to a single constraint in 1, which in turn is equivalent to 16 with a new decision variable Ľ defined in 17. Additionally, compared with 1 the Ǧ and Ȟ matrices in 16 are now identities. B. Reformulation of 6a and 16 to a classical minimum rank matrix approximation problem in 19 With the equivalence between 6b, 6c and 16 described in the previous subsection, the proposed problem in 6a, 6b, 6c is equivalent to the following optimization problem in terms of a new decision variable Ľ. Dk C minimize rank k Ľ P 1 Ľ P 18 subject to σ ˇF + Ľ < 1 with ˇF defined in 17 and the original decision variable ˆL being parameterized by Ľ as Dk C ˆL = k P 1 Ľ P where P 1 and P are defined in 17. Then, left and right multiplying the objective function in 18 with appropriate invertible matrices and introducing an extra dummy decision variable Ľ1, the problem in 18 becomes Ľ1 minimize rank Ľ Ľ 1,Ľ Dk C k P 1 Ľ1 subject to σ ˇF < 1 Ľ 1 = D k C k P 1 19 This minimum rank matrix approximation problem can be solved via SVD by an extension of the theorem of Eckart- Young-Mirsky. See reference 15 by Golub et al. for details. Finally, from the above descriptions it can be verified that proposed problem can be solved by an On+n k 3 algorithm. V. NUMERICAL EXAMPLES A. Reduction of a Youla optimized controller In this example the feedback setup in Fig. 1 is considered. A stable third order open loop plant P is taken from 11 p.1 and p.77. In 11, P is converted into discrete-time and then a stable controller is designed using the Youla optimized method in 1, 13. The controller has 15 states and it includes an integrator. To ensure that the reduced controller also has an integrator, the plant P in Fig. is modified by adding the controller integrator as a feedback. Then a reduced controller is designed with the modified plant, and after that the controller integrator is added back to form the final reduced controller. In applying the proposed procedure, the parameters in 6b and 6d are α =.99 and V =.5I. In the end, a 5th order minimal reduced controller is found. As a comparison, two other reduced controllers of order 5 are obtained. One is by directly applying balanced truncation BT see, e.g. 7 pp.7-8 to the full controller K, and the other is by frequency weighted balanced truncation FWBT 7 pp with the input and output weights being I P,K 1 P,1 and I P,K 1 P 1, respectively to minimize first order closed loop difference. It happens that the closed loop system with the BT controller is stable. Fig. 3 shows the magnitude Bode diagram of the error of closed loop systems due to different reduced controllers. In Fig. 3, the magnitude Bode diagram of the full order closed loop system is also given as a reference. In the frequency range of importance i.e. ω.1, π the approximation quality of all reduced controllers are comparable, while the quality of balanced truncated reduced controllers are better in lower frequencies. B. Reduction of an almost closed loop unstable controller In this example, the setup in Fig. 1 is again considered. This example is constructed with the intention that the closed Ľ

5 5 closed loop error + full order closed loop, input 1 to output controller error Magnitude db SVD error FWBT error BT error full Frequency rad/sec Fig. 3. Magnitude Bode diagrams of the closed loop system error. Solid: error with SVD reduced controller. Dash: error with FWBT reduced controller. Dot: error with BT reduced controller. Dash-dot: full order closed loop for reference. Errors in the frequency range of importance ω.1, π of all reduction methods are comparable. loop system is almost unstable. First an open loop plant P is constructed as the feedback interconnection of a low order but large gain system and a high order but small gain system. Then a H optimal controller is designed for the plant. The H controller is reducible because of the structure of P. Finally, the H controller is multiplied with a scalar gain to form the controller K so that the closed loop system is almost unstable. The MATLAB code to generate the setup in this example can be obtained from the authors upon request. Applying the proposed reduction procedure with parameters α =.99 and V = I results in another 5th order reduced controller. Again, two other 5th order reduced controllers are obtained using BT and FWBT. However, this time the closed loop system with the BT with MATLAB default MatchDC option reduced controller has a unstable pole with magnitude 1.1. On the other hand, the maximum magnitudes of the closed loop poles due to the proposed and FWBT reduced controllers are both Fig. 4 shows the magnitude Bode diagrams of the controller errors. The proposed method has a better controller matching in general. Also, the controller errors for the proposed method and FWBT both have dips around the closed loop resonance peak ω π. In contrast, without any knowledge of the plant BT would not do anything special for any particular frequency. Finally, Fig. 5 shows the magnitude Bode diagrams of the closed loop system errors due to different reduced controllers. The proposed reduced controller fares better than the other controllers in most of the frequency range shown except for the resonance peak. VI. CONCLUSION In this paper a new way to obtain reduced controllers preserving closed loop stability was described. The proposed method uses the closed loop state covariance matrix due to the full controller to formulate sufficient conditions to solve for a good but not necessarily optimal reduced controller. It was shown that the proposed problem could be formulated as a minimum rank generalized matrix approximation Magnitude db SVD FWBT BT Frequency rad/sec Fig. 4. Magnitude Bode diagrams of the errors of the reduced controllers. Solid: SVD reduced controller. Dash: FWBT reduced controller. Dot: BT reduced controller. The proposed SVD controller is closed loop stable, and it has better approximation quality than other reduced controllers except at the dip around ω = π. Magnitude db closed loop error + full order closed loop, input 1 to output 1 SVD error FWBT error BT error full Frequency rad/sec Fig. 5. Magnitude Bode diagrams of the closed loop system error. Solid: error with SVD reduced controller. Dash: error with FWBT reduced controller. Dot: error with BT reduced controller. Dash-dot: full order closed loop for reference. Error due to the proposed controller is better than the error due to both BT and FWBT in most cases. problem, to which an efficient SVD based solution method was proposed. Finally, the generality of the proposed method was demonstrated by some numerical application examples. However, there remains an open question. A theoretical justification of how restrictive the proposed sufficient conditions are has not been fully answered in this paper. ACKNOWLEDGEMENT The authors would like to thank Olof Garpinger for providing an example and Aivar Sootla for helpful discussions. A. Proof of Theorem 1 VII. APPENDIX For convenience, denote r rankˆl. The inequality order ˆK r holds trivially when r ˆn k. Therefore, for the rest of the proof it is assumed that r < ˆn k. Since ˆB Â is a sub-matrix of ˆL, rank ˆB Â r. Therefore, there exists e.g. by SVD an orthonormal matrix S Rˆn k ˆn k

6 such that S T S = Iˆnk and S T ˆBk  k = S T ˆBk S T  k = Bk1 à k1 B A where the zero matrices B and A have at least ˆn k r rows. Thus, ˆK defined by ˆL is a similarity transform, with transformation matrix S, of à k1 S Bk1 A B Ĉ k S ˆDk It can be seen from that K has at least ˆnk r uncontrollable states, and hence order K r. Finally, since ˆK and K have the same order as they are similarity transforms, it is concluded that order ˆK r = rankˆl. B. Proof of Theorem Note that in 6d the matrix Ā + B L C is the A matrix of P K which is assumed to be stable. Therefore, the matrix X in 6d is positive definite since V in 6d is positive definite. Together with the assumption that α >, 6b can be summarized as X Ā + B ˆL C X Ā + B ˆL C, X. A quadratic Lyapunov function argument shows that the matrix Ā + B ˆL C, which is the A matrix of P ˆK, is stable. Therefore, P ˆK is stable. C. Proof of Theorem 3 First define the controllability Grammian ˆX of the closed loop system P ˆK by the equation ˆX = Ā + B ˆL C ˆX Ā + B ˆL C + B1 + B ˆL D1 B1 + B 1 ˆL D1 1 can be rewritten as α B1 + B ˆL D1 B1 + B ˆL D1 = α ˆX α Ā + B ˆL C ˆX Ā + B ˆL C Substituting in the right-hand-side of 6b yields X α ˆX Ā + B ˆL C X α ˆXĀ + B ˆL C 3 Since ˆL satisfies 6b, Theorem implies that the matrix Ā + B ˆL C is stable. Hence, 3 concludes that X α ˆX. Note that X in 6d can be decomposed into X = X v + X w in which X v was defined in 7 and X w = Ā + B L C Xw Ā + B L C + B1 + B L D 1 B1 + B L D 1 4 Then the inequality X α ˆX yields ˆX 1 α X w + 1 α X v 5 On the other hand, it can be verified that the closed loop matrices C cl and D cl of P K are, respectively C cl = C 1 + D 1 D k C D 1 C k D cl = D 11 + D 1 D k D 1 and an analogous expression holds for Ĉcl and ˆD cl in the case of P ˆK. Therefore, 6c guarantees that C cl = Ĉcl and D cl = ˆD cl. Consequently, by noting that X w in 4 and ˆX in 1 are the controllability Grammians of P K and P ˆK respectively, 5 implies P ˆK = Tr Ĉcl ˆXĈ cl T + ˆD cl ˆDT cl < 1 α Ĉcl Tr X w Ĉcl T + ˆD cl ˆDT cl + 1 α Ĉcl Tr X v Ĉcl T = 1 α C Tr cl X w Ccl T + D cldcl T + 1α Tr C cl X v Ccl T = 1 α P K + 1 α C Tr cl X v Ccl T = + 1 α C Tr cl X v Ccl T In the inequality above, 5 and the fact that α,1 are used. The last equality is due to P K =. REFERENCES 1 A. Vandendorpe and P. Van Dooren, On Model Reduction of Interconnected Systems, Proccedings International Symposium Mathematical Theory of Networks and Systems, 4. H. Sandberg and R.M. Murray, Model Reduction of Interconnected Linear Systems, Optimal Control Applications and Methods, vol. 3, no. 3, 8, pp D. Enns, Model Reduction with Balanced Realizations: An Error Bound and a Frequency Weighted Generalization, Proceedings of 3rd Conference on Decision and Control, December B.D.O. Anderson and Y. Liu, Controller Reduction: Concepts and Approaches, IEEE Transactions on Automatic Control, vol. 34, no. 8, August, P.J. Goddard and K. Glover, Controller Reduction: Weights for Stability and Performance Preservation, Proccedings of 3nd IEEE Conference on Decision and Control, pp K. Zhou, Frequency Weighted Model Reduction with L Error Bounds, Systems and Control Letters, vol. 1, pp G. Obinata and B.D.O. Anderson, Model Reduction for Control System Design, Springer, 1. 8 P. Gahinet and P. Apkarian, A Linear Matrix Inequality Approach to H Control, International Journal of Robust and Nonlinear Control, vol. 4, 1994, pp T. Iwasaki and R.E. Skelton, All Controllers for the General H Control Problem: LMI Existence Conditions and State Space Formulas, Automatica, vol. 3, no. 8, 1994, pp D. Ankelhed, A. Helmersson and A. Hansson, Suboptimal model reduction using LMIs with convex constraints, Technical report, Department of Electrical Engineering, Linköping University, O. Garpinger, Design of Robust PID Controllers with Constrained Control Signal Activity, Licentiate Thesis, Department of Automatic Control, Lund University, Sweden, 9. 1 S. Boyd, C. Barratt and S. Norman, Linear Controller Design: Limits of Performance Via Convex Optimization, Proceedings of the IEEE, vol. 78, no. 3, 199, pp A. Wernrud, QTool.1 Reference Manual, Department of Automatic Control, Lund University, Sweden, K.C. Sou, On the Minimum Rank of the Generalized Matrix Approximation Problem in the Maximum Singular Value Norm, the 19th International Symposium on Mathematical Theory of Networks and Systems, June, G. Golub, A. Hoffman and G. Stewart, A Generalization of the Eckart-Young-Mirsky Matrix Approximation Theorem, Linear Algebra and its Applications, vol. 88/89, pp , 1987.