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1 A further remark on the problem of reliable stabilization using rectangular dilated LMIs Journal: IMA Journal of Mathematical Control and Information Manuscript ID: IMAMCI-0-0.R Manuscript Type: Original Article Date Submitted by the Author: n/a Complete List of Authors: Befekadu, Getachew; University of Notre Dame, Department of Electrical Engineering; University of Notre Dame, Department of Electrical Engineering Gupta, Vijay; University of Notre Dame, Department of Electrical Engineering Antsaklis, Panos J.; University of Notre Dame, Department of Electrical Engineering Keywords: reliable control, decentralized control, stabilization, dilated LMIs

2 Page of 0 Manuscripts submitted to IMA Journal of Mathematical Control and Information IMA Journal of Mathematical Control and Information Page of doi:0.0/imamci/dnnxxx A further remark on the problem of reliable decentralized stabilization using rectangular dilated LMIs GETACHEW K. BEFEKADU, VIJAY GUPTA AND PANOS J. ANSTAKLIS Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN, USA. [Received on 0 June 0] In this brief paper, an extension of the result of Fujisaki & Befekadu (00) concerning the problem of reliable decentralized stabilization for generalized multi-channel systems is given. We specifically provide a relaxed sufficient condition for the stability of multi-channel system using a rectangular dilated LMIs when all of the controllers work together and when one of the controllers is extracted due to a failure. Keywords: reliable control; decentralized control; stabilization; dilated LMIs.. Introduction Recently, the problem of reliable decentralized stabilization for generalized multi-channel systems with a single failure in any of the control channels has been addressed by Fujisaki & Befekadu (00) via dilated LMIs and unknown disturbance observers. In this brief paper, we extend this result for a multichannel system using a rectangular dilated LMIs technique, where the new extension can be looked as a sufficiently decoupling framework (i.e., separating the design variables from the system data) that provides a tractable (and also less-conservative) design technique for reliable stabilization of multichannel system. This brief paper is organized as follows. In Section, we present the main result where the problem of reliable stabilization for a generalized multi-channel system is formally restated. Here a relaxed sufficient condition (which is a verifiable condition) is given in terms of a set of rectangular dilated LMIs for the reliable decentralized stabilization of multi-channel system. Notation. For a matrix A R n n, He(A) denotes a hermitian matrix ined by He(A) =(A+A T ), where A T is the transpose of A. For a matrix B R n p with r = rankb, B R (n r) n denotes an orthogonal complement of B, which is a matrix that satisfies B B=0 and B B T 0. S n + denotes the set of strictly positive inite real matrices and C denotes the set of complex numbers with negative real parts, that is C = {s C R{s} < 0}. σ(a) denotes the spectrum of a matrix A R n n, i.e., σ(a) ={λ C rank(a λi)<n} and GL n (R) denotes the general linear group consisting of all real nonsingular n n matrices. Note that the problem of reliable stabilization is essentially equivalent to a strong stabilization problem that involves an intractable problem (e.g., see references Vidyasagar & Viswanadham (), Nemirovsk () and Blondel & Tsitsiklis ()). The author 0. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

3 Page of of G. B. BEFEKADU, V. GUPTA AND P. J. PANOS. Main result Consider the following finite-dimensional generalized multi-channel system ẋ(t)=ax(t)+ B j u j (t), x(0)=x 0, (.) j N where x(t) X R n is the state of the system, u j (t) U j R r j is the control input to the jthchannel of the system and N denotes the number of input/output channels. A R n n, B j R n r j, and N ={,,...,N}. Let us also introduce some additional notation that will be useful in the sequel r=r 0 = r j, j N r j = r i, i N j K = K 0 =(K j ) j N K, K j =(K i ) i N j K j, K = K 0 = K j R r n, K j j N = K i R r j n, i N j where the sets N j are ined by N j = N \{ j} for j=,,...,n and N 0 = N with cardinality of N 0 =N and N j =N, respectively. For the generalized multi-channel system in (.), we restrict the set K to be the set of all linear, time-invariant (reliable) stabilizing state-feedback controllers that satisfies { } K (K,K,...,K N ) K j σ(a+b j K j ) C, j N {0}, (.) j N where B 0 = ( ) B i i N, B j = ( ) B i and B i N j K j = j i N j B i K i for j= 0,,...,N. REMARK. In this brief paper, we consider the stability of the closed-loop system ( A+B j K j ) under nominal operation condition (i.e., when j = 0) as well as under a possible single-channel controller/agent failure (i.e., when j N ). Let us ine the following matrices that will be later used in Theorem.. DEFINITION. For j= 0 E 0 = [ ] {}}{ I n n I n n I n n, X0,X 0 =blockdiag{x 0, X 0,X 0,...,X 0 }, }{{}}{{} ( N 0 +) times ( N 0 +) times {}}{ [A,B] U0,L 0 =[ AU 0 B L B L B N L N ], U 0,W 0 =blockdiag{u 0, W,W,...,W }{{ N }}, } ( N 0 +) times =X 0 =W 0

4 Page of 0 Manuscripts submitted to IMA Journal of Mathematical Control and Information and for j N A FURTHER REMARK ON THE PROBLEM OF RELIABLE DECENTRALIZED STABILIZATION of E j = [ ] {}}{ I n n I n n I n n I n n, Xj,X j =blockdiag{x j, X j,...,x j,x j,...,x j }, }{{}}{{} ( N j +) times ( N j +) times [A,B] Uj,L j =[ AU j B L B j L j B j+ L j+ B N L N ], =W j {}}{ U j,w j =blockdiag{u j, W,...,W j,w j+,...,w N }. }{{} ( N j +) times Next we can characterize the set K using a new class of dilated LMIs (i.e., rectangular dilated LMIs) as follow. THEOREM. Suppose the pairs(a,b j ) are stabilizable for all j N {0}. Then, there exist X j S n +, ε j > 0, U j GL n (R), j= 0,,...,N, W i GL n (R) and L i R r i n, i=,,...,n such that [ ] 0 n n E j X j,x j X j,x j E T j 0 ( N j +)n ( N j +)n ([ ] [A,B j ] + He Uj,L [ ] ) j E T j ε j I ( N U j,w j j +)n ( N j +)n 0, (.) Furthermore, for any family of N-tuples ( L, L,..., L N ) and ( W, W,..., W N ) as above, and setting K i = L i Wi for each i =,,...,N, then ( ) A+B j K j are Hurwitzian matrices for all j N {0}, i.e., σ(a+b j K j ) C, j N {0}. Proof of Sufficiency. Note that X j,x j E T j = ET j X j and [ [ [A,B] Uj,L j U j,w j E j ε j I ( N j +)n ( N j +)n ] =[ I n n [A,B] Uj,L j U j,w j ], =X j = [ I n n (A+B j K j ) ], (.) ] = [ ] ε j I ( N j +)n ( N j +)n E j, (.) for j= 0,,...,N. Then, eliminating U j,w j from (.) by using these matrices, we have the following matrix inequalities [ ][ ] I n n [A,B] Uj,L j U j,w j 0 n n E j X j,x j X j,x j E T j 0 ( N j +)n ( N j +)n [ ] I n n ( U j,w j ) T [A,B] T U j,l j = He((A+B j K j )X j ) 0, (.)

5 Page of of G. B. BEFEKADU, V. GUPTA AND P. J. PANOS [ ] [ ] 0 ε j I ( N j +)n (N+)n E n n E j X j,x j j X j,x j E T j 0 ( N j +)n ( N j +)n [ ] ε j I ( N j +)n ( N j +)n E T = ε j ( N j +)X j 0. (.) j Hence, we see that equations (.) and (.) exactly state the Lyapunov stability condition with X j S n + and state-feedback gains K i = L i Wi for i=,,...,n. Proof of Necessity. Suppose the system in (.) is stable with state-feedback gains K i = L i Wi for W i GL n (R), i=,,...,n. Then, there exist sufficiently small ε j > 0 for j= 0,,...N that satisfy He ( ) (A+B j K j )X j + ε [ j A,B X ]X j,l j,x j [ A,B ] T 0, (.) j X j,l j where [A,B] Xj,L j = [ AX j B L B j L j B j+ L j+ B N L N ] for j =,,...,N and [A,B] X0,L 0 =[ AX 0 B L B L B N L N ]. Note that X j,x j 0 and X j,x j E T j = ET j X j, employing the Schur complement for (.), then we have [ He ( (A+B j K j )X j ) ] [ ] ε j [A,B] Xj,L j X j,x j 0 n n E j X j,x j ε j X j,x j ([A,B] Xj,L j ) T = ε j X j,x j X j,x j E T j 0 ( N j +)n ( N j +)n ( [ [A,B]Xj,L + He U j j,w j ] X j,x j [ E T ] ) j ε j I ( N j +)n ( N j +)n 0. (.) I n n Thus, the above expression, i.e., equation (.), implies that (.) holds with U j,w j = X j,x j for U j GL n (R) and j N {0}. REMARK. We remark that the above dilated LMIs framework stated in Theorem. is useful in the context of reliable control for a system with generalized multi-channel configurations, since the framework effectively separates the design variables from the system data. Note that Theorem. is a generalization of the square dilated LMIs technique that has been considered by Fujisaki & Befekadu (00) in the context of reliable stabilization for multi-channel systems (e.g., see Geromel, De Oliveira & Hsu (), Fujisaki & Befekadu (00), Pipeleers et al. (00) and references therein for a review of square dilated LMIs technique). In fact, if we multiply equation (.) from the left side by and from the right side by [ ( N j +)I n n 0 M j = n ( N j +)n 0 n n E j [ M T ( N j +)I n n 0 n n j = 0 ( N j +)n n E T j ], (.0) ]. (.) Making use of the following relation E j E T j =( N j +)I and setting W i W for i=,,,n and U j W for j = 0,,,N (which also gives us the following condition W,W j E T j = ET jw), then

6 Page of 0 Manuscripts submitted to IMA Journal of Mathematical Control and Information A FURTHER REMARK ON THE PROBLEM OF RELIABLE DECENTRALIZED STABILIZATION of (.) reduces to [ 0 ( N j +)X j ( N j +)X j 0 ] ([ (AW + B + He j L j ) W ] [ ( N j +)I n n ε j I n n ] ) 0, (.) which is basically the square dilated LMIs condition, i.e., if we let further ε j ( N j +)ε j for all j N {0}, then we have [ 0 Xj X j 0 ] ([ ] (AW + B + He j L j ) [ W In n ε j I n n ] ) 0. (.) Moreover, for any W GL n (R) and a family of N-tuples( L, L,..., L N ) as above, and setting K i = L i W for each i=,,...,n, then ( ) A+B j K j are Hurwitzian matrices for all j N {0}. We remark that equation (.) describes a set of dilated LMIs conditions in terms of W i GL n (R), L i R ri n, i =,,...,N, and U j GL n (R), X j S n +, j = 0,,...,N. Note also that a common set of {W i, L i } N i= matrix variables is used for all failure modes, i.e., for all j N {0}. This is because we need a set of reliable state-feedback gains K i R ri n that works well for all possible closed-loop systems. However, it should be noted that, since we use a rectangular dilated LMIs framework, this does not require to employ a common quadratic Lyapunov stability certificate X S n + as in the case of quadratic Lyapunov technique or a common W GL n (R) and {L i } N i= as in the case of square dilated LMIs technique (where the latter is based on the conditions in (.)) for all possible failure modes. In this sense, the new extension, which is based on Theorem., is not as conservative as the quadratic Lyapunov technique or the square dilated LMIs technique. Note that although we have considered a reliable state-feedback stabilization problem, the problem of reliable stabilization via multi-controller configuration that was actually explored by Fujisaki & Befekadu (00) can be treated in the same way. In fact, that paper also presented a tractable design method which covers a class of plants that can be stabilized reliably using dynamic output feedback controllers (with fixed order of the controllers). Acknowledgements This work was supported in part by the National Science Foundation under Grant No. CNS-0; G. K. Befekadu acknowledges support from the Moreau Fellowship of the University of Notre Dame. REFERENCES BLONDEL, V. & TSITSIKLIS, J. (), NP-hardness of some linear control design problems, SIAM J. Control Optim.,,. FUJISAKI, Y. & BEFEKADU, G.K. (00), Reliable decentralized stabilization of multi-channel systems: a design method via dilated LMIs and unknown disturbance observers, Int. J. Contr., (), FUJISAKI, Y. & BEFEKADU, G.K., Reliable decentralised stabilization via dilated LMIs and unknown disturbance observers, in Proc. 00 IEEE Conf. Contr. Applications, Singapore, 00, pp.. GEROMEL, J.C., DE OLIVEIRA, M.C. & HSU, L. (), LMI characterization of structural and robust stability, Linear Alg. Appl., (-), 0. NEMIROVSKI, A. (), Several NP-hard problems arising in robust stability analysis, Math. Contr. Sign. Syst., (), 0.

7 Page of of G. B. BEFEKADU, V. GUPTA AND P. J. PANOS PIPELEERS G., DEMEULENAERE, B. SWEVERS, J., & VANDENBERGHE, L. (00), Extended LMI characterizations for stability and performance of linear systems, Syst. Control Lett. (), 0. VIDYASAGAR, M. & VISWANADHAM, N. (), Reliable stabilization using a multi-controller configuration, Automatica, (), 0.

8 Page of 0 Manuscripts submitted to IMA Journal of Mathematical Control and Information IMA Journal of Mathematical Control and Information Page of doi:0.0/imamci/dnnxxx A further remark on the problem of reliable stabilization using rectangular dilated LMIs GETACHEW K. BEFEKADU, VIJAY GUPTA AND PANOS J. ANSTAKLIS Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN, USA. [Received on 0 May 0] In this brief paper, an extension of the result of Fujisaki & Befekadu (00) concerning the problem of reliable stabilization for generalized multi-channel systems is given. Specifically, we use a rectangular dilated LMIs framework to provide a relaxed sufficient condition for the reliable stabilization of a multichannel system both when all of the controllers work together and when one of the controllers ceases to function due to a failure. Keywords: reliable control; decentralized control; stabilization; dilated LMIs.. Introduction Recently, the problem of reliable stabilization for generalized multi-channel systems with a single failure in any of the control channels has been addressed by Fujisaki & Befekadu (00) via dilated LMIs and unknown disturbance observers. In this brief paper, we extend this result for a multi-channel system using a rectangular dilated LMIs framework. The new extension can be looked as a sufficiently decoupling framework (i.e., separating the design variables from the system data) that provides a tractable (and also less-conservative) design technique for reliable stabilization of multi-channel system. This brief paper is organized as follows. In Section, we present the main result where the problem of reliable stabilization for a generalized multi-channel system is formally restated. Specifically, a relaxed and verifiable sufficient condition is given in terms of a set of rectangular dilated LMIs for the reliable stabilization of multi-channel system. Notation. For a matrix A R n n, He(A) denotes a hermitian matrix ined by He(A) =(A + A T ), where A T is the transpose of A. For a matrix B R n p with r = rankb, B R (n r) n denotes an orthogonal complement of B, which is a matrix that satisfies B B = 0 and B B T 0. S n + denotes the set of strictly positive inite n n real matrices and C denotes the set of complex numbers with negative real parts, that is C = {s C Re{s} < 0}. Sp(A) denotes the spectrum of a matrix A R n n, i.e., Sp(A) = {λ C rank(a λ I) < n} and GL n (R) denotes the general linear group consisting of all n n real nonsingular matrices. Note that the problem of reliable stabilization is essentially equivalent to a strong stabilization problem that involves an intractable problem (e.g., see references Vidyasagar & Viswanadham (), Nemirovsk () and Blondel & Tsitsiklis ()). The author 0. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

9 Page of of G. B. BEFEKADU, V. GUPTA AND P. J. PANOS. Main result Consider the following finite-dimensional generalized multi-channel system ẋ(t)=ax(t)+ B j u j (t), x(0)=x 0, (.) j N where x(t) X R n is the state of the system, u j (t) U j R r j is the control input to the jth-channel of the system and N denotes the number of input channels. A R n n, B j R n r j and N = {,,...,N}. For the above system, we restrict the set K to be the set of all linear, time-invariant (reliable) stabilizing state-feedback gains that satisfies K (K,K,...,K N ) K i R r i n Sp A + B j K j C, j N {0} i N i N, (.) where the sets N 0 = N and N j are ined by N j = N \{ j} for j =,,...,N with cardinality of N 0 = N and N j = N, respectively. Moreover, B 0 = B i i N, B j = B i and i N j B j K j = i N j B i K i for j N {0}. REMARK. In this brief paper, we consider the stability of the closed-loop system A + B j K j under nominal operation condition (i.e., when j = 0) as well as under any single-channel controller failure (i.e., when j N ). Let us ine the following matrices that will be later used in Theorem.. DEFINITION. For j = 0 E 0 = I n n I n n I n n, X0,X 0 = blockdiag{x 0, X 0,X 0,...,X 0 }, ( N 0 +) times ( N 0 +) times [A,B] U0,L 0 =[ AU 0 B L B L B N L N ], U 0,W 0 = blockdiag{u 0, W,W,...,W N }}, ( N 0 +) times ( N 0 +) times and for j N E j = I n n I n n I n n I n n, Xj,X j = blockdiag{x j, X j,...,x j,x j,...,x j }, ( N j +) times ( N j +) times [A,B] Uj,L j =[ AU j B L B j L j B j+ L j+ B N L N ], ( N j +) times =W j U j,w j = blockdiag{u j, W,...,W j,w j+,...,w N }, ( N j +) times where X j S n +, U j GL n (R) for j = 0,,...,N, W i GL n (R) and L i R r i n for i =,,...,N. =X 0 =W 0 =X j

10 Page of 0 Manuscripts submitted to IMA Journal of Mathematical Control and Information A FURTHER REMARK ON THE PROBLEM OF RELIABLE STABILIZATION of REMARK. Notice that the above set of matrices allows us to introduce a common set of matrix variables {L i, W i } i N that will be useful for the main result of this section. Next we can characterize the set K using a new-class of dilated LMIs (i.e., rectangular dilated LMIs) as follow. THEOREM. Suppose there exist X j S n +, ε j > 0, U j GL n (R), j = 0,,...,N, W i GL n (R) and L i R r i n, i =,,...,N such that 0 n n E j X j,x j [A,B j ] X j,x j E T + He Uj,L j j 0 ( N j +)n ( N j +)n U j,w j E T j ε j I ( N j +)n ( N j +)n 0, j N {0}. (.) For any family of N 0 -tuples (L, L,...,L N ) and (W, W,...,W N ) as above, if we set K i = L i Wi for each i =,,...,N, then the matrices A + B j K j are Hurwitz for all j N {0}, i.e., Li Wi i N K. Proof of Sufficiency. Note that X j,x j E T j = ET j X j and [A,B] Uj,L j U j,w j E j ε j I ( N j +)n ( N j +)n = I n n [A,B] Uj,L j U j,w j, = I n n (A + B j K j ), (.) = ε j I n n E j, (.) for j = 0,,...,N. Then, eliminating U j,w j from (.) by using these matrices, we have the following matrix inequalities I n n [A,B] Uj,L j U j,w j 0 n n E j X j,x j X j,x j E T j I n n (U j,w j ) T [A,B] T U j,l j 0 n n E j X j,x j ε j I n n E j X j,x j E T j 0 ( N j +)n ( N j +)n 0 ( N j +)n ( N j +)n = He((A + B j K j )X j ) 0, (.) ε j I n n E T j = ε j ( N j + )X j 0. Hence, we see that equations (.) and (.) state the Lyapunov stability condition with X j S n + and state-feedback gains L i Wi i N K. Notice that the theorem is solvable only if all of the pairs (A,B j ) for j N are stabilizable. Moreover, the stabilizability of one of the pairs implies stabilizability of (A,B 0 ), thus we do not have to assume this explicitly. (.)

11 Page 0 of of G. B. BEFEKADU, V. GUPTA AND P. J. PANOS Proof of Necessity. Suppose the system in (.) is reliably stable with state - feedback gains Li Wi i N K and W i GL n (R), i =,,...,N. Then, there exist sufficiently small ε j > 0 for j = 0,,...N that satisfy He (A + B j K j )X j + ε j A,B X U j,l j,x j A,B T 0, (.) j U j,l j with U j GL n (R) for j = 0,,...,N. Note that X j,x j = X j,x j T 0, (A+B j K j )= A,B U U j,l j,w j E T j j and E T j X j = X j,x j E T j, employing the Schur complement for (.), then we have He (A + B j K j )X j ε j [A,B] Uj,L j X j,x j ε j X j,x j T [A,B] Uj,L j ε j X j,x j He A,B U U = j,l j,w j E T j j X j ε j [A,B] Uj,L j + E j E j X j,x j X j,x j ε j ([A,B] Uj,L j ) T + E T j ET j ε j X j,x j [A,B]Uj,L = + He U j j,w j X j,x j 0 n n E j X j,x j X j,x j E T j 0 ( N j +)n ( N j +)n I ( N j +)n ( N j +)n E T j ε j I ( N j +)n ( N j +)n, 0. (.) Thus, the above expression (i.e., equation (.)) implies that (.) holds with (U j + U T j )=X j and (W j +W T j )=X j for j N {0}. REMARK. We remark that the above extended LMI framework stated in Theorem. is useful in the context of reliable control for systems with generalized multi-channel configurations, since the framework effectively separates design variables such as X j from the system data (A,B j ) for all j N {0}. Note that Theorem. is a generalization of the square dilated LMIs technique that has been considered by Fujisaki & Befekadu (00) in the context of reliable stabilization for multi-channel systems (e.g., see Geromel, De Oliveira & Hsu (), Ebihara & Hagiwara (00), Fujisaki & Befekadu (00), Pipeleers et al. (00) and references therein for a review of square dilated LMIs technique). In fact, if we multiply equation (.) from the left-side by ( N j + )I n n Γ j = 0 n n E j 0 n ( N j +)n, (.0) and from the right-side by the transpose matrix Γ T j. Finally, making use of the relation E je T j = ( N j + )I and setting W i W for i =,,,N and U j W for j = 0,,,N (which also gives Note that the parameters ε j for all j N {0} can be chosen with a line-search (or bisection) method.

12 Page of 0 Manuscripts submitted to IMA Journal of Mathematical Control and Information A FURTHER REMARK ON THE PROBLEM OF RELIABLE STABILIZATION of us the condition W,W j E T j = ET jw), then (.) reduces to 0 ( N j + )X j ( N j + )X j 0 (AW + B + He j L j ) W ( N j + )I n n ε j I n n 0, (.) which is basically the square extended LMI condition presented in Fujisaki & Befekadu (00), i.e., if we let further ε j ( N j + )ε j for all j N {0}, we then have 0 Xj (AW + B + He j L j ) X j 0 W In n ε j I n n 0. (.) Moreover, for any family of N 0 -tuple (L, L,..., L N ) and W GL n (R) as above, if we set K i = L i W for each i =,,...,N, then the matrices A+B j K j are Hurwitz for all j N {0}. We remark that equation (.) describes a new-class of extended LMI conditions in terms of W i GL n (R), L i R ri n, i =,,...,N, and U j GL n (R), X j S n +, j = 0,,...,N. Note also that a common set of matrix variables {L i, W i } i N is used for all failure modes, i.e., for all j N {0}. This is because we need an N 0 -tuple state-feedback gain K =(K K...K N ) with K i K i for i N that ensures stability for all possible closed-loop systems. However, it should be noted that, since we use a new-class of extended LMI framework, we do not require either a common quadratic Lyapunov stability certificate X S n + as in the case of quadratic Lyapunov technique or a common W GL n (R) and {L i } i N that will be needed in the case of square extended LMI technique for all possible failure modes (c.f. equation (.)). In this sense, the new extension, which is based on Theorem., is not as conservative as the quadratic Lyapunov technique or the square dilated LMIs technique. Note that although we have considered a reliable state-feedback stabilization problem, the problem of reliable stabilization via multi-controller configuration that was actually explored by Fujisaki & Befekadu (00) can be treated in the same way. In fact, that paper also presented a tractable design method which covers a class of plants that can be stabilized reliably using dynamic output feedback controllers (with fixed order of the controllers). Acknowledgements This work was supported in part by the National Science Foundation under Grant No. CNS-0; G. K. Befekadu acknowledges support from the Moreau Fellowship of the University of Notre Dame. REFERENCES BLONDEL, V.& TSITSIKLIS, J. (), NP-hardness of some linear control design problems, SIAM J. Control Optim.,,. EBIHARA, E.& HAGIWARA, T. A dilated LMI approach to robust performance analysis of linear time-invariant uncertain systems, Automatica, (), pp.. FUJISAKI, Y.&BEFEKADU, G.K. (00), Reliable decentralized stabilization of multi-channel systems: a design method via dilated LMIs and unknown disturbance observers, Int. J. Contr., (), FUJISAKI, Y.& BEFEKADU, G.K., Reliable decentralised stabilization via dilated LMIs and unknown disturbance observers, in Proc. 00 IEEE Conf. Contr. Applications, Singapore, 00, pp.. GEROMEL, J.C.,DE OLIVEIRA, M.C.&HSU, L. (), LMI characterization of structural and robust stability, Linear Alg. Appl., (-), 0.

13 Page of of G. B. BEFEKADU, V. GUPTA AND P. J. PANOS NEMIROVSKI, A. (), Several NP-hard problems arising in robust stability analysis, Math. Contr. Sign. Syst., (), 0. PIPELEERS G., DEMEULENAERE, B. SWEVERS, J.,& VANDENBERGHE, L. (00), Extended LMI characterizations for stability and performance of linear systems, Syst. Control Lett. (), 0. VIDYASAGAR, M.& VISWANADHAM, N. (), Reliable stabilization using a multi-controller configuration, Automatica, (), 0.

14 Page of 0 Manuscripts submitted to IMA Journal of Mathematical Control and Information Response to Reviewer Comments Manuscript ID IMAMCI-0-0: A further remark on the problem of reliable stabilization using rectangular dilated LMIs by Getachew K. Befekadu, Vijay Gupta and Panos J. Antsaklis, Department of Electrical Engineering, University of Notre Dame We thank all of the reviewers and the Associate Editor for their time and valuable comments on our brief paper. We have amended the paper in response to their suggestions. We believe that the paper has improved as a result, and hope that the concerns expressed by the reviewers and the Associate Editor have been addressed. Please find below itemized, point-by-point detailed responses to all the questions and comments of the reviewers and the Associate Editor. For your convenience, the reviewers comments are provided in italics. RESPONSE TO COMMENTS OF REVIEWER This is an interesting paper where the authors use the dilated LMIs of Geromel, de Oliveira and Hsu to handle the problem of reliable stabilization with a single input channel failure. The results are interesting and seem to be technically correct. The notation is a bit on the heavy side and required some digesting. My main concerns are with respect to the terminology. The authors mention that the result is for a decentralized control. However, the gains K i are not block-diagonal, that is, they feedback full state information. Of course one could ask for W i, L i to be block-diagonal at the expense of losing the necessity part in the Theorem.. We thank the reviewer for his comments. We would like to mention that the system is not an interconnected system which is composed of several subsystems. That is, there is no subsystem in our formulation. The input of system is partitioned into N channels, while the state is not partitioned. Although this partition is not useful in the context of standard stabilization, introduction of such partition is indeed important in the context of reliable

15 Page of stabilization, where K i plays actually an important role. Moreover, N is a small number in the context of reliable control since it corresponds to redundancy of the control system. To clarify, we have rephrased the term decentralized throughout the paper. I was also disappointed by the lack of a simple numerical example. It would be very helpful for one looking to decipher the paper. I would recommend publication after terminology is clarified. An example would be a plus. We agree with the reviewer that additional numerical example can be used to demonstrate the approach, which is currently under consideration in combination with other techniques like the dissipativity-based certification (e.g., G. K. Befekadu, V. Gupta and P. J. Antsaklis, Robust/Reliable stabilization of multi-channel systems via dilated LMIs and dissipativitybased certifications, in Proc. th IEEE Mediterranean Conf. Contr. Automation, Corfu, Greece, 0, pp. 0.) Therefore, we prefer that our brief paper, which is mainly focusing on reliable stabilization for a multi-channel system, to be considered as a theoretical contribution. We would also like to mention that Equation (.) describes a new-class of extended LMI conditions in terms of W i GL n (R), L i R ri n, i =,,...,N, and U j GL n (R), X j S n +, j =0,,...,N. Notice that a common set of matrix variables {L i,w i } i N is used for all failure modes, i.e., for all j N {0}. However, it should be noted that, since we use a new-class of extended LMI framework, we do not require a common W GL n (R) together with {L i } i N as in the case of square extended LMI technique for all possible failure modes (c.f. Equation (.) in the revised paper). In this sense, the new extension, which is based on Theorem., is not as conservative as the quadratic Lyapunov technique or the square dilated LMIs technique. RESPONSE TO COMMENTS OF REVIEWER In the paper the authors discuss reliable decentralized stabilizing controller synthesis using the dilated LMI technique. Unfortunately, this reviewer is not impressed by the paper mainly because the main result, Theorem., seems not valid. The problem stems from the misleading notation in the fourth line of Page on U j,w j where W j in practice shares W,W,...,W N in common over j =,,...,N. Therefore, the replacement U j,w j = X j,x j at the end of the proof of Theorem., where X j is in general

16 Page of 0 Manuscripts submitted to IMA Journal of Mathematical Control and Information different for each index j, is not possible. We thank the reviewer for his comments. In order to clarify the above, we have revised Theorem. and the statements around this theorem as well as its proof. In particular, we have revisited Equation (.) where such a condition always holds true if the system in Equation (.) is reliably stable with the state-feedback gains (L i W i ) i N K. Moreover, we have also added one more middle expression as part of Equation (.) which clarifies further the main result of the theorem. Notice that Equation (.) implies that Equation (.) holds if (U j + U T j )=X j and (W j + W T j) =X j for j N {0}. Note that we will prefer to use X j X j S n, without losing much. S n + for all j N {0} in our paper, although one could also use RESPONSE TO COMMENTS OF REVIEWER Major Comments: My major comments relate to both the statement and the proof of Theorem.. The current statement seems to suggest that (i) implies (ii) (i) The pairs (A, B j ) are stabilizable. (ii) There exist X j S n +, U j GL n (R), W i ingl n (R) and L i R r i n such that LMI (.) holds for all j. However, this implication is not proved. I m not even sure it holds. Suppose N =. Then, (i) implies that there exists K, K, K, K such that the matrices A + B K, A + B K and A + B K are stable. However, it is not obvious that (i) is not sufficient to guarantee that a solution with K = K and K = K exists; and it is certainly not obvious that (i) is sufficient for the even more restrictive condition (ii) to hold. We thank this reviewer for his thorough review and several constructive comments. We agree with the reviewer about this fact. Notice that Theorem. is solvable only if the condition in (i) holds true, i.e., if all of the pairs (A, B j ) for j N are stabilizable. Moreover, the stabilizability of one of the pairs implies stabilizability of (A, B 0 ). In order to clarify, we have revised the statements around this theorem (where the result of this theorem is aimed on the problem of reliable stabilization) and also we have stated this fact more clearly in the revised manuscript (see also the revised theorem as well as the footnotes on Page of the revised paper). The only relation that holds is (ii) (iii), where (iii) K =(L i W i ) i N K

17 Page of The proof of this implication is more or less given in the paper in the paragraph Proof of Sufficiency. However, in the lines above (.) the authors state that U j,w j are eliminated, but this is initely not the case. The fact that (.) implies (.) follows from multiplying (.) on the left with the orthogonal complement derived on (, ), and on the right with the transpose of this matrix. Similarly, (.) implies (.). As (.) already implies X j 0, it is ok to use X j S n instead of X j S n + in statement (ii). In the paragraph Proof of Necessity the authors seem to prove (iii) (ii), but this proof is incorrect. The result of their proof is that U j,w j = X j,x j satisfies (.). However, these equations do not have a solution in general. Let N =, then these equations amount to: j =0: U 0 = X 0, W = X 0, W = X 0, j =: U = X, W = X, j =: U = X, W = X, Hence, unless X 0 = X = X, these equations fail. In order to clarify the above fact more clearly, we have revised the statements around the Proof of Necessity part of this theorem. In particular, we have revisited Equation (.) where such a condition always holds true if the system in Equation (.) is reliably stable with the state-feedback gains (L i W i ) i N K. Moreover, we have also added one more middle expression as part of Equation (.) which clarifies further the main result of the theorem. Notice that Equation (.) implies that Equation (.) holds if (U j + U T j )=X j and (W j + W T j) =X j for j N {0}. Finally, we keep to use X j S n + for all j N {0} instead of X j S n in our manuscript. As the result is just a sufficient synthesis condition, numerical experiments analyzing this conservatism and comparing it with other results from the literature are essential. We agree with the reviewer that additional numerical example can be used to demonstrate the approach, which is currently under consideration in combination with other techniques like the dissipativity-based certification (e.g., G. K. Befekadu, V. Gupta and P. J. Antsaklis, Robust/Reliable stabilization of multi-channel systems via dilated LMIs and dissipativitybased certifications, in Proc. th IEEE Mediterranean Conf. Contr. Automation, Corfu, Greece, 0, pp. 0.) Therefore, we prefer that our brief paper, which is mainly

18 Page of 0 Manuscripts submitted to IMA Journal of Mathematical Control and Information focusing on reliable stabilization for a multi-channel system, to be considered as a theoretical contribution. We would also like to mention that Equation (.) describes a new-class of extended LMI conditions in terms of W i GL n (R), L i R ri n, i =,,...,N, and U j GL n (R), X j S n +, j =0,,...,N. Notice that a common set of matrix variables {L i,w i } i N is used for all failure modes, i.e., for all j N {0}. However, it should be noted that, since we use a new-class of extended LMI framework, we do not require either a common quadratic Lyapunov stability certificate X S n + as in the case of quadratic Lyapunov technique or a common W GL n (R) and {L i } i N as in the case of square extended LMI technique for all possible failure modes (c.f. Equation (.) in the revised paper). In this sense, the new extension, which is based on Theorem., is not as conservative as the quadratic Lyapunov technique or the square dilated LMIs technique. Minor Comments: Page, line below Eq. (.): As state-feedback is considered, I think it is more appropriate to replace N denotes the number of input/output channels by N denotes the number of inputs. We have revised this statement. Note that the input of system is partitioned into N channels, while the state is not partitioned. Although this partition is not useful in the context of standard stabilization, introduction of such partition is indeed important in the context of reliable stabilization, where K i plays actually an important role. Moreover, N is a small number in the context of reliable control since it corresponds to redundancy of the control system. Page, Eq. (.): As the presented design conditions does not allow enforcing particular structure in K j, I don t see the merit of the set K j. The only set that matters is K, ined as K (K,K,...,K N ) K j σ(a + B j K j ) C, j N {0}, j N We agree with this reviewer. In fact, we have revised the statements around Theorem. and also the notations just above this theorem. Also note that σ(a + B j K j ) C is used instead of σ(a + B j K j ) C in (.). We have revised this.

19 Page of To emphasize that (.) corresponds to many LMIs, j N {0} should be added after... 0 We have revised this. The inition of K can be used to simplify the last lines of Theorem..: (L i W i ) K. We have revised this theorem and, in fact, we have added a new line that points to the inition of the set K, i.e., (L i W i ) i N K. Eq. (.): the orthogonal complement is [ j I n n E T j]. Thanks, we have revised this typo. RESPONSE TO COMMENTS OF REVIEWER The paper is well written and technically sound. As commented above the results are not considered as major by the reviewer. Discussion about rectangular and square dilated LMIs is not clear. A short numerical example could advantageously illustrate the conservatism reduction and how is chosen. We thank the reviewer for his comments. In order to clarify the above fact more clearly, we have revised Theorem. and the statements around this theorem as well as its proof part. In particular, we would also like to remark that Equation (.) describes a new-class of extended LMI conditions in terms of W i GL n (R), L i R r i n, i =,,...,N, and U j GL n (R), X j S n +, j =0,,...,N. Notice that a common set of matrix variables {L i,w i } i N is used for all failure modes, i.e., for all j N {0}. However, it should be noted that, since we use a new-class of extended LMI framework, we do not require a common W GL n (R) and {L i } i N that will be needed in the case of square extended LMI technique for all possible failure modes (c.f. Equation (.) in the revised paper). In the revised paper, we is sense, the new extension, which is based on Theorem., is not as conservative as the quadratic Lyapunov technique or the square dilated LMIs technique. Moreover, regarding the revisions, please refer to our reply to items and under major comments provided by Reviewer. Moreover, we have added a remark (see the footnotes on Page ) that states these parameters j, j N {0} can be, in general, chosen with line-search and/or bisection methods.

20 Page of 0 Manuscripts submitted to IMA Journal of Mathematical Control and Information RESPONSE TO COMMENTS OF ASSOCIATE EDITOR The statement and proof of Theorem. should be fixed as pointed out by reviewers and. We have revised our manuscript in response to all comments provided by reviewers and. Moreover, regarding the revisions, please refer above itemized, point-by-point responses to all comments/suggestions of reviewers and. The discussion about rectangular and square dilated LMIs (Reviewer ). We have revised our paper in response to all comments provided by reviewer. Moreover, regarding the revisions, please refer above itemized, point-by-point responses to all comments/suggestions of reviewer. Conservatism of the results (Reviewer ). We have revised our paper in response to all comments provided by reviewer. Moreover, regarding the revisions, please refer above itemized, point-by-point responses to all comments/suggestions of reviewer. The notation is a bit on the heavy side and needs to be rationalized (Reviewer ). We have revised our paper in response to all comments provided by reviewer. In fact, we have revisited the statements around Definition. our nations by adding further remarks to clarify and Moreover, regarding the revisions, please refer above itemized, point-by-point responses to all comments/suggestions of reviewer. The terminology decentralized control, as questioned by Reviewer. We have revised our paper in response to all comments provided by reviewer. To clarify this, we have rephrased the term decentralized throughout the paper. An example should be added to the paper to illustrate the conservatism reduction that can be achieved with the proposed results and how the scalars j are chosen (Reviewers and ). We have revised our paper in response to all comments provided by reviewers and. In fact, we have added a remark (see the footnotes on Page ) that states these parameters j, j N {0} can be, in general, chosen with line-search and/or bisection methods. The paper may be published after a major revision and subject to a round of positive evaluation.

21 Page 0 of We have revised our paper in response to their suggestions. Moreover, regarding the revisions, please refer above itemized, point-by-point responses to all comments and suggestions of the reviewers.