On the reliable decentralised stabilisation of n MIMO systems

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1 This article was downloaded by: [University of Notre Dame] On: 12 December 2014 At: 01:15 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House Mortimer Street London W1T 3JH UK International Journal of Control Publication details including instructions for authors and subscription information: On the reliable decentralised stabilisation of n MIMO systems Getachew K. Befekadu a Vijay Gupta a & Panos J. Antsaklis a a Department of Electrical Engineering University of Notre Dame Notre Dame IN USA Accepted author version posted online: 10 Jan 2014.Published online: 26 Feb To cite this article: Getachew K. Befekadu Vijay Gupta & Panos J. Antsaklis 2014 On the reliable decentralised stabilisation of n MIMO systems International Journal of Control 87: DOI: / To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information the Content contained in the publications on our platform. However Taylor & Francis our agents and our licensors make no representations or warranties whatsoever as to the accuracy completeness or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses actions claims proceedings demands costs expenses damages and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with in relation to or arising out of the use of the Content. This article may be used for research teaching and private study purposes. Any substantial or systematic reproduction redistribution reselling loan sub-licensing systematic supply or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at

2 International Journal of Control 2014 Vol. 87 No On the reliable decentralised stabilisation of n MIMO systems Getachew K. Befekadu Vijay Gupta and Panos J. Antsaklis Department of Electrical Engineering University of Notre Dame Notre Dame IN USA Received 18 July 2013; accepted 24 December 2013 In this paper we consider the problem of reliable decentralised stabilisation for n multi-input multi-output MIMO systems when some of the shared controllers among these systems are faulty in the sense that they fail to operate properly due to subsystem module failures that may occur in actuators sensors or controllers. Specifically we present a design framework using a dilated linear matrix inequalities LMIs technique for deriving reliable stabilising state-feedback gains for all n MIMO systems; while a set of decentralised unknown disturbance observers UDOs that are shared by all n MIMO systems is used for extending the result to the output-feedback case. Moreover a sufficient condition for the solvability of the set of decentralised UDOs is given in terms of a minimum-phase condition of each subsystem and a lower-bounding condition on the number of outputs of each channel. We also present a numerical example that illustrates the applicability of the proposed technique. Keywords: dilated LMI; multi-input multi-output systems; reliable decentralised stabilisation; unknown disturbance observers 1. Introduction The problem of reliable decentralised stabilisation for multi-input multi-output MIMO systems with multicontroller configurations has been addressed by a number of authors in the past e.g. see references Fujisaki & Befekadu 2009; Gündeş1988; Šiljak 1980; Shor Perkins & Medanić 1992; Vidyasagar & Viswanadham In a multi-controller configuration the objective is to enhance robustness and/or guarantee certain stability and performance criteria of the closed-loop system against possible subsystem module failures that may occur in actuators sensors or controllers. It should also be noted that several interesting theoretical results have been achieved in the context of reliable stabilisation for MIMO systems e.g. see Befekadu Gupta & Antsaklis 2013; Ghosh 2013; Lavaei & Aghdam 2007; Tan Šiljak & Ikeda 1992; Ünyelioǵlu Ózgüler & Khargonekar 1992 and the references therein though such a problem is in general a very difficult problem. This is because reliable decentralised stabilisation is essentially equivalent to a strong stabilisation problem which involves an intractable problem e.g. see Blondel & Tsitsiklis 1997; Ohta Maeda & Kodama 1991; Vidyasagar & Viswanadham In this paper without attempting to give a literature review we consider the problem of reliable decentralised stabilisation for n MIMO systems with multi-controller configurations. Specifically we present a design framework for reliable decentralised stabilising controllers using a combination of dilated linear matrix inequalities LMIs technique and a set of decentralised unknown disturbance observers UDOs. Initially using a dilated LMIs technique we design a set of state-feedback gains that reliably and simultaneously stabilises all n MIMO systems when there is a single-controller failure in any of the control channels. Then we extend the result to the output-feedback case using a set of decentralised UDOs together with the set of reliable stabilising state-feedback gains. We also provide an algebraic procedure on how to determine this set of decentralised UDOs where such decentralised UDOs are shared by all n MIMO systems and correspondingly they also provide asymptotic estimations of the states and/or input signals of each subsystem. Furthermore a sufficient condition for the solvability of this set of decentralised UDOs is given in terms of a minimum-phase condition of each subsystem and a lower-bounding condition on the number of outputs of each channel. In general such a design framework can be looked as a sufficiently decoupling framework that is separating the design variables from the system data which provides a tractable and also lessconservative design technique for reliable stabilising of all n MIMO systems with multi-controller configurations. This paper is organised as follows. In Section 2 we present preliminary result on the problem of reliable statefeedback stabilisation for n MIMO systems with multicontroller configurations. Section 3 presents the main results where we also provide a solvability condition for Corresponding author. gbefekadu1@nd.edu C 2014 Taylor & Francis

3 1566 G.K. Befekadu et al. the problem of reliable output-feedback stabilisation for all n MIMO systems. In Section 4 we provide a numerical example which illustrates the applicability of the proposed technique and Section 5 contains concluding remarks. 2. Preliminaries 2.1 Notation For a matrix A R d d HeA denotes a Hermitian matrix ined by He A = A + A T where A T is the transpose of A. For a matrix B R d p with r = rankb B R d r d denotes an orthogonal complement of B which is a matrix that satisfies B B = 0 and B B T 0. S + d denotes the set of strictly positive inite d d real matrices and C denotes the set of complex numbers with negative real parts that is C ={s C Re{s} < 0}. SpA denotes the spectrum of a matrix A R d d that is SpA ={s C ranka si <d} and GL d R denotes the general linear group consisting of all d d real nonsingular matrices. 2.2 System description Consider the following continuous-time n MIMO systems each with N input/output channels: ẋ k t = A k x k t + B ki u ki t Sk : i N y ki t = C ki x k t i N k N where x k t R d is the state of the kth system u ki t R r i is the control input to the ith channel and y k i t is the output of ith channel of the kth system. A k R d d B ki R d r i C ki R mi d for each k N and N ={1 2...N} represents the set of input/output channels for each MIMO system and N ={1 2...n} represents the set of n MIMO systems. For the above n MIMO systems we restrict the set K to be a set of all linear time-invariant reliable stabilising state-feedback gains that satisfies 1 { K K 1 K 2...K N R r i d i N Sp A k + B ki K i C i N j } j N {0} k N 2 where the sets N 0 and N j are ined by N 0 = N and N j = N \{j} for j = N respectively. Remark 1: In this paper we consider the stability of all closed-loop systems A k + i N 0 B ki K i under nominal operating condition that is when j = 0 and for all k N and A k + i N j B ki K i under any singlechannel controller failure that is when j N and for any k N inanyofthemimosystems. 2.3 Reliable simultaneous stabilising state-feedback controllers In the following we state a lemma whose proof for n = 1 is given in Fujisaki & Befekadu 2009; Befekadu et al. 2013; see also Ebihara & Hagiwara 2005; de Oliveira Bernussou & Geromel 1999 for related results which is used for characterising the set of reliable stabilising state-feedback gains K for all n MIMO systems in Equation 1. Lemma 2.1: Suppose there exist X kj S + d ɛ k j > 0 j= Nk= nW GL d R and L i R ri d i = N such that [ ] [ 0d d X kj Ak W + i N j + He B ] kil i W X kj 0 d d [ I d d ɛ kj I d d ] 0 j N {0} k N. Then the system matrices A k + i N j B ki K i for all j N {0} and for all k N are Hurwitz when we set K i = L i W 1 for each i = N that is L 1 W 1 L 2 W 1... L N W 1 K. 1 Remark 2: We remark that the above lemma is solvable only if the pairs A k B k j for all j N {0} are stabilisable with B k 0 = [ ] B k1 B k2 B kn and B k i = [ ] B k1 B ki 1 B ki+1 B kn i N k N. Remark 3: The above systems in Equation 1 can be stabilised individually each with its own set of reliable stabilising state-feedback gains. However this does not necessarily imply that the whole n MIMO systems can be stabilised with a common set of reliable stabilising state-feedback gains. Next we assume that the following statement holds for all n MIMO systems in Equation 1. Assumption 2.2: There are no unstable decentralised fixed modes DFMs with respect to triplets of C k j A k B k j for all j N {0} with 2 C k 0 = [ Ck1 T CT k2 T kn] CT and C k i = [ Ck1 T CT ki 1 CT ki+1 T kn] CT i N k N. 3

4 International Journal of Control 1567 Remark 4: Note that the above assumption is required for synthesising the main result of the paper which also includes the problem of decentralised stabilisation with respect to triplets of C k j A k B k j for all j N {0} and for each k N. The necessary and sufficient condition for decentralised stabilisation can be characterised in terms of the fixed modes of the system e.g. see Wang & Davidson Main results In this section we first consider a set of decentralised UDOs that provide asymptotic estimations of the states and/or input signals of each subsystem. Later these decentralised UDOs together in combination with the set of reliable stabilising state-feedback gains will be used to extend the result to the output-feedback case for the n MIMO systems. Note that we also provide a sufficient condition for the solvability of this set of decentralised UDOs in terms of the minimum-phase condition of each subsystem and a lower-bounding condition on the number of outputs of each channel. 3.1 A set of decentralised unknown disturbance observers Consider the following decentralised UDOs that are shared by all n MIMO systems: Oi : ξ i t = A πi ξ i t + B πki u ki t + L πki y ki t û i t = C πi ξ i t + D πki y ki t i N where the matrix A πi R d i d i with SpA πi C and appropriate matrices B πki R d i r i C πi R r i m i D πki R r i m i L πki R d i m i for i N and for k N. Next let us ine the state estimation errors as 4 e i t = ξ i t Z π ki x k t i N 5 where Z πki R d i d for i N and for all k N. For each k N let us rewrite Equation 1 as ẋ k t = A k x k t + B ki u ki t + B kl u kl t Sk : l N i y ki t = C ki x k ti N. Then we have the following equations for the estimation error dynamics: ė i t = ξ i t Z πki ẋ k t = A πi ξ i t + B πki u ki t + L πki y ki t Z πki A k x k t Z πki B ki u ki t Z πki B kl u kl t l N i = A πi e i t + Aπi Z πki Z π A ki k + L C πki ki x k t }{{} Equation 8 + Bπki Z B πki ki K i x k t }{{} Equation 9 Z πki B kl u kl ti N 6 l N i }{{} Equation 10 and let us also introduce the following estimation error for ith input signals of the MIMO system: ũ i t = û i t ω π ki u kit = C πi ξ i t + D πki C ki x k t ωki π K ix k t = C πi e i t + Cπi Z πki + D C πki ki ωki π K i }{{} Equation 11 x k ti N 7 with ωki π > 0 and ωπ ki = 1 for all i N. Next we state the following result. Theorem 3.1: Suppose that the N-tuple of state-feedback gains K 1 K 2...K N K that satisfy the conditions in Lemma 2.1 is given. Moreover let the matrices A πi R d i d i with SpA πi C and B πki R d i r i C πi R r i m i D πki R r i m i L πki R d i m i Z πki R di d and ωki π > 0 with ωπ ki = 1 and d i d fori N and for k N satisfy the following set of equations: Aπi Z πki Z A πki k + L C πki ki = 0 8 Bπki Z B πki ki = 0 9

5 1568 G.K. Befekadu et al. Z πki B kl = 0 10 l N i Cπi Z πki + D C πki ki ωki π K i = Then there exists a set of decentralised UDOs shared by all n MIMO systems that asymptotically estimate the states and the corresponding input signals of each subsystem. Proof. The proof follows the same lines of argument as that of Fujisaki and Befekadu Suppose that the statefeedback gains K 1 K 2...K N K satisfy Lemma 2.1. Note that the decentralised UDOs in Equation 4 will asymptotically estimate the states and also the input signals of each MIMO system only if the following conditions hold: lim t ξ i t Z πki x k t = 0 x k0 ξ i0 u ki. u k i. i N k N 12 with u k i. = [u T k1. ut ki 1.uT ki+1. ut kn.]t for i N. That is the state estimation errors will exponentially decrease to zero for all i N. Hence the matrices A πi i N must be Hurwitz with the eigenvalues at least to the left of all eigenvalues of the closed-loop systems that is { { { { }} max Re Sp Aπi min Re Sp i N i N A k + }} B kl K l k N. 13 l N i Moreover from Equation 6 we have the following conditions: Aπi Z πki Z A πki k + L C πki ki = 0 Bπki Z B πki ki = 0 l N i Z πki B kl = 0 i N. On the other hand from Equation 7 and noting the conditions in Equation 13 we have also the following additional condition: lim t = lim t û i t ω π ki u kit û i t ωki π K ix k t = 0 x k0 ξ i0 u ki. u k i. y ki. i N k N 14 which further implies the following: Cπi Z πki + D C πki ki ωki π K i = 0 i N. This completes the proof. Remark 5: We remark that the algebraic equations in Equations 8 11 have more matrix variables than the number of equations. In order to solve these equations it is necessary to assign values to some of these matrix variables. For instance the matrices A πi for i N can be fixed provided that they satisfy the additional conditions in Equation 13. Later in Section 3.2 we will give a generic sufficient condition for the solvability of these algebraic equations cf. Lemma 3.5 and Corollary 3.6. Let us introduce the following matrices that will be used in the sequel. Definition 3.2: and A D = block diag{a 1 A 2...A n } B D 0 = block diag{b 1 0 B B n 0 } C D 0 = block diag{c 1 0 C C n 0 } K D 0 = [ K1 T T KT 2 N]...KT B D i = block diag{b B i 1 0 B i B n 0 } C D i = block diag{c C i 1 0 C i C n 0 } K D i = [ K1 T T...KT i 1 KT i+1 N]...KT for i N. Definition 3.3: and for i N A π D 0 = block diag{a π1 A π2...a πn } BD π 0 = block diag{ B π1 B π2... B πn } CD π 0 = block diag{c π1 C π2...c πn } DD π 0 = block diag{ D π1 D π2... D πn } L π D 0 = block diag{ L π1 L π2... L πn } Z π 0 = [ Z T π 1 Z T π 2 Z T π N ] T π 0 = [ T π 1 T π 2 T π N ] T A π D i = block diag{a π1...a πi 1 A πi+1...a πn } BD π i = block diag{ B π1... B πi 1 B πi+1... B πn } CD π i = block diag{c π1...c πi 1 C πi+1...c πn } DD π i = block diag{ D π1... D πi 1 D πi+1... D πn } L π D i = block diag{ L π1... L πi 1 L πi+1... L πn } Z π i = [ Z T π 1... Z T π i 1 Z T π i+1... Z T π N ] T π i = [ T π 1... T π i 1 T π i+1... T π N ] T

6 with for i N. B πi = [ ] B π1i B π2i B πni D πi = [ ] D π1i D π2i D πni L πi = [ ] L π1i L π2i L πni Z πi = [ Zπ T 1i Zπ T 2i Zπ T ] T ni πi = [ ω1i π ωπ 2i ] ωπ ni International Journal of Control 1569 Then the rest of the proof follows a standard state-space transformation of the closed-loop systems. To this end let us introduce the following state-space transformations: [ ] [ ] xd t Ɣ π xd t ξ j t j j N {0} 18 ξ j t with nonsingular matrices 3.2 Reliable decentralised stabilising output-feedback controllers In the following using the results of Lemma 2.1 and Theorem 3.1 we present our main result where we provide a set of reliable decentralised simultaneous stabilising outputfeedback controllers for the n MIMO systems. Theorem 3.4: Suppose that Theorem 3.1 holds then there exist F i G i H i J i i N such that [ AD + B Sp J D j D C j D j B H ] D j D j G D j C D j F D j C j N {0}. 15 Moreover the decentralised output-feedback controllers that achieve reliable stabilisation are given by where C i s = H i si F i 1 G i + J i i N 16 F i = A πi + B πki C πi G i = [G 1i G 2i... G ni ] with G ri = B πki D πri + L πri r N H i = C πi J i = D πi 17 for i N. Proof. For j N let us rewrite Equations 8 9 and 11 as ZD π j A D = A π D j ZD π j + L π D j C D j ZD π j B D j = BD π j K D j π j = Cπ D j ZD π j + DD π j C D j and also Equation 17 as for all j N {0}. F D j = A π D j CD π j G D j = L π D j DD π j H D j = CD π j J D j = DD π j Ɣ π j = [ In d n d Z π j 0 n d N j d I N j d N j d GL n+ N j drj N {0} where x D t = [ x1 T t xt 2 t xt n t] T ξ 0 t = [ ξ1 T t ξ2 T t ξ N T t] T and ξ i t = [ξ1 T T t ξi 1 tξ i+1 T t ξn T t]t for i N. Further N j denotes the cardinality of N j for j N {0}. Then we obtain the following set of transformed systems: Ɣ π j [ AD + B J D j D C j D j B H ] D j D j Ɣ G D j C D j F j π 1 D j [ ] I 0 = Z j π I [ AD + B D j DD π C j D j B D j CD π ] j L π D j DD π j C D j A π D j CD π j [ ] I 0 Z j π I [ AD + B D j K D j B D j C π ] D = j 0 A π j N {0}. D j ] 19 Since all of A k + i N j B ki K i forj N {0} and for k N and A πi for i N are Hurwitz matrices that is SpA D + B D j K D j SpA π D j C for j N {0} then we immediately see that the statement of the theorem holds. We remark that the solvability condition of Theorem 3.1 is given by the following lemma that is the minimumphase together with the relative-degree condition of each subsystem 3 see also Corollary 3.6 for an additional condition. Lemma 3.5: The set of Equations 8 11 are solvable with Hurwitz matrices A πi for all i N if the following hold:

7 1570 G.K. Befekadu et al. i the relative degree condition rankc ki B k i = rankb k i = r i = j N i r j ii the minimum-phase condition rank i N k N 20 [ ] Ak si B k i = d + r C ki 0 i i N k N 21 for all s C 0 ={s C Re{s} 0} and iii the lower-bounding condition on the number of outputs of each channel r i m i i N k N. 22 Remark 6: Note that if m i < rankc ki B k i for any i N and/or for any k N then one cannot design a stable decentralised UDO that satisfies the conditions of Theorem 3.1 see also Kudva Viswanadham & Ramakrishna 1980 for related discussions. Finally the following corollary provides a bound on the number of MIMO systems with respect to the input/output channels of each subsystem that can be reliably and simultaneously stabilised. Corollary 3.6: Suppose that Equation 22 holds then the following condition is sufficient for reliable and simultaneous stabilisation of all n MIMO systems: max m j r i n i N. 23 j N Remark 7: We remark that we assume the degrees of the decentralised UDOs to be d i where d i d for each i N. Then the condition of Corollary 3.6 states that the number of MIMO systems that could be reliably and simultaneously stabilised is bounded from above see also Ghosh & Byrnes 1983 Theorem 1.1; and Vidyasagar & Viswanadham 1982 for a similar statement but in a different context. [ ] [ ] [ ] A 2 = B = B 0 22 = 2 C 21 = [ 21] C 22 = [3 2]. If we employ Lemma 2.1forɛ 1 j = 1 and ɛ 2 j = 1 with j {0 1 2} then we have the following solutions: {[ ] [ {X 1j } j=0 = {[ ] {X 2j } j=0 = 0 5 [ ] 1 1 W = 1 1 ] [ {L i } 2 i=1 ={[ 2 2 ] [ 2 2 ]} [ ]} ] [ ]} that satisfy all conditions in Equation 3. 4 Note also that we obtain the following state-feedback gains: K 1 = [ 2 0] K 2 = [ 0 2 ] that achieve reliable stabilisation. In fact these statefeedback gains satisfy the following: Sp A k + i {12}\{j} j {0 1 2} k ={1 2}. B ki K i C That is all matrices are Hurwitz with the following eigenvalues: 1 {λ 12 } 2 j=0 = {{ 2 ± ı} { 1 1} { 1 1}} for k = 1 and 2 {λ 12 } 2 j=0 = {{ 3 ± ı 6} { 1 ± ı 2} { 1 ± ı 2}} for k = 2. Moreover if we choose the following matrices noting that the conditions in both Lemma 3.5 and Corollary 3.6 are satisfied A π1 = 4 B π11 = 1 B π12 = 1 C π1 = 2 D π11 = 1 D π12 = 1 L π11 = 9 2 L π 12 = 25 4 A π2 = 4 B π21 = 2 B π22 = 2 C π2 = 2 4. Numerical example Consider the following two two input/output channels: MIMO systems each with [ ] [ ] [ ] A 1 = B = B 0 12 = 1 C 11 = [21] C 12 = [ 12] D π21 = 1 D π22 = 1 L π21 = 3 2 L π 22 = 7 4 Z π11 = [ 12] Z π12 = [ 21] Z π21 = [ 1 1 ] Z π22 = [ 1 1] then we see that Equations 8 11 are satisfied with SpA πk C for k = 1 2 cf. Equation 13. Note that we also obtain the following coefficient matrices for the

8 International Journal of Control 1571 decentralised controllers: [ ] 15 9 F 1 = 10 G 1 = H 1 = 2 J 1 = [ 11] 2 2 [ ] F 2 = 10 G 2 = H 2 = 2 J 2 = [ 11] 4 4 that achieve reliable and simultaneous stabilisation. In fact all system matrices corresponding to the closed-loop systems are Hurwitz that is [ AD + B D j DD π Sp C j D j B D j CD π ] j L π D j DD π j C D j A π D j CD π j C j {0 1 2}. Moreover using Equation 16 of Theorem 3.4 see also Equation 17 the reliable decentralised output-feedback controllers C i s = H i si F i 1 G i + J i for i = 1 2 are given by [ ] s 5 s + 1 C 1 s = and s + 10 s + 10 [ ] 2s 17 2s + 1 C 2 s =. 2s s Concluding remarks In this paper we considered the problem of reliable and simultaneous stabilisation for a set of n MIMO systems with multi-controller configurations. A sufficient condition for the solvability of such a design problem is also derived in terms of a set of dilated LMI conditions; while a set of decentralised UDOs that are shared by all n MIMO systems whose asymptotic estimation error dynamics satisfying certain conditions related with the state and/or the control input signals of each subsystem is used for extending the result to the output-feedback case. Acknowledgements The first author acknowledges support from the College of Engineering University of Notre Dame. Funding This work was supported in part by the National Science Foundation NSF [CPS grant number CNS ]. Notes 1. Here we remark that a common set of {L i W} i N is used for all failure modes that is for all j N {0} and for all k N. This is because we need state-feedback gains K 1 K 2...K N that ensure stability for all possible closedloop systems. 2. For systems with unstable DFMs the problem of decentralised stabilisation using time-varying controllers has been addressed by a number of authors e.g. see Anderson & Moore 1981; Lavaei & Aghdam 2007; Ünyelioǵlu et al and the references therein. 3. This is a direct interpretation of Theorem 3.1 where the existence condition for this class of decentralised observers entails strong* detectability conditions e.g. see Hautus Note that one cannot design reliable stabilising state-feedback gains for such systems based on a common Lyapunov function. References Anderson B.D.O. & Moore J.B Time-varying feedback laws for decentralized control. IEEE Transactions on Automatic Control Befekadu G.K. Gupta V. & Antsaklis P.J On reliable stabilization via rectangular dilated LMIs and dissipativitybased certifications. IEEE Transactions on Automatic Control Blondel V. & Tsitsiklis J NP-hardness of some linear control design problems. SIAM Journal of Control and Optimization Ebihara E. & Hagiwara T A dilated LMI approach to robust performance analysis of linear time-invariant uncertain systems. Automatica Fujisaki Y. & Befekadu G.K Reliable decentralized stabilization of multi-channel systems: A design method via dilated LMIs and unknown disturbance observers. International Journal of Control Ghosh A Decentralized simultaneous stabilization of a class of two MIMO systems using a continuous-time periodic controller. Automatica Ghosh B. & Byrnes C Simultaneous stabilization and simultaneous pole-placement by nonswitching dynamic compensation. IEEE Transactions on Automatic Control Gündeş A.N Reliable decentralized stabilization of linear systems. IEEE Transactions on Automatic Control Hautus M.L.J Strong detectability and observers. Linear Algebra and Its Applications Kudva P. Viswanadham N. & Ramakrishna A Observers for linear systems with unknown inputs. IEEE Transactions on Automatic Control Lavaei J. & Aghdam A.G Simultaneous LQ control of a set of LTI systems using constrained generalized sampled-data hold functions. Automatica Ohta Y. Maeda H. & Kodama S Unit interpolation in H : Bounds of norm and degree of interpolants. Systems and Control Letters de Oliveira M.C. Bernussou J. & Geromel J.C A new discrete-time robust stability condition. Systems and Control Letters Shor M.H. Perkins W.R. & Medanić J.V Design of reliable decentralized controllers: A unified continuous/discrete formulation. International Journal of Control Šiljak D.D Reliable control using multiple control systems. International Journal of Control Tan X.-L. Šiljak D.D. & Ikeda M Reliable stabilization via factorization methods. IEEE Transactions on Automatic Control

9 1572 G.K. Befekadu et al. Ünyelioǵlu K.A. Ózgüler A.B. & Khargonekar P.P Decentralized simultaneous stabilization and reliable control using periodic feedback. Systems and Control Letters Vidyasagar M. & Viswanadham N Algebraic design techniques for reliable stabilization. IEEE Transactions on Automatic Control Vidyasagar M. & Viswanadham N Reliable stabilization using a multi-controller configuration. Automatica Wang S.H. & Davidson E.J On stabilization of decentralized control systems. IEEE Transactions on Automatic Control