Output-Feedback Model-Reference Sliding Mode Control of Uncertain Multivariable Systems

Transcription

1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 23 1 Output-Feedback Model-Reference Sliding Mode Control of Uncertain Multivariable Systems José Paulo V. S. Cunha, Liu Hsu, Ramon R. Costa and Fernando Lizarralde Abstract This paper considers the robust output tracking problem using a model-reference sliding mode controller for linear multivariable systems of relative degree one. It is shown that the closed loop system is globally exponentially stable and the performance is insensitive to bounded input disturbances and parameter uncertainties. The strategy is based on output-feedback unit vector control to generate sliding mode. The only required a priori information about the plant high frequency gain matrix K p is the knowledge of a matrix S p such that K ps p is Hurwitz which relaxes the positive definiteness requirement usually needed by other methods. Index Terms sliding mode, model-reference control, multivariable systems, output-feedback, unit vector control. I. INTRODUCTION One classical approach to the robust control of multivariable systems is variable structure control (VSC). A primary problem is robust stabilization by output feedback which is the subject of several works, e.g., 1, 2, 3, 4, 5. A more challenging problem is the robust output tracking of a reference signal using only output measurement. In this case a standard approach is to specify the desired closed loop response using a reference model. In this framework, variable structure control approach has been applied in 6 for single-input-single-output (SISO) plants and in 7, 8, 9, 1 for multi-input-multi-output (MIMO) plants. The model-reference controller proposed in 9, 1 is based on a state space description of the plant and a nonlinear observer to estimate the plant state. In contrast, the present paper, likewise 7, 8, relies on the plant transfer function matrix formulation and follows the model-reference adaptive control (MRAC) approach without explicit state observers 11. Our approach allows the demonstration of global exponential stability properties, which are not analyzed in 7, 8, 9, 1. This paper proposes a unit vector model-reference sliding mode controller (UV-MRAC) for MIMO plants with relative degree one. The only required a priori information about the high frequency gain (HFG) matrix K p of the plant is the knowledge of a matrix S p such that K ps p is Hurwitz. This relaxes the positive definiteness property usually needed by other methods 7, 8. In previous sliding mode schemes 9, 1, 4, 5 there is no explicit restriction on the plant HFG matrix and the input distribution matrix may be uncertain B p =Bp nom + B p, where Bp nom is the nominal input matrix and B p is the uncertainty, but the output distribution matrix C p is assumed to be known. In 9, 1, 5 the uncertainty is matched, i.e., B p = Bp nom F (t) with the matrix F (t) being bounded by F (t) k f < 1. A similar restriction on B p is assumed in 4. In contrast, the UV-MRAC does not need the boundedness of B p which, also, can be an unmatched uncertainty. Indeed, for a given nominal K p, say Kp nom = Cp nom, and some S p such B nom p This work was partially supported by FAPERJ and CNPq, Brazil. J. P. V. S. Cunha is with the Department of Electronics and Telecommunication Engineering, State University of Rio de Janeiro, Rua São Francisco Xavier 524, sala 536A Rio de Janeiro, Brazil, jpaulo@ieee.org. L. Hsu and R. R. Costa are with the Department of Electrical Engineering, COPPE/Federal University of Rio de Janeiro, P.O. Box Rio de Janeiro, Brazil, s: liu,ramon@coep.ufrj.br. F. Lizarralde is with the Department of Electronics and Computer Engineering, Federal University of Rio de Janeiro, Brazil, fernando@coep.ufrj.br. that K p nom S p is Hurwitz, there exists an open neighborhood around Kp nom such that K ps p is Hurwitz. Such a neighborhood is not necessarily bounded or satisfy a matching condition, i.e., more general uncertainties can be coped with. The following notation and basic concepts are employed in this paper: The maximum and minimum eigenvalues of a symmetric matrix P are denoted as λ max(p ) and λ min(p ), respectively. x denotes the Euclidean norm of a vector x and A = σ max(a) denotes the corresponding induced norm of a matrix A, i.e., the maximum singular value of A. The set of matrices with p rows and m columns whose elements are rational functions of s with real coefficients is denoted R p m (s). The set of polynomial matrices of dimension p m is denoted R p m s. Mixed time domain and Laplace transform domain (operator) representations will be adopted. The output signal of a linear time invariant system with transfer function H(s) and input u is written as H(s)u. Pure convolution operations h(t) u(t), h(t) being the impulse response from H(s), will be written as H(s) u. Filippov s definition for the solution of discontinuous differential equations is assumed 12. II. PROBLEM STATEMENT This paper considers the model-reference control of an observable and controllable MIMO linear time-invariant plant described by ẋ p = A px p + B pu + d(t), y = C px p, (1) where x p R n is the state, u R m is the input, d R m is an unmeasurable input disturbance, and y R m is the output. The corresponding input-output model is y = G(s)u + d(t), (2) where G(s)=C p(si A p) 1 B p R m m (s) is a strictly proper transfer function matrix. We assume that the parameters of the plant model are uncertain, i.e., only known within certain finite bounds. The following assumptions regarding the plant are taken as granted: (A1) The transmission zeros of G(s) have negative real parts; (A2) G(s) has relative degree 1 (i.e., det(c pb p) ) and full rank; (A3) The observability index ν of G(s) is known; (A4) For the high frequency gain matrix K p = C pb p it is assumed that a matrix S p R m m is known such that K ps p is Hurwitz; (A5) The disturbance d(t) is piecewise continuous and a bound d(t) is known such that d(t) d(t) d sup <+, t. The minimum phase assumption (A1) is essential in MRAC schemes 7, 8, 13. Assumption (A2) focuses the simplest case amenable to Lyapunov design. It is verified in practical applications such as helicopter control 9, furnace control 1 and, fault tolerant control of a trailer chain 14. The case det(c pb p) = is fairly more complex as can be seen in some preliminary works 8, 15. Assumption (A3) can be weakened to require only the knowledge of an upper bound on ν, as in 13, which, however, would increase the order of the filters and the number of parameters. A remarkable feature of the proposed method is the Hurwitz condition required in assumption (A4). It relaxes the much more restrictive requirement of positive definiteness and symmetry of K ps p in 7, 8, 13, Section Symmetry is a non generic property. It can be easily destroyed by arbitrarily small uncertainties in the HFG matrix. Moreover, if K ps p is positive definite, then this implies that K ps p is Hurwitz but the converse is not true. This advantage becomes evident in the example in Sec. VI and in the fault tolerant control system presented in 14.

2 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 23 2 The reference model is defined by y M = W M (s)r, W M (s) = diag { 1 s + γ 1,..., } 1, (3) s + γ m with r, y M R m, γ i >, (i = 1,..., m). The reference signal r(t) is assumed piecewise continuous and uniformly bounded. W M (s) has the same relative degree as the plant and its high frequency gain is the identity matrix (i.e., lim s sw M (s) = I). A state space realization of the reference model is given by ẏ M = A M y M + r, A M = diag {γ 1,..., γ m}. (4) The control objective is to achieve asymptotic convergence of the output error (e(t) := y(t) y M (t)) to zero for arbitrary piecewise continuous and uniformly bounded reference signals r(t). III. UNIT VECTOR CONTROL The unit vector control (UVC) law has the form U = ρ(x, t) v(x) v(x), (5) where x R n is the state vector, U R m is the control signal, v : R n R m is a smooth vector function of the state of the system and ρ : R n R R +. We refer to ρ( ) as the unit vector modulation function, which is designed to induce a sliding mode on the manifold v(x) =. To have a complete definition of the control law we will henceforth assume that U = if v(x)=. The following lemma is instrumental for the controller synthesis and stability analysis. We use LI to denote locally integrable in the sense of Lebesgue. Lemma 1: Consider the MIMO system ė(t) = A M e(t) + K U + d U (t) + π(t), (6) U = ρ(e, t) e e, (7) where A M, K R m m ; d U (t) and ρ are LI. The signal π(t) is LI and exponentially decreasing, i.e., π(t) R exp( λt), t, for some positive scalars R and λ. If K is Hurwitz and where it is emphasized that the term c e e(t) in the modulation function (8) is equivalent to the proportional feedback term c ee(t) in (1). The term Kc ee can be added to A M e in (6) resulting in the transfer function matrix (si A M +c ek) 1 K, which is stable with scalar gain feedback U = ke, k R +, provided that K be Hurwitz and c e be large enough. It is noteworthy observing that SPR transfer functions are also stable under scalar gain feedback, which is a common property of SISO and MIMO sliding mode MRACs. IV. CONTROL PARAMETERIZATION If the plant is perfectly known and free of input disturbances (d(t) ), then a control law which achieves matching between the closedloop transfer matrix and W M (s) is given by 18 u = θ T ω, (11) where the parameter matrix θ and the regressor vector ω(t) are given by θ T = θ T 1 θ T 2 θ T 3 θ T 4, ω = ω T 1 ω T 2 y T r T T, (12) ω 1 = A(s)Λ 1 (s)u, ω 2 = A(s)Λ 1 (s)y, (13) A(s) = Is ν 2 Is ν 3... Is I T, Λ(s) = λ(s)i, (14) ω 1, ω 2 R m(ν 1) ; θ 1, θ 2 R m(ν 1) m ; θ 3, θ 4 R m m and, λ(s) is a monic Hurwitz polynomial of degree ν 1. If an input disturbance is present it can be canceled by the additional signal W d (s) d(t) included in the control law (11) as follows u = θ T ω W d (s) d(t), W d (s) = I θ T 1 A(s)Λ 1 (s). (15) The model-reference control scheme is depicted in Fig. 1, where the effect of the disturbance cancellation signal becomes clear. ρ(e, t) δ + c e e(t) + (1 + c d ) d U (t), t, (8) where c e, c d are appropriate constants, and δ is an arbitrary constant, then k 1, k 2, λ 1 > such that e(t) (k 1 e() + k 2R) exp( λ 1t), t. (9) Therefore, the system is globally exponentially stable when π(t). Moreover, if δ >, then the sliding mode at e= is reached after some finite time t s. (Proof: see Appendix.) Corollary 1: If A M = γ M I, γ M >, then Lemma 1 is valid with c e = in (8). (Proof: see Appendix.) Remark 1: Lemma 1 considers MIMO systems with transfer function matrix W M (s)k, where W M (s) = (si A M ) 1 and K R m m is Hurwitz. For SISO systems, if A M is stable, this result can be extended through the Kalman-Yakubovich-Meyer Lemma to any W M (s) strictly positive real (SPR), since W M (s)k (K R) is SPR for any K > 16. In the MIMO case if W M (s) R m m (s) is an SPR transfer function matrix and K is a generic Hurwitz matrix, then W M (s)k may not be SPR, as can be concluded from 17, Lemma 1.1. Remark 2: Equations (7) (8) can be rewritten as U = c ee(t) ρ d (t) e e, (1) ρ d (t) δ + (1 + c d ) d U (t), t, Fig. 1. Model-reference control structure and parameterization. Matching conditions: Consider a right matrix fraction description of the plant G(s) = N R(s)D 1 R (s), NR(s), DR(s) Rm m s. Assuming that no input disturbance is present, the transfer function matrix from r to y is W M (s) if and only if the following Diophantine equation is satisfied 18 { W M (s) = N R(s) Λ(s) θ1 T A(s) D R(s) } 1 θ2 T A(s) + θ3 T Λ(s) N R(s) Λ(s)θ T 4. (16) If the control parameterization is given by (11) (14) then θ such that (16) is satisfied 18, Proposition This matching condition requires that θ4 T = Kp 1. The uniqueness of θ is not guaranteed by this proposition.

3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 23 3 The error equations can be developed following the usual approach for SISO MRAC 19. Define X := x T p ω1 T ω2 T T. Let {A c, B c, C o} be a nonminimal realization of W M (s) with state vector X M. Then, the state error (X e :=X X M ) and the output error satisfy 15 Ẋ e = A cx e + B ck p u θ T ω + W d (s) d(t), (17) e = C ox e. The error equation can be rewritten in input-output form as e = (si A M ) 1 K p u θ T ω + W d (s) d(t). (18) From the control parameterization described above, we now make the following assumption on the class of admissible control laws: (A6) The control law satisfies the inequality sup τ t u(τ) k rd + k ω sup τ t ω(τ), (k rd, k ω > ). This assumption guarantees that no finite time escape occurs in the system signals. Indeed, the system signals will be regular and therefore can grow at most exponentially 18. V. DESIGN AND ANALYSIS OF THE UV-MRAC The UV-MRAC (Unit Vector MRAC) stems from the VS-MRAC (Variable Structure MRAC) structure developed for SISO plants in 6 and generalized to the MIMO case in 8. Compared to the results of 8, the main new features are: (a) global exponential stability properties can be demonstrated and; (b) less restrictive assumption on the plant high frequency gain is required. From the error equation (18), and according to Lemma 1, the proposed UVC law is u = u nom + S pu, U = ρ e e, unom = θ nomt ω, (19) where S p R m m is a design matrix which verifies assumption (A4) and θ nom is some nominal value for θ. The nominal control u nom allows the reduction of the modulation function amplitude if the parameter uncertainty θ θ nom is small. From Lemma 1, exponential convergence of the output error e is achieved if the modulation signal ρ satisfies the inequality ρ δ + c e e + (1 + c d ) (θ nom θ ) T ω + W d (s) d(t), (2) Sp 1 where c e, c d are appropriate constants which satisfy the inequalities (33) given in the Appendix and δ is an arbitrary constant. Noting that d(t) is bounded and W d (s) is proper and stable, an alternative modulation function which satisfies (2) and (A6) is ρ = δ + c 1 ω + c 2 e + c 3 ˆd(t), (21) ˆd(t) = d(t) + c4 s + γ d(t) (22) d I θ1 T A(s)Λ 1 (s) d(t). The upper bound (22) is obtained through the application of Lemma 3 (see Appendix) with γ d > being the stability margin of A(s)Λ 1 (s) (see Lemma 3). The plant transfer function matrix G(s) belongs to some given class P which is a subset of R m m (s). Each element of P satisfies assumptions (A1) (A4) with some fixed S p and ν. The implementation of (21) (22) needs the following assumption: (A7) Values for the constants c i (i = 1,..., 4) and γ d > are known such that inequality (2) is satisfied for any G(s) P with some corresponding θ verifying (16). Details on the computation of these constants are given in Sec. VI. Remark 3: The uniqueness of θ is not required in (A7). A parameterization which satisfies the uniqueness condition found in 2 results in minimal order filters for the generation of ω 1 and ω 2. However, this requires a priori knowledge of all the observability indices of the plant. Remark 4: We note that the signals ω 1 and ω 2 can be expressed as filtered signals of the output y obtained with stable causal filters, see 13, eq. (6.4.12) for the SISO case. From Lemma 3, ω 1 and ω 2 can be bounded by a signal ρ y satisfying ρ y = c 5ρ y +c 6 y, c 5, c 6 >. This result recovers the modulation function found in 2, eq. (6), except that the UV-MRAC does not require the knowledge of K p to be used for the design of the control law and that the switching function is here the unit vector instead of the vector sign function as loc. cit.. However, such a simplified bound may lead to large modulation function. We are now ready to state the main stability result. Theorem 1: Consider the system (17) and (19). If assumptions (A1) (A7) hold, then the UV-MRAC strategy is globally exponentially stable. Moreover, if δ >, then the output error e(t) becomes zero after some finite time. Proof: Throughout the proof k i (i N) denotes appropriate positive constant. The error equation (17) can be represented by the Kalman decomposition Ā 11 Ā 12 B 1 X e = Ā 31 Ā 22 Ā 32 Ā 33 Ā 34 Ā 42 Ā 44 X e + B 3 u u, e = C1 C2 Xe, (23) using some appropriate linear transformation T X e = X e. The Kalman decomposition partitions the system into observable, non observable, controllable and non controllable sub-systems. Since the transfer function matrix of (23) is given by e= C B1u 1(sI Ā11) 1 u which is equal to the error transfer function matrix (18), we have that the plant HFG matrix is given by K p = C 1 B1, where the square matrices C 1 and B 1 are nonsingular. The nonsingularity of these matrices allows the application of the transformation e C 1 C2 x 2 x 3 = T X e, T = I B B I, (24) x 4 I resulting in the regular form 21 ė A 11 A 12 ẋ 2 ẋ 3 = A 22 A 31 A 32 A 33 A 34 ẋ 4 A 42 A 44 e x 2 x 3 + x 4 K p u u. (25) Comparing the transfer function matrix of (25) (e = (si A 11) 1 K pu u ) with the error transfer function matrix (18), we conclude that A 11 = A M. Furthermore A ii (i = 1,..., 4) are Hurwitz matrices since A c is Hurwitz. Applying Lemma 1 to the closed-loop system (19) (2) and (25) with K = K ps p, K being Hurwitz by assumption (A4), π(t) = K 1 A 12 exp(a 22t)x 2() and d U (t)=sp 1 u nom u, we have that the output error is bounded by e(t) (k 1 e() + k 2 x 2() ) exp( λ 1t), (26)

4 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 23 4 t. Moreover, if δ >, then the sliding mode at the point e = starts after some finite time t s. Now, based on the stability of the matrices A ii it is easy to show that x i(t) k 3 exp( λ 5t) X e(), (i = 2, 3, 4), (27) t, where < λ 5 < min{λ 1, λ 2, λ 3, λ 4}, λ i = min j{re( λ ij)} > and {λ ij} is the spectrum of A ii (i = 2, 3, 4). From the bounds (26) and (27), we conclude that X e(t) k 4 exp( λ 5t) X e(), t, which proves that the system is globally exponentially stable. Remark 5: The Hurwitz condition on the matrix K appears to be the least restrictive condition on the plant high frequency gain matrix, since it is a necessary and sufficient condition for the existence of sliding mode in unit vector control systems, according to 22, Theorem 1, 15. Remark 6: The knowledge of a fixed matrix S p such that K ps p is Hurwitz is not needed in the context of adaptive stabilization. In 23, the assumption about S p is weaker. Only a finite set of matrices containing one suitable S p, referred to as spectrum-unmixing set for K p, is required. A mechanism is provided for cycling through the elements of the spectrum-unmixing set. However, the algorithm proposed loc. cit. is not globally exponentially stable and some signals which would theoretically remain finite are prone to become exceedingly large as a consequence of measurement noise. In contrast, the UV-MRAC is globally exponentially stable and has better noise immunity. VI. DESIGN EXAMPLE To illustrate the design of the UV-MRAC, we consider a third order system described by G α(s) = s s α 2 K p, K p =, (28) 2 α where the constant α.3, 4 is uncertain. All other parameters are known. Consequently the plant belongs to the class P = {G α(s) :.3 α 4}. This plant has poles at s = {1, 1, 1}, a transmission zero at s = 1.8 and observability index ν = 2. The input disturbance is uniformly bounded by d(t) 5. If we choose S p = I, K ps p is Hurwitz if and only if 1 < α <.25 or α >. Then the UV-MRAC can be applied. However, in order to keep K ps p positive definite to allow the application of VSC schemes such as 7, 8 the uncertain parameter should satisfy.525 < α < 1.49, which is clearly a more restrictive condition. Moreover, it can be verified that S p R 2 2, det(s p), such that K ps p =(K ps p) T, α.3, 4. Design: The chosen reference model is W M (s)=(s + 2) 1 I. The state filters are chosen with λ(s) = s + 1. A nominal parameter matrix is computed for α nom = 1 which results in (29), where p 1 and p 2 are arbitrary constants which span the complete set of θ satisfying the Diophantine equation (16) for α = 1. We choose θ nom = θ with p 1 = p 2 = which gives a least squares solution of the Diophantine equation. The modulation function (21) should be designed in view of (A7) and aiming at keeping the unit vector control amplitude small in a suboptimal sense (see 24). The constant c 1 is computed to satisfy c 1 c 1 = 2 P K p (θ nom θ ) T, (3) λ min(q) which was developed from (2) and (33) but using a less conservative upper bound P K p(θ nom θ ) T P K ps p Sp 1 (θ nom θ ) T that can be found from (31). The matrix P = P T > satisfies the Lyapunov equation P K ps p+(k ps p) T P =Q> for a given Q which is a free design parameter. We have chosen Q=I. In (3), P, K p and θ depend on the plant uncertain parameter α. The plot of c 1 versus α presented in Fig. 2 let us conclude that c 1 = 17 satisfies inequality (2) for any plant that belongs to P. The modulation function can be simplified since the reference model is such that A M = γ M I, γ M >, then c 2 =, c.f. Corollary 1. The constant c 3 should satisfy c 3 c 3 = 2 P K p /λ min(q) and can be computed through a procedure similar to that applied in the choice of c 1, which gives c 3 = 6.9. Since the disturbance is uniformly bounded, we have ˆd(t) w dc d, where w dc max.3 α 4 ( W d () ) and W d () is the DC gain of W d (s) which depends on the parameter α. Here w dc = 2.8. The constant δ =.1 guarantees finite-time convergence of the output error. Fig. 2. Modulation function coefficient c 1 as a function of the plant uncertain parameter α. Simulation results: The reference signals are r 1(t) = 8 sqw(24t) and r 2(t)=4 sin(18t), where sqw( ):=sgnsin( ) is a square wave. The input disturbance is d(t) = 2, 2 T sqw(3t). Fig. 3 displays the simulation results obtained with the value of the plant parameter being α =.35. The remarkable behavior of the UV-MRAC becomes evident from the fast convergence of the output signals to the reference trajectories. Fig. 3. System (y 1, y 2 ) and model (y M1, y M2 ) outputs. VII. CONCLUSION This paper proposes an output-feedback model-reference sliding mode controller (UV-MRAC) for multivariable linear systems based on the adaptive control formulation and on the unit vector control approach. The high frequency gain (HFG) matrix of the plant (K p) is not assumed to be known. The main result states that the system is globally exponentially stable and can be designed such that the sliding mode surface (e = ) is reached in finite time provided that K ps p be Hurwitz for some known matrix S p. This is less restrictive than the assumption of positive definiteness of K ps p required in previous works that assume not necessarily small uncertainties of the HFG. The extension of the proposed controller for systems of arbitrary relative degree is a current research topic 15.

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 23 5 θ T = p1 p 2 1, 3, 2, 2, 1, 1,,. (29) APPENDIX Proof of Lemma 1: Since K is Hurwitz, there exists P = P T > such that K T P + P K = Q >. Thus, consider the quadratic form V (e)=e T P e which has time derivative V = ρ et Qe e +et (A T MP +PA M )e+2e T P K(d U +π) (31) ρλ min(q) e +λ max(a T MP +PA M ) e σ max(p K) e ( d U + π ). (32) Now, choosing ρ as in (8) with { } λmax(a T M P +P A M ) c e max + λ δ,, min(q) c d 2 σmax(p K) λ min(q) 1, (33) where the constant δ > provides some desired stability margin, we obtain V λ min(q) δ+ δ e (1+c d )R exp( λt) e. (34) Now, in view of the Rayleigh-Ritz inequality λ min(p ) e 2 V (e) λ max(p ) e 2, and denoting c Q1 = λ min(q)/ λ max(p ) (> ), c Q2 = λ min(q)/λ max(p ) (> ) and c D = (1 + c d )λ min(q)/ λ min(p ) (> ), inequality (34) can be rewritten as V δc Q1 V δ cq2v + c DR exp( λt) V. Then, defining r := V, one obtains 2ṙ δc Q1 δ c Q2 r+c DR exp( λt). Thus from Lemma 2, we can conclude that r(t) r()+cr exp( λ 1t), where c > and λ 1 < min(λ, δ c Q2/2). Applying the Rayleigh- Ritz inequality we finally obtain inequality (9). If δ > in (8), from Lemma 2, one can further conclude that t 1 <+ such that r(t), t > t 1, hence, the sliding mode at e = starts in some finite time t s, t s t 1. Proof of Corollary 1: With A M = γ M I, γ M >, we have that λ max(a T M P +P A M ) = 2γ M λ min(p ) and inequality (34) is verified with c e = and δ = 2γ M λ min(p )/λ min(q). The proof can be completed following the proof of Lemma 1. Lemma 2: Let r(t) be an absolutely continuous scalar function. Suppose r(t) is nonnegative and while r > it satisfies ṙ δ γr + R exp( λt), where δ, γ, λ, R are nonnegative constants. Then, one can conclude that: (a) r(t) is bounded by r(t) r() + cr exp( λ 1t), t, where c > is an appropriate constant and λ 1 < min(λ, γ); (b) if δ > then t s < + such that r(t), t t s. Proof: The proof presented in 15, Lemma 3 is based on the Comparison Theorem 12, Theorem 7. Lemma 3: Consider a stable system with strictly proper transfer function matrix W (s) R p m (s). Let γ be the stability margin of W (s), i.e., < γ < min j Re(p j), where {p j} are the poles of W (s). Let d(t) be an instantaneous upper bound of the signal d(t), i.e., d(t) d(t), t. Then, c 1 > such that the impulse response w(t) satisfies w(t) c 1e γt and the inequality w(t) d(t) c 1e γt d(t)= c 1 d(t) holds. s+γ Proof: The proof follows from a direct extension of the scalar case in 25. REFERENCES 1 B. L. Walcott and S. Żak, Combined observer-controller synthesis for uncertain dynamical systems with applications, IEEE Trans. Syst. Man and Cyber., vol. 18, no. 1, pp , S. V. Emelyanov, S. K. Korovin, A. L. Nersisian, and Y. Y. Nisenzon, Discontinuous output feedback stabilizing an uncertain MIMO plant, Int. J. Contr., vol. 55, no. 1, pp , S. Oh and H. K. 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