Output-feedback model-reference sliding mode control of uncertain multivariable systems

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 2003

Output-Feedback Model-Reference Sliding Mode Control of Uncertain Multivariable Systems Jos´e 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 Kp is the knowledge of a matrix Sp such that −Kp Sp 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. I NTRODUCTION 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], [10] for multi-input-multi-output (MIMO) plants. The model-reference controller proposed in [9], [10] 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], [10]. 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 Kp of the plant is the knowledge of a matrix Sp such that −Kp Sp is Hurwitz. This relaxes the positive definiteness property usually needed by other methods [7], [8]. In previous sliding mode schemes [9], [10], [4], [5] there is no explicit restriction on the plant HFG matrix and the input distribution matrix may be uncertain Bp = Bpnom + ∆Bp , where Bpnom is the nominal input matrix and ∆Bp is the uncertainty, but the output distribution matrix Cp is assumed to be known. In [9], [10], [5] the uncertainty is matched, i.e., ∆Bp = Bpnom F (t) with the matrix F (t) being bounded by kF (t)k ≤ kf < 1. A similar restriction on ∆Bp is assumed in [4]. In contrast, the UV-MRAC does not need the boundedness of k∆Bp k which, also, can be an unmatched uncertainty. Indeed, for a given nominal Kp , say Kpnom = Cpnom Bpnom , and some Sp such 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 S˜ao Francisco Xavier 524, sala 5036A — 20550-900 — Rio de Janeiro, Brazil, e-mail: [email protected]. L. Hsu and R. R. Costa are with the Department of Electrical Engineering, COPPE/Federal University of Rio de Janeiro, P.O. Box 68504 — 21945-970 — Rio de Janeiro, Brazil, e-mails: liu,[email protected]. F. Lizarralde is with the Department of Electronics and Computer Engineering, Federal University of Rio de Janeiro, Brazil, e-mail: [email protected].

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that −Kpnom Sp is Hurwitz, there exists an open neighborhood around Kpnom such that −Kp Sp 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 λmax (P ) and λmin (P ), respectively. • kxk denotes the Euclidean norm of a vector x and kAk = σmax (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 Rp×m (s). The set of polynomial matrices of dimension p × m is denoted Rp×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. P ROBLEM S TATEMENT This paper considers the model-reference control of an observable and controllable MIMO linear time-invariant plant described by x˙ p = Ap xp + Bp [u + d(t)] , n

m

y = C p xp ,

(1)

m

where xp ∈ R is the state, u ∈ R is the input, d ∈ R is an unmeasurable input disturbance, and y ∈ Rm is the output. The corresponding input-output model is y = G(s)[u + d(t)] ,

(2)

where G(s) = Cp (sI−Ap )−1 Bp ∈ Rm×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(Cp Bp ) 6= 0) and full rank; (A3) The observability index ν of G(s) is known; (A4) For the high frequency gain matrix Kp = Cp Bp it is assumed that a matrix Sp ∈ Rm×m is known such that −Kp Sp is Hurwitz; (A5) The disturbance d(t) is ¯ is known such that kd(t)k ≤ piecewise continuous and a bound d(t) ¯ ≤ d¯sup < +∞, ∀t ≥ 0. d(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 [10] and, fault tolerant control of a trailer chain [14]. The case det(Cp Bp ) = 0 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 Kp Sp in [7], [8], [13, Section 9.7.3]. Symmetry is a non generic property. It can be easily destroyed by arbitrarily small uncertainties in the HFG matrix. Moreover, if Kp Sp is positive definite, then this implies that −Kp Sp 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].

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 2003

The reference model is defined by yM = WM (s)r ,

WM (s) = diag



1 1 ,..., s + γ1 s + γm



, (3)

with r, yM ∈ Rm , γi > 0, (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., lims→∞ sWM (s) = I). A state space realization of the reference model is given by y˙ M = AM yM + r ,

AM = −diag {γ1 , . . . , γm } .

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The control objective is to achieve asymptotic convergence of the output error (e(t) := y(t) − yM (t)) to zero for arbitrary piecewise continuous and uniformly bounded reference signals r(t). III. U NIT V ECTOR C ONTROL

where it is emphasized that the term ce ke(t)k in the modulation function (8) is equivalent to the proportional feedback term −ce e(t) in (10). The term −Kce e can be added to AM e in (6) resulting in the transfer function matrix (sI − AM + ce K)−1 K, which is stable with scalar gain feedback U = −ke, ∀k ∈ R+ , provided that −K be Hurwitz and ce 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. C ONTROL PARAMETERIZATION If the plant is perfectly known and free of input disturbances (d(t) ≡ 0), then a control law which achieves matching between the closedloop transfer matrix and WM (s) is given by [18] u∗ = θ∗T ω ,

(11)

where the parameter matrix θ ∗ and the regressor vector ω(t) are given by

The unit vector control (UVC) law has the form U = −ρ(x, t)

2

v(x) , kv(x)k

(5)

where x ∈ Rn is the state vector, U ∈ Rm is the control signal, v : Rn → Rm is a smooth vector function of the state of the system and ρ : Rn × 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) = 0. To have a complete definition of the control law we will henceforth assume that U = 0 if v(x) = 0. 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 e(t) ˙ = AM e(t) + K [U + dU (t) + π(t)] , e U = −ρ(e, t) , kek

(6)

θ∗T = [θ1∗T θ2∗T θ3∗T θ4∗T ] , ω1 = A(s)Λ A(s) = [Is

−1

ν−2

(s)u ,

Is

ν−3

ω = [ω1T ω2T y T rT ]T ,

(12)

−1

(s)y ,

(13)

Λ(s) = λ(s)I ,

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ω2 = A(s)Λ T

. . . Is I] ,

ω1 , ω2 ∈ Rm(ν−1) ; θ1∗ , θ2∗ ∈ Rm(ν−1)×m ; θ3∗ , θ4∗ ∈ Rm×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 Wd (s)∗d(t) included in the control law (11) as follows u∗ = θ∗T ω−Wd (s) ∗ d(t) ,

Wd (s) = I −θ1∗T 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.

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where AM , K ∈ Rm×m ; dU (t) and ρ are LI. The signal π(t) is LI and exponentially decreasing, i.e., kπ(t)k ≤ R exp(−λt), ∀t ≥ 0, for some positive scalars R and λ. If −K is Hurwitz and ρ(e, t) ≥ δ + ce ke(t)k + (1 + cd )kdU (t)k ,

∀t ≥ 0 ,

(8)

where ce , cd ≥ 0 are appropriate constants, and δ ≥ 0 is an arbitrary constant, then ∃k1 , k2 , λ1 > 0 such that ke(t)k ≤ (k1 ke(0)k + k2 R) exp(−λ1 t) ,

∀t ≥ 0 .

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Therefore, the system is globally exponentially stable when π(t) ≡ 0. Moreover, if δ > 0, then the sliding mode at e = 0 is reached after some finite time ts ≥ 0. (Proof: see Appendix.) Corollary 1: If AM = −γM I, γM > 0, then Lemma 1 is valid with ce = 0 in (8). (Proof: see Appendix.) Remark 1: Lemma 1 considers MIMO systems with transfer function matrix WM (s)K, where WM (s) = (sI − AM )−1 and −K ∈ Rm×m is Hurwitz. For SISO systems, if AM is stable, this result can be extended through the Kalman-Yakubovich-Meyer Lemma to any WM (s) strictly positive real (SPR), since WM (s)K (K ∈ R) is SPR for any K > 0 [16]. In the MIMO case if WM (s) ∈ Rm×m (s) is an SPR transfer function matrix and −K is a generic Hurwitz matrix, then WM (s)K may not be SPR, as can be concluded from [17, Lemma 10.1]. Remark 2: Equations (7)–(8) can be rewritten as e , (10) U = −ce e(t) − ρd (t) kek ρd (t) ≥ δ + (1 + cd )kdU (t)k ,

∀t ≥ 0 ,

Fig. 1. Model-reference control structure and parameterization.

Matching conditions: Consider a right matrix fraction description of −1 the plant G(s) = NR (s)DR (s), NR (s), DR (s) ∈ Rm×m [s]. Assuming that no input disturbance is present, the transfer function matrix from r to y is WM (s) if and only if the following Diophantine equation is satisfied [18] nh i WM (s) = NR (s) Λ(s) − θ1∗T A(s) DR (s) h i o−1 − θ2∗T A(s) + θ3∗T Λ(s) NR (s) Λ(s)θ4∗T . (16) If the control parameterization is given by (11)–(14) then ∃θ ∗ such that (16) is satisfied [18, Proposition 6.3.3]. This matching condition requires that θ4∗T = Kp−1 . The uniqueness of θ ∗ is not guaranteed by this proposition.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 2003

The error equations can be developed following the usual approach for SISO MRAC [19]. Define X := [xTp ω1T ω2T ]T . Let {Ac , Bc , Co } be a nonminimal realization of WM (s) with state vector XM . Then, the state error (Xe := X − XM ) and the output error satisfy [15] h i X˙ e = Ac Xe + Bc Kp u − θ ∗T ω + Wd (s) ∗ d(t) , (17) e = C o Xe .

The error equation can be rewritten in input-output form as h i e = (sI − AM )−1 Kp u − θ ∗T ω + Wd (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 sup0≤τ ≤t ku(τ )k ≤ krd + kω sup0≤τ ≤t kω(τ )k , (krd , kω > 0). 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. D ESIGN AND A NALYSIS 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 = unom + Sp U , e U = −ρ , unom = θnomT ω , kek

(19)

m×m

where Sp ∈ R is a design matrix which verifies assumption (A4) and θ nom is some nominal value for θ ∗ . The nominal control unom allows the reduction of the modulation function amplitude if the parameter uncertainty kθ ∗ − θnom k is small. From Lemma 1, exponential convergence of the output error e is achieved if the modulation signal ρ satisfies the inequality ρ ≥ δ + ce kek + (1 + cd )×

h i

T × Sp−1 (θnom − θ∗ ) ω + Wd (s) ∗ d(t) ,

(20)

where ce , cd ≥ 0 are appropriate constants which satisfy the inequalities (33) given in the Appendix and δ ≥ 0 is an arbitrary constant. Noting that d(t) is bounded and Wd (s) is proper and stable, an alternative modulation function which satisfies (20) and (A6) is ˆ , ρ = δ + c1 kωk + c2 kek + c3 d(t) c 4 ¯ ˆ = d(t) ¯ + ∗ d(t) d(t) s + γd

h

i

≥ I − θ1∗T A(s)Λ−1 (s) ∗ d(t) .

(21) (22)

The upper bound (22) is obtained through the application of Lemma 3 (see Appendix) with γd > 0 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 Rm×m (s). Each element of P satisfies assumptions (A1)–(A4) with some fixed Sp and ν. The implementation of (21)–(22) needs the following assumption: (A7) Values for the constants ci ≥ 0 (i = 1, . . . , 4) and γd > 0 are known such that inequality

3

(20) 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 [20] results in minimal order filters for the generation of ω1 and ω2 . However, this requires a priori knowledge of all the observability indices of the plant. Remark 4: We note that the signals ω1 and ω2 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, kω1 k and kω2 k can be bounded by a signal ρy satisfying ρ˙ y = −c5 ρy +c6 kyk , c5 , c6 > 0 . This result recovers the modulation function found in [2, eq. (6)], except that the UV-MRAC does not require the knowledge of Kp 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 > 0, then the output error e(t) becomes zero after some finite time. Proof: Throughout the proof ki (i ∈ N) denotes appropriate positive constant. The error equation (17) can be represented by the Kalman decomposition     ¯1 A¯11 A¯12 0 0 B ¯22  0  0 A 0 0 ˙ ¯e =   ¯   [u − u∗ ] , X ¯3  A¯31 A¯32 A¯33 A¯34  Xe + B 0 A¯42 0 A¯44 0   ¯2 0 0 X ¯1 C ¯e , (23) e= C ¯ e . The using some appropriate linear transformation T¯Xe = X Kalman decomposition partitions the system into observable, non observable, controllable and non controllable sub-systems. Since the ¯1 (sI − A¯11 )−1 B ¯1 [u− transfer function matrix of (23) is given by e = C u∗ ] which is equal to the error transfer function matrix (18), we have ¯1 , where the square ¯1 B that the plant HFG matrix is given by Kp = C ¯ ¯ matrices C1 and B1 are nonsingular. The nonsingularity of these matrices allows the application of the transformation    ¯2 0 0 ¯1 C C e  x2  0 I 0 0 ¯e , ,   = TX (24) T = ¯ −1 ¯3 B −B x3  0 I 0 1 x4 0 0 0 I

resulting in the regular form [21]        e˙ A11 A12 0 0 e Kp x˙ 2   0   x2   0  A 0 0 22 ∗  =     x˙ 3  A31 A32 A33 A34  x3  +  0  [u − u ] . x˙ 4 0 A42 0 A44 x4 0 (25)

Comparing the transfer function matrix of (25) (e = (sI − A11 )−1 Kp [u − u∗ ]) with the error transfer function matrix (18), we conclude that A11 = AM . Furthermore Aii (i = 1, . . . , 4) are Hurwitz matrices since Ac is Hurwitz. Applying Lemma 1 to the closed-loop system (19)–(20) and (25) with K = Kp Sp , −K being Hurwitz by assumption (A4), π(t) = K −1 A12 exp(A22 t)x2 (0) and dU (t) = Sp−1 [unom − u∗ ], we have that the output error is bounded by ke(t)k ≤ (k1 ke(0)k + k2 kx2 (0)k) exp(−λ1 t) ,

(26)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 2003

∀t ≥ 0. Moreover, if δ > 0, then the sliding mode at the point e = 0 starts after some finite time ts ≥ 0. Now, based on the stability of the matrices Aii it is easy to show that kxi (t)k ≤ k3 exp(−λ5 t)kXe (0)k ,

(i = 2, 3, 4) ,

(27)

∀t ≥ 0, where 0 < λ5 < min{λ1 , λ2 , λ3 , λ4 }, λi = minj {Re(−λij )} > 0 and {λij } is the spectrum of Aii (i = 2, 3, 4). From the bounds (26) and (27), we conclude that kX e (t)k ≤ k4 exp(−λ5 t)kXe (0)k , ∀t ≥ 0, 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 Sp such that −Kp Sp is Hurwitz is not needed in the context of adaptive stabilization. In [23], the assumption about Sp is weaker. Only a finite set of matrices containing one suitable Sp , referred to as spectrum-unmixing set for Kp , 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. D ESIGN E XAMPLE To illustrate the design of the UV-MRAC, we consider a third order system described by " s+0.6   −1.2 # 1 2α2 s2 −1 s2 −1 Gα (s) = K K , (28) p, p = s+2.2 0.4 −2 α 2 2 s −1

s −1

where the constant α ∈ [0.3, 4] is uncertain. All other parameters are known. Consequently the plant belongs to the class P = {Gα (s) : 0.3 ≤ 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 ¯ ≡ 5. is uniformly bounded by d(t) If we choose Sp = I, −Kp Sp is Hurwitz if and only if −1 < α < −0.25 or α > 0. Then the UV-MRAC can be applied. However, in order to keep Kp Sp positive definite to allow the application of VSC schemes such as [7], [8] the uncertain parameter should satisfy 0.525 < α < 1.490, which is clearly a more restrictive condition. Moreover, it can be verified that @Sp ∈ R2×2 , det(Sp ) 6= 0, such that Kp Sp = (Kp Sp )T , ∀α ∈ [0.3, 4]. Design: The chosen reference model is WM (s) = (s + 2)−1 I. The state filters are chosen with λ(s) = s + 1. A nominal parameter matrix is computed for αnom = 1 which results in (29), where p1 and p2 are arbitrary constants which span the complete set of θ ∗ satisfying the Diophantine equation (16) for α = 1. We choose θ nom = θ∗ with p1 = p2 = 0 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 c1 is computed to satisfy

2

T nom (30) − θ∗ ) , c1 ≥ c¯1 =

P Kp (θ λmin (Q)

which was developed from (20) and (33) but using a less conservative upper bound kP Kp (θnom − θ∗ )T k ≤ kP Kp Sp kkSp−1 (θnom − θ∗ )T k that can be found from (31). The matrix P = P T > 0 satisfies the

4

Lyapunov equation P Kp Sp+(Kp Sp )T P = Q > 0 for a given Q which is a free design parameter. We have chosen Q = I. In (30), P , Kp and θ∗ depend on the plant uncertain parameter α. The plot of c¯1 versus α presented in Fig. 2 let us conclude that c1 = 17 satisfies inequality (20) for any plant that belongs to P. The modulation function can be simplified since the reference model is such that AM = −γM I, γM > 0, then c2 = 0, c.f. Corollary 1. The constant c3 should satisfy c3 ≥ c¯3 = 2kP Kp k/λmin (Q) and can be computed through a procedure similar to that applied in the choice of c1 , which gives c3 = 6.9. Since ¯ where ˆ ≡ wdc d, the disturbance is uniformly bounded, we have d(t) wdc ≥ max0.3≤α≤4 (kWd (0)k) and Wd (0) is the DC gain of Wd (s) which depends on the parameter α. Here wdc = 2.8. The constant δ = 0.1 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 r1 (t) = 80 sqw(24t) and r2 (t) = 40 sin(18t), where sqw(·) := sgn[sin(·)] is a square wave. The input disturbance is d(t) = [2, 2]T sqw(30t). Fig. 3 displays the simulation results obtained with the value of the plant parameter being α = 0.35. The remarkable behavior of the UV-MRAC becomes evident from the fast convergence of the output signals to the reference trajectories.

Fig. 3. System (y1 , y2 ) and model (yM 1 , yM 2 ) outputs.

VII. C ONCLUSION 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 (Kp ) 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 = 0) is reached in finite time provided that −Kp Sp be Hurwitz for some known matrix Sp . This is less restrictive than the assumption of positive definiteness of Kp Sp 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].

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θ∗T =



−0.99 −0.23

0.57 −0.11

−0.42 −0.34

−0.42 −0.34

−0.39 −1.03

A PPENDIX

T

e Qe V˙ = −ρ +eT (ATMP +PAM )e+2eT P K(dU +π) kek ≤ −ρλmin (Q)kek+λmax (ATMP +PAM )kek2 + 2 σmax (P K)kek (kdU k+kπk) .

(31)

(32)

Now, choosing ρ as in (8) with λmax (ATM P +P AM ) ¯ ce ≥ max +δ, 0 λmin (Q) σmax (P K) − 1, cd ≥ 2 λmin (Q)

0.2 0.4

   p1 −0.4 [1, −3, −2, −2, 1, 1, 0, 0] . + p2 0.2

(29)

R EFERENCES

Proof of Lemma 1: Since −K is Hurwitz, there exists P = P T > 0 such that K T P + P K = Q > 0 . Thus, consider the quadratic form V (e) = eT P e which has time derivative



1.41 −0.43

5



, (33)

where the constant δ¯ > 0 provides some desired stability margin, we obtain   ¯ V˙ ≤ −λmin (Q) δ+ δkek−(1+c d )R exp(−λt) kek .

(34)

2 Now, in view of the Rayleigh-Ritz inequality λmin (P p) kek ≤ 2 V (e) ≤ λmax (P ) kek , and denoting cQ1 = λmin (Q)/ λmax (P ) (> 0), cQ2 p= λmin (Q)/λmax (P ) (> 0) and cD = (1 + λmin (P ) (> 0), inequality (34) cd )λmin (Q)/ √ √ can be rewritten ¯ ˙ as V √≤ −δcQ1 V − δ cQ2 V + cD R exp(−λt) V . Then, defining r := V , one obtains 2r˙ ≤ −δcQ1 − δ¯ cQ2 r +cD R exp(−λt) . Thus from Lemma 2, we can conclude that r(t) ≤ [r(0)+cR] exp(−λ1 t) , where c > 0 and λ1 < min(λ, δ¯ cQ2 /2). Applying the RayleighRitz inequality we finally obtain inequality (9). If δ > 0 in (8), from Lemma 2, one can further conclude that ∃t1 < +∞ such that r(t) ≡ 0, ∀t > t1 , hence, the sliding mode at e = 0 starts in some finite time ts , 0 ≤ t s ≤ t1 .  Proof of Corollary 1: With AM = −γM I, γM > 0, we have that λmax (ATM P +P AM ) = −2γM λmin (P ) and inequality (34) is verified with ce = 0 and δ¯ = 2γM λmin (P )/λmin (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 > 0 it satisfies r˙ ≤ −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(0) + cR] exp(−λ1 t) , ∀t ≥ 0 , where c > 0 is an appropriate constant and λ1 < min(λ, γ); (b) if δ > 0 then ∃ts < +∞ such that r(t) ≡ 0 , ∀t ≥ ts . 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) ∈ Rp×m (s). Let γ be the stability margin of W (s), i.e., 0 < γ < minj |Re(pj )|, where {pj } are the poles ¯ be an instantaneous upper bound of the signal of W (s). Let d(t) ¯ d(t), i.e., kd(t)k ≤ d(t), ∀t ≥ 0. Then, ∃c1 > 0 such that the impulse response w(t) satisfies kw(t)k ≤ c1 e−γt and the inequality ¯ holds. ¯ = c1 ∗ d(t) kw(t)∗d(t)k ≤ c1 e−γt ∗ d(t) s+γ Proof: The proof follows from a direct extension of the scalar case in [25]. 

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