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nom Output- Model-Reference Sliding Mode Control that Kp Sp is Hurwitz, there exists an open neighborhood around nom− of Uncertain Multivariable Systems Kp such that KpSp is Hurwitz. Such a neighborhood is not nec- essarily bounded−or satisfy a matching condition, i.e., more general Jose´ Paulo V. S. Cunha, Liu Hsu, Ramon R. Costa uncertainties can be coped with. and Fernando Lizarralde The following notation and basic concepts are employed in this pa- per: Abstract—This paper considers the robust output tracking problem us- • The maximum and minimum eigenvalues of a symmetric matrix ing a model-reference sliding mode controller for linear multivariable sys- P are denoted as λmax(P ) and λmin(P ), respectively. tems of relative degree one. It is shown that the closed loop system is glob- • x denotes the Euclidean norm of a vector x and A = ally exponentially stable and the performance is insensitive to bounded in- k k k k σmax(A) denotes the corresponding induced norm of a matrix put disturbances and parameter uncertainties. The strategy is based on output-feedback unit to generate sliding mode. The only re- A, i.e., the maximum singular value of A. quired a priori information about the plant high frequency gain matrix Kp • The set of matrices with p rows and m columns whose ele- is the knowledge of a matrix Sp such that −KpSp is Hurwitz which relaxes ments are rational functions of s with real coefficients is denoted the positive definiteness requirement usually needed by other methods. Rp×m(s). The set of polynomial matrices of dimension p m is p×m × Index Terms—sliding mode, model-reference control, multivariable sys- denoted R [s]. tems, output-feedback, unit vector control. • Mixed time domain and Laplace transform domain (operator) representations will be adopted. The output signal of a linear time invariant system with transfer H(s) and input u is writ- I. INTRODUCTION ten as H(s)u. Pure convolution operations h(t) u(t), h(t) being ∗ One classical approach to the robust control of multivariable sys- the impulse response from H(s), will be written as H(s) u. tems is variable structure control (VSC). A primary problem is robust ∗ • Filippov’s definition for the solution of discontinuous differential stabilization by output feedback which is the subject of several works, equations is assumed [12]. e.g., [1], [2], [3], [4], [5]. A more challenging problem is the robust output tracking of a reference signal using only output measurement. II. PROBLEM STATEMENT In this case a standard approach is to specify the desired closed loop re- sponse using a reference model. In this framework, variable structure This paper considers the model-reference control of an observable control approach has been applied in [6] for single-input-single-output and controllable MIMO linear time-invariant plant described by

(SISO) plants and in [7], [8], [9], [10] for multi-input-multi-output x˙ p = Apxp + Bp[u + d(t)] , y = Cpxp , (1) (MIMO) plants. n m m The model-reference controller proposed in [9], [10] is based on a where xp R is the state, u R is the input, d R is an unmeasur- ∈ ∈ m ∈ state space description of the plant and a nonlinear observer to esti- able input disturbance, and y R is the output. The corresponding ∈ mate the plant state. In contrast, the present paper, likewise [7], [8], input-output model is relies on the plant transfer function matrix formulation and follows the y = G(s)[u + d(t)] , (2) model-reference adaptive control (MRAC) approach without explicit −1 m×m state observers [11]. Our approach allows the demonstration of global where G(s)=Cp(sI Ap) Bp R (s) is a strictly proper trans- properties, which are not analyzed in [7], [8], [9], fer function matrix. W−e assume that∈ the parameters of the plant model [10]. are uncertain, i.e., only known within certain finite bounds. The fol- This paper proposes a unit vector model-reference sliding mode lowing assumptions regarding the plant are taken as granted: (A1) The controller (UV-MRAC) for MIMO plants with relative degree one. transmission zeros of G(s) have negative real parts; (A2) G(s) has The only required a priori information about the high frequency gain relative degree 1 (i.e., det(CpBp) = 0) and full rank; (A3) The ob- 6 (HFG) matrix Kp of the plant is the knowledge of a matrix Sp such servability index ν of G(s) is known; (A4) For the high frequency m×m that KpSp is Hurwitz. This relaxes the positive definiteness prop- gain matrix Kp = CpBp it is assumed that a matrix Sp R − ∈ erty usually needed by other methods [7], [8]. In previous sliding is known such that KpSp is Hurwitz; (A5) The disturbance d(t) is − mode schemes [9], [10], [4], [5] there is no explicit restriction on piecewise continuous and a bound d¯(t) is known such that d(t) k k ≤ the plant HFG matrix and the input distribution matrix may be un- d¯(t) d¯sup <+ , t 0. nom nom ≤ ∞ ∀ ≥ certain Bp =Bp + ∆Bp, where Bp is the nominal input matrix The minimum phase assumption (A1) is essential in MRAC and ∆Bp is the uncertainty, but the output distribution matrix Cp is schemes [7], [8], [13]. Assumption (A2) focuses the simplest case assumed to be known. In [9], [10], [5] the uncertainty is matched, amenable to Lyapunov design. It is verified in practical applications nom i.e., ∆Bp = Bp F (t) with the matrix F (t) being bounded by such as helicopter control [9], furnace control [10] and, fault tolerant F (t) kf < 1. A similar restriction on ∆Bp is assumed in control of a trailer chain [14]. The case det(C B ) = 0 is fairly more k k ≤ p p [4]. In contrast, the UV-MRAC does not need the boundedness of complex as can be seen in some preliminary works [8], [15]. Assump- ∆Bp which, also, can be an unmatched uncertainty. Indeed, for tion (A3) can be weakened to require only the knowledge of an upper k k nom nom nom a given nominal Kp, say Kp = Cp Bp , and some Sp such bound on ν, as in [13], which, however, would increase the order of the filters and the number of parameters. This work was partially supported by FAPERJ and CNPq, Brazil. A remarkable feature of the proposed method is the Hurwitz condi- J. P. V. S. Cunha is with the Department of Electronics and Telecommu- nication Engineering, State University of Rio de Janeiro, Rua Sao˜ Francisco tion required in assumption (A4). It relaxes the much more restrictive Xavier 524, sala 5036A — 20550-900 — Rio de Janeiro, Brazil, e-mail: requirement of positive definiteness and symmetry of KpSp in [7], [email protected]. [8], [13, Section 9.7.3]. Symmetry is a non generic property. It can be L. Hsu and R. R. Costa are with the Department of Electrical Engineering, easily destroyed by arbitrarily small uncertainties in the HFG matrix. COPPE/Federal University of Rio de Janeiro, P.O. Box 68504 — 21945-970 — Moreover, if KpSp is positive definite, then this implies that KpSp Rio de Janeiro, Brazil, e-mails: liu,[email protected]. − F. Lizarralde is with the Department of Electronics and Computer is Hurwitz but the converse is not true. This advantage becomes evi- Engineering, Federal University of Rio de Janeiro, Brazil, e-mail: dent in the example in Sec. VI and in the fault tolerant control system [email protected]. presented in [14]. IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 2003 2

The reference model is defined by where it is emphasized that the term ce e(t) in the modulation func- k k tion (8) is equivalent to the proportional feedback term cee(t) in 1 1 − yM = WM (s)r , WM (s) = diag , . . . , , (3) (10). The term Kcee can be added to AM e in (6) resulting in the s + γ1 s + γm − −1   transfer function matrix (sI AM +ceK) K, which is stable with m − + with r, yM R , γi > 0, (i = 1, . . . , m). The reference signal r(t) scalar gain feedback U = ke, k R , provided that K be Hurwitz ∈ − ∀ ∈ − is assumed piecewise continuous and uniformly bounded. WM (s) has and ce be large enough. It is noteworthy observing that SPR transfer the same relative degree as the plant and its high frequency gain is the functions are also stable under scalar gain feedback, which is a com- identity matrix (i.e., lims→∞ sWM (s) = I). A state space realization mon property of SISO and MIMO sliding mode MRACs. of the reference model is given by IV. CONTROL PARAMETERIZATION y˙M = AM yM + r , AM = diag γ1, . . . , γm . (4) − { } If the plant is perfectly known and free of input disturbances (d(t) ≡ The control objective is to achieve asymptotic convergence of the 0), then a control law which achieves matching between the closed- output error (e(t) := y(t) yM (t)) to zero for arbitrary piecewise loop transfer matrix and is given by [18] − WM (s) continuous and uniformly bounded reference signals r(t). u∗ = θ∗T ω , (11) III. UNIT VECTOR CONTROL where the parameter matrix θ∗ and the regressor vector ω(t) are given The unit vector control (UVC) law has the form by v(x) U = ρ(x, t) , (5) ∗T ∗T ∗T ∗T ∗T T T T T T − v(x) θ = [θ1 θ2 θ3 θ4 ] , ω = [ω1 ω2 y r ] , (12) k k −1 −1 where x Rn is the state vector, U Rm is the control signal, v : ω1 = A(s)Λ (s)u , ω2 = A(s)Λ (s)y , (13) n ∈m ∈ R R is a smooth vector function of the state of the system and A(s) = [Isν−2 Isν−3 . . . Is I]T , Λ(s) = λ(s)I , (14) →n ρ : R R R+. We refer to ρ( ) as the unit vector modulation × → · m(ν−1) ∗ ∗ m(ν−1)×m ∗ ∗ m×m function, which is designed to induce a sliding mode on the manifold ω1, ω2 R ; θ , θ R ; θ , θ R and, λ(s) is ∈ 1 2 ∈ 3 4 ∈ v(x) = 0. To have a complete definition of the control law we will a monic Hurwitz polynomial of degree ν 1. If an input disturbance is − henceforth assume that U =0 if v(x)=0. present it can be canceled by the additional signal Wd(s) d(t) included ∗ The following lemma is instrumental for the controller synthesis and in the control law (11) as follows stability analysis. We use “LI” to denote locally integrable in the sense ∗ ∗T ∗T −1 u = θ ω Wd(s) d(t) , Wd(s) = I θ A(s)Λ (s) . (15) of Lebesgue. − ∗ − 1 Lemma 1: Consider the MIMO system The model-reference control scheme is depicted in Fig. 1, where the

e˙(t) = AM e(t) + K [U + dU (t) + π(t)] , (6) effect of the disturbance cancellation signal becomes clear. e U = ρ(e, t) , (7) − e k k m×m where AM , K R ; dU (t) and ρ are LI. The signal π(t) is LI ∈ and exponentially decreasing, i.e., π(t) R exp( λt), t 0, for k k ≤ − ∀ ≥ some positive scalars R and λ. If K is Hurwitz and − ρ(e, t) δ + ce e(t) + (1 + cd) dU (t) , t 0 , (8) ≥ k k k k ∀ ≥ where ce, cd 0 are appropriate constants, and δ 0 is an arbitrary ≥ ≥ constant, then k1, k2, λ1 >0 such that ∃ e(t) (k1 e(0) + k2R) exp( λ1t) , t 0 . (9) k k ≤ k k − ∀ ≥ 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 Fig. 1. Model-reference control structure and parameterization. − ce =0 in (8). (Proof: see Appendix.) Remark 1: Lemma 1 considers MIMO systems with transfer func- Matching conditions: Consider a right matrix fraction description of −1 −1 m×m tion matrix WM (s)K, where WM (s) = (sI AM ) and K the plant G(s)=NR(s)DR (s), NR(s), DR(s) R [s]. Assum- m×m − − ∈ ∈ R is Hurwitz. For SISO systems, if AM is stable, this result ing that no input disturbance is present, the transfer function matrix can be extended through the Kalman-Yakubovich-Meyer Lemma to any from r to y is WM (s) if and only if the following Diophantine equa- WM (s) strictly positive real (SPR), since WM (s)K (K R) is SPR tion is satisfied [18] ∈m×m for any K > 0 [16]. In the MIMO case if WM (s) R (s) is ∈ ∗T an SPR transfer function matrix and K is a generic Hurwitz ma- WM (s) = NR(s) Λ(s) θ1 A(s) DR(s) − − trix, then WM (s)K may not be SPR, as can be concluded from [17, −1 ∗T nh ∗T i ∗T Lemma 10.1]. θ2 A(s) + θ3 Λ(s) NR(s) Λ(s)θ4 . (16) − Remark 2: Equations (7)–(8) can be rewritten as h i o If the control parameterization is given by (11)–(14) then θ∗ such e ∃ U = cee(t) ρd(t) , (10) that (16) is satisfied [18, Proposition 6.3.3]. This matching condition − − e ∗T −1 ∗ k k requires that θ4 = Kp . The uniqueness of θ is not guaranteed by ρd(t) δ + (1 + cd) dU (t) , t 0 , ≥ k k ∀ ≥ this proposition. IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 2003 3

The error equations can be developed following the usual approach (20) is satisfied for any G(s) with some corresponding θ∗ veri- T T T T ∈ P for SISO MRAC [19]. Define X := [x ω ω ] . Let Ac, Bc, Co fying (16). Details on the computation of these constants are given in p 1 2 { } be a nonminimal realization of WM (s) with state vector XM . Then, Sec. VI. ∗ the state error (Xe :=X XM ) and the output error satisfy [15] Remark 3: The uniqueness of θ is not required in (A7). A pa- − rameterization which satisfies the uniqueness condition found in [20] ∗T X˙ e = AcXe + BcKp u θ ω + Wd(s) d(t) , (17) results in minimal order filters for the generation of ω and ω . How- − ∗ 1 2 ever, this requires a priori knowledge of all the observability indices e = C X . h i o e of the plant. The error equation can be rewritten in input-output form as 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, −1 ∗T see [13, eq. (6.4.12)] for the SISO case. From Lemma 3, ω1 and e = (sI AM ) Kp u θ ω + Wd(s) d(t) . (18) k k − − ∗ ω2 can be bounded by a signal ρy satisfying ρ˙y = c5ρy +c6 y , h i k k − k k From the control parameterization described above, we now make c5, c6 > 0 . This result recovers the modulation function found in [2, the following assumption on the class of admissible control laws: 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 (A6) The control law satisfies the inequality sup0≤τ≤t u(τ) k k ≤ function is here the unit vector instead of the vector sign function as krd + kω sup0≤τ≤t ω(τ) , (krd, kω > 0). This assumption guar- antees that no finite timek escapek occurs in the system signals. Indeed, loc. cit.. However, such a simplified bound may lead to large modula- the system signals will be regular and therefore can grow at most ex- tion function. ponentially [18]. 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 V. DESIGN AND ANALYSIS OF THE UV-MRAC stable. Moreover, if δ > 0, then the output error e(t) becomes zero af- The UV-MRAC (Unit Vector MRAC) stems from the VS-MRAC ter some finite time. (Variable Structure MRAC) structure developed for SISO plants in [6] Proof: Throughout the proof ki (i N) denotes appropriate positive and generalized to the MIMO case in [8]. Compared to the results of constant. The error equation (17) can∈ be represented by the Kalman [8], the main new features are: (a) global exponential stability prop- decomposition erties can be demonstrated and; (b) less restrictive assumption on the plant high frequency gain is required. A¯11 A¯12 0 0 B¯1 ¯ From the error equation (18), and according to Lemma 1, the pro- ˙ 0 A22 0 0 0 ∗ X¯e =   X¯e +   [u u ] , posed UVC law is A¯31 A¯32 A¯33 A¯34 B¯3 −  0 A¯42 0 A¯44  0  u = unom + S U ,     p ¯ ¯ ¯    e e = C1 C2 0 0 Xe , (23) U = ρ , unom = θnomT ω , (19) − e   ¯ ¯ k k using some appropriate linear transformation T Xe = Xe. The m×m Kalman decomposition partitions the system into observable, non ob- where Sp R is a design matrix which verifies assumption (A4) nom ∈ ∗ nom servable, controllable and non controllable sub-systems. Since the and θ is some nominal value for θ . The nominal control u al- −1 transfer function matrix of (23) is given by e= C¯1(sI A¯11) B¯1[u lows the reduction of the modulation function amplitude if the param- ∗ − − ∗ nom which is equal to the error transfer function matrix (18), we have eter uncertainty θ θ is small. From Lemma 1, exponential u ] k − k that the plant HFG matrix is given by ¯ ¯ , where the square convergence of the output error e is achieved if the modulation signal Kp = C1B1 matrices ¯ and ¯ are nonsingular. The nonsingularity of these ma- ρ satisfies the inequality C1 B1 trices allows the application of the transformation

ρ δ + ce e + (1 + cd) ≥ k k × e C¯1 C¯2 0 0 −1 nom ∗ T Sp (θ θ ) ω + Wd(s) d(t) , (20) x2 ¯ 0 I 0 0 × − ∗   = T Xe , T =  −1  , (24) x3 B¯3B¯ 0 I 0 h i − 1 where ce, cd 0 are appropriate constants which satisfy the inequal- x   0 0 0 I ≥  4   ities (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 alter- resulting in the regular form [21] native modulation function which satisfies (20) and (A6) is e˙ A11 A12 0 0 e Kp

ρ = δ + c1 ω + c2 e + c3dˆ(t) , (21) x˙ 2 0 A22 0 0 x2 0 ∗ k k k k   =     +   [u u ] . c x˙ 3 A31 A32 A33 A34 x3 0 − dˆ(t) = d¯(t) + 4 d¯(t) (22) x˙ 4  0 A42 0 A44 x4  0  s + γd ∗                 (25) I θ∗T A(s)Λ−1(s) d(t) . ≥ − 1 ∗ h i Comparing the transfer function matrix of (25) (e = (sI The upper bound (22) is obtained through the application of Lemma 3 −1 ∗ − A11) Kp[u u ]) with the error transfer function matrix (18), we −1 − (see Appendix) with γd >0 being the stability margin of A(s)Λ (s) conclude that A11 = AM . Furthermore Aii (i = 1, . . . , 4) are (see Lemma 3). Hurwitz matrices since Ac is Hurwitz. Applying Lemma 1 to the The plant transfer function matrix G(s) belongs to some given class closed-loop system (19)–(20) and (25) with K = KpSp, K being which is a subset of Rm×m(s). Each element of satisfies as- −1 − P P Hurwitz by assumption (A4), π(t) = K A12 exp(A22t)x2(0) and sumptions (A1)–(A4) with some fixed and . The implementation −1 nom ∗ Sp ν dU (t)=Sp [u u ], we have that the output error is bounded by of (21)–(22) needs the following assumption: (A7) Values for the con- − stants ci 0 (i = 1, . . . , 4) and γd > 0 are known such that inequality e(t) (k1 e(0) + k2 x2(0) ) exp( λ1t) , (26) ≥ k k ≤ k k k k − IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 2003 4

T t 0. Moreover, if δ > 0, then the sliding mode at the point e = 0 Lyapunov equation P KpSp+(KpSp) P =Q>0 for a given Q which ∀ ≥ starts after some finite time ts 0. Now, based on the stability of the is a free design parameter. We have chosen Q=I. In (30), P , Kp and ≥ ∗ matrices Aii it is easy to show that 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 xi(t) k3 exp( λ5t) Xe(0) , (i = 2, 3, 4) , (27) (20) for any plant that belongs to . The modulation function can be k k ≤ − k k P simplified since the reference model is such that AM = γM I, γM > t 0, where 0 < λ5 < min λ1, λ2, λ3, λ4 , λi = − ∀ ≥ { } 0, then c2 = 0, c.f. Corollary 1. The constant c3 should satisfy c3 minj Re( λij ) > 0 and λij is the spectrum of Aii (i = ≥ { − } { } c¯3 = 2 P Kp /λmin(Q) and can be computed through a procedure 2, 3, 4). From the bounds (26) and (27), we conclude that Xe(t) k k k k ≤ similar to that applied in the choice of c1, which gives c3 =6.9. Since k4 exp( λ5t) Xe(0) , t 0, which proves that the system is glob- the disturbance is uniformly bounded, we have dˆ(t) wdcd¯, where ally exponentially− k stable.k ∀ ≥  ≡ wdc max0.3≤α≤4( Wd(0) ) and Wd(0) is the DC gain of Wd(s) Remark 5: The Hurwitz condition on the matrix K appears to be ≥ k k − which depends on the parameter α. Here wdc = 2.8. The constant the least restrictive condition on the plant high frequency gain matrix, δ =0.1 guarantees finite-time convergence of the output error. since it is a necessary and sufficient condition for the existence of slid- ing mode in unit vector control systems, according to [22, Theorem 1], [15]. Remark 6: The knowledge of a fixed matrix Sp such that KpSp 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 con- taining 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 Fig. 2. Modulation function coefficient c1 as a function of the plant uncertain parameter α. globally exponentially stable and has better noise immunity.

Simulation results: The reference signals are r1(t) = 80 sqw(24t) VI. DESIGN EXAMPLE and r2(t)=40 sin(18t), where sqw( ):=sgn[sin( )] is a square wave. · T · To illustrate the design of the UV-MRAC, we consider a third order The input disturbance is d(t) = [2, 2] sqw(30t). Fig. 3 displays the system described by simulation results obtained with the value of the plant parameter being . The remarkable behavior of the UV-MRAC becomes evi- s+0.6 −1.2 α = 0.35 2 s2−1 s2−1 1 2α dent from the fast convergence of the output signals to the reference Gα(s) = Kp , Kp = , (28) 0.4 s+2.2 2 α trajectories. " s2−1 s2−1 # −  where the constant α [0.3, 4] is uncertain. All other parameters are ∈ known. Consequently the plant belongs to the class = Gα(s) : P { 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 − is uniformly bounded by d¯(t) 5. ≡ If we choose Sp = I, KpSp is Hurwitz if and only if 1 < − − α < 0.25 or α > 0. Then the UV-MRAC can be applied. How- − ever, in order to keep KpSp positive definite to allow the application of VSC schemes such as [7], [8] the uncertain parameter should sat- isfy 0.525 < α < 1.490, which is clearly a more restrictive condition. 2×2 Moreover, it can be verified that @Sp R , det(Sp) = 0, such that T ∈ 6 KpSp =(KpSp) , α [0.3, 4]. Fig. 3. System (y1, y2) and model (yM1, yM2) outputs. ∀ ∈ −1 Design: The chosen reference model is WM (s)=(s + 2) I. The state filters are chosen with λ(s)=s + 1. A nominal parameter matrix nom is computed for α = 1 which results in (29), where p1 and p2 are arbitrary constants which span the complete set of θ∗ satisfying the VII. CONCLUSION nom ∗ Diophantine equation (16) for α = 1. We choose θ = θ with This paper proposes an output-feedback model-reference sliding p1 = p2 = 0 which gives a least squares solution of the Diophantine mode controller (UV-MRAC) for multivariable linear systems based equation. on the adaptive control formulation and on the unit vector control ap- The modulation function (21) should be designed in view of (A7) proach. The high frequency gain (HFG) matrix of the plant (Kp) is and aiming at keeping the unit vector control amplitude small in a sub- not assumed to be known. The main result states that the system is optimal sense (see [24]). The constant c1 is computed to satisfy globally exponentially stable and can be designed such that the sliding mode surface (e = 0) is reached in finite time provided that KpSp 2 nom ∗ T − c1 c¯1 = P Kp (θ θ ) , (30) be Hurwitz for some known matrix Sp. This is less restrictive than ≥ λmin(Q) − the assumption of positive definiteness of KpSp required in previous which was developed from (20) and (33) but using a less conservative works that assume not necessarily small uncertainties of the HFG. The nom ∗ T −1 nom ∗ T upper bound P Kp(θ θ ) P KpSp Sp (θ θ ) extension of the proposed controller for systems of arbitrary relative k − k≤k kk T − k that can be found from (31). The matrix P = P > 0 satisfies the degree is a current research topic [15]. IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 12, DECEMBER 2003 5

∗T 0.99 0.57 0.42 0.42 0.39 1.41 0.2 0.4 p1 θ = − − − − − + [1, 3, 2, 2, 1, 1, 0, 0] . (29) 0.23 0.11 0.34 0.34 1.03 0.43 0.4 0.2 p2 − − − − − − − − −   

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