Unit vector control of multivariable systems

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2002 IFAC World Congress (Barcelona, ESP)

UNIT VECTOR CONTROL OF MULTIVARIABLE SYSTEMS ✂✁



Liu Hsu 1 José Paulo Vilela Soares da Cunha ✂✁ ✄✄✂✁ Ramon R. Costa 1 Fernando Lizarralde 1

COPPE/Federal University of Rio de Janeiro, Brazil ([email protected]) ✂ State University of Rio de Janeiro, Brazil ([email protected]) ✄✂ Dept. of Electronic Eng./Federal University of Rio de Janeiro, Brazil

Abstract: This paper presents a unit vector based output-feedback model-reference adaptive control (UV-MRAC) for uncertain multi-input-multi-output (MIMO) linear systems. Some features of this new algorithm are: (i) Less restrictive a priori knowledge about the high frequency gain matrix of the plant is required. (ii) It applies to plants of any uniform relative degree. (iii) The closed loop system is exponentially stable with respect to some small residual set. (iv) The controller is free of peaking. Keywords: Multivariable systems, variable structure control, sliding mode control, model reference adaptive control, output feedback.

1. INTRODUCTION In (Ambrosino et al., 1984; Bartolini and Zolezzi, 1988) the structure of Model-Reference Adaptive Controllers (MRAC) was introduced in the variable structure control (VSC) theory for single-input-singleoutput (SISO) plants. Following this approach, new controllers were proposed in (Hsu et al., 1994) for SISO plants, where adaptation is achieved through signal synthesis, instead of parameter adjustment. An early reference on output-feedback adaptive VSC of MIMO systems is (Tao and Ioannou, 1989) where, however, VSC was used as an auxiliary control signal to achieve robustness and disturbance rejection. In this paper, an output-feedback based unit vector MRAC (UV-MRAC) for MIMO plants is proposed. While the controller of (Spurgeon et al., 1996) uses a state space description of the plant and a nonlinear observer, the approach here, likewise (Tao and Ioannou, 1989; Chien et al., 1996), is based on the plant transfer function matrix and follows the development of MRAC without explicit state observers. The UVMRAC is an extension of the scheme developed in (Hsu et al., 1994) for SISO systems and generalized 1

Partially supported by CNPq/Brazil.

to the MIMO case in (Chien et al., 1996). Compared to the results of (Chien et al., 1996), the main new features are: (i) global exponential stability can be demonstrated, and (ii) less restrictive assumption on the plant high frequency gain matrix is required due to the use of unit vector control. Moreover, the peaking phenomena, a flaw in high gain observer based VSC schemes (Esfandiari and Khalil, 1992), does not occur in the UV-MRAC. 2. PRELIMINARIES ☎✝✆

x ✆ denotes the Euclidean norm of a vector x and A ✆✟✞ σmax ✠ A ✡ denotes the corresponding induced norm of a matrix A. The ☛ ∞e norm of the signal x ✠ t ✡✌☞ Rn is defined as ✆ xt ✆ ∞ : ✞ sup0 ✍ τ ✍ t ✆ x ✠ τ ✡ ✆✏✎ Mixed time-domain and Laplace transform domain (operator) representations will be adopted. The norm of an operator H ✠ s ✡ is defined as ✆ H ✠ s ✡ ✆ : ✞ ✆ h ✠ t ✡ ✆ 1 ✞✒✑✏0✓ ∞ ✆ h ✠ τ ✡ ✆ d τ . The following holds for y ✠ t ✡ ✞ h ✠ t ✡✕✔ u ✠ t ✡ : ✆ yt ✆ ∞ ✖ ✆ h ✠ t ✡ ✆ 1 ✆ ut ✆ ∞ ✎ We assume t ☞ R ✓ so that ✗ t means ✗ t ✘ 0. Filippov’s definition for the solution of discontinuous differential equations is assumed (Filippov, 1964).













☎ Let G s ✡ be a strictly proper and nonsingular m ✠

m rational transfer function matrix. The interactor matrix associated to G ✠ s ✡ has the form (Wolovich and Falb, 1976)

✁ ξ ✠ s ✡ ✞ H ✠ s ✡ diag sn1✂ ✄ sn2✂ ✄✆☎✝☎✞☎✟✄ snm✂ ✠



(1)

where H ✠ s ✡ is a unit lower triangular polynomial matrix, such that Kp ✞

lim ξ ✠ s ✡ G ✠ s ✡ ✡

s ∞

(2)

is finite and nonsingular. The matrix K p ☞ Rm ☛ m is called high frequency gain (HFG) matrix and ✠ n1 ✄ n2 ✄☞☎✝☎✞☎✟✄ nm ✡ is the vector relative degree of G ✠ s ✡ . If ni ✞ n , ✠ i ✞ 1 ✄✝☎✞☎✝☎✌✄ m ✡ , we say that G ✠ s ✡ has uniform vector relative degree n .

approach of (Chien et al., 1996) which employs a precompensator to render ✁ the relative degree uniform and equal to n ✞ maxi ni ✠ . Assumption (A5) is a considerable reduction in the amount of a priori knowledge concerning the plant HFG matrix required. In (Tao and Ioannou, 1988; Tao and Ioannou, 1989; Chien et al., 1996) the more restrictive assumption of positive definiteness of K p S p (and also symmetry in some approaches) is needed. The reference model is defined by yM ✞ WM ✠ s ✡ r ✄ WM ✠ s ✡ ✞ diag

We consider an observable and controllable MIMO linear time-invariant plant described by y ✞ G ✠ s ✡✎✍ u ✏ d ✠ t ✡✒✑



(3)

where G ✠ s ✡ is an m m transfer function matrix, u is the input, d is an unmeasurable input disturbance, and y is the output. We assume that the parameters of the plant model are uncertain, i.e., only known within finite bounds. The following assumptions regarding the plant are taken as granted: (A1) G ✠ s ✡ is minimum phase. (A2) G ✠ s ✡ has full rank and is strictly proper. (A3) The observability index ν of G ✠ s ✡ is known. (A4) The interactor matrix ξ ✠ s ✡ is diagonal and G ✠ s ✡ has known uniform vector relative degree n (i.e., ξ ✠ s ✡ ✞ sn ✂ I). (A5) A matrix S p is known such that ✓ K p S p is Hurwitz. (A6) The disturbance d ✠ t ✡ is piecewise continuous and a bound d¯✠ t ✡ is known such that ✆ d ✠ t ✡ ✆ ✖ d¯✠ t ✡ ✖ d¯sup ✔ ✏ ∞ ✄ ✗ t. Assumptions (A1) to (A3) are usual in MIMO adaptive control. Some prior knowledge of the interactor is usually assumed in MIMO adaptive/VSC literature (Tao and Ioannou, 1988; Tao and Ioannou, 1989; Chien et al., 1996). Assumption (A4) may look too strong, however, it can be argued that a diagonal interactor can be achieved by means of an appropriate precompensator. Indeed, in most cases (in a generic sense), Lemma 2.6 in (Tao and Ioannou, 1988) guarantees that there exists a precompensator Wp ✠ s ✡ so that G ✠ s ✡ Wp ✠ s ✡ has diagonal interactor matrix. Moreover, Wp ✠ s ✡ does not depend on the plant parameters. Once the interactor is known to be diagonal and if the relative degree of each element of G ✠ s ✡ (or of G ✠ s ✡ Wp ✠ s ✡ ) is known, then ξ ✠ s ✡ can be determined without any prior knowledge about the transfer function parameters (Wolovich and Falb, 1976). In order to achieve uniform vector relative degree, one can follow the

r✄ yM ☞ Rm ✄

(4)

✠ s✡ ✄

(5)

1

s ✏ γj

L ✠ s ✡ ✞ L1 ✠ s ✡ L2 ✠ s ✡

Li ✠ s ✡ ✞ 3. PROBLEM STATEMENT





L✗

1

☎✞☎✝☎ LN ✠ s✡ ✄

✠ s ✏ αi ✡ ✄

(6) (7)

γ j ✘ 0 ✄ ✠ j ✞ 1 ✄✞☎✝☎✞☎✟✄ m ✡ ✄ αi ✘ 0 ✄ ✠ i ✞ 1 ✄✝☎✞☎✞☎✌✄ N ✡ ✄ and N ✞ n ✓ 1. The reference signal r ✠ t ✡ is assumed piecewise continuous and uniformly bounded. WM ✠ s ✡ has the same n as G ✠ s ✡ and its HFG is the identity matrix. The control objective is to achieve asymptotic convergence of the output error e ✠ t ✡ ✞ y ✠ t ✡✙✓ yM ✠ t ✡ to zero, or to some small residual neighborhood of zero in the error space, as t ✚✛✏ ∞.

4. UNIT VECTOR CONTROL Here, we adopt the unit vector control law u✞

v x✡ ✓ ρ ✠ x✄ t ✡ ✆ ✠ ✆ ✄ v ✠ x✡

(8)

where x is the state vector, v ✠ x ✡ is a vector function of the state of the system and ρ ✠ x ✄ t ✡ ✘ 0, ✗ x, ✗ t. We refer to ρ ✠ ☎ ✡ as the modulation function, which is designed to induce a sliding mode on the manifold v ✠ x ✡ ✞ 0. We will henceforth assume that u ✞ 0 if v ✠ x ✡ ✞ 0. Some lemmas regarding the application of the unit vector control into the MRAC framework are presented in (Hsu et al., 2002). These lemmas are instrumental for the controller synthesis and stability analysis.

5. CONTROL PARAMETERIZATION If the plant is perfectly known, then a control law which achieves matching between the closed-loop transfer matrix and WM ✠ s ✡ is given by u ✞ θ

T

ω



Wd ✠ s ✡✕✔ d ✠ t ✡

where the parameter matrix θ vector ω ✠ t ✡ are given by



(9)

and the regressor

T ✞

θ

✍ θ1



ω

T ✍ ω1



B ✠ s✡

ω2T yT rT ✑ T



(10)



(11)

B ✠ s✡ B ✠ s✡ u ✄ ω2 ✞ y✄ Λ ✠ s✡ Λ ✠ s✡ ν 2 ν 3 ✎ ✎✂✎ Is ✗ Is I ✑ T ✄ ✍ Is ✗ B ✠ s✡ ✄ I ✓ θ1 T Λ ✠ s✡ ✞

ω1

Wd ✠ s ✡

θ2 T θ3 T θ4 T ✑

T



(12) (13) (14)

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. We consider a minimal realization of the plant (3) with x p ☞ Rn being the state. Let X ✞ ✍ xTp ω1T ω2T ✑ T . Then, the regressor vector can be expressed as ✞

ω

Ω1 X ✏ Ω2 r ✎

Theorem 1. Consider the system (16), (17), and (19). If n ✞ 1 and (A1)–(A7) and inequality (20) are verified, then the system is globally exponentially stable. Moreover, if δ ✘ 0, then the output error e ✠ t ✡ becomes zero after some finite time. Proof: see (Hsu et al., 2002).

(15)

6.2 The case of higher relative degree For higher uniform relative degree the unit vector control strategy cannot be applied directly. Similarly to the SISO case, to overcome this difficulty, the controller structure is now modified according to Figs. 1 and 2 (Hsu et al., 1994; Hsu et al., 1997). A key idea

where Ω1 and Ω2 are appropriate constant matrices. Defining the error state as Xe , the error equation can be written as (Hsu et al., 2002) X˙e ✞ Ac Xe ✏ Bc ✠ θ4 T ✡ e ✞ Co Xe ✄ ✗

1

✍u✓

u

✑ ✄

(16) (17)

or in input-output form e ✞ WM ✠ s ✡ K p ✍ u ✓ u ✑



(18)

We now make the following assumption on the class of admissible control laws. (A7) The control law satisfies the inequality ✆ ut ✆ ∞ ✖ Kω ✆ ωt ✆ ∞ ✏ Krd ✄ where Kω ✄ Krd are positive constants.

Fig. 1. UV-MRAC for n



2

In this case the system signals will be regular which guarantees that no finite time escape occurs in the system signals (Sastry and Bodson, 1989).

6. DESIGN AND ANALYSIS

Fig. 2. Implementation of the operator ☛

The UV-MRAC stems from the variable structure model-reference adaptive controller (VS-MRAC) developed for SISO plants in (Hsu et al., 1994).

6.1 The case of relative degree one For n ✞ 1, we have L ✠ s ✡ law is (Hsu et al., 2002) e u ✞ unom ✓ S pρ ✆ ✆ ✄ e



I. The proposed control unom



θ nomT ω ✄

(19)

where S p ☞ Rm ☛ m is a design matrix which verifies assumption (A5) and θ nom is some nominal value for θ . Considering the error equation (18) and the control law (19), exponential stability is achieved if the modulation signal satisfies

ρ

δ ✘ ✎✄✂



✂ S p✗ ✂

cε ✆ e ✆ 1





θ



nom ✓



1 ✏ cd ✡

θ ✡

T



ω ✏ Wd ✠ s ✡✕✔ d ✠ t ✡✝✆ ✂✂ ✄ ✂

where cε ✄ cd ✘ 0 are appropriate constants and δ is an arbitrary constant.

for the controller generalization is the introduction of the prediction error eˆ ✞ WM ✠ s ✡ L ✠ s ✡ K nom ✞ U0 ✓ L ✗

1

✠ s✡

UN ✟



(21)

where K nom is a nominal value of K ✞ K p S p and the operator L ✠ s ✡ , as given by (6), is such that G ✠ s ✡ L ✠ s ✡ and WM ✠ s ✡ L ✠ s ✡ have uniform vector relative degree one. The operator L ✠ s ✡ is noncausal but can be approximated by the unit vector lead filter ☛ shown in Fig. 2. The averaging filters Fi✗ 1 ✠ τ s ✡ in Fig. 2 are lowpass filters with transfer function given by Fi✗ 1 ✠ τ s ✡ ✞ 1 favi ✗ ✠ τ s ✡ ✄ with f avi ✠ τ s ✡ being Hurwitz polynomials in τ s such that the filter has unit DC gain ( f avi ✠ 0 ✡ ✞ 1), e.g., favi ✠ τ s ✡ ✞ τ s ✏ 1. If the time constant τ ✘ 0 is sufficiently small, the averaging filters give an approximation of the equivalent control signals ✠ Ui 1 ✡ eq ✠ ✗ Fi✗ 1 ✠ τ s ✡ Ui 1 (Utkin, 1992). ✗

(20) ✘

0

6.2.1. Error equations Here, we present the expressions for the auxiliary error signals which are convenient for the controller design and stability anal-

ysis (Hsu et al., 1997). From (18) and (21), using u ✞ θ nomT ω ✓ S pUN , K ✞ K p S p and U¯ : ✞

K nom ✡ ✠



Wd ✠ s ✡

1





Kp ✍ ✠ θ ✞

the auxiliary error ε0 ✞

ε0

✁I

✑ ✓

d✠ t✡

θ nom ✡ T ω ✓ ✓



K ✠

nom

1

✡ ✗





K UN ✄

(22)

e ✓ eˆ can be rewritten as

WM ✠ s ✡ L ✠ s ✡ K

nom



1

U0 ✓ L ✗

¯ ✠ s✡ U



εi

1

Fi✗

τ s ✡ Ui ✠



1✓

Li✗

1

✠ s✡



(24)





Ui ✓ F1✗ ✁ i1 ✠ τ s ✡ Li✗ ✓ ✓

✠ s✡

πei ✓ π0i ✄ ✓



εN

1

Li✗



i ✞ 1✄

✎✂✎ ✎ ✄

1 1 ✁ N ✠ s✡



✓ ✆



N ✓ 1✡

(25)



1



LN✗ ✠ s ✡ ✠ K I✓





nom

K nom ✡

1



✡ ✗



τ s ✡ Ud

F1✗ ✁ N1 ✠

K UN ✏

K βuN ✓ πeN





1





π0N ✄ ✓

(26)



(Li ✁ j ✠ s ✡ 1 if j ✔ i), Fi ✁ j ✠ τ s ✡ where Li ✁ j ✠ s ✡ is defined in similar way and (by convention, πe1 0) ✞

∏kj i Lk ✠ s ✡





S p✗ 1 ✠ θ

Ud

βuN



πei



π0i



θ nom ✡ T ω ✓ Wd ✠ s ✡ ✔ d ✠ t ✡ ✓

✄ ✆

(27)



F1 ✁ N ✠ τ s ✡

I F1✗ ✁ N1 ✠ τ s ✡ LN✗ 1 ✠ s ✡ UN ✄ ✓

(28)





Li ✗

1 ✠ s✡

Fi✗

1



τ s ✡ πe ✁ i ✗

εi

1✏

1







WM ✠ s ✡ F1 ✁ i ✠ τ s ✡ Li ✁ N ✠ s ✡ K nom

✄ ✆

(29)

1





πei ✠ t ✡

Xε ✠ t ✡



ε0 ✎

Π0 ✄ ✖

τ I✓ ✖





K

nom

(33) ✡ ✗

1



✆ ✄ ✆



π0i ✠ t ✡



K KeN C ✠ t ✡ ✂

Π✄ ✏

(34)





βuN ✠ t ✡

where C✠ t✡ ✄



with some Mθ Mred

These auxiliary errors can be rewritten as

εi

εN ✠ t ✡

(23)



Ui



✆ ✄ ✆

εi ✠ t ✡ ✆

The auxiliary errors in the lead filters are given by ✞

errors εi , ✠ i ✞ 0 ✄✞☎✞☎✝☎✌✄ N ✓ 1 ✡ , tend to zero at least exponentially. Moreover,

i ✞ 1✄



Π0 ✄ ✖

τ KβN C ✠ t ✡

Mθ ✆ ωt ✆

∞✏



✎✂✎ ✎ ✄

Π

N✄

(35)

0✄

(36)

Mred ✄

(37)

0. Proof: see Appendix A. ✘

Remark 3. In the above theorem the Hurwitz condition on ✓ K nom and ✓ ✠ K nom ✡ ✗ 1 K could be satisfied choosing K nom ✞ knom I, with knom ☞ R, knom ✘ 0. In particular, with K nom ✞ knom I the condition on the HFG is simply ✓ K Hurwitz. 6.2.3. Error system stability Theorem 4. For N ✞ n ✓ 1 ✘ 1, assume that ✓ K nom and ✓ ✠ K nom ✡ ✗ 1 K are Hurwitz, and that the modulation functions satisfy (32). Then, for sufficiently small τ ✘ 0, the error system (16), (23), (25) and (26) with state z as defined in (31) is globally exponentially stable with respect to a residual set of order τ . Proof: see Appendix B.

(30) 7. SIMULATION RESULTS

6.2.2. Bounds for the auxiliary errors Consider the error system (16)–(17), (23), (25), and (26). Let Xε denote the state vector of (23) and x0FL denote the transient state corresponding to the following operators: L ✗ 1 in (23), F1✗ ✁ i1 Li✗ ✓ 11 ✁ N in (25) and all the remaining operators associated with βuN ✄ πei ✄ π0i in (28)– (30). Since all these operators are stable, there exist positive constants KFL and aFL such that ✆ x0FL ✠ t ✡ ✆ ✖ KFL e ✗ aFLt ✆ x0FL ✠ 0 ✡ ✆ ✎ In order to fully account for the initial conditions, the following state vector z is used T ✞

z

0 T T T ✍ ✠ z ✡ ✄ εN ✄ Xe ✑ ✄ ✂ ✎ T T T ✎ ✎ ✄ ✍ Xε ✄ ε1 ✄ ε2 ✄



z0 ✡

T ✞

εNT ✗

✄ 1 ✠

x0FL ✡

T ✎



(31)

In what follows, all K’s and a’s denote generic positive constants, and “Π" and “Π0 " denote any term of the form K ✆ z ✠ 0 ✡ ✆ e ✗ at and K ✆ z0 ✠ 0 ✡ ✆ e ✗ at , respectively. Theorem 2. For N ✞ n ✓ 1 ✘ 1, consider the auxiliary errors (23), (25) and (26). If ✓ K nom and ✓ ✠ K nom ✡ ✗ 1 K are Hurwitz, and the relay modulation functions satisfy

ρ0 ✘

ρi ✘

ρN ✞



1 ✏ cd0 ✡ ✆ L ✗ ✠

1 ✏ cdi ✡ ✠



1 ✏ cdN ✡





cε 0 ✆ ε0 ✆ 1 1 ¯✡ ✆ ✠ F1✗ ✁ i Li✗ ✓ 1 ✁ N ✡✕✔ ✠ U 1



F1✗ ✁ N1 ✔

U¯ ✆

Ud







(32)

✆ ✄

✠ i 1 ✄✝☎✞☎✝☎✌✄ N ✓ 1 ✡ , ✗ t, with some appropriate constants cε 0 ✘ 0 and cdi ✘ 0 for i ✞ 0 ✄ ✎✂✎ ✎ ✄ N, then the auxiliary

An articulated suspension system is described in Fig. 3. The objective is to make the load position (y2 ) and orientation angle (φ ) track the reference model output through the control of two linear displacement actuators. The control signals are the actuator forces ¯u1 and ¯u2 . The plant is linearized in the neighborhood of φ ✞ 0 rad, resulting in the input-output representation 1 1 (38) y ✞ diag 2 ✄ 2 K p ✍ ¯u ✏ d ✑ ✄ s s l1 J ✓ l2 J 0 ✄ ✄ Kp ✞ d ✞ K p✗ 1 (39) g 1 m 1 m

✟✞ ✠ ✠





✠ ✡



☛☞

where the plant output vector is y ✞ ✍ φ y2 ✑ T and the control vector is ¯u ✞ ✍ 1¯u ¯u2 ✑ T ✞ u ✓ d nom . The gravity compensation term d nom ✞ ✍ 50 50✑ T N allows the reduction of the amplitude of the control signal u. The plant parameters are: load mass m ✞ 10 kg, load moment of inertia J ✞ 1 kg m2 , g ✞ 9 ✎ 81 m s2, l ✞ 4 m and l2 ✞ 3 m. These parameters are not available for the UV-MRAC design. The reference model is WM ✠ s ✡ ✞ ✍ ✠ s ✏ 1 ✡ ✠ s ✏ 5 ✡✒✑ ✗ 1 I. The reference signals are r1 ✞ 0 ✎ 1 sin ✠ t ✡ and r2 ✞ 0 ✎ 1sqw ✠ 2t ✡ (square wave). The controller parameters are: θ nom ✞ 0, L ✠ s ✡ ✞ Λ ✠ s ✡ ✞ s ✏ 5, τ ✞ 0 ✎ 003 s and K nom ✞ 0 ✎ 1I. The convergence of the error signals observed in Fig. 4 is exponentially fast. However, quite large peaks are observed during



0 τ for some small enough τ . This implies that the UV-MRAC preserves global stability and is free 0, unlike the HGO case. In the of peaking as τ other hand, experiments on underwater vehicle control have shown the robustness with respect to input disturbances, unmodeled dynamics and measurement noise of a model reference sliding mode controller similar to UV-MRAC (Cunha et al., 1995). Future works include the robustness analysis to unmodeled dynamics and chattering alleviation which can be extended from the SISO case. ✄







Fig. 3. Diagram of the suspension system

9. REFERENCES

Fig. 4. Output error signals the transient due to unfavorable initial conditions. The Hurwitz condition required to apply the UV-MRAC is satisfied for S p I if and only if l1 J m, which is always true. However, if the control law requires that K p S p 0, such as in (Tao and Ioannou, 1989; Chien et al., 1996), then the necessary and sufficient conditions are l1 J m and l22 2Jm 1 l2 J 2 m 2 1 4Jlm 0. In this example the load position should be kept within l2 1 164 m, otherwise, an appropriate S p matrix should be chosen to make K p S p positive definite. It should be stressed that the positive definiteness of K p S p implies that K p S p be Hurwitz, but the converse is not true. This suggests that the UV-MRAC can applied to a broader class of plants. ✞







































8. CONCLUSION This paper proposes a model-reference sliding mode control strategy for uncertain linear MIMO systems. An important stability condition is that K p S p should be Hurwitz. It does not seem to be overly restrictive since it is known to be a necessary and sufficient condition for the existence of sliding modes in unit vector control systems as may be concluded from (Baida, 1993, Theorem 1). ✓

The UV-MRAC employs averaging filters with sufficiently small time constant τ . This time constant is similar to the small parameter ε which characterizes high gain observers (HGO) in sliding mode controllers. In the latter, peaking of control and observer signals may arise as ε 0. In both types of controllers, tracking errors tend to zero as the small parameters tend to zero. However, an important feature of the UV-MRAC is that it possesses global exponential properties uniformly with respect to τ ✚



Ambrosino, G., G. Celentano and F. Garofalo (1984). Variable structure MRAC systems. Int. J. Contr. 39(6), 1339–1349. Baida, S. V. (1993). Unit sliding mode control in continuous- and discrete-time systems. Int. J. Contr. 57(5), 1125–1132. Bartolini, G. and T. Zolezzi (1988). The V.S. approach to the model reference control of nonminimal phase linear plants. IEEE Trans. Aut. Contr. 33(9), 859–863. Chien, C.-J., K.-C. Sun, A.-C. Wu and L.-C. Fu (1996). A robust MRAC using variable structure design for multivariable plants. Automatica 32(6), 833–848. Cunha, J. P. V. S., R. R. Costa and L. Hsu (1995). Design of a high performance variable structure position control of ROV’s. IEEE J. Oceanic Eng. 20(1), 42–55. Esfandiari, F. and H. K. Khalil (1992). Output feedback stabilization of fully linearizable systems. Int. J. Contr. 56, 1007–1037. Filippov, A. F. (1964). Differential equations with discontinuous right-hand side. American Math. Soc. Translations 42(2), 199–231. Hsu, L., A. D. Araújo and R. R. Costa (1994). Analysis and design of I/O based variable structure adaptive control. IEEE Trans. Aut. Contr. 39(1), 4–21. Hsu, L., F. Lizarralde and A. D. Araújo (1997). New results on output-feedback variable structure adaptive control: design and stability analysis. IEEE Trans. Aut. Contr. 42(3), 386–393. Hsu, L., J. P. V. S. Cunha, R. R. Costa and F. Lizarralde (2002). Multivariable output-feedback sliding mode control. In: Variable Structure Systems: Towards the 21st Century (X. Yu and J.-X. Xu, Eds.). Chap. 12. Springer-Verlag. Sastry, S. S. and M. Bodson (1989). Adaptive Control: Stability, Convergence and Robustness. PrenticeHall. Spurgeon, S. K., C. Edwards and N. P. Foster (1996). Robust model reference control using a sliding mode controller/observer scheme with application to a helicopter problem. In: Proc. IEEE Workshop on Variable Struct. Sys.. pp. 36–41.

Tao, G. and P. A. Ioannou (1988). Robust model reference adaptive control for multivariable plants. Int. J. Adaptive Contr. Signal Process. 2, 217– 248. Tao, G. and P. A. Ioannou (1989). A MRAC for multivariable plants with zero residual tracking error. In: Proc. IEEE Conf. on Decision and Control. Tampa. pp. 1597–1600. Utkin, V. I. (1992). Sliding Modes in Control Optimization. Springer-Verlag. Wolovich, W. A. and P. L. Falb (1976). Invariants and canonical forms under dynamic compensation. SIAM J. Contr. Optim. 14(6), 996–1008.

Ac Xe Bc K nom e˙ˆN αN eˆN , where eˆN : ✞ εN ✠ β¯uN πeN π0N ✡ . To eliminate the derivative term e˙ˆN , a variable transformation X¯e : ✞ Xe Bc K nom eˆN is performed yielding (B.2) X¯˙e ✞ Ac X¯e ✠ Ac α I ✡ Bc K nom eˆ ✎ ✏

















N



N

Since Ac is Hurwitz, and from (34), one has ✆ X¯e ✠ t ✡ ¯ ✠ t ✡ Π. Moreover, τ KC

✆ ✖





Xe ✠ t ✡ ✆ and ✆ e ✠ t ✡ ✆ ✖ τ K0C ✠ t ✡ ✆ ωt ✆ ∞ ✖ τ K1C ✠ t ✡ K2 ✆ z ✠ 0 ✡ ✆ Kred K4 ✆ z ✠ 0 ✡ ✆ ✎ ✟ C✠ t✡ ✖ 1 τ K3

Π Km

(B.3) (B.4)













(B.5)



Indeed, inequalities (B.3) follow from X¯e ✞ Xe Bc K nom eˆN . From the relation (15), it follows that ✆ ω ✆ ✖ KM KΩ ✆ Xe ✆ and, then from (B.3) we obtain (B.4), where K2 ✆ z ✠ 0 ✡ ✆ comes from the initial value of the term Π in (B.3). Now, from (37) and (B.4), ✟ , whereby, after an C ✠ t ✡ ✖ τ K3C ✠ t ✡ K4 ✆ z ✠ 0 ✡ ✆ Kred algebraic manipulation one obtains (B.5), which is valid for τ K3 1 . Now, as explained below, we can also write ✓

Appendix A. PROOF OF THEOREM 2



By (Hsu et al., 2002, Corollary 1), ε0 and its state Xε in (23) converge to zero, at least exponentially, if the signal ρ0 satisfies (32). Note that the transient part of L 1 is represented by π ✠ t ✡ which is thus bounded by Π0 . Consequently, one concludes ✆ ε0 ✠ t ✡ ✆ ✖ Π0 . From (30) one can write π0i ✞ Hi ✠ s ✡ ε0 ✞ hi ✠ t ✡ ✔ ε0 ✠ t ✡ π0i0 ✠ t ✡ . Since ✆ π0i0 ✆ ✖ Π0 and ✆ hi ✠ t ✡ ✔ ε0 ✠ t ✡ ✆ ✖ Π0 , then ✆ π0i ✆ ✖ Π0 . Now, for N 1 and i ✞ 1, since πe1 0, equation (25) results in ✆ ε1 ✠ t ✡ ✆ ✖ Π0 , from (Hsu et al., 2002, Lemma 2) with β ✠ t ✡ 0. With similar argument, we recursively conclude from (25), (29) and Lemma 2 that ✆ πei ✠ t ✡ ✆ ✖ Π0 and ✆ εi ✠ t ✡ ✆ ✖ Π0 ✠ i ✞ 2 ✎ ✎✂✎ N 1 ✡ . Now consider εN (see (26)). Note that, from assumption (A7) (see Sect. 5), Mθ and Mred can be chosen so that ✆ UN ✠ t ✡ t ✆ ∞ ✖ C ✠ t ✡ . Since πe ✁ N 1 and εN 1 are bounded by Π0 , so is πeN , i.e., ✆ πeN ✠ t ✡ ✆ ✖ Π0 . From (28) one has ✗











✆ ✆























βuN ✂

✄✁ ✂

✆ t ∞ ✖

0 βuN ✡ ✓

F1 ✁ N ✠ τ s ✡



τ Kβ N C ✠ t ✡

☎✝✆







































Appendix B. PROOF OF THEOREM 4 It is convenient to rewrite (26) as ε ✞ L 1 U U¯ β¯ π N



N ✗

N





uN





eN ✓

π0N ✄

(B.1)

where β¯uN ✞ ✠ F1 ✁ N I ✡ F1 ✁ N1LN 1U¯ can be bounded by C ✠ t ✡ (see (37)) similarly as βuN in (A.1), i.e., ✆ ✠ β¯uN ✞ 0 ✡ ✆ ¯ β¯uN unom t ∞ ✖ τKβ NC ✠ t ✡ . Remembering that u 1 S pUN , we note that LN in (B.1) operates on the same signal UN as the one in (16). From (16) and (26), the model following error can be rewritten as X˙e ✞ ✓















z0 ✠ 0 ✡



Π ✎ (B.7)

O✠ τ✡ ✏



























ze ✠ t ✡



τ K5 K6 e ✖

✆ ✆

z ✠ t✡

ze ✠ tk ✓ 0



0

z ✠ tk ✓

1✡ 1✡





ze ✠ tk✡ Kz e



λ ✖







a t tk





✖ ✆











ze ✠ 0 ✡

(B.6) ✄













τ K5

✆ ✞



z0 ✠ 0 ✡





(A.1) ✄



0 t ✡ is bounded by Π0 . Then, applying where βuN ✠ Lemma 2 to equation (26), which can be rewritten as εN ✞ LN 1 ✠ K nom ✡ 1 K UN F1 ✁ N1Ud ✠I nom 1 0 nom 1 0 K ✡ βuN βuN K ✡ βuN πeN ✡ ✡ ✠K ✠I ✠K π0N , one readily concludes (34). Since ✆ ✠ βuN ✆ 0 ✡ ✆ 0 ✆ ✘ ✆ β ✆ βuN βuN βuN Π0 , then (36) folt ∞ ✘ uN lows from (A.1). ✍

Kz e

Indeed, the variables Xε εi 0 ✁ ✠ ✠ ✠ ✁ N 1 x0FL are bounded by Π0 in Theorem 2. Therefore, with the partition zT ✞ ✠ z0 ✡ T zTe , where zTe : ✞ εNT XeT one gets (B.6), where only the initial condition on z0 appears. Now, from (34), (B.3) and (B.5) follows (B.7), where O ✠ τ ✡ is independent of the initial conditions. Noting that the initial time is irrelevant in deriving the above expressions, we can write, for arbitrary t ✘ tk ✘ 0 ✠ k ✞ 0 1 ✎✂✎ ✎ ✡ and some T1 ✞ tk ✓ 1 tk 0







az t ✆

















ze ✠ t ✡



I ✂ F1 ✁ N1 ✠ τ s ✡ LN 1 ✠ s ✡ C ✠ t ✡ ✓

z0 ✠ t ✡



✆ ✏

az t tk ✆

ze ✠ tk ✡

λ z ✠ tk ✡



O ✠ τ✡ ✏

z ✠ tk ✡





z ✠ tk ✡



(B.9) ✄

0

✆✏✎

(B.8) ✄



0



0





z0✠ tk✡ ✁









O✠ τ ✡ ✏





(B.10) (B.11)

Equations (B.10) and (B.11) are obtained from (B.8) and (B.9) as follows: for τ K5 1 , there exists T1 0 such that λ ✞ max ✠ τ K5 K6 e aT1 Kz e az T1 ✡ 1. Then, the linear recursive inequalities (B.10) and (B.11) hold and lead to the conclusion that, for τ small enough, the error system is globally exponentially stable with respect to a residual set of order τ . ✗















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