Global tracking sliding mode control for a class of nonlinear systems via variable gain observer

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int. J. Robust. Nonlinear Control 2011; 21:177–196 Published online 9 April 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.1584

Global tracking sliding mode control for a class of nonlinear systems via variable gain observer Alessandro Jacoud Peixoto1,∗, † , Tiago Roux Oliveira2 , Liu Hsu2 , Fernando Lizarralde2 and Ramon R. Costa2 1 Department 2 Department

of Electrical Engineering/CEFET-RJ, Federal Center of Technology, Rio de Janeiro, Brazil of Electrical Engineering/COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

SUMMARY A novel output-feedback sliding mode control strategy is proposed for a class of single-input single-output (SISO) uncertain time-varying nonlinear systems for which a norm state estimator can be implemented. Such a class encompasses minimum-phase systems with nonlinearities affinely norm bounded by unmeasured states with growth rate depending nonlinearly on the measured system output and on the internal states related with the zero-dynamics. The sliding surface is generated by using the state of a high gain observer (HGO) whereas a peaking free control amplitude is obtained via a norm observer. In contrast to the existing semi-global sliding mode control solutions available in the literature for the class of plants considered here, the proposed scheme is free of peaking and achieves global tracking with respect to a small residual set. The key idea is to design a time-varying HGO gain implementable from measurable signals. Copyright 䉷 2010 John Wiley & Sons, Ltd. Received 12 March 2009; Revised 28 January 2010; Accepted 29 January 2010 KEY WORDS:

sliding mode control; uncertain nonlinear systems; output-feedback; high gain observer; norm observer; global practical tracking

1. INTRODUCTION Several approaches to deal with the tracking problem by output-feedback sliding mode (OFSM) control for arbitrary relative degree uncertain systems have been proposed in the literature [1–5], where strategies using high gain observers (HGOs) [6, 7] represent a particular important design class. Exact output tracking can be achieved via higher order sliding mode control based on robust exact differentiators [8]. However, stability and/or convergence of the overall control system is guaranteed only locally. Most available OFSM designs achieve global results only under rather stringent assumptions, such as linearly or uniformly globally bounded vector fields [4, 5, 7]. More general nonlinear plants are dealt within [5, 6, 9, 10], but only semi-global tracking was achieved. This is not surprising since, as shown in [11], for systems with polynomial nonlinearities

∗ Correspondence

to: Alessandro Jacoud Peixoto, Departamento de Engenharia El´etrica (DEPEL/CEFET-RJ), Maracan˜a 229, Bloco E, Andar 1, Maracan˜a, Rio de Janeiro, Brazil. † E-mail: [email protected] Contract/grant sponsor: CNPq Contract/grant sponsor: FAPERJ Contract/grant sponsor: CAPES Copyright 䉷 2010 John Wiley & Sons, Ltd.

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in the unmeasured states [6, 12], the global stabilization/tracking problem via continuous outputfeedback may not be solvable. Beyond the sliding mode control, several approaches to solve the global tracking problem by output feedback have been proposed based on backstepping-like designs, time-varying high gain techniques (HGO with variable gain) [13–17], homogeneity in the bi-limit [18, 19] and some kind of adaptation [20]. In contrast, no global tracking results are available for the class of systems dealt here in the domain of OFSM control, where robustness and good transient properties can be significant advantages. We believe that the class of system considered in this note is in the state of the art of global output-feedback control framework commonly considered by other authors [13, 16, 18, 19, 21, 22]. We deal with time-varying minimum-phase nonlinear plants, affine in the control, transformable to a normal form and for which a norm state estimator can be implemented. Such a class encompasses the standard output feedback form, the parametric strict feedback form, lower triangular systems with linear growth condition in the unmeasured states and growth rate possibly depending on the inverse dynamics unmeasured state, on the system output and time. Strong polynomial nonlinearities in the inverse dynamics state and the output system are also allowed. In the recent years Praly and several others have shown that, by using dynamic observer gain, global results can be achieved without invoking the global Lipschitz conditions or ‘output-feedback’ forms. Time-varying HGOs have also been used to cope with the effect of measurement noise and to establish the connections with the Extended Kalman Filter [17, 23]. In this paper, we extend the applicability of [24] to a wider class of nonlinear plants. The main result is to show that an OFSM control based on an HGO with dynamic observer gain can also be used for a state-of-the-art class of nonlinear systems to guarantee global practical tracking. Differently from most of the existing schemes, the HGO gain is not updated through a Riccati equation [13, 18, 22] but, instead, we use simple functions (e.g. polynomials) based on measurable signals and norm domination techniques [16, 18, 19]. To the best of our knowledge, this is the first global OFSM tracking scheme for the class of plants considered here. A well-known drawback of HGO-based control strategies is the peaking phenomenon [25], which can degrade the system performance or even lead to instability. Peaking avoidance through control saturation has already been proposed by Oh and Khalil [6] and Esfandiari and Khalil [9], but such an approach leads only to semi-global results. Here, following [7], we circumvent control peaking by using measurable signals and estimates not based on high gain to generate the control law magnitude while the HGO is used in the proposed sliding mode scheme only to generate the sliding surface. Global asymptotic stability with respect to a compact set and ultimate exponential convergence to a small residual set in the error space are obtained. Two academic examples illustrate the class of systems and the time-varying behaviour of the HGO gain.

2. PRELIMINARIES The following notations and terminology are used: • The 2-norm (Euclidean) of a vector x = [x 1 x2 . . . xn ]T and the corresponding induced norm of a matrix A are denoted by |x| and |A|, respectively. The symbol [A] denotes the spectrum of A and m [A] = − maxi {Re{[A]}}. • The L∞e norm of a signal x(t) ∈ Rn is defined as xt  := sup0t |x()|. • Classes of K, K∞ functions are defined according to [26, p. 144]. ISS, OSS and IOSS mean Input-State-Stable (or Stability), Output-State-Stable (or Stability) and Input-Output-StateStable, respectively [27]. • (i)  denotes class-K functions; (ii)  denotes class-K∞ functions; (iii)  denotes class-KL functions; (iv)  denotes known class-K functions; and (v) , ¯ denotes known non-negative functions. Copyright 䉷 2010 John Wiley & Sons, Ltd.

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Consider the single-input single-output (SISO) nonlinear systems of the form x˙ = f (x, t)+ g(x, t)u,

(1)

y = h(x, t),

(2)

where u ∈ R is the control input (discontinuous), y ∈ R is the measured output, x is the state and the uncertain functions f (·, ·), g(·, ·) and h(·, ·) are smooth enough to ensure local existence and uniqueness of the solution through every initial condition (x 0 , t0 ). For each solution of (1) there exists a maximal time interval of definition given by [0, t M ), where t M may be finite or infinite. Thus, finite-time escape is not precluded, a priori. Filippov’s definition of solution is adopted [28] and the extended equivalent control concept‡ is used [4, Section 2.3] [29]. We denote the equivalent control signal (piecewise continuous) simply by u(t). Our output-feedback strategy relies on the implementation of a norm observer for the plant state x. In the following definition let: (i) u be the plant input, (ii) y be the plant output, (iii) o be a smooth function and (iv) o (·, ·, t) and ¯ o (·, t) be the non-negative functions, piecewise continuous and upper-bounded in t (as defined in [22]) and continuous in their other arguments. Definition 1 A norm observer for system (1)–(2) is a m-order dynamic system of the form: 1 ˙ 1 = − 1 +u,

(3)

˙ 2 = o ( 2 )+2 o ( 1 , y, t), 2

(4)

with states 1 ∈ R, 2 ∈ Rm−1 and positive constants 1 , 2 such that for t ∈ [0, t M ): (i) if |o | is uniformly bounded by a constant co >0, then | 2 | can escape at most exponentially and there exists ∗2 (co ) such that the 2 -dynamics is BIBS (Bounded-Input Bounded-State) stable w.r.t. o for 2 ∗2 and (ii) for each x(0), 1 (0), 2 (0), there exists ¯ o such that |x(t)| ¯ o ( (t), t)+o (t),

:= [ 1 T2 y]T ,

where o := o (| 1 (0)|+| 2 (0)|+|x(0)|)e−o t with some o ∈ K∞ and positive constant o .

(5) 

3. PROBLEM STATEMENT We consider the global tracking problem of systems of the form (1)–(2) transformable into the normal form [26]:

˙ = f 0 (x, t),

(6)

˙ = A +b k p (x, t)[u +d(x, t)],

y = c ,

(7)

where the transformed state is defined as x¯ := [ T T ]T = T (x, t).

(8)

The -subsystem represents the inverse dynamics with ∈ Rn− and the state of the external dynamics () is given by  := [y y˙ . . . y ( −1) ]T .

(9)

‡ In general the equivalent control is defined only when the sliding surface is reached. The extended concept is valid

also during the reaching phase. Copyright 䉷 2010 John Wiley & Sons, Ltd.

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The pair (A , b ) is in Brunovsky’s canonical ⎡ 0 1 ··· ⎢. . . ⎢ .. .. .. A = ⎢ ⎢ ⎣0 0 · · · 0 0 ···

controllable form, i.e. ⎤ ⎡ ⎤ 0 0 ⎢ .⎥ .. ⎥ ⎢ .⎥ .⎥ ⎥ ; b = ⎢ . ⎥ ⎥ ⎢ ⎥ 1⎦ ⎣ 0⎦ 0 1

(10)

and c = [1 0 · · · 0] and d(x, t) is regarded as a nonlinear matched disturbance and k p (x, t) is the plant high frequency gain (HFG) assumed to be bounded away from zero. Note that, it is thus assumed that the plant (1)–(2) has a strong§ relative degree . Remark 1 (Normal form) For time invariant plants, the uniform relative degree assumption [26, 31] is a necessary and sufficient condition for the existence of a local change of coordinates (local diffeomorphism) which transforms (1)–(2) into (6)–(7). Here, we do not require mapping T (x, t) (8) to be invertible, but it should be a global transformation. One sufficient condition to assure that the time-varying plant (1)–(2) is transformable to the normal form is given in Appendix A.1. In the following assumption, we formulate the restrictions imposed on T (x, t), k p (x, t) and d(x, t), where the dependence on y = h(x, t) is explicitly given to allow the implementation of less conservative upper bounds. First of all, for i = 1, 2, 3, let: (a) i (|x|, y, t) are non-negative functions continuous and increasing in |x|, continuous in y, piecewise continuous and upper bounded in t; (b) ¯ i (y, t) are non-negative functions continuous in y and piecewise continuous and upper bounded in t and (c) i (|x|) are locally Lipschitz class-K functions. Assumption 1 There exist known functions i ,  ¯ i , i and a known positive constant c p such that the following inequalities hold ∀x, y, ∀t ∈ [0, t M ): T (|x|)+T (y, t)  |T (x, t)|1 (|x|, y, t), 00 is constant, K m ∈ R1× is such that Am is Hurwitz and r (t) is assumed piecewise continuous and uniformly bounded. 3.2. Reducing tracking to regulation Subtracting (12) from (7) one has ˙ e = Am e +b k p [u +de ],

T e = cm e ,

(13)

T = [1 0 · · · 0] (so e =  − = y − y ) and the where e := −m is the state tracking error, cm m 1 m1 error input disturbance de is defined by

k p de (x, , t) := k p d(x, t)− K m −km r.

(14)

Then, the tracking problem can be formulated as a regulation problem which consists in finding an OFSM control law u such that the output e is regulated to a neighborhood of zero, i.e. for all initial conditions x(0), 1 (0), 2 (0): (i) the solutions of (3), (4), (6) and (7) are uniformly bounded and (ii) the output e = 1 −m1 of (13), i.e. the tracking error (11), tends to a neighbourhood of zero as t → ∞. 3.3. Auxiliary upper bounds via norm observer The following available upper bounds for , k p and d are obtained, modulo exponentially decaying term, by using the bounding functions given in Assumption 1 and the norm observer given in Definition 1 (for details, see Appendix A.3): ||  1 ( , t)+1 ,

(15)

k p (x, t)  2 ( , t)+1 ,

(16)

|d(x, t)|  3 ( , t)+1 ,

(17)

Copyright 䉷 2010 John Wiley & Sons, Ltd.

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where i ( , t) := i (2 ¯ o , y, t)+  ¯ i (y, t) (i = 1, 2, 3) and 1 = 1 (| (0)|+|x(0)|)e−o t with some 1 ∈ K∞ and o in Definition 1. Then, with c p in Assumption 1 and from (14) one can verify that |de ||d|+(|K m |||+km |r |)/c p . Moreover, from (15) and (17) the following upper bound holds: |de (x, , t)|+( , t)+2 ,

(18)

where  is an arbitrary non-negative constant, ( , t) := 3 +(|K m | 1 +km |r |)/c p +,

(19)

and 2 := |K m |1 /c p +1 .

4. OUTPUT-FEEDBACK SLIDING MODE CONTROL When only y is available for feedback, we choose ˆ := S ˆ e = 0,

ˆ ˆ e := − m,

(20)

as the sliding surface, where S is such that (Am , b , S) is strictly positive real and ˆ is an estimate of  (9) provided by an HGO. The control law u is given by u = −( , t) sgn((t)). ˆ

(21)

Then, defining the estimation error as ˆ ˜ e := e − ˆ e = − ,

(22)

the following lemma can be stated. Lemma 1 (ISS property from |˜ e | to e ) Consider the e -dynamics (13) with output ˆ = Se − S ˜ e , u given in (21),  in (19) and de in (14). Then, (13) is ISS with respect to ˜ e and the following inequality holds |e (t)|ke |˜ e (t)|+e , where e := e (| (0)|+|x(0)|+|e (0)|)e−e t , e ∈ K∞ , 00 is an appropriate constant. Proof See Appendix A.4.



Our goal is to provide an estimate ˆ by an HGO (with variable gain) such that the observer error norm |˜ e (t)| is an arbitrarily small, modulo exponentially decaying term, and use Lemma 1 to conclude global practical tracking. As in [7], an eventual peaking [25] in ˆ is blocked by the sgn(·) function in (21) and the control signal u is peaking free since ( , t) is implemented using only the well conditioned (without peaking) signals. The proposed scheme is depicted in Figure 1.

5. HIGH GAIN OBSERVER WITH VARIABLE GAIN The HGO is given by ˙ ˆ ˆ ˆ = A +b u + H L o (y −c ), Copyright 䉷 2010 John Wiley & Sons, Ltd.

(23)

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Figure 1. Global OFSM control using an HGO to generate (t). ˆ

where L o and H are given by L o := [l1 . . . l ]T

H := diag(−1 , . . . , − ).

and

(24)

The observer gain L o is such that s +l1 s −1 + . . .+l is Hurwitz. In this paper, instead of using a constant , we introduce a variable parameter  = (t) = 0, ∀t ∈ [0, t M ), of the form ( , t) :=

¯ , 1+  ( , t)

(25)

where  , named domination function, is a non-negative function (to be designed later on) continuous in its arguments and >0 ¯ is a design constant. For each system trajectory,  is absolutely continuous and . ¯ Note that  is bounded for t in any finite sub-interval of [0, t M ). Therefore, ( , t) ∈ [, ], ¯

∀t ∈ [t∗ , t M ),

(26)

for some t∗ ∈ [0, t M ) and  ∈ (0, ). ¯ Thus, considering the SISO nonlinear plant (1)–(2) transformable into the normal form (6)–(7) under Assumptions 1–3, control law (21), with  given by (19) and HGO (23) with  be given by (25) and appropriate domination function  . Then, for sufficiently small constants 2 , >0, ¯ GAS of the error system with respect to the compact set and ultimate exponential convergence of error system state to a residual set of order ¯ are guaranteed, with both sets being independent of the initial conditions. Moreover, all signals in the closed loop system are uniformly bounded. A detailed stability analysis and a formal statement of the main result (Theorem 1) will be presented later in Section 6. 5.1. High gain observer error dynamics The transformation [6, 7]  := T ˜ e ,

T := [ H ]−1 ,

(27)

is fundamental to represent the ˜ e -dynamics in convenient coordinates to allow us to show that ˜ e is an arbitrarily small, modulo exponentially decaying term. First, from (10), (24) and (27), note that: T (A − H L o c )T−1 =

Copyright 䉷 2010 John Wiley & Sons, Ltd.

1 Ao , 

(28)

T b = b ,

(29)

˙ T˙ T−1 = , 

(30)

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where Ao := A − L o c and  := diag(1− , 2− , . . . , 0). Then, subtracting (23) from (7) and applying the above relationships, the dynamics of ˜ e (22) in the new coordinates  (27) is given by: ˙ ˙ = [Ao + (t)]+b [],

(31)

 := (k p −1)u +k p d.

(32)

where

5.2. Domination function design The HGO gain is inversely proportional to the small parameter  which is time varying due to the domination function  ( , t) in (25). In this section, our task is to establish properties for  ( , t) so that || and ˙ can be bounded by a constant of order O(), ¯ at least after a finite time interval, where ¯ in (25) is a design constant. With such properties, we can prove that the estimation error ˜ e is ultimately small provided ¯ is chosen sufficiently small. Then, by means of Lemma 1 we can conclude global practical tracking (Theorem 1). 5.2.1. Auxiliary upper bounds. Note that, from the definition of u (21), one has |u(t)|( , t). Thus, from the upper bounds (16) and (17), the signal  (32) satisfies ||  ( , t)+3 ,

(33)

where  :=  2 ++2 + 2 3 + 22 + 23 is known and 3 := 321 . Then, from (25) and (33), one can write ||

 + ¯ 3. 1+ 

(34)

In order to develop an upper bound for || ˙ we need an upper bound for | |. ˙ From (9), one has | y˙ ||| and, from (15), one can verify that | y˙ | 1 ( , t)+1 . Moreover, from Definition 1 and (21), ˙ 1 and ˙ 2 satisfy 1 | ˙ 1 || 1 |+( , t) and 2 | ˙ 2 ||o ( 2 )|+2 |o |, respectively. Then, one concludes that | | ˙ ( , t)+1 ,

(35)

where ( , t) := 1 +| 1 |/1 +/1 +|o |/2 +|o | is known. Then, multiplying (25) and (35), one gets | | ˙

+ ¯ 1. 1+ 

(36)

5.2.2. Domination function properties. We start by choosing the domination function  in (25) so that the following property holds with  in (33) and in (35): (P0)  , c0 (1+  ), ∀t ∈ [0, t M ) where c0 0 is a known constant. If  satisfies (P0) then, from (34) and (36), || and | | ˙ can be bounded by ||  O()+ ¯ 3,

(37)

| | ˙  O()+ ¯ 1.

(38)

In order to obtain a norm bound for , ˙ ˙ can be calculated differentiating (25):

(t) ˙ =−

2 ¯



*  *

+ ˙

Copyright 䉷 2010 John Wiley & Sons, Ltd.

*  *t



* 

* 

*t . ˙ = − *  − 1+  1+ 

(39)

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GLOBAL TRACKING SLIDING MODE CONTROL

Note that, ˙ is a piecewise continuous time signal which can be upper bounded by





*

*











*t

*

| |+ ˙ . |(t)| ˙ 1+  1+ 

185

(40)

Our strategy is to design  ( , t) such that the following additional property holds: (P1) |*  /* |, |*  /*t|c1 (1+  ), ∀t ∈ [0, t M )

where c1 0 is a known constant.

This property is trivially satisfied by polynomial  with positive coefficients (see Section 5.2.3). Now, with  satisfying (P1), one has that: ˙ |(t)|c ˙ 1 | |+c 1 .

(41)

Therefore, from (41), (37) and (38) the following holds: |(t)|, ˙

||O()+ ¯ 4,

(42)

where 4 := c1 1 +3 . Note that, from (5) and Assumption 1, if any closed loop system signal escapes in some finite time, then also escapes not latter than that. Indeed, according to Assumption 3, the system possesses an unboundedness observability property [33]. We will use this fact to design  ( , t) so that if escapes in some finite time then  ( , t) also escapes not later than this time. From (25), this will ensure that the second term on the right-hand side of (42) will be of order O(), ¯ before any eventual finite time escape. To this end, we design  to satisfy the property: (P2)  t e− t   ( , t), ∀ , ∀t ∈ [0, t M ) where  is a design positive constant. The exponential term with rate  acts like a forgetting factor which allows a less conservative

 design. Reminding that 4 can be written as 4 = 4 (| (0)|+|x(0)|)e−4 t , with some 4 ∈ K∞ and some positive constant 4 , then if  satisfies (P2), the following holds 4 ¯

4  (| (0)|+|x(0)|)e−4 t ¯ 4 , 1+  1+ t e− t

(43)

¯ ∀t ∈ [0, t M ). We can show that (see Appendix A.3) the right-hand side of (43) is bounded by , at least after some finite time (t 0). Finally, if  is designed so that (P0)–(P2) hold, then from (42) and (43) one can verify that there exists a finite t ∈ [0, t M ) such that: | |, ||  5 (| (0)|+|x(0)|+|(0)|) ∀t ∈ [0, t ) |(t)|, ˙ ||  O() ¯

∀t ∈ [t , t M ),

(44) (45)

with some 5 ∈ K∞ . To see that (44) and (45) hold, refer to Appendix A.3. 5.2.3. One specific variable gain () design. The following assumption is useful to determine at least one specific class of time-varying  satisfying the aforementioned properties, at the expense of some conservatism: Assumption 4 There exists a polynomial p¯  (| |) in | |, with positive real coefficients, such that the functions o ,  ¯ o (Assumption 3) and the bounding functions i , ¯ i (Assumption 1) satisfy (i = 1, 2, 3): |o ( 2 )|, |o ( 1 , y, t)|  p¯  (| |), ¯ o ( , t), y, t),  ¯ i (y, t)  p¯  (| |). i (2 Copyright 䉷 2010 John Wiley & Sons, Ltd.

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This assumption is not so restrictive since only polynomial growth condition is imposed on o ,  ¯ o, o , i ,  ¯ i . Now, recall that  ( , t) in (33) and ( , t) in (35) are given by  ( , t) =  2 ++ 2 + 2 3 + 22 + 23 and ( , t) = 1 +| 1 |/1 +/1 +|o |/2 +|o |, respectively, where i = i (2 ¯ o , y, t)+  ¯ i (y, t) (i = 1, 2, 3) in (15)–(17). Then, with Assumption 4, one can easily obtain a polynomial p (| |) in | |, with positive real coefficients, such that:

 ,

 p (| |).

(46)

We choose  as:

 ( , t) := p (| |)+ t e− t ,

(47)

where  >0 is a design constant. It is not difficult to verify that (47) satisfies (P0) and (P2). To see that (P1) also holds, see Appendix A.3.

6. STABILITY ANALYSIS AND MAIN RESULTS In order to account for all initial conditions involved in the error system (13) and (31), let: z T (t) := [z 0 (t), Te (t), T (t)],

z 0 (t) := z 0 (0)e−t ,

(48)

where z 0 (0) := [ T (0) T (0)] and >0 is a generic constant. The stability analysis is carried out through the following steps: Step 1. First, we demonstrate that |z(t)| is uniformly bounded by a class-K∞ function of |z(0)|, ∀t ∈ [0, t ). Step 2. Then, for t ∈ [t , t M ) we prove that the observer error norm is bounded by |˜ e (t)|z1 (|z (0)|)e−z1 t +O(), ¯ where z1 >0 is a constant and z1 ∈ K∞ , provided ¯ is chosen sufficiently small and (P0)–(P1) hold. Step 3. Applying Lemma 1, one can also verify that |e |, |z(t)|z2 (|z(0)|)e−z2 t +O(), ¯ where z2 >0 is a constant and z2 ∈ K∞ . Moreover, z(t) cannot escape in finite time. Step 4. Finally, we verify that no closed loop signal can escape in finite time and, moreover, are uniformly bounded ∀t, provided 2 (in Definition 1) is chosen sufficiently small. The following theorem summarizes the main result. Theorem 1 Consider the SISO nonlinear plant (1)–(2) transformable into the normal form (6)–(7) under Assumptions 1–3. Let the control law be given by (21), with  given by (19) and consider HGO (23) with  be given by (25) and domination function  designed such that the properties (P0)–(P2) hold. Then, for sufficiently small constants 2 , >0, ¯ there exist z (·) ∈ K∞ and positive constants a, b such that the complete error state z (48) satisfies |z(t)|[z (|z(0)|)+b]e−at +O(), ¯

(49)

∀t0 and ∀z(0), i.e. GAS of the error system with respect to the compact set {z : |z|b} and ultimate exponential convergence of z(t) to a residual set of order O() ¯ are guaranteed, with both sets being independent of the initial conditions. Moreover, all signals in the closed loop system are uniformly bounded. Proof See Appendix A.4.



Finite frequency chattering is avoided and an ideal sliding mode is produced thanks to the ideal sliding loop (ISL) formed around the relay function (see Figure 1), according to the following corollary. Copyright 䉷 2010 John Wiley & Sons, Ltd.

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Corollary 1 (Ideal sliding mode) Additionally to the assumptions of Theorem 1, if |K m ||m |+|km ||r |+ with >0 then ˆ ≡ 0 is reached in finite time. Proof See Appendix A.4.



Remark 2 The importance of the existence of an ideal sliding mode has been discussed in several works, e.g. [34, p.210], [35, 36]. This is because, in the absence of noise, the frequency of chattering can be arbitrarily increased by reducing the sampling period in practical real time computer implementation of the control scheme. In many applications, such as in electrical drives or converters, sufficiently high frequency chattering is acceptable and allows the advantages of sliding mode control to be preserved. In contrast, when using linear causal differentiating filters to restore the states required in the switching function , ˆ inevitable small lags are introduced in the high frequency loop and this usually leads to chattering with limited frequency, independently of the sampling period, thus deteriorating the sliding mode control performance. Remark 3 (Absence of peaking) In addition, one can conclude that e is peaking free by noting that (13) is ISS with respect to u and that the sgn(·) function in u (21) blocks the eventual peaking present in ˆ to u.

7. CONTROLLER ALGORITHM The complete controller is summarized in Table I. The design parameters can be obtained as follows. First, we design a norm observer for the plant state x, according to Definition 1, and transform the original system to the normal form. From the bounding functions  ¯ o , i , ¯ i (i = 1, 2, 3), given in Assumptions 1–3, we obtain: the bounding functions i , the modulation function (19)  and bounding functions and  . Then, we design the domination function  to satisfy the properties (P0)–(P2). The constant  0 is arbitrary and L o is such that s +l1 s −1 +· · ·+l is Hurwitz. The HGO can be implemented from (23). Thus, the control law (21) is implemented with the sliding surface (20) chosen so that (Am , b , S) is strictly positive real. Table I. Proposed algorithm for achieving global tracking with a peaking free control signal. Reference model (12) Output error (11) Norm observer (4) Auxiliary upper bounds Modulation function (19)

˙ m = Am m +b km r,

( −1) T m := [ym y˙m . . . ym ]

e = y − ym ˙ 1 = − 1 +u and 2 ˙ 2 = o ( 2 )+ 2 o ( 1 , y, t) (see Definition 1) 1

i ( , t) := i (2¯ o , y, t)+ ¯ i (y, t) (i = 1, 2, 3), with i , ¯ i in Assumption 1 ( , t) := 3 ( , t)+|K m | 1 ( , t)+km |r |+ .

Domination functions

HGO (23) (25) Sliding surface (20) Control law (21) Copyright 䉷 2010 John Wiley & Sons, Ltd.

 ( , t) :=  2 +  + 2 + 2 3 + 22 + 23 ,

( , t) := 1 +| 1 |/1 + /1 +|o |/2 +|o |,

 ( , t) designed to satisfy (P0)(P1)(P2) ˙

ˆ = A ˆ +b u + H L o (y −c ˆ ) L o = [l1 . . . l ]T , H = diag(−1 , . . . , − ), ( , t) := ¯ /1+  ( , t), where ¯ is a design constant ˆ := S(ˆ −  ) = 0 with (Am , b , S) strictly positive real m

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From the functions o , o and the constant 1 given in Assumption 3 we implement the norm observer. Finally, by simulation, we start with not so small values of , ¯ 2 and then decrease ¯ until acceptable tracking error is obtained, which is guaranteed in the stability analysis. Then, we decrease 2 to assure 2 boundedness (see Definition 1). 8. AN ILLUSTRATIVE CLASS OF NONLINEAR PLANTS We can tackle plants (1)–(2) of the form

˙ = 0 (x, y, t),

(50)

v˙1 = v2 +1 (x, y, t), .. .

(51)

v˙ −1 = v + −1 (x, y, t), v˙ = ku u + (x, y, t), y = v1 , transformable to the normal form (see Remark 1) and satisfying Assumption 1. The state x is partitioned as x T := [ T v T ], with v ∈ R , and ku >0 being a constant. Note that this system is neither in the triangular form nor time-invariant such as in [21]. Now, we formulate sufficient conditions on T = [1 . . .  ] such that (50)–(51) satisfies the minimum phase (Assumption 2) and the norm observer existence (Assumption 3). First, as in [21, 37, 38], we consider that: (C0) (Triangularity condition) For i = 1, . . . , : |i |r (| |, y, t)(|v1 |+ . . .+|vi |)+v (| |, y, t), ¯ r, ¯ v are known non-negative functions continuous in y and piece∀t ∈ [0, t M ), where  wise continuous and upper bounded in t satisfying r (| |, y, t)r (| |)+  ¯ r (y, t) and v (| |, y, t)v (| |)+  ¯ v (y, t) with known r , v ∈ K locally Lipschitz functions. Then, for the -subsystem, we assume that one can obtain a storage function V ( ) satisfying ¯ 2 and , ¯ known so that the following condition (| |)V ( )¯(| |), with () = 2 , ¯ () =  holds: (C1) There exist a known non-negative function  (y, t), continuous in y, piecewise continuous and upper bounded in t and a known  ∈ K such that ∀t ∈ [0, t M ): *V ( ) 0 −(| |)+ (y, t), *

(52)

where the class-K function ◦ ¯ −1 is stiffening¶ in the interval (0, ∞). Note that, (C1) implies Assumption 2. Moreover, (C0) and (C1) allow us to implement the following three-order norm observer for x: 1 ˙ 1 = − 1 +u,

(53)

˙ 21 = −c0 21 +1 (y, t),

(54)

˙ 22 = −(1−e− 22 )+2 2 ( 21 )+2 3 (y, t), 2 ¶ As

(55)

in [39], we say that 1 () is stiffening if for every >0, there exists >0 such that  ⇒ 1 ().

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which is in agreement with Definition 1. The 21 -dynamics provides a norm bound for whereas the 22 -dynamics provides a norm bound for v, so that: |x|4 ( 1 , 21 , y, t, 2 )+c1 ec2 | 22 | +,

(56)

where c0 , c1 , c2 are non-negative constants,  := 0 (| (0)|+|x(0)|)e−o t and 1 , 2 , o are positive design constants. In Appendix A.2 we give the steps needed to obtain the norm observer functions 1 , 2 , 3 and 4 . Now, we illustrate the class of system by the following nontrivial academic example. Example 1 (Class of systems) Consider the four-order nonlinear plant with = 3:

˙ = − 5 −|y| 2 + y(t),

(57)

v˙1 = v2 + y , 2

v˙2 = v3 +

v32 4+4v32

v˙3 = u + 2 y

sin(v2 )+ 2 yv2 ,

2/3 1/3

v2 v3 ,

y = v1 ,

(58)

where (t) is a uniformly bounded time-varying function. The nonlinear terms in the vdynamics satisfy (C0) with r = 2 |y| and v = | |y 2 +0.25 and the inverse dynamics, adapted from [40, Ex. 1], satisfies (C1) with V ( ) = 2 /2,  = | |6 /4 and  = y 2 [1+2 ]/2+0.51/3 . 2/3 1/3

The cross term v2 v3 was inspired from [37, 38, Ex. 2.4]. Note that the system is nontriangular but transformable to the normal form (6)–(7). Moreover, by computing the time ... derivatives y˙ , y¨ , y one can obtain T (x, t), k p (x, t) and d(x, t) satisfying Assumption 1. Assumption 3 also holds and the steps to construct the norm observer (53)–(55) are given in Appendix A.2.  In the next example, we focus only on the time varying behavior of (t). Example 2 (Simulation results) We consider the simple academic case, with no zeroes dynamics and relative degree 2 ( = 2), where (50)–(51) is reduced to: v˙1 = v2 , v˙2 = ku u −1 v2 +2 y 2 +3 sin(24 t), y = v1 . The plant is already in the normal form (6)–(7) (with T = I ), where  = x, k p = ku and k p d = −1 2 +2 y 2 +3 sin(24 t). Assumption 1 is satisfied with: 1 = T =  ¯ 1 = 1, 1 = T = 0, c p = 1, 2 = ¯ 2 = 2, 2 = 0, 3 = 3|x|,  ¯ 3 = 3y 2 +2 and 3 = 3 +  ¯ 3. The uncertain parameters are: 1ku 2, 11 , 2 , ∀>, for any >0 and 00 and an arbitrary column vector L satisfying ATL P +PA L = −I , where A L = A −Lc is a Hurwitz matrix. Now, with T := diag(1, ε, ε 2 , . . . , ε −1 ) and any given constant ε>0, the following properties can be checked: (i) TA T −1 = ε −1 A , (ii) c T −1 = c and (iii) T b = b ε −1 . Then, adding and subtracting the term (εT )−1 Lc v¯ to the v-dynamics, ¯ one can write v˙¯ = [A −(εT )−1 Lc ]v¯ + b ku 1 ++(εT )−1 L y. Moreover, applying the transformation ϑ = T v¯ and the above properties (i)–(iii), one can also write ϑ˙ = ε −1 A L ϑ+b ε −1 ku 1 +ε −1 Ly+ T . The key step is to note that, due to the triangularity condition (C0): |T |kϑ r |ϑ|+ϑ , where kϑ is ε independent. Then, by using the Dini derivative and the bounding function v given in (C0), the time derivative of V := (ϑT Pϑ)1/2 along the solution of the ϑ-dynamics satisfies c1 ¯ 1 ( 21 , 1 , y, t, ε)+1 , V˙ − V +c2 r V +  ε where 1 is a exponentially decaying term and the non-negative function ¯ 1 and the non-negative constants c1 , c2 are all known and satisfy c1 1/(2 M [P]), c2 |P|kϑ /m [P] and [|b ε −1 ku 1 + √ ε −1 L y|+ϑ ]c3  ¯ 1 +1 , with c3 |P|/ m [P]. Therefore, given any V , either c1 V  ¯ 1 or V˙ − V +c2 r V + V +1 . (A2) ε Now, let  ¯ 4 ( 21 , y, t) := 2 ( 21 )+3 (y, t),

(A3)

with the non-negative functions 2 , 3 in (55) to be determined. Then, (55) can be written as ˙ 22 = −

1 ( 22 )+  ¯ 4, 2

(A4)

with () := 1−e− . Hence, by using the bounding function r , given in (C0), we must choose ¯ 4 in (A3) (and the functions 2 , 3 ) in order to satisfy: ¯ 4 ( 21 , y, t). c2 r +1 Norm bound for v: The norm bound for the v-subsystem can be obtained considering two cases for the growth rate r (| |, y, t): r >kr and r kr , where kr = 3/(c2 2 ) and 2 is the positive design constant in (A4). Case 1: In this case, one has 3/2 c2 kr +1c2 r +1¯ 4 . Thus, one can verify that ¯ 4 −1, ()22 

∀.

(A5)

 To

avoid the Dini derivative we could have used the relationship aba 2 +b2 , valid ∀a, b>0, at the expense of some conservatism.

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Now, let W := ln(V +1) [22]. Then, W˙ = V˙ /(V +1) and, from (A2), one can write V ¯ 3

1 W˙ − (W )+  ¯ 4 +1 , 2

or

(A6)

with ε = c1 2 ,  ¯ 3 :=  ¯ 1 |ε=c1 2 and noting that V /(V +1), 1/(V +1)1. Now, given any W , we have two possibilities: W < 22 or W  22 . Considering the latter case, one can write −( 22 )−(W ), since  is a increasing function. Therefore, from (A6) and (A4), one has ˙ 22 W˙ −1 . In addition, from (A5), ˙ 22 also satisfies ˙ 22 1/2 . Consequently, adding the last two inequalities one has 1 W˙ −2 ˙ 22 − +1 . 2 Now, recall that 1 = 1 e−1 t and let W¯ = W +1 /1 , for some positive constant 1 and some ˙¯ −2 ˙ 22 − 12 , from which one can conclude that, W¯ 2 22 −t/2 + 1 ∈ K∞ . Then, one has W |W¯ (0)−2 22 (0)|. Note that, it is always possible to find an exponential decaying term which is an upper bound for the above affine time function, i.e. −t/2 +|W¯ (0)−2 22 (0)|2 , where 2 := 2 (|W¯ (0)|+| 22 (0)|)e−2 t , with 2 ∈ K∞ and some constant 2 >0. Finally, given W , one can conclude that W 2| 22 |+2 +1 /1 and, by using comparison theorems [26] and recalling that V = e W −1 one can write V e2| 22 | +3 ,

(A7)

where 3 is an exponential decaying term. Case 2: Assume now that r kr and set ε = c1 /(c2 kr +2) in (A2). Then, one can write: V  ¯2

or

V˙ − V +1 ,

(A8)

where  ¯ 2 = ¯ 1 |ε=c1 /(c2 kr +2) . In this case, adding the two upper bounds obtained from (A8) one can write V ¯ 2 +4 ,

(A9)

where 4 is an exponential decaying term. Then, from (A7) and (A9) one has V e2| 22 | +  ¯ 1 ( 21 , 1 , y, t, ε)+5 ,

(A10)

with ε = c1 /(3/2 +2) and using the Rayleigh’s inequality one can obtain an upper bound for v. Finally, putting together the norm bounds for v and we obtain the non-negative function 4 and the non-negative constants in (56). A.3. Auxiliary proofs Proof of Inequalities (15), (16) and (17): From the plant state estimator, one has |x|¯ o ( , t)+o . Note that, for any increasing function : R+ → R+ , one can write (a +b) (2a)+ (2b), ∀a, b ∈ R+ . Since i (i = 1, 2, 3) are non-negative and increasing functions in their first arguments, then follows that i (|x|, y, t)i (2 ¯ o , y, t)+i (2o , y, t). Moreover, one can further conclude from Assumption 1 that i (2o , y, t)i (2o )+  ¯ i (y, t) and, since i ∈ K are locally Lipschitz functions, i (2o )1 , where 1 = 1 (| (0)|+|x(0)|)e−o t with some 1 ∈ K∞ . Therefore, one can write i (|x|, y, t) i ( , t)+1 , where i ( , t) := i (2 ¯ o , y, t)+  ¯ i (y, t). Recalling that [ T T ]T = T (x, t), then |||T (x, t)|. Hence, from Assumption 1, one has ||1 (|x|, y, t). Therefore, , k p and d satisfy (15), (16) and (17). Proof of Inequalities (44) and (45): If 4 (| (0)|+|x(0)|)1 or t M is infinite it is trivial due to the vanishing exponential e−4 t . Now, consider that 4 (| (0)|+|x(0)|)>1 and t M is finite. Then, one has: (i) e− t e− t M , ∀t ∈ [0, t M ) and (ii) ∃t1 ∈ [0, t M ) such that  t , ∀t ∈ [t1 , t M ), where  is an arbitrary constant. Hence, from (i) and (ii) and taking (4 −1)e t M , one also has that the right-hand side of (43) is bounded by . ¯ In addition, during the interval [0, t ), by definition of Copyright 䉷 2010 John Wiley & Sons, Ltd.

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t , one has that | (t)|5 (| (0)|+|x(0)|). By noting that (i) e t can be bounded by a class-K function of | (0)|+|x(0)| and (ii)  (31) escapes at most exponentially, one can conclude that || and | | can be bounded by some class-K function of | (0)|+|x(0)|+|(0)|. Proving that (47) Satisfies (P1): First note that for any absolutely continuous function g(t), gt  = |g(t)| or gt  is a positive constant. Thus, |dgt /d|g||1, almost everywhere, consequently, |*  /*| |||d p (||)/d|||+e− t . Moreover, since d p(a)/dak1 p(a), where p(a) is any polynomial in a with positive real coefficients and k1 is an appropriate constant, one can also write |*  /*| ||k1 p (| |)+e− t . In addition, since |*| |/* |1, then |*  /* ||*  /*| || and one can conclude that (P1) holds, since *  /*t = −  t e− t . A.4. Main proofs Proof of Lemma 1 First, applying the coordinate transformation en = Tn e , where Tn := [I S T ]T , system (13) can be rewriting into the normal form and one can conclude that (13) is OSS w.r.t. the output Se , i.e. e satisfies |e |k1 |Se |+1 , where k1 is a positive constant and 1 = 1 (|e (0)|)e−m t , with some 1 ∈ K∞ and 00 is the solution of ATm P +PAm = −I , one can conclude that the time derivative of V along the solutions of (13) satisfies V˙ −|e |2 −2k p |Se |[−|de |]. Thus, since  in (19) satisfies (18), i.e. >|de |, then one has V˙ −|e |2 , which leads to the conclusion that |Se ||S ˜ e |+2 and, consequently, the e -dynamics is ISS w.r.t. ˜ e .  Proof of Theorem 1 [STEP1] : From Definition 1, Assumption 1 and (44), one can verify that |z(t)|1 (|z(0)|)+ k1 , ∀t ∈ [0, t ], where 1 ∈ K∞ and k1 0 is a constant. [STEP2]: Consider the -dynamics (31) and the storage V = T P, where P = P T >0 is the solution of ATo P +PAo = −I . Then, the time derivative of V along the solutions of T (31) satisfies V˙ = −||2 +()[2 ˙ P]+()[2T Pb ]. Now, designing  to satisfy (P0)–(P2), (45) holds and the following inequality is valid ∀t ∈ [t , t M ): V˙ − ||2 +O()k ¯ 1 ||2 +O()k ¯ 2 ||, where k1 := 2|P||| and k2 := 2|P||b |. Moreover, since 2 ab0. Now, either V 2O()/ ¯ 1 or V˙ −1 V /2. Consider the latter case. Since 0 and some 2 ∈ K∞ . In the last inequality, the norm bound for ˜ e was obtained by noting that ˜ e = T−1  implies |˜ e |||, since |T−1 |1 for 0, k3 0 and some k3 ]e−3 t +O(), 3 ∈ K∞ . Thus, |z(t)| cannot escape in finite time and it is uniformly bounded in I := [0, t M ) (UBI). Copyright 䉷 2010 John Wiley & Sons, Ltd.

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[STEP4]: Since z(t) is UBI, then e ,  = Se ,  and  = e +m are UBI and, from Assumption 2, , x¯ are also UBI. In addition, according to the lower bound for |T (x, t)| given in Assumption 1 one has that x UBI. Thus, the bounding functions given in Assumption 1 guarantee that d, k p , de are also UBI. Now, rewriting (13) into the normal form one can write ˙ = −4 +k4 (u +de ), for some constants 4 , k4 >0. Moreover, by linearity of the solution of the last equation, one can further write  = 1 +2 , where ˙ 1 = −4 1 +k4 u and ˙ 2 = −4 2 +k4 de , with appropriate initial conditions. Thus, since  and de are UBI so are 2 and 1 . Then, any signal satisfying ˙ 3 = −5 3 +k5 u, where 5 , k5 >0 are constants, is also UBI, in particular, 1 defined in (3). Since y, 1 is UBI and o is piecewise continuous in its arguments then the 2 -dynamics, in Definition 1, cannot escape in finite time. Finally, one can conclude that all system signals cannot escape in finite time, i.e. t M → ∞. Now, from STEP 3, one can directly verify that the error system is GAS with respect to the compact set {z : |z|b} and ultimate exponential convergence of z(t) to a residual set of order O(). ¯ Closed loop signals boundedness: One can further conclude, subsequently, that ||, y, | |, |x|, 1 and 1 converge to a set of order O(|r |+k5 ) after some finite time, where k5 is a positive constant depending on the time-varying disturbances. Then, there exists 2 sufficiently small and independent of the initial conditions, which assures that 2 is bounded after some finite time. Finally, one can conclude that all system signals are UB ∀t.  Proof of Corollary 1 Recalling that A = Am −b K m , ˆ = ˆ e +m , ˆ = e +m − ˜ e , ˆ e = e − ˜ e and ˜ e = T−1 , then ˙ from (23) one can write ˆ e = Am ˆ e +b u +m +e , where m = −b (K m m +km r ) and e = (b K m + H L o c )(˜ e −e )+ H L o e. Note that, from Theorem 1, all system signals are uniformly bounded and z(t) → O(). ¯ Then, there exists a finite time T1 >0 such that |e |1 , ∀tT1 , for T any 1 >0. Now, consider the storage function V = ˆ e P ˆ e , where P = P T >0 is the solution of ATm P +PAm = −Q, where Q = Q T >0 and Pb = S T (recall that (Am , b , S) is strictly positive real). Then, computing V˙ along the solutions of the ˆ e -dynamics, one can verify that the condition ˙ˆ for the existence of sliding mode ˆ 0 is an arbitrary constant.  REFERENCES 1. Yu X, Xu J-X (eds). Variable Structure Systems: Towards the 21st Century. Springer: Berlin, 2002. 2. Edwards C, Colet EF, Fridman L (eds). Advances in Variable Structure and Sliding Mode Control. Springer: Berlin, 2006. 3. Sabanovic A, Fridman L, Spurgeon SK (eds). Variable Structure Systems: From Principles to Implementation (IEE Control Engineering). Academic Press: New York, 2004. 4. Hsu L, Cunha JPVS, Costa RR, Lizarralde F. Multivariable output-feedback sliding mode control. In Variable Structure Systems: Towards the 21st Century, Yu X, Xu J-X (eds). Springer: Berlin, 2002; 283–313. 5. Hsu L, Peixoto AJ, Cunha JPVS, Costa RR, Lizarralde F. Output feedback sliding mode control for a class of uncertain multivariable systems with unmatched nonlinear disturbances. Advances in Variable Structure and Sliding Mode Control. Springer: Berlin, 2006; 195–225. 6. Oh S, Khalil HK. Nonlinear output-feedback tracking using high-gain observer and variable structure control. Automatica 1997; 33(10):1845–1856. 7. Cunha JPVS, Hsu L, Costa RR, Lizarralde F. Sliding mode control of uncertain linear systems based on a high gain observer free of peaking. Preprints of the 16th IFAC World Congress, Prague, Czech Republic, July 2005. 8. Levant A. Higher-order sliding modes, differentiation and output-feedback control. International Journal of Control 2003; 76(9):924–941. 9. Esfandiari F, Khalil HK. Output feedback stabilization of fully linearizable systems. International Journal of Control 1992; 56:1007–1037. 10. Oliveira TR, Peixoto AJ, Nunes EVL, Hsu L. Control of uncertain nonlinear systems with arbitrary relative degree and unknown control direction using sliding modes. International Journal of Adaptive Control and Signal Processing 2007; 21:692–707. 11. Mazenc F, Praly L, Dayawansa W. Global stabilization by output feedback: examples and counterexamples. Systems and Control Letters 1994; 23:119–125. Copyright 䉷 2010 John Wiley & Sons, Ltd.

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