Properties of a Combined Adaptive/Second-Order Sliding Mode Control Algorithm for some Classes of Uncertain Nonlinear Systems

Transcription

1 1334 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 7, JULY 2000 results that include the well-known matrix equality condition [1] as a special case, still allowing arbitrary state weighting matrices. Theorem 6 and Corollary 4 are also new results that weaken the condition on the state weighting matrix. It is known that the terminal weighting matrices presented in this paper can be represented as LMI forms and computed by using existing semi-definite programming methods [21] numerically. The monotonicity of the optimal cost obtained in this paper may be useful for the closed-loop stability of constrained linear systems. REFERENCES [1] W. H. Kwon and A. E. Pearson, A modified quadratic cost problem and feedback stabilization of a linear system, IEEE Trans. Automat. Contr., vol. 22, pp , May [2] W. H. Kwon and D. G. Byun, Receding horizon tracking control as a predictive control and its stability properties, Int. J. Contr., vol. 50, no. 5, pp , [3] M. V. Kothare, V. Balakrishnan, and M. Morari, Robust constrained model predictive control using linear matrix inequalities, Automatica, vol. 32, pp , [4] H. Michalska and D. Q. Mayne, Robust receding horizon control of constrained nonlinear systems, IEEE Trans. Automat. Contr., vol. 38, pp , Nov [5] H. Chen and F. Allgower, A quasiinfinite horizon nonlinear model predictive control scheme with guaranteed stability, Automatica, vol. 34, pp , [6] J. Richalet, A. Rault, J. Testud, and J. Papon, Model predictive heuristic control: Applications to industrial processes, Automatica, vol. 14, no. 5, pp , [7] C. R. Cutler and B. L. Ramaker, Dynamic matrix control A computer control algorithm, in Proc. AIChE National Meeting, Houston, TX, [8] G. De Nicolao and S. Strada, On the stability of receding-horizon LQ control with zero-state terminal constraint, IEEE Trans. Automat. Contr., vol. 22, pp , Feb [9] D. L. Kleinman, An easy way to stabilize a linear constant system, IEEE Trans. Automat. Contr., vol. 15, p. 692, Dec [10], Stabilizing a discrete, constant, linear system with application to iterative methods for solving the Riccati equation, IEEE Trans. Automat. Contr., vol. 19, pp , June [11] W. H. Kwon and A. E. Pearson, On feedback stabilization of timevarying discrete linear systems, IEEE Trans. Automat. Contr., vol. 23, pp , Mar [12] S. Keerthi and E. Gilbert, Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: Stability and moving-horizon approximation, J. Optimization Theory Appl., vol. 57, no. 2, [13] J. B. Rawlings and K. R. Muske, The stability of constrained receding horizon control, IEEE Trans. Automat. Contr., vol. 38, pp , Oct [14] A. Bemporad, L. Chisci, and E. Mosca, On the stabilizing property of SIORHC, Automatica, vol. 30, pp , [15] G. De Nicolao and R. R. Bitmead, Fake Riccati equations for stable receding horizon control, in Proc. 4th European Control Conf., Bruxelles, BE, [16] W. H. Kwon, A. M. Bruckstein, and T. Kailath, Stabilizing state-feedback design via the moving horizon method, Int. J. Contr., vol. 37, pp , [17] M. A. Poubelle, R. R. Bitmead, and M. R. Gevers, Fake algebraic Riccati techniques and stability, IEEE Trans. Automat. Contr., vol. 33, pp , Apr [18] R. R. Bitmead, M. Gevers, and V. Wertz, Adaptive Optimal Control: The Thinking Man s GPC. Englewood Cliffs, NJ: Prentice-Hall, [19] J. W. Lee, W. H. Kwon, and J. H. Choi, On stability of constrained receding horizon control with finite terminal weighting matrix, Automatica, vol. 34, no. 12, pp , [20] L. Magni and R. Sepulchre, Stability margin of nonlinear receding horizon control via inverse optimality, Syst. Contr. Lett., vol. 32, pp , [21] L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev., vol. 38, pp , Properties of a Combined Adaptive/Second-Order Sliding Mode Control Algorithm for some Classes of Uncertain Nonlinear Systems Giorgio Bartolini, Antonella Ferrara, Luisa Giacomini, and Elio Usai Abstract In this paper, a combined adaptive/variable structure control approach is presented that exploits the good properties of the backstepping procedure and of a second-order sliding-mode control algorithm. This algorithm enables one to attain the conditions =0, _ =0(second-order sliding mode) in a finite time, =0being a predefined sliding manifold. The combined approach retains the stability and convergence features of the original adaptive strategy. In addition, it allows one to deal with systems with uncertainties of more general types, ensuring robustness, as well as a reduction in the computational load. Index Terms Backstepping, nonlinear systems, second-order sliding-mode. I. INTRODUCTION The research activity on adaptive control in the early 1990 s has been massively focused on the study of uncertain nonlinear systems, especially those with the full state available for feedback design (see, for instance, [5], [9], [10], [12], [14], [15], and [18]). In particular, two classes of nonlinear systems, characterized by uncertainties expressible through a linear parametric dependence, have received the attention of the researchers, i.e., systems that can be transformed into the so-called parametric-strict feedback form and parametric-pure feedback form. For such systems, an adaptive control procedure, named backstepping, has been developed, that is capable of ensuring global regulation and tracking properties in the former case, and an estimate of the region of attraction in the latter. The backstepping strategy is characterized by a step-by-step procedure that interlaces, at each step, a coordinate transformation with the design of a virtual control, via a classical Lyapunov technique, through the definition of a tuning function. As a result, at the last step, the true control expression and the actual update law are obtained [8]. Moreover, it avoids the necessity for overparametrization, in contrast with previous adaptive controllers, ensuring enhanced stability and parameter convergence properties [8]. Together with the development of efficient adaptive control techniques for nonlinear systems, the control community has witnessed great advances in the research on robust nonlinear feedback control. Most of the control strategies belonging to this area can be viewed as alternatives in nature to the adaptive control ones. Yet, some of the former strategies have proved to be efficaciously combinable with the latter to fully exploit the advantages usually exhibited by the single approaches. This is undoubtedly the case with variable structure control (VSC). A VSC strategy can be designed in two steps: the choice of a manifold such that, if the system trajectory is confined to lying upon it, then the Manuscript received December 12, 1996; revised November 21, Recommended by Associate Editor, E. Yaz. This work was supported in part by Contract MAS3CT AMADEUS of the European Community and under Contract IST PROTECTOR of the European Community. G. Bartolini and E. Usai are with the Department of Electrical and Electronic Engineering, University of Cagliari, Cagliari, Italy ( {giob@diee.unica.it; eusai@diee.unica.it). A. Ferrara is with the Department of Computer Engineering and Systems Science, University of Pavia, Pavia, Italy ( ferrara@conpro.unipv.it). L. Giacomini is with the School of Engineering and Applied Science, Aston University, Birmingham, U.K. ( l.giacomini@aston.ac.uk). Publisher Item Identifier S (00) /00$ IEEE

2 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 7, JULY system exhibits the desired behavior; the determination of a control law (discontinuous on the manifold) capable of forcing the system trajectory to reach the manifold and remain on it, featuring a so-called sliding mode [19], [21], in spite of possible matched disturbances and parameter uncertainties with known upper and lower bounds. In the literature, numerous combined adaptive/vsc schemes have appeared for both linear and nonlinear systems (see, among others, [1], [13], [16], and [22]). In particular, in [22], a class of linearizable nonlinear systems with unknown parameters is considered, and an adaptive sliding manifold S(t) =0is designed, relying on the application of n steps of a standard backstepping procedure (n being the system order). Then, a sliding mode is generated by simply choosing the actual control to be discontinuous on the adaptive sliding manifold, thus ensuring the same stability and convergence properties as those of the original backstepping technique and, in addition, robustness to unknown perturbation inputs. The idea pursued in this paper has been inspired by [22]. It relies on the possibility of combining with the backstepping procedure not a standard VSC strategy but a VSC strategy enforcing a second-order sliding mode (SOSM), that is, a motion confined to the manifold S(t) =0, with _ S(t) =0. The justification of this choice is based on the positive features of SOSM control which have been outlined in the recent literature [2] [4], [11]. Among them, the major one is that SOSM control enables one to overcome the limit of conventional VSC related to the necessity of a complete availability of the system state. The proposed combined control procedure retains n 0 2 steps of the basic backstepping technique, coupling them with the construction of an auxiliary second-order uncertain nonlinear system to which the aforesaid SOSM control applies. The states of the auxiliary system are steered to zero in finite time. The remainder n 0 2 equations of the transformed system turn out to be an autonomous system for which the same stability and convergence properties as those of the standard backstepping adaptive control scheme are obtained. The advantages of the proposed combined procedure are numerous; among them: it allows the presence of nonparametric uncertainties in the last two equations of the system and increases robustness. Further, it reduces the computational load, as compared with the standard backstepping strategy, as well as with the previous proposals of combined backstepping/vsc schemes relying on first order sliding mode. The paper is organized as follows: a short outline of the backstepping procedure presented in [8] is given in Section II. The second-order VSC algorithm is described in Section III, followed by the combined procedure. Finally, a couple of simulation examples are provided in Section IV. II. THE BACKSTEPPING PROCEDURE The backstepping procedure is a well-established technique [5] [9], [22], that can be used to solve the problems of stabilization and tracking control for a class of nonlinear systems via state feedback. The procedure is applicable to feedback linearizable nonlinear systems that can be transformed into a parametric-pure feedback form and a parametric-strict feedback form [12], [17], [20]. For the sake of simplicity, we shall focus on the latter form, i.e., _x i(t) =x i+1(t) + T 8 i(x 1; 111;x i); i =1; 111; _x n(t) =8 0(x(t)) + T 8 n(x(t)) + 0(x(t))u(t) where x(t) = [x 1(t); 111;x n(t)] T 2 n, = [ 1; 111; p] T 2 p is a vector of constant unknown parameters, and 8 i (x(t)) 2 p, (1) and 0(x(t)) 2 are known smooth nonlinear functions. The goal is, for instance, to stabilize the system around the equilibrium point x e 1 = 0, x e i+1 = 0 T 8 e i := 0 T 8 i (0; 0 T 8 e 1, 111; 0 T 8 e i01), i =1; 111;. Note that the solution to the tracking problem is not conceptually different, but, for the sake of clarity, only the stabilization problem is considered here. The backstepping procedure consists of the step-by-step construction of a transformed system with the state z i = x i 0 i01, i = 1; 111;n, where i is the so-called virtual control signal at the design step i. The i s are computed at step i + 1to drive z = [z 1 ; 111;z n ] T to the equilibrium state (0; 111; 0). This state is proved to be stable through a standard Lyapunov analysis. The Lyapunov functions computed step by step are used to determine i and the so-called tuning function i, which is a partial formula for the adaptive update of the parameter vector. The last stabilizing function n is the true control u(t), which is applied directly to the original system. In the following, the basic backstepping procedure [8] is briefly outlined for the reader s convenience, since it is the starting point of the development of the combined backstepping/second-order sliding mode procedure proposed in this paper. Step i: Set z i+1 = x i+1 0 i (i = 0; 111;; 0 = 0) and substitute it into _x i = x i+1 + T 8 i so as to obtain _z i = z i+1 + i + ^ T 8 i 0 i01 (@i01=@x k)(x k+1 + T ) 0(@ i01=) _^. The partial system with z k coordinates, where k = 1; 111;i, is stabilized with respect to the Lyapunov function V i = (1=2) z2 k i +(1=2)( 0 ^) T 0 01 ( 0 ^), where ^(t) is the estimate of the unknown parameter vector and 0 2 p2p is a constant positive definite weighting matrix. Lyapunov stability is achieved by setting i = 0z i01 0 c i z i + + i =0 i l=1 i02 i01 x k+1 i ^ T 8 i 0 z l 8 l 0 l01 = i01 +0z i 8 i 0 l01 i01 i01 (3) where i is the tuning function at step i. If this were the last step, with _^ = i and z i+1 =0, then _ V i = 0 i c kz 2 k 0. But, in general, this is not the last step. Thus, _ V i contains spurious terms that will be eliminated at step i +1. Step n: _V n = 0 c k z 2 k + + z n z + 0 u (^ 0 ) T 0 01 _^ 0 0 _^ + ^ T 8 n 0 n l=1 0 x k+1 z l 8 l l01 : (4)

3 1336 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 7, JULY 2000 Setting and u = 1 0 n l01 _^ = n =0 z l 8 l 0 l=1 0z 0 c nz n n 2 8 n x k ^ T one obtains _ V n = 0 n c kz 2 k. This means that, for the system in (z i (t); ^) the equilibrium point [z 1 (t); 111;z n (t)] T = [0; 111; 0] T, ^ = =[ 1; 111; p] T is globally stable. Then, x e 1 = z e 1 =0, x e 2 = z e 2 + e 1 = e 1 = 0 T 8 e 1; 111, and x e n = z e n + e = 0 T 8 e, so the control objective is attained. Moreover, it can be proved that, if the matrix [8 e 1; 111; 8 e n] has a rank equal to p, the equilibrium point is globally asymptotically stable because the estimate ^ converges to its true value [9]. III. VARIABLE-STRUCTURE CONTROL APPLIED TO BACKSTEPPING In the literature, some possibilities of slightly enlarging the class of systems tractable with the backstepping technique, relying on the use of suitable sliding manifolds and related discontinuous control signals, have been reported. In particular, Zinober and Rios Bolivar [22] have increased robustness to bounded disturbances over the input channel. The system considered in their work is of the type _x i(t) =x i+1(t) + T 8 i(x 1; 111;x i) i =1; 111;n0 1 _x n(t) =8 0(x(t)) + T 8 n(x(t)) + 0(x(t))u(t)+(t) (5) where x(t) =[x 1 (t); 111;x n (t)] T, and (t) can be viewed as a general uncertainty in the 8 n (x(t)), 8 0 (x(t)), 0 (x(t)) functions. Then, the authors recall that, according to [16], the full expression of z n = x n 0 could be interpreted as an adaptive sliding surface in the original state-space, and that a sliding mode can be generated on it by replacing the term z n in the expression of u(t), determined in the previous section, with the term sign(z n ). More precisely, denoting the sliding manifold by S, one has S = z n. Thus, by choosing u = 1 0 0z 0 k sign S n 0 n x k ^ (6) one obtains _ V n = 0 j=1 c jz 2 j 0 kjsj. The adaptive controller cooperates in ensuring the attractiveness of the sliding surface S = z n. Simulation tests show a satisfactory behavior of the combined adaptive/variable structure scheme in the presence of disturbances. The convergence properties of the state and the parameter vector estimates remain unchanged with respect to those of the standard backstepping procedure. Taking into account [22], the idea underlying our contribution can be summarized as follows: if, instead of choosing S = z n, one could identify a sliding manifold dependent only on the variables z i, with i =1; 111;n0 2, and a control u(t) capable of forcing S = _ S =0 in a finite time, then the stability of the equilibrium point would be guaranteed with a significantly reduced computational load. A control of such a type can be named second-order sliding-mode control. Its role will be illustrated, in more details, in the remainder of this section. A. Second-Order Sliding-Mode Control In order to clarify our idea relevant to a different way of exploiting the VSC philosophy within a backstepping framework, let us consider the following second-order system: _x 1 (t) =f 1 (x 1 ;x 2 ) (7) _x 2(t) =f2(x1; x 2)+g(x1; x 2)u(t) where f 1 (x 1 ;x 2 ) 2, continuous and differentiable with respect to x 1, x 2, f 2(x1; x 2), g(x1; x 2) 2, are assumed to be uncertain, and x 2 is not available for measurements. u(t) 2 is the control input. By setting y 1 (t) =x 1 (t) and y 2 (t) = _x 1 (t), (7) can be rewritten as where and _y 1 (t) =y 2 (t) _y 2 (t) =H(x 1 ;x 2 ;y 2 )+d(x 1 ;x 2 )u(t) H(x 1;x 2;y 1 (x 1 ;x 2 1 y 2 1 (x 1 ;x 2 2 f 2(x1; x 2) d(x 1 ;x 2 2 g(x 1 ;x 2 ): The following assumptions relevant to H(x 1;x 2;y 2) and d(x1; x 2) can be made (8) jh(x 1 (t); x 2 (t); y 2 (t))j <H; 0 <D 1 d(x1(t); x2(t)) D2: (9) The problem consists of steering both y 1(t) and the unknown y2(t) to zero in a finite time. By analogy to the well-known solution to the time optimal control problem, the control u(t) can be chosen as a bang bang control switching between two extreme values, 0U Max and +U Max. The classical switching logic for a double integrator (H(x 1(t); x2(t); y2(t)) = 0, D1 = D2 =1)is u(t) = 0U Max y 1 (t) > y 1(t) =0 1 2 y 2 (t)jy 2 (t)j U Max y 2 (t)jy 2 (t)j U Max y 1(t) < 0 1 y 2 (t)jy 2 (t)j +U Max y 1(t) < 0 2 U Max y 1 (t) =0 1 2 y 2(t)jy2(t)j U Max y 1 (t) > 0 : (10) Such a switching logic, instead of being based on the signs of y 1(t) + (y 2 (t)jy 2 (t)j)=(2u Max ) and y 1 (t), and therefore instead of being dependent on both y 1 (t) and y 2 (t), could be expressed only in terms of

4 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 7, JULY y 1(t), which, by assumption, is available for measurement. Indeed, it is easy to verify the following: consider a point belonging to a segment of the switching line y 1 (t) +(y 2 (t)jy 2 (t)j)=(2u Max ) = 0. At such a point, the switching-line segment intersects a parabola (representing the past trajectory of the controlled system) that has an opposite concavity with respect to it. Then, the modulus of the component y 1 (t) of the point belonging to the switching line is equal to one half of the modulus of the component y 1 (t) of the extreme value of that parabola. Assume that the extreme value can be evaluated along each parabolic trajectory, and denote its abscissa by y Max. Then, the foregoing considerations can be summarized by the following algorithm, which is equivalent to the optimal one in the case H[1; 1; 1] =0, D 1 = D 2 =1, z 1(0)z 2(0) > 0, 3 =1, where 3 is a parameter used in the algorithm. Algorithm 1: i) Set 3 2 (0; 1] \ (0; (3D 1=D 2)). ii) Set y Max = y 1 (0). Repeat the following steps for any t>0. iii) If [y 1 (t) 0 (1=2)y Max ][y Max 0 y 1 (t)] > 0, then set = 3, else set =1. iv) If y 1(t) is an extreme value, then set y Max = y 1(t). v) Apply the control law u(t) = 0U Max signfy 1 (t) 0 (1=2)y Maxg until the end of the control time interval. This algorithm proves to be valid also for H[1; 1; 1] 6= 0, D 1 6= D 2 6=1, and for y 1 (0)y 2 (0) not necessarily positive, in the sense that it allows the origin of the y 1 (t), y 2 (t) state space to be reached in a finite time. Indeed, the following result can be proved. Theorem 1: Given the state equation (8) with bounds as in (9) and given y 2 (t) not available for measurements, then, if the extreme value of y 1(t) is evaluated with ideal precision for any y 1(0) and y 2(0), the suboptimal control strategy defined by Algorithm 1, with the additional constraint U Max > max H 3 D 1 ; 4H 3D D 2 (11) causes the generation of a sequence of states with coordinates (y Max ; 0) featuring the contraction property jy Max j < jy Max j, i = 1; Moreover, the convergence of the system trajectory to the origin of the state plane takes place in a finite time (i.e., a SOSM is enforced). Proof Part 1: Proof of the contraction property. Consider (8), with ju(t)j U Max and bounds as in (9). Moreover U Max > H 3 D 1 (12) and 3 as in i) (Algorithm 1), since, by assumption, condition (11) is satisfied. Depending on the initial condition y 1 (0) and y 2 (0), one can distinguish the following cases. Case 1: [y 1 (0) = y Max > 0, y 2 (0) = 0, i.e., the initial point lies on the right side of the abscissa axis.] In this case, u(t) =0 3 U Max and by integration of system (8) it is trivial to show that when the commutation occurs at the time instant t c such that y 1 (t c )=(1=2)y Max, according to Algorithm 1, the corresponding value of y 2 (t c ) belongs to the interval Starting from any point over this interval, for t>t c, and integrating system (8) with u(t) =U Max ( =1), one can easily show that the state trajectory crosses the abscissa axis over the interval 0 1 ( 3 D 2 0 D 1 )U Max +2H y Max ; 2 D 1 U Max 0 H 1 (D D 1)U Max +2H y Max : 2 D 2U Max + H The right extreme of this interval is nearer the origin than the considered starting point; then, to assess the contraction, it is sufficient that the modulus of the left bound of the previous interval be less than y Max. Considering also (12), this sufficient condition and i) (Algorithm 1), can be expressed by the following system of inequalities: 3 1 \ 3 D 1 U Max >H \ (3 D 2 0 D 1 )U Max +2H < 2: D 1 U Max 0 H (13) By means of simple computations, one can find the interval solutions to (13) with respect to 3 and U Max, that is U Max > H 3 D 1 ; if 3 2 0; 4H 3D D 2 ; if 3 2 \ 3D 1 4D 1 + D 2 3D 1 4D 1 + D 2 ; 1 3D 1 4D 1 + D 2 ; 3D1 D 2 : (14) It is also plain to verify that, if 3 > (3D 1=4D 1 + D 2) then (H= 3 D 1 ) < (4H=3D D 2 ),if 3 < (3D 1 =4D 1 + D 2 ) then (H= 3 D 1 ) > (4H=3D D 2 ), and that the intersection between the two segments of the limiting curve for U Max occurs at 3 = (3D 1 =4D 1 + D 2 ), according to condition (11), which is true by assumption. Case 2: [y 1(0) = y Max < 0, y 2(0) = 0, i.e., the initial point lies on the left side of the abscissa axis.] The proof is the same as for Case 1 but with reverse extremes of the related intervals. Case 3: [y 1(0)y 2(0) > 0, y 1(0)y 2(0) < 0, y 1(0) = 0; y 2(0) 6= 0, i.e., all the other possible initial conditions.] It is trivial to see that, after at most a finite time interval, the system trajectory reaches a point of the types considered in Cases 1 and 2. Part 2: Proof of the fact that the convergence to the origin takes place in a finite time. Algorithm 1 defines a sequence ft Max g of time instants at each of which an extreme value of y 1 (t) occurs. It can be proved that each term of this sequence is upperbounded by the corresponding term of the sequence (D D 2)U Max ^t Max = ^t Max + (D 1U Max 0 H) p 3 D 2U Max + H 2 jz Max j: (15) Recursively, from (15) one can derive (D D 2 )U Max ^t Max = (D 1 U Max 0 H) p 3 D 2 U Max + H k 2 j=1 jy Max j + t Max 0 y Max( 3 D 2U Max + H); 0 y Max( 3 D 1U Max 0 H) : = k j=1 jy Max j + t Max (16)

5 1338 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 7, JULY 2000 t Max being the time interval from t =0to the time instant when the first extreme value occurs. The previous relationships yield jy Max j < 1 ( 3 D 2 0 D 1 )U Max +2H 2 D 1 U Max 0 H j01 jy Max j (17) so that, in compact form, with implicit definitions of the symbols, one can write t Max < j01 j=1 k = 0 k j=1 jy Max j + t Max j01 + t Max : (18) By assumption, (11) is true and, obviously, < 1. Therefore, from (17), lim k!1 y Max =0, and from (18), lim k!1 t Max < ( 0 =1 0 ) +t Max, which concludes the proof. The convergence of the sequence fy Max g in a finite time implies the convergence of the phase trajectories to zero, since, over any time interval [t Max ;t Max ], the maximum value of jy 2(t)j is bounded by a function of jy Max j, which becomes zero in a finite time. B. How to Apply the Previous Result In order to apply the SOSM control strategy described in Section III-A within a backstepping framework, let us choose S = c z + z as a sliding surface. Indeed, setting y 1(t) =S and y 2 (t) = S, _ the following auxiliary system is obtained where y 2(t) is not available due to the presence of the unknown vector in the first equation of (19). Since the same form as in (8) is obtained, Algorithm 1 can be applied to steer y 1 (t) and y 2 (t) to zero, after suitably choosing the various upper bounds required. The whole procedure can be expressed in algorithmic form as follows. Algorithm 2: iv) Stop the backstepping procedure at step n-2 and compute the quantities, z,, and set _^ =. v) Compute S = c z + z. vi) Compute the upper bounds of the relevant functions in (19). vii) Apply Algorithm 1 with U Max as in (11). Now it is necessary to study the stability and convergence properties of the transformed system _z 1 = 0c 1z 1 + z 2 +( 0 ^) T! 1 _z 2 = 0z 1 0 c 2z 2 + z 3 +( 0 ^) T! z k! k _z i = 0z i01 0 c i z i + z i+1 +( 0 ^) T! i. i01 0 k=i+1 0z k! k + 0! i i =1; 111;n0 3 (21) _y 1(t) =y 2(t) :=x n + + T 3 _z = 0z n03 0 2c z +( 0 ^) T! where _y 2 (t) = x k T 8 n + + T n03 = c x 3 8 T n03 x k+1 3 x k+1 n03 _^ (19) n04 + _x = x n + T 8 (x(t)) _x n =8 0 (x(t)) + T 8 n (x(t)) + 0 (x(t))u(t) _^ =0 z l w l l=1 where! i := 8 i 0 i01 j=1 (@i01=@xj)8j. Relying on the nature of Algorithm 1, the condition S = 0with S _ = 0is reached in a finite time. One can use this fact to show the system stability (through the design of a Lyapunov function V ) and the convergence properties of the parameter estimates. Let V = (1=2) z2 k, then, the expression for V _ is 3 n03 = c x +8 : _^ n03 _V = 0 c k z 2 k + n04 n03 2 z n03 + z + 0 n03 _^ + ^ T 8 0 n03 0 n03 x n03 + z System (19) can be rewritten as +(^ 0 ) T 0 01 _^ 0 n03 : (22) _y 1(t) =y 2(t) _y 2 (t) =8 0 (x(t)) + F 1 (x(t)) + T F 2 (x(t)) + T F 3(x(t)) + 0(x(t))u(t) (20) Setting _^ = and observing that, on S =0, c z + z = 0, z = 0c z, one obtains V _ = 0 c kzk. 2 Then, asymptotically, z 1 = 0 ) x 1 = 0; 111, z = 0 ) x =

6 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 7, JULY Fig. 1. Example 1: backstepping procedure (left), combined procedure (right). 0 T 8 e n03, _^ = 0;x = 0 T 8 e, x n = 0 T 8 e. The parameter convergence is attained if the matrix [8 1 ; 111; 8 ] has a row rank greater than or equal to the number of parameters that actually appear in the first n 0 2 equations of (1). As in the original backstepping procedure, the state convergence is not affected by the parameter mismatch. IV. SIMULATION EXAPMLES To complement the theoretical discussion, two examples are presented in this section. A comparison is made with the standard backstepping technique. As a first example, we use a classical benchmark system modified by Zinober [22] to ensure the convergence of the parameter estimates, i.e., _x 1 = x 2 + (x 1 +0:5) 2 _x 2 = x 3 _x 3 = u: (23) Assuming that the ideal (uncertain) value of is 1, the equilibrium point of the system is fx 1;x 2;x 3g = f0; 00:25; 0g. We refer to [22] for the full expressions for the backstepping quantities, and limit ourselves to showing the plottings of the temporal evolutions of the vectors fx 1;x 2;x 3; ^g obtained via the basic backstepping procedure [Fig. 1 (left)]. In considering our proposal, the sliding surface y 1 = S = x 1 + x 2 +^(x 1 +0:5) 2 and the law _^ = x1 (x 1 +0:5) 2 are chosen. The control is determined with reference to the auxiliary system _y 1 = _ S = _x 1 + _x 2 + _^(x1 +0:5) 2 +2^(x 1 +0:5) _x 1 = x 2 + (x 1 +0:5) 2 1+2^(x 1 +0:5) + _^(x1 +0:5) 2 + x 3 (24) _y 2 = S =(x 3 +2(x 1 +0:5)) 1+2^(x 1 +0:5) + (x 1 +0:5) 3 x 2(5x 1 +0:5) + u +2 ^x 2 + x 2 1(x 1 +0:5) 2 + (x 1 +0:5) 2 (x 1(x 1 +0:5) 3 +2 ^x 2 + x 2 1(x 1 +0:5) 2 ) + 2 ^(x1 +0:5) 4 = F 1 (x 1 ;x 2 ;x 3 ; ^) +F 2 (x 1 ;x 2 ;x 3 ; ^) + 2 F 3 (x 1 ;x 2 ;x 3 ; ^) +u (25) Fig. 2. Example 2: detail of the state trajectories by the combined procedure (solid line) against the results of the basic backstepping (dashed line). according to Algorithm 2. Upper bounds of the quantities in the above expression can be easily computed off-line, relying on the knowledge of the bounds of x 1, x 2, x 3,. In the example, U max =20. In Fig. 1 (right), the asymptotic convergences of the state vector and of the parameter estimates obtained via the proposed combined approach are shown. In this example, the parameter convergence is a bit slower than that obtained via the basic backstepping procedure. A simple extension of the above example is the following system _x 1 = x 2 0 1(x 1 +0:3) 2 _x 2 = x sin(x 2 0 0:5) _x 3 = x 4 _x 4 = 0(x)u + 0(x): (26) Assuming that the ideal (uncertain) values of 1 and 2 are 5 and 2, respectively, the equilibrium point of the system is fx 1;x 2;x 3;x 4g = f0; 0:45; 0:099; 0g. To build the control according to the basic backstepping procedure, it is supposed that 0 (x) =1and 0 (x) =0, whereas, to design the control according to the proposed combined procedure, we let 0 (x) and 0 (x) to be uncertain with known bounds, i.e., 0 < 0 (x) 10 and j 0 (x)j < 20 [note that, under this assumption, 0(x) in particular can be viewed as a bounded disturbance acting on the system]. Further, we choose S = z 3 + z 2 =[2+5(x 1 +0:3) 4 ] [x 2 +x 1 0 ^ 1 (x 1 +0:3) 2 ]+x 1 0x 2 +5(x 1 +0:3) 4 x 1 +^ 1 (x 1 +0:3) 2 0^ 2 sin(x 2 0 0:5), _^ 1 = 05(x 1 0 z 2 (@ 1 =@x 1 )) (x 1 +0:3) 2, _^ 2 = 100 z 2 sin(x 2 0 0:5). Figs. 2 and 3, dashed line, refer to the behavior obtained with the control designed via the backstepping procedure,

7 1340 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 7, JULY 2000 V. CONCLUSIONS Fig. 3. Example 2: parameter estimates by the combined procedure (solid line) against the results of the basic backstepping (dashed line). Fig. 4. Example 2 with uncertain (x) and disturbance: combined procedure (parameters). Fig. 5. Example 2 with uncertain (x) and disturbance: combined procedure (state trajectories). while Figs. 2 and 3, solid line, refer to the behavior obtained with the control designed via the proposed combined procedure, in the case in which 0(x) =1, 0(x) =0(the control amplitude is equal to 1000). In Figs. 4 and 5, the behavior relevant to the case in which the bounded disturbance is present, i.e., 0(x) = 0:2 sin(x1), and the control is that designed via the proposed combined procedure, is illustrated. The performances of the combined adaptive/sosm control procedure are still satisfactory. Some chattering effect is apparent in x4. It could be reduced by applying some modifications to the VSC part of the design according, for instance, to [4]. In this paper, an algorithm that retains n 0 2 steps of the classical backstepping procedure and applies a SOSM control has been presented. The use of the second-order variable-structure procedure ensures robustness to disturbances. This allows the presence of nonparametric uncertainties in the last two system equations. Moreover, the combined algorithm results in a significant reduction in the number of computations needed to design the control, as compared with the standard backstepping algorithm. Indeed, the major burden at every backstepping step i lies in the computations of the 2 h, i.e., those necessary to compute _ i01. At the last step, one needs the evaluation 2 h. In Algorithm 2, one needs S, which is tied to, hence a number of computations equal to that required by the evaluation of _. REFERENCES [1] G. Bartolini and A. Ferrara, Model-following VSC using an inputoutput approach, in Variable Structure and Lyapunov Control, A. S. I. Zinober, Ed. London, U.K.: Springer-Verlag, 1994, pp [2] G. Bartolini, A. Ferrara, and E. Usai, Applications of a suboptimal discontinuous control algorithm for uncertain second order systems, Int. J. Robust Nonlin. Contr., vol. 7, pp , [3], Output tracking control of uncertain nonlinear second-order systems, Automatica, vol. 33, no. 12, pp , [4], Chattering avoidance by second-order sliding mode control, IEEE Trans. Automat. Contr., vol. 43, pp , Feb [5] I. Kanellakopoulos, P. V. Kokotovic, and A. S. Morse, Systematic design of adaptive controllers for feedback linearizable systems, IEEE Trans. Automat. Contr., vol. 36, pp , [6] M. Krstić and P. V. Kokotović, Adaptive nonlinear output-feedback schemes with Marino Tomei controller, IEEE Trans. Automat. Contr., vol. 41, pp , [7], Observer-based schemes for adaptive nonlinear state-feedback control, Int. J. Contr., vol. 59, pp , [8] M. Krstić, P. V. Kokotović, and I. Kanellakopoulos, Transient-performance improvement with a new class of adaptive controllers, Syst. Contr. Lett., vol. 21, pp , [9] M. Krstić, I. Kanellakopoulos, and P. V. Kokotović, Adaptive nonlinear control without overparametrization, Syst. Contr. Lett., vol. 19, pp , [10] P. V. Kokotović, I. Kanellakopoulos, and A. S. Morse, Adaptive feedback linearization of nonlinear systems, in Foundations of Adaptive Control, P. V. Kokotović, Ed. New York: Springer-Verlag, 1991, pp [11] A. Levant and L. Fridman, Higher order sliding modes as a natural phenomenon in control theory, in Robust Control Via Variable Structure and Lyapunov Techniques, E. Garofalo and L. Glielmo, Eds. London, U.K.: Springer-Verlag, 1996, pp (Lecture Notes in Control and Optimization). [12] K. Nam and A. Arapostathis, A model reference adaptive control scheme for pure-feedback nonlinear systems, IEEE Trans. Automat. Contr., vol. 33, pp , [13] K. S. Narendra and J. D. Boskovic, A combined direct, indirect and variable structure method for robust adaptive control, IEEE Trans. Automat. Contr., vol. 37, pp , [14] L. Praly, A. Bastin, J.-B. Pomet, and Z. P. Jiang, Adaptive stabilization of nonlinear systems, in Foundations of Adaptive Control, P.V.Kokotović, Ed. New York: Springer-Verlag, 1991, pp [15] D. Seto, A. M. Annaswamy, and J. Baillieul, Adaptive control of a class of nonlinear systems with a triangular structure, IEEE Trans. Automat. Contr., vol. 39, pp , [16] H. Sira-Ramirez, M. Zribi, and S. Ahmad, Adaptive dynamical feedback regulation strategies for linearizable uncertain systems, Int. J. Contr., vol. 57, pp , [17] R. Su and L. R. Hunt, A canonical expansion for nonlinear systems, IEEE Trans. Automat. Contr., vol. 31, pp , [18] D. G. Taylor, P. V. Kokotović, R. Marino, and I. Kanellakopoulos, Adaptive regulation of nonlinear systems with unmodeled dynamics, IEEE Trans. Automat. Contr., vol. 34, pp , [19] V. I. Utkin, Sliding Modes in Control and Optimization. Berlin, Germany: Springer-Verlag, 1992.

8 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 7, JULY [20] W. Zhan and L. Y. Wang, Coordinate transformation in backstepping design for a class of nonlinear systems, Int. J. Adapt. Contr. Signal Proc., vol. 9, pp , [21] A. S. I. Zinober, Ed., Variable Structure and Lyapunov Control. London, U.K.: Springer-Verlag, [22] A. S. I. Zinober and E. M. Rios-Bolivar, Sliding mode control for uncertain linearizable nonlinear systems: A backstepping approach, in Proc. IEEE Workshop Robust Control Via Variable Structure and Lyapunov Techniques, Benevento, Italy, 1994, pp A Dual Method for Parameter Identification Under Deterministic Uncertainty A. L. Dontchev, M. P. Polis, and V. M. Veliov Abstract In this note, we propose a numerical method for reduction of a priori bounds on the values of uncertain plant parameters. The method successively eliminates parts of the parameter domain which are inconsistent with the plant measurement. The size of the eliminated parts is determined by a measure of inconsistency evaluated with the help of the support functions of the set of feasible states and of the set of possible measurements. We specify our approach on an identification problem for a linear control system with bounds on the magnitude and/or on the energy for the disturbances in the dynamics and in the measurement. Index Terms Bounded uncertainty, deterministic parameter identification, set estimation, uncertain control systems. I. INTRODUCTION In this note, we describe a numerical method for deterministic identification of parameters by successive elimination of parts of the a priori region of parameter values that are inconsistent with measurements of the output. The goal is to reduce the parameter region P given a priori by technological constraints, for the purposes of, for instance, robust control design. Roughly, the method can be described as follows. Consider a plant model described by relations involving a parameter vector p with values from some given set P and suppose that certain variables are available for measurement in the presence of some uncertainty. The inconsistency of a value, p 2P, of the parameter vector with the measurement is expressed by the condition of void intersection of two sets, X (p) and Y(p), where X (p) is a set of vectors obtained from the plant model for p and Y(p) is a corresponding set of vectors which match (are consistent with) the measurement. The precise definition of the sets X (p) and Y(p) for a general model will be given in Section II. To measure the inconsistency of the value p we use a kind of distance between X (p) and Y(p). Then we find a neighborhood of p where the values of the parameter vector are still inconsistent with the measurement. Taking away this neighborhood, we successively reduce the size of the region of possible values of the parameter vectors. To evaluate Manuscript received May 12, 1998; revised May 17, 1999 and November 18, Recommended by Associate Editor, J. Chen. A. L. Dontchev is with the Mathematical Reviews, Ann Arbor, MI USA. M. P. Polis is with the School of Engineering and Computer Science, Oakland University, Rochester, MI USA. V. M. Veliov is with the Institute of Mathematics and Informatics, Bulgarian Academy of Science, 1113 Sofia, Bulgaria, and also with the University of Technology, A-1040 Vienna, Austria. Publisher Item Identifier S (00) the inconsistency of a parameter value we solve an auxiliary optimization problem in a space which is dual to the space of plant vectors, in contrast to the approaches where optimization is carried out directly on the space of the plant or measurement variables. We note that the so-called consistency set P y, which contains only the points from P that are consistent with the measurement y, is a known concept; it is the same (up to differences in the mathematical model) as the membership set in [2] and [9], and the posterior feasible set in [11]. A detailed presentation and comprehensive bibliography on deterministic identification can be found in [5] and [11]. In Section II, we present the method and give conditions for its convergence, for a general model in a set-membership framework. The general model of the identification problem is similar to the one in [4]; however, our aim here is not to select a single point that is consistent with the measurements, but to find a set which contains the consistency set P y and is an acceptable approximation of it, in a sense specified later in the note. For that purpose we use tools from set-valued analysis and optimization. Related work on approximation of the consistency set P y by using interval analysis is presented in [3] and [6] (see Section IV for a discussion). In Section III, we apply our method to the linear control system _x(t) =A(p)x(t) +B(p)u(t) +R(p)w(t); t 2 [0; T] (1) y(t) =C(p)x(t)+D(p)u(t)+E(p)v(t): (2) Here x 2 n is the state vector, the final time T > 0 is fixed, u(1): [0; T] 7! m is a fixed (known) control function, y 2 n is the measurement vector, w(1): [0; T] 7! W represents a disturbance in the dynamics and v(1): [0; T] 7! V describes measurement noise. The initial condition is assumed unknown. The matrices A, B, R, C, D, E are continuous functions of an unknown parameter vector p with values from a given set P q. The sets V r and W r of the disturbance values are given closed convex sets and V is bounded. We also take onto account possible bounds on the energy of the disturbance w represented by the integral constraint E(w) = where Q given positive definite symmetric matrix, 3 given positive number, denotes transposition. 0 T w(t) 3 Qw(t) dt 2 (3) For this model, the identification (set-estimation) problem reads as follows. Given measured output values y 0 = y(t 0); 111;y N = y(tn ) at a finite number of times 0 t 0 < 111 < t N T, find (enclose, approximate) the set P y of points p 2 P that are consistent with the measurement. Here the set P y consists of all points p 2Pfor which there exist an initial state x(0), a Lebesgue measurable function w: [0;T] 7! W with E(w) 2, and a function v: [0;T] 7! V, such that the corresponding solution of (1) and (2) satisfies y(t k ) = y k ; k =0; 111;N. In [7], a particular case of this problem was considered without disturbance in the measurement; then the problem of finding a feasible value of the parameter vector is reduced to an optimization problem with equality constraints. In contrast, our approach is based on solving an optimization problem which is the dual, in the sense of Fenchel Moreau (see, e.g., [8]), to the problem in [7]. In Section IV we present numerical examples /00$ IEEE