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of Proceedings the 42nd IEEE Conference on Decision and Control Maui, Hawaii USA, December 2003 FrE09-5 Integrated design of discrete-time controller for an active suspension system Dina Shona Laila Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville 3010, Victoria, Australia. d.laila@pgrad.unimelb.edu.au Abstract A novel approach to solve a stabilization problem of an active suspension system using a quarter car model is presented. We apply a combination of our results for the framework of the approximate based direct discrete-time design and the Euler based discrete-time backstepping technique. This stabilization problem is very interesting since utilizing a simple quadratic Lyapunov function brings the system into a LaSalle type stability, which makes the design more complicated. To handle this problem, we use our result on changing supply rates lemma for LaSalle type stability condition, to construct a composite Lyapunov function that can be used for the design within our framework. LaSalle s invariance principle is in most cases not applicable when approximate based discrete-time design is used, since semiglobal type of stability is usually achieved. In this situation, we apply our result from [6] to construct a composite Lyapunov function that can be used to characterize stability property of the system. II. PRELIMINARIES The set of real and natural numbers (including 0) are denoted respectively by R and N. SN denotes the class of smooth nondecreasing functions q : R 0 R 0 , which satisfy q(t) > 0 for all t > 0. A function : R 0 R 0 is of class G if it is continuous, nondecreasing and zero at zero. It is of class K if it is of class G and strictly increasing; and it is of class K if it is of class K and unbounded. Functions of class K are invertible. A function : R 0 R 0 R 0 is of class KL if ( , t) is of class K for each t 0 and (s, ) is decreasing to zero for each s > 0. Given two functions ( ) and ( ), we denote their composition and multiplication respectively as ( ) and ( ) ( ). We consider a parameterized family of discrete-time nonlinear systems of the following form: x(k + 1) = FT (x(k), u(k)), y(k) = h(x(k)) , (1) I. INTRODUCTION Most control systems nowadays are sampled-data in nature. A controller is usually implemented digitally and it is inter-connected with a continuous-time plant via ADC and DAC. In this paper, we study the problem of stabilizing an active suspension system, which is used to enable a car to run smoothly on a rough road for comfortable driving. Presently, active suspension systems are controlled using a hydraulic controller. In view of space limitation in a vehicle, it is most appropriate to use digital device to control the active suspension system, as it requires much less space. Since the active suspension module itself is a mechanical - therefore analog - plant, designing a digital controller for this system is a sampled-data system design. Recently, a general uni ed framework for controller design based on approximate discrete-time models was presented in [10] and further generalized in [7] for the input to state stabilization problem. In particular, the results provide suf cient conditions for the continuous-time plant model, the controller and the approximate discrete-time model, to guarantee that the controller input-to-state stabilizes the exact discrete-time plant model, provided it stabilizes the approximate discretetime plant model. We design a discrete-time controller to asymptotically stabilize the active suspension system, using the Euler based backstepping technique [9]. Backstepping is a popular technique in nonlinear control design (see [4]). It is then shown that the Euler based discrete-time controller outperforms the emulation controller. This active suspension design problem is very interesting and motivating since the system enjoys a LaSalle type stability when using a simple quadratic Lyapunov function. In [3], where continuous-time stabilization for the same system was considered, stability analysis was done using LaSalle s invariance principle. Unfortunately, where x Rn , u Rm , y Rl are respectively the state, input and output of the system. Note that the input u can be a control signal or an exogenous disturbance. It is assumed that FT is well de ned for all x, u and suf ciently small T , FT (0, 0) = 0 for all T for which FT is de ned, h(0) = 0 and FT and h are continuous. T > 0 is the sampling period, which parameterizes the system and can be arbitrarily assigned. The following de nitions are used to state results presented later in this section. De nition 2.1: The system (1) is semiglobally practically input-output to state stable (SP-IOSS), if there exist functions , , K , and , G, and for any triple of strictly positive real numbers ( x , u , ), there exists T > 0 and for all T (0, T ) there exists a smooth function VT : Rn R 0 such that for all |x| x , |u| u the following holds: (|x|) VT (x) (|x|) VT (FT (x, u)) VT (x) T (|x|) + T (|y|) + T (|u|) + T . (3) (2) 0-7803-7924-1/03/$17.00 2003 IEEE 6406 The function VT is called a SP-IOSS Lyapunov function. If the system is SP-IOSS with = 0, we say that the system is semiglobally practically input to state stable (SP-ISS) and VT is called a SP-ISS Lyapunov function. If = 0 and the system (1) is an input-free system ( = 0), the system is semiglobally practically asymptotically stable (SP-AS) and VT is called a SP-AS Lyapunov function. Moreover, for SPISS, if the argument of ( ) is the norm of the output y, which consists of only partial states, we have semiglobal practical quasi ISS (SP-qISS). De nition 2.2: [9] Let T > 0 be given and for each T (0, T ) let the functions VT : Rn R 0 and uT : Rn R be de ned. We say that the pair (uT , VT ) is a semiglobally practically asymptotically (SPA) stabilizing pair for FT if there exist , , K , such that for any pair of strictly positive real numbers ( , ) there exists a triple of strictly positive real numbers (T , L, M ), with T T , such that for all x, z Rn with max{|x| , |z|} , and T (0, T ) we have: (|x|) VT (x) (|x|) VT (FT (x, u)) VT (x) T (|x|) + T . |VT (x) VT (z)| L |x z| |uT (x)| M . (4) (5) (6) (7) with given w(k), u(k) and x0 = x(k), over the sampling interval [kT, (k + 1)T ]. If we denote by x(t) the solution of the initial value problem (9) at time t with given x0 = x(k), u(k) and w(k), then the exact discrete-time model of (8) can be written as: (k+1)T x(k + 1) = x(k) + kT f (x( ), u(k), w(k))d (10) e =: FT (x(k), u(k), w(k)) . e FT Since is not known in most cases (see [7]), we use an approximate discrete-time model of the plant a x(k + 1) = FT (x(k), u(k), w(k)) . (11) to design a discrete-time controller for the original plant (8). For instance, the Euler approximate model is x(k + 1) = x(k) + T f (x(k), u(k), w(k)). We consider a family of dynamic feedback controllers z(k + 1) = GT (x(k), z(k)) u(k) = uT (x(k), z(k)) , (12) III. DESIGN TOOLS A. Framework for approximate based direct discrete-time design In this subsection we present a result from [7] on input to state stabilization via approximate discrete-time models. Consider a continuous-time nonlinear plant x(t) = f (x(t), u(t), w(t)), y(t) = h(x(t)) , (8) where x Rnx , u Rm , w Rp and y Rl are respectively the state, control input, disturbance and output. We assume that for any given x0 , u( ) and w( ) the differential equation in (8) has a unique solution de ned on its maximal interval of existence [0, tmax ). This may be guaranteed, for instance, by requiring f in (8) to be locally Lipschitz. The control is taken to be a piecewise constant signal u(t) = u(kT ) =: u(k), t [kT, (k + 1)T ), k N, where T > 0 is the sampling period, and we suppose that the disturbance w( ) is constant during sampling intervals, that is w(t) = w(k), t [kT, (k + 1)T ). We assume that some combination (output) or all of the states (x(k) := x(kT )) are available at sampling instant kT, k N. The exact discretetime model for the plant (8), which describes the plant behavior at sampling instants kT , is obtained by integrating the initial value problem x(t) = f (x(t), u(k), w(t)) , (9) where z Rnz . We emphasize that if the controller (12) input to state stabilizes the approximate model (11) for all small T , this does not guarantee that the same controller would input to state stabilize the exact model (10) for all small T (see [1], [2], [10]). The following result provides a framework for controller design via approximate discretetime models. Theorem 3.1: [7] Suppose that there exist , , K and K, and for any strictly positive real numbers ( 1 , 2 , 3 , ) there exist K , strictly positive real numbers T , L, M such that for all T (0, T ) there exists a function VT : Rnx +nz R 0 such that for all |(x, z)| 1 , |u| 2 , |w| 3 , T (0, T ) we have: 1. SP-ISS Lyapunov conditions for closed-loop a e approximate; 2. consistency between FT and FT ; 3. uniform local boundedness of uT (see [7] for detail de nitions). Then, there exists KL, G such that for any strictly positive real numbers ( 1 , 2 , ) there exists T > 0 such that for all |(x(0), z(0))| 1 , w 2 and T (0, T ) the solutions of (10), (12) satisfy SP-ISS of closed-loop exact. We emphasize that the consistency condition in Theorem e 3.1 is checkable although FT is not known in general. De nitions and lemmas that give suf cient conditions for consistency condition are stated in [7]. B. Euler based discrete-time backstepping design In this subsection, a result from [9] is cited. The Euler model is used, since it preserves the strict feedback structure of the plant that is needed for a backstepping design and it satis es the consistency property required by Theorem 3.1. Consider a continuous-time plant of the strict feedback form: x = f (x) + g(x) =u . (13) (14) 6407 The Euler approximate model of (13),(14) is: x(k + 1) = x(k) + T (f (x(k)) + g(x(k)) (k)) (k + 1) = (k) + T u(k) . (15) (16) x3 ms u u mus x1 x4 x2 Under certain properties and conditions (see [9]), there exists a SPA stabilizing pair (uT , VT ) for the Euler model (15),(16). In particular, we can take: W T T + , uT = c( T (x)) T T (17) d Fig. 1. The quarter car suspension model where c > 0 is arbitrary, = T (x) asymptotically stabilizes (13) and T = T (x + T (f + g )) T (x) W T = W T ( T (x)) , T WT (x + x A linear time invariant dynamic model of the system is represented as follows: x1 = x2 d, x2 = 2 x1 + u, x3 = x2 + x4 x4 = u (23) (18) = T (x) (19) (20) T (f + g ))g, = T (x) W T = WT (x(k + 1)) WT (x + T (f + g T )) 1 and the Lyapunov function VT = WT + 2 ( T (x))2 . C. A LaSalle criterion for SP-ISS The result from [6] on changing supply rates for SP-ISS discrete-time systems, provides a recipe for constructing a composite Lyapunov function to solve LaSalle type stability problem in sampled-data system. Consider the system (1). Using Corollary 5.1 of [6], we show that if the functions V1T : Rn R 0 and V2T : Rn R 0 are respectively a SP-qISS Lyapunov function and a SP-IOSS Lyapunov function of the system (1), and lim sup 2 (s) < + , 1 (s) s + (21) where x1 - tire de ection (m), x2 - unsprung mass velocity (m/sec), x3 - suspension de ection (m) and x4 - sprung mass velocity (m/sec). The parameter is the unsprung mass natural frequency, is the sprung to unsprung mass ratio and assume that the travel limit of the suspension is D. In other words, as long as the suspension de ection x3 satis es D < x3 < D, the suspension will not bottom out. Following [3], we use the parameters =2 10 rad/sec, := ms /mus =10, D =0.1 m. B. Discrete-time backstepping controller design To obtain a strict feedback form, the state equations are reordered using the following diffeomorphism: z1 = x1 + +1 x3 , 1 z2 = +1 x2 + +1 x4 , z3 = x3 z4 = x2 + x4 The model is then rewritten in the following form z1 = z2 d 2 2 z2 = z1 + z3 +1 ( + 1)2 z3 = z4 2 z4 = 2 z1 z3 (1 + )u = u +1 (24) (25) (26) (27) Then, the function VT : Rn R 0 that satis es VT = V1T + (V2T ) . s (22) where (s) := 0 q( )d , with q SN and K , is a SP-ISS Lyapunov function of the system (1). IV. CONTROL OF AN ACTIVE SUSPENSION SYSTEM A. Car suspension system modeling We use the quarter car model as the mathematical description of the suspension system, following model the used in [3]. The schematic diagram of the model is shown in Figure 1. In this model, the suspension actuator is taken to be a force actuator acting between the car body (the sprung mass) and the axle of the car. The tire is an ideal, undamped spring between the axle and the ground. Finally, the axle and wheel assemblies are represented as a mass (the unsprung mass) connected to the ground via the tire spring. As shown in Figure 1, the suspension force also reacts against the unsprung mass. The Euler model of the system in a strict feedback form is written as follow: z1 (k + 1) = z1 (k) + T (z2 (k) d) 2 z1 (k) 2 z3 (k) + ) z2 (k + 1) = z2 (k) + T ( +1 ( + 1)2 z3 (k + 1) = z3 (k) + T z4 (k) 2 z3 (k) z4 (k + 1) = z4 (k) + T ( 2 z1 (k) +1 (1 + )u(k)) = z4 (k) + T u(k) (28) (29) (30) (31) In the design, the disturbance d is taken to be zero, which is a reasonable approach since the disturbances affecting the system are nearly impulsive and thus correlate to nonzero 6408 initial conditions. Therefore, the problem is simpli ed to an asymptotic stabilization problem. We follow similar design steps to those done in [3], applying the Euler based backstepping design [9] as cited in Subsection 3.2. Due to space limitation, some trivial steps are omitted. Step 1: From the continuous-time model, it can be seen that if z3 0, then subsystem (28), (29) is marginally stable. We design a virtual feedback control law z3d (z1 , z2 ) which is bounded between D and D and renders the origin of the closed-loop (z1 , z2 ) subsystem SP-AS. A control that satis es this is k1 z2 ), k1 > 0 . (32) z3d = D tanh( D Unfortunately, the candidate Lyapunov function 1 2 2 1 2 z+ z, (33) 2 +1 1 2 2 which was used in the continuous-time design [3], gives V0T1 (z1 , z2 ) = V0T1 T M z2 tanh(z2 ) + T 01 (34) with some M, M1 , M3 > 0. Hence, V0T2 is a SP-IOSS Lyapunov function for the rst two subsystems. From (34) and (36), it is obvious that the condition (21) is satis ed, and hence all conditions of Corollary 5.1 of [6] holds. Hence, we can conclude that for some K , the function V0Ta that satis es V0Ta = V0T1 + (V0T2 ) (37) is a SP-AS Lyapunov function for the rst two subsystems. The surface plots of V0T2 and V0T2 are shown in Figure 3. Suppose we are given a set of initial conditions, such that the SP-AS property of the subsystem (28), (29) is guaranteed with T = 0.001 sec. For a x = 0.1, choosing an appropriate K , then we can use formula (37) to combine Figure 2 and Figure 3 after scaling V0T2 with the function , to show the SP-AS Lyapunov surface V0Ta and the SP-AS difference V0Ta of the subsystem (28), (29), for the given set of initial conditions. Choosing ( ) = Id( ) 1 with z3 = z3d , which is negative semide nite with small offset 01 > 0. While we can apply LaSalle Invariance Principle for the continuous-time case, we cannot do the same for the sampled-data design when semiglobal stability condition occurs. The Euler based backstepping [9] we use does not facilitate this condition, and the candidate Lyapunov function V0T1 does not satisfy the rst condition of Theorem 3.1. To 1.5 1 z1 0.5 0.01 z2 0.5 0.01 0.02 0.03 2 1 0.04 0.05 1 1.5 0.02 0.8 0.6 0.0004 0.4 z1 0.4 0.2 1400 1200 1000 800 600 40 20 z2 1 2 400 200 z1 20 40 1 2 0.8 1 0.6 0.0002 0.0003 0.0004 0.0005 0.0006 z2 0.0002 0.0004 0.0006 0.0008 0.001 Fig. 3. Surface plots for V0T2 (l) and V0T2 (r), with T = 0.001 sec and = 0.1. 1400 1200 1000 800 600 40 20 z2 1 2 400 200 z1 20 40 0.02 0.01 results in a Lyapunov function V0Ta = V0T1 + V0T2 (38) Fig. 2. Surface plots for V0T1 (l) and V0T1 (r), with T = 0.001 sec. solve this problem, we apply Corollary 5.1 of [6] to construct a SP-AS Lyapunov function for subsystem (28), (29). It has been shown earlier that V0T1 (z1 , z2 ) is in fact a SP-qISS Lyapunov function for the subsystem. The surface plots of V0T1 and V0T1 are shown in Figure 2. To show that the subsystem is SP-AS, we introduce another function 1 2 2 1 2 z1 z2 V0T2 (z1 , z2 ) = z + z + . (35) 2 2 +1 1 2 2 (1 + z1 )3/4 For small sampling period T > 0 and small > 0, the Lyapunov difference of V0T2 satis es 2 V0T2 T M z2 tanh(z2 ) T M1z1 2 + T M3 (z2 + tanh2 (z2 )) + T 02 for the subsystem (28), (29), and we can immediately see that with z3 = z3d the Lyapunov difference V0Ta = V0T1 + V0T2 is negative de nite and hence the subsystem (28), (29) is SP-AS. Remark 4.1: We emphasize that choosing = Id in our case is possible since we are dealing with a semiglobal practical property. The V0Ta obtained does not hold globally, because of the saturation coming from the tanh function in z3d . We also have to choose a small to guarantee that V0Ta does not become positive for quite large z2 . To continue the design procedure, for the purpose of simpler computation, we choose to use V0T = 1 1 V0Ta = (V0T1 + V0T2 ) , 2 2 (39) (36) since it is allowable to scale a Lyapunov function with a constant. We also x = 0.1 in this design. It is obvious that V0T satis es the rst condition of Theorem 3.1. 6409 Step 2: We de ne 1 = z3 z3d (z2 ) and denote 2 := 1 to obtain the following third order Euler model: z1 (k + 1) = z1 + T z2 z2 (k + 1) = z2 + T 2 2 z1 + (z3d + 1 ) +1 ( + 1)2 1 (k + 1) = 1 + T 2 . (40) 2 1 where 2 := +1 z1 + 2 2 ( +1)2 z3 , and + + W T = V1T ( 1 + T 2 ) V1T ( 1 + T 2d ) k2 2 2 = (2T 1 2 2T 1 2d + T 2 2 T 2 2d ). 2k4 Moreover, using (19) we have W T k2 k2 T = 1 + ( 2 + 2d ) . T k4 2k4 (45) Using a candidate Lyapunov function 2 and choosing the stabilizing controller as 2d V1T (z1 , z2 , 1 ) = V0T (z1 , z2 ) + k2 , k2 > 0 (41) Hence, we obtain u = uT by substituting (44),(45) to k5 W T T uT = ( 2 2d ) + , k5 > 0. k4 T T (46) 1 2 z1 = (z + ) k3 1 , 22 k2 ( + 1) (1 + (z1 + T z2 )2 )3/4 2 V1T = V0T |z3d T k2 k3 1 + T 1 , with k3 > 0, we have (42) which is negative de nite with a small offset 1 > 0. Hence, the equilibrium (z1 , z2 , 1 ) = (0, 0, 0) is SP-AS. Since z3d (0) = 0 we can conclude that the origin (z1 , z2 , z3 ) = (0, 0, 0) is also SP-AS. Step 3: Backstepping 2 through an integrator results in the dynamical system, whose Euler model can then be written as follow: z1 (k + 1) = z1 + T z2 z2 (k + 1) = z2 + T ( 2 2 z1 + (z3d + 1 )) +1 ( + 1)2 1 (k + 1) = 1 + T 2 (43) 2 (k + 1) = 2 + T u . It can be shown that implementing uT to the system results in V2T negative de nite with small offset 2 > 0. This means that the equilibrium (z1 , z2 , 1 , 2 ) = (0, 0, 0, 0) is SP-AS. Since z3d (0) = 0, then the origin (z1 , z2 , z3 , z4 ) = (0, 0, 0, 0) is also SP-AS. We have seen earlier that V1T satis es the SP-ISS (in this case SP-AS) Lyapunov condition of Theorem 3.1. It is then obvious that with V2T that the rst condition of Theorem 3.1 still holds. Step 4: Following exactly as in the continuous-time design, the resulting control law u = uT that SPA stabilizes the Euler model (28)-(31) has form uT = 1 ( uT 3d + 2 x1 ) +1 (47) At this step, we consider a candidate Lyapunov function k4 V2T (z1 , z2 , 1 , 2 ) = V1T (z1 , z2 , 1 ) + ( 2 2d )2 , 2 with k4 > 0. We apply the formula (17) to obtain uT using the following terms: 2 z1 ) k3 1 (z2 + k2 ( + 1)2 (1 + (z1 + T z2 )2 )3/4 k2 2 WT = , 2k4 1 (it turns out that T := 2d ) and get T = T 2d (k + 1) 2d (k) = T T 1 2 z1 = 2 + k2 ( + 1)2 (1 + (z1 + 2T z2 + T 2 2 )2 )3/4 1 2 1 z1 k2 T ( + 1)2 (1 + (z1 + 2T z2 + T 2 2 )2 )3/4 1 k3 z4 (1 + (z1 + T z2 )2 )3/4 1 k3 D(tanh(z2 + T 2 ) tanh(z2 )) , (44) T with 3d := z3d . Finally, by substituting the appropriate terms, we have uT as a nonlinear control law parameterized by the sampling period T and ve positive tuning parameters k1 , k2 , k3 , k4 and k5 . Expanding uT in series representation, we can show that uT satis es the third condition of Theorem 3.1. Since all conditions of Theorem 3.1 hold, we can guarantee that uT SPA stabilizes the closed-loop approximate model, and also stabilize the closed-loop exact model. We further use results from [8], to conclude the SP-AS for the sampled data system (23), (47). C. Comparing the Euler-based controller with the Emulation controller We have designed a discrete-time backstepping controller (47) that SPA stabilizes the active suspension system. Now, we implement the controller (47), and observe the performance of the closed-loop sampled-data system with the designed controller, and compare it with a controller that has form: 1 u= ( u 3d + 2 x1 ) (48) +1 k5 2 where u = 2d kk4 1 k4 ( 2 2d ) , with 2d := 2d . The controller (48) is obtained via emulation design, by holding the continuous control constant during every sampling 6410 We rst run the Simulation Road-1 (see Figure 4), when setting T = 0.001 sec, initial states (0.01 0 0.01 0)T and bump height A = 0.01 m. 15 x 10 3 period (using zero order hold). By applying Corollary 5.1 of [5], we can show that the discrete-time emulation controller (48) also SPA stabilizes the continuous-time plant (23). We study the condition when there are small offsets to the initial states, in other words, when allowing nonzero initial states. We observe the responses of the system to bumps of different heights and compare the performance of each controller. The shape of the isolated bump 1 is chosen to be in the form that gives rise to the following velocity input: 0, t 0 (49) d(t) = 10 A sin(20 t), 0 < t 0.1 0, t > 0.1. (a) 0.06 0.04 1 0.02 0 x4 x2 x1 0 2 (b) 0.02 1 0.04 0.06 0 0.1 0.2 0.3 0.4 2 0 0.1 0.2 0.3 0.4 (c) 0.04 0.02 0 x3 50 0.02 0.04 0.06 0 u 150 (d) 100 0 0.1 0.2 t (sec) 0.3 0.4 50 0 0.01 0.02 t (sec) 0.03 0.04 Fig. 5. Simulation Road-2 for a high bump condition, emulation and Euler. (a) 0.5 (b) 10 x4 x2 0 5 0.5 0 1 5 0 0.1 0.2 0.3 0.4 1.5 0 0.1 0.2 0.3 0.4 12 10 8 6 x3 x 10 3 (c) 150 (d) 100 50 4 2 0 2 0 0.1 0.2 t (sec) 0.3 0.4 50 0 0.01 0.02 t (sec) 0.03 0.04 0 Fig. 4. Simulation Road-1 for a low bump condition, emulation and Euler. In Simulation Road-2 (see Figure 5), we set A = 0.1 meter, which is considered as a high bump. We set T = 0.001 sec and initial states (0.01 0 0.01 0)T . From the two sets of simulation, we see that the system can always achieve better performance with the Euler based controller than with the emulation controller. The authors gratefully acknowledge the contribution of National Research Organization and reviewers comments. References are important to the reader; therefore, each citation must be complete and correct. If at all possible, references should be commonly available publications. V. REFERENCES [1] J.P. Barbot and S. Monaco and D. Normand-Cyrot, A sampled normal form for feedback linearization, Math. Contr. Sig. Sys., vol. 9, 1996, pp 162-188. 1 this kind of bump is a haversine of height A m and length l = 2 m, while assuming the vehicle is traversing the road at a speed of v = 20 m/s. [2] F. Esfandiari and H.K. Khalil, On the robustness of sampled-data control to unmodelled high frequency dynamics, IEEE TAC, vol. 34, 1989, pp 900-903. [3] N. Karlsson, A. Teel and D. Hrovat, A backstepping approach to control of active suspensions , Proc. IEEE CDC, Orlando, FL, 2001, pp 4170-4175. [4] M. Krsti , I. Kanellakopoulos and P. V. Kokotovi , c c Nonlinear and Adaptive Control Design, Wiley; 1995. [5] D.S. Laila, D. Ne i and A.R. Teel, Open and closed sc loop dissipation inequalities under sampling and controller emulation, European Journal of Control, vol. 8, 2002, pp 109-125. [6] D.S. Laila and D. Ne i , Changing supply rates for sc input-output to state stable discrete-time nonlinear systems with applications, Automatica, vol. 39, 2003, pp 821-835. [7] D. Ne i and D.S. Laila, A note on input to state sc stabilization for nonlinear sampled-data systems, IEEE TAC., vol. 47, 2002, 1153-1158. [8] D. Ne i A.R. Teel and E.D. Sontag, Formulas relating sc KL stability estimates of discrete-time and sampleddata nonlinear systems, SCL, vol. 38, 1999, pp 49-60. [9] D. Ne i and A.R. Teel, Backstepping on the Euler sc approximate model for stabilization of sampled-data nonlinear systems , Proc. IEEE CDC, Orlando, FL, 2001, pp 1737-1742. [10] D. Ne i and A.R. Teel and P. Kokotovi , Suf cient sc c conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations, SCL, vol. 38, 1999, pp 259-270. x1 u 6411

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