OUTPUT-FEEDBACK CONTROL FOR NONHOLONOMIC SYSTEMS WITH LINEAR GROWTH CONDITION

Transcription

1 J Syst Sci Complex 4: OUTPUT-FEEDBACK CONTROL FOR NONHOLONOMIC SYSTEMS WITH LINEAR GROWTH CONDITION Guiling JU Yuiang WU Weihai SUN DOI:.7/s z Received: 8 December 8 / Revised: 5 June 9 c The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg Abstract This paper deals with the stabilization of the nonholonomic systems with strongly nonlinear uncertainties. The objective is to design an output feedback law such that the closed-loop system is globally asymptotically regulated at the origin. The systematic strategy combines the input-state scaling techniue with the backstepping techniue. A novel switching control strategy based on the output measurement of the first subsystem is employed to make the subsystem far away from the origin. The simulation demonstrates the effectiveness of the proposed controller. Key words Backstepping, global asymptotic stabilization, input-state scaling, nonholonomic system with uncertainties, switching control strategy. Introduction Over the past decade, the control and stabilization of nonholonomic system have been studied by many researchers [ 4]. The flow of research activity has been mainly triggered by the well-known Brockett s work [5], where a necessary condition for asymptotic ability is stated. One of the conseuences of the necessary condition is that the nonholonomic systems is not stabilizable by stationary continuous state feedback. To overcome this impossibility, a number of approaches have been proposed for the problem, which can be classified as a Discontinuous time-invariant stabilization [6] ; b Time-varying stabilization [7 8] ; c Hybrid stabilization [9] ;see the survey paper [ 4] for more details and references therein. In real industrial or physical systems, it is usually difficult to measure full states. Moreover, most of systems have various uncertainties such as unknown parameters, disturbances and unknown structure. Hence, it is important and necessary to consider the output feedback control problem for uncertain nonholonomic systems. The output feedback issue has been studied for asympotic and exponential stability properties for the chained systems with uncertainties [ 3,5]. The output feedback control problem for uncertain nonholonomic systems are much difficult and still largely open. Guiling JU Weihai SUN Department of Mathematics, Armored Force Engineering Institute, Beijing 7, China. guilingju@63.com; whsun@mail.ustc.edu.cn. Yuiang WU Institute of Automation, Qufu Normal University, Qufu 7365, China. wy@fnu.edu.cn. This research is supported by the National Natural Science Foundation of China under Grant No This paper was recommended for publication by Editor Jinhu LÜ.

2 CONTROL FOR NONHOLONOMIC SYSTEMS 863 In this paper, we consider the output feedback control problem for a class of uncertain nonholonomic systems [5] dominated by a linearly growing triangular condition, which is, reinforced than the assumption in [5]. Meanwhile, the nonlinear function in the first subsystem makes it impossible to deal with the problem with the method used in []. To the best of our knowledge, there is still no design tool for this problem. By combing discontinuous change of coordinate [6] and the design of observer similar to [3], we give the recursive design [7 8] of the controllers and render the closed-loop system globally asymptotically regulated. In particular, to make the state scaling effective and to prevent the finite time escape phenomenon from happening, the switching control strategy based on the state measurement of the first subsystem is employed to achieve the asymptotic stabilization. The rest of the paper is organized as follows. In Section, problem formulation is presented. In Section 3, the output feedback controller is proposed based on the backstepping method. In Section 4, a switching controller is designed based on the output measurement of the first subsystem. An example is provided in Section 5 to illustrate the effective of this design. We conclude the paper in Section 6. Problem Formulation Consider the following nonholonomic system [5] : ẋ = u + x φ t, x, ẋ = x u + φ d t, u,x,x,dt, ẋ i = x i+ u + φ d i t, u,x,x,dt, i n, ẋ n = u + φ d nt, u,x,x,dt, y =x,x T, where x R, x =x,x,,x n T R n,u =u,u T R,y R are the system states, input and output, respectively. The functions φ d i,i =,,,n are continuous that involve uncertainty and may not be precisely known, and φ t, x is a smooth function. Clearly, when φ, φ d i =, the system becomes a standard chained form system that has been widely studied in [3]. And φ, φ d i are assumed to satisfy the following reuirement. Assumption For every i n, there exist a continuous function α i x, an unknown constant θ> and a smooth nonnegative function ϕ x such that φ d i t, u,x,x,dt α i x x + x + + x i θ, φ t, x ϕ x. 3 Remark Assumption implies that the origin is the euilibrium set of the system and imposes the uncertain nonlinearities to satisfy a linearly growing triangular condition, which is more reinforced than the assumption in [5], and it is difficult for the problem of out-feedback stabilization to be solved. Remark The system is similar to that in [], but the x -subsystem is different from that. Here system contains the nonlinear function φ t, x. To deal with the inherently triangular structure of system, we design the control input u and u in two separate strategy. We give the explicit formulation of the controllers such that the control u globally exponentially stabilizes the x -subsystem while the second input u make all states of the rest in system converges to zero.

3 864 GUILING JU YUQIANG WU WEIHAI SUN The control u is taken as u = λ x x ϕ x, λ >. 4 Conseuently, for the x -subsystem of, the following lemma can be estabilished. Lemma For any initial condition t,x t, wheret, ifu is given as 4, then the corresponding solution x t exists for any t t and satisfies that if x t,thenx t as t t. Proof Substituting 4 into the first euation of, we have Introduce the Lyapunov function ẋ = λ x x ϕ x +x φ t, x. 5 V x = x. By the assumption condition 3, the time derivative of V x along the solution of 5 satisfies V = λ x x ϕ x +x φ t, x λ x x ϕ x +x ϕ x = λ x. 6 This shows that x t tends to zero exponentially. In the following, we will conclude that x t dose not become zero in any time constant. In fact, for any bounded x t, since ϕ x,φ t, x are smooth function with respect to their arguments, there exists a positive constant M, such that ϕ x M, φ t, x M for x. This guarantees that as x, the following ineuality V = λ x x ϕ x +x φ t, x ρx 7 holds, where ρ =M + λ. Thisimpliesthatas x, x t converges to zero with rate less than a certain constant ρ. Lemma For given control signal u in 4, there exists a known smooth nonnegative function ϕ x such that for t, u u ϕ x. Proof By 4, we have u = λ ϕ x x ϕ x u + x φ. This results in u = λ + ϕ x +x ϕ u x φ t, x. λ + ϕ x Then, it is easy to see u = λ + ϕ x + x + ϕ x + ϕ x ϕ λ + ϕ x x. 8 u The proof is completed. For system, we introduce a state-input scaling discontinuous transformation [6] defined by z i = x i u n i, i n. 9

4 CONTROL FOR NONHOLONOMIC SYSTEMS 865 Under the new z-coordinates, the x-subsystem is transformed into where ż i = z i+ + φ i t, u,x, z,dt, i n, ż n = u + φ n t, u,x, z,dt, φ i = φ i t, u,x, z, dt = φd i t, u,x,x,dt u n i n iz i u u. For given u as in 4, the transformed system 9 is defined for x t. By Assumption and Lemma, one can see that φ i α ix x + x + + x i θ u n i +n i z i ϕ x n i z i ϕ x +α i x u i z + u i z + + z i θ. This together with the fact that x is bounded implies that there exists an unknown constant γ>such that Let φ i γ z + z + + z i, i n. 3 z i z i = p i, i n, u = u, 4 pn where p is a rescaling factor to be determined later. Under the new z-coordinate, the uncertain system can be expressed as ż i = pz i+ + φ i t, u,x,z,dt, i n, ż n = u + φ n t, u,x,z,dt, 5 where φ i = φ i t, u,x,z,dt = φ i t,u,x,z,dt p, i n. By 3, we have i φ i γ z + z + + z i p i γ z + z + + z i. 6 3 Controller Design The differential Euation 5 can be written into the compact form ż = paz + bu + Φ, 7 where φ A = , b =., Φ = φ... φ n

5 866 GUILING JU YUQIANG WU WEIHAI SUN In this section, we first design the control input u subject to x t. Thecasefor x t = will be treated in next subsection. The design of the control input u is based on the backstepping method to the transformed system 7. The recursive procedure stops once the true system input u occurs. We choose a vector L =[l,l,,l n ] T such that A = A Lc T is Hurwitz, where c = [,,, ] T. Define the observer as where y = z. Let e = z ẑ, then we obtain ẑ = paẑ + bu + ply c T ẑ, 8 ė = pa e + Φ. 9 Step Start with the e-system. This step can be views as the initial assignation of the entire design procedure. We consider the uadratic Lyapunov function candidate as V = n et Pe, where is a positive design parameter to be specified later. P is a positive define solution of the euation The time derivative of V along 9 satisfies V e = n et P pa e + Φ A T P + PA = I. p n et e + n e P φ + φ + + φ n p n et e + γ n e P z +z + z + + nz + z + + z n p n et e + γ nn + e P z n + z + + z n. nn+ For convince, we note δ =γ P. By ineuality x + x + + x n x + x + + x n, we obtain V e p n et e + δ n e z + z + + z n p n et e + δ n e ẑ + e + ẑ + e + + ẑ n + e n nδ p n et e + n n n et e + δ n e ẑ + ẑ + + ẑ n nδ 4 e T e + n ẑ + ẑ + + ẑ n.

6 CONTROL FOR NONHOLONOMIC SYSTEMS 867 Step Define ξ = ẑ,ξ = ẑ α with α being a virtual control function and ξ is a new variable. Take the Lyapunov function V e, ξ =V e+ p ξ. It is easily proved that with 8 and, the derivative of V satisfies V p nδ n n nδ p n n Choose the virtual controller nδ 4 e T e + nδ 4 e T e + + α + ξ ξ + ξ α + e + l 4 ξ. n ẑ + ẑ + + ẑ n +ξ ẑ + l e n ẑ 3 + ẑ ẑ n +ξ + ξ then, we get α = k ξ, k = n + l 4 +, V n n nδ 4 e T e + n ẑ 3 + ẑ4 + + ẑn n ξ + ξ ξ + ξ. 3 Step Define ξ 3 = ẑ 3 α, and take the Lyapunov function as V e, ξ,ξ =V e, ξ + p ξ, 4 where ξ = ẑ α = ẑ + k ẑ. Then, it is easy to get d dt p ξ = ξ ẑ 3 + l e + k ẑ + l e = ξ ξ3 + α +l + k l e k ξ + k ξ ξ ξ3 + α + a e + a ξ + k ξ, 5 where a = l + k l,a = k. With 3 and 5, we have V n n nδ 4 n kξ + e T e + n ξ a ξ e +a +ξ ξ + k + ξ. n ẑ 4 + ẑ ẑ n n α + ξ ξ 3 + ξ α

7 868 GUILING JU YUQIANG WU WEIHAI SUN By the above assumption, we have, then V n n nδ 4 e T e + n ẑ 4 + ẑ5 + + ẑn n k ξ + 4 ξ α + ξ ξ 3 + ξ α + a 4 ξ + e + a + ξ 4 + ξ + k + ξ. Take the virtual controller α = k ξ, k = n + a 4 + a + + k +. 4 It can be obtained the following ineualities V n n nδ 4 e T e + n ẑ 4 + ẑ ẑ n n k ξ n k ξ + ξ ξ ξ 3. 6 Step i 3 i n Suppose at Step i, we have designed the smooth Lypunov function V i e, ξ,,ξ i which is positive definite, the new variable ξ j and the virtual controller α j, j =, 3,,i satisfying V i n n nδ 4 i e T e + n ẑ i+ + ẑi+ + + ẑn i n + i k j j ξ j + ξ i ξ i i + i ξ i, 7 j= ξ j = ẑ j α j, α j = k j ξ j, j =, 3,,i. Now, consider the Lyapunov function V i e, ξ,ξ,,ξ i =V i e, ξ,ξ,,ξ i + p i ξ i. 8 It is easy to see that ξ i = ẑ i + k i ẑ i + k i k i ẑ i + + i k i k i k ẑ. Then, a direct calculation gives d dt p i ξ i = ξ i ξ i i i ẑ i+ + l i e + ẑ j+ + l j e ẑ j = ξ i i ẑ i+ + l i e + j= i j= ξ i i ξ i+ + α i + i b e + i b ξ i j k i k j ξ j+ k j ξ j + l j e + i b ξ + + b i ξ i + k i ξ i, 9

8 CONTROL FOR NONHOLONOMIC SYSTEMS 869 where b,b,,b i,k i are independent of. With 7 and 9, we can get V i n n nδ 4 i e T ẑ e + n i+ + ẑi ẑn + n ξ i+ + i n α i n + i k j j ξ j + ξ iξ i i+ j= + ξ b iα i i + ξ i i e + b i ξ + + b i + ξ i i + k i + i 3 ξ i n n nδ 4 i e T ẑ e + n i+ + ẑi ẑn + i ξ i+ + i α i i n + i k j j ξ j + ξ iξ i i+ + ξ iα i i + b 4 i 3 ξ i j= + e + b 4 i 3 ξ i + ξ + + b i + 4 i 3 ξi + i 5 ξ i + k i + i 3 ξi. Taking the virtual controller α i = k i ξ i, k i = n i ++ b 4 + b b i + + k i +, 4 we can get V i n n nδ 4 i e T ẑ e + n i+ + ẑi ẑ n i j= j n + i k j ξ j + i ξ iξ i+ + i ξ i+. 3 Step n From Step i, the following ineuality V n n n nδ 4 n n e T e j k j ξj j= + ξ n ξ n n + n ξ n 3 holds. Take the Lyaponov function for the whole system and the new variable ξ n as V n = V n + By the inductive step, we can see that ξ n = ẑ n u. p n ξ n, 3 ξ n = ẑ n + k n ẑ n + k n k n ẑ n + + n k n k n k ẑ.

9 87 GUILING JU YUQIANG WU WEIHAI SUN So, it is easy to get d dt p n ξ n = = n ξ n n p u ξ n + l n e + ẑ j+ + l j e ẑ j= j n ξ n n p u + l n e + n j k n k j ξ j+ k j ξ j + l j e j= ξ n n p u + n c e + n c ξ + n c ξ + + c n ξ n + k n ξ n, 33 where c,c,,c n,k n are independent of. Following 3 and 33, we get V n n n nδ 4 n n e T e j= j k j ξ j + p n ξ nu c +ξ n n e + c n ξ + + c n + ξ n n + k n + n 3 ξ n n n nδ 4 n n e T e j k j ξ j + p ξ nu n j= Choose + c 4 n 3 ξ n + e + c 4 n 3 ξ n + ξ + + c n + 4 n 3 ξ n + n 5 ξ n + k n + n 3 ξ n. u = pk n ξ n, k n =+ c 4 + c c n + + k n Then, by 4 we have u = p n k n ξ n 35 and V n n n nδ 4 n n e T e j k j ξ j n 3 ξ n. 36 j= Theorem For the system, under Assumption, if the control strategy 4 and 35 are applied with an appropriate design parameters, then the system is globally asymptotically stabilized and the state x is regulated at the origin as long as the initial condition x t is nonzero. Proof Take the Lyapunov function V n for the whole control system as 3. Taking the proper parameters, following the ineuality 36, we can get that e, ξ,ξ,,ξ n is bounded. From LaSalle s Invariant Theorem, it further concludes that e, ξ,ξ,,ξ n ast.by ξ = ẑ,wegetẑ ast, while from ξ i = ẑ i α i,α i = k i ξ i,i=, 3,,n, and ξ n = ẑ n u,u = k n ξ n, we can obtain ẑ i ast, i =, 3,,n. From e = z ẑ,

10 CONTROL FOR NONHOLONOMIC SYSTEMS 87 we have z ast. By transformation 9 and 4 and x ast,wecansee that x i ast. In the next section, in order to handle the singular case when x t =, a switching strategy based on the magnitude of the state of the x -subsystem rather than the time length is derived to prevent the singular phenomenon of uncontrollability. 4 Switching Controller Design Without loss of generality, it can be assumed that t =. When the initial state x, we have given the controller expressions 4 and 35 for u and u of system. Now, we consider the design of thecontrollawsu and u when x =. In the absence of the disturbances, most commonly used control strategy is to use constant control u = u in time interval [,t s. Then, at a time instant t s, the constant feedback law is switched to an exponential regulator which is also based on discontinuous coordinate transformation of the form 9 and 4. However, if there exist nonlinear functions φ, φ d i n, the choice of a constant feedback for u and u may lead to a finite time escape for the system. Some solutions blow up before the given switching time t s.to prevent this phenomenon from happening, for parametric uncertain system, a kind of nonlinear switching schedule based on the value x was used. In the next, we will give the switching control strategy for control input u by the use of the state of the first subsystem in. When x =, choose u as At x =, we know that u = u, u >. 37 ẋ = u + x φ,x = u >. Since x + x φ,x =<u and φ t, x is a known smooth function, there is a small neighborhood Ω of x =, such that Suppose that x satisfies In Ω, x t is increasing until x + x φ t, x u. x + x φ t, x = u. x + x φ t, x >u. Now, we define the switching control law u as u = u, u >, when x x, 38 u = λ x x ϕ x, when x > x. 39 During the time period satisfying x x, we apply the backstepping-based nonlinear feedback u = u x, ẑ, ẑ,, ẑ n which follows from an application of the procedure in the above section to the original x-system of while u as defined in 37 in stead of 4. Then, it concluded that the x-system of cannot blow up during x x,atthistimex is not zero x > x, then, we switch the control inputs u and u to 4 and 35, respectively.

11 87 GUILING JU YUQIANG WU WEIHAI SUN Now, the above analysis is summarized into the following theorem. Theorem Under Assumption, if the above switching control strategy is applied to with an appropriate choice of the design parameters, uncertain system is globally asymptotically regulated at the origin. 5 An Example Consider the three-dimensional non-holonomic systems: ẋ = u + x x, ẋ = x u + x x, ẋ = u + dtx, 4 where dt θ, θ is an unknown constant. The purpose is to design u and u basedononly y =[x,x ] T, such that [x t,x t,x t] ast. If x =, u and u are taken as Section 4 in interval [,t such that x t,thenwe can adopt the controls developed in Section 4. Therefore, without loss of generality, we assume that x. Take Let u = λ x x 3. 4 z = x, u z = x p, u = u p. 4 Then by 4 and 4, we have It is easy to see ż = pz + x z + λ λ +3x λ + x z, ż = u + dtz. 43 φ i θ +3λ + x z + z =:γ z + z, i =,. Design observer to reconstruct z and z of system 43 as follows ẑ = pẑ + pl z ẑ, λ =, ẑ = u + pl z ẑ, λ =. 44 Then, the estimation error e =[z ẑ,z ẑ ] T satisfies the euation ė = pa e + Φ, 45 where A = [ ] λ λ +3x, Φ = λ + x. dt Solving the matrix matrix euation A T P + PA = I,weobtain P = 5 6.

12 CONTROL FOR NONHOLONOMIC SYSTEMS 873 The eigenvalues of P are.937 and.396, thus P =.937. Define ξ = ẑ,ξ = ẑ α x, ẑ. Then, by the backstepping strategy in Section 3, we get α = k ξ, k =+ l 4 +, u = k ξ, k =+ l + k l Then, the control signal is expressed as 4 + k 4 + k +. u = p k ξ. 46 In simulation, we take dt =θ sinx. The design parameters are chosen as λ =.5, θ =.5, l =,l =, = 33, p = 53. The initial values are set to x =.5, x =.875, x = 76.5, ẑ =, ẑ =. The simulation results are shown in Figures 5 given below. It is clear from these figures that all of the states converge to zero. And the control laws also tend to zero x t x t.. x t Figure state x t Figure state x t Figure 3 state x t 3 x 6. u t..3 ut.4 5 Figure 4 control law u t...4 Figure 5 control law ut 6 Conclusion This paper investigates the problem of output feedback stabilization for the nonholonomic systems with strongly uncertainties bounded by a linearly growing condition. By combing stateinput scaling techniue and the backstepping method, we design an output feedback controller

13 874 GUILING JU YUQIANG WU WEIHAI SUN which can globally regulate the nonholonomic systems. In order to avoid the uncontrollability, a novel switching control strategy is suggested that is based on the output measurement of the first subsystem. The advantage of the switching strategy is that the extremely large control law in the small neighborhood of the starting point can be overcome. The simulation demonstrates the efficiency of the proposed method. References [] C. C. Wit and O. J. Sordalen, Expunential stabilization of mobile robots with nonholonomic constraints, IEEE Trans. Automat. Control, 99, 37: [] R. Mccolockey and R. Murray, Exponential stabilization of driftless nonlinear control systems using homogeneous feedback, IEEE Trans. Automat. Control, 997, 45: [3] R. Murray and S. Sastry, Nonholonomic motion planning steering using sinusoids, IEEE Trans. Automat. Control, 993, 385: [4] Z. D. Sun, S. S. Ge, W. Huo, et al., Stabilization of nonholonomic chained systems via nonregular feedback linerization, Systems and Control Letters,, 444: [5] W. Brockett, Asymptotic Stability and Feedback Stabilization, R. W. Brockett, R. S. Millman, H.J. Sussman Eds., Differential Geometric Control Theory, 983. [6] A. Astolfi, Discontiouous control of nonholonomic systems, Systems and Control Letters, 996, 7: [7] G. C. Walsh and L. G. Bushnell, Stabilization of multiple input chained form control systems, Systems and Control Letters, 995, 63: [8] Y. P. Tian and S. H. Li, Exponential stabilization of nonholonomic systems by smooth time varying control, Automatica,, 387: [9] O. J. Sordalen and C. C. Wit, Exponential stabilization of nonholonomic chained systems, IEEE Trans. Automat. Control, 995, 4: [] Z. Xi, G. Feng, Z. P. Jiang, et al., A switching algorithm for global exponential stabilization of uncertain chained systems, IEEE Trans. Automat. Control, 3, 48: [] Y. G. Liu and J. F. Zhang, Output-feedback adaptive stabilization control design for nonholonomic systems with strong nonlinear drifts, International Journal of Control, 5, 787: [] Q. D. Wang, C. L. Wei, and S. Y. Zhang, The output feedback control for uncertain nonholonomic systems with uncertainties, Journal of Control Theorey and Application, 6, 3: 8 3. [3] S. S. Ge, Z. P. Wang, and T. H. Lee, Adaptive stabilization of uncertain nonholonomic systems by atate and output feedback, Automatica, 3, 398: [4] Z. P. Jiang and H. Nijmeijer, A recursive techniue for tracking control of nonholonomic systems in chained form, IEEE Trans. Automat. Control, 999, 44: [5] Z. Xi, G. Feng, Z. P. Jiang, et al., Output feedback exponential stabilization of uncertain chained systems, Journal of Franklin Institute, 7, 344: [6] K. D. Do and J. Pan, Adaptive global stabilization of nonholonomic systems with strong nonlinear drifts, Systems and Control Letters,, 463: [7] Z. P. Jiang, Iterative design of time-varying stabilizers for multi-input systems in chained form, Systems and Control Letters, 996, 85: [8] C. Qian and W. Lin, Output feedback control of a class of nonlinear systems: A nonseparation principle paredigm, IEEE Trans. Automat. Control,, 47: 7 75.