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基于 Barrier Lyapunov 函数的严格反馈系统的 http://www.paper.edu.cn 全状态约束控制 王春晓,武玉强 曲阜师范大学工学院,日照,276862 摘要: 本文研究了非对称时变全状态约束的严格反馈非线性系统的自适应控制问题。时变非 对称障碍李雅普诺夫函数的引入保证了状态约束条件的满足。时变的障碍李雅普诺夫函数可以 随着跟踪轨迹的改变而做相应调整从而放松了对于初始条件的要求,所以时变非对称的障碍李 雅普诺夫函数在控制设计中更具灵活性。首次通过变量代换及虚拟误差的界限消除了由反步法 所带来的高阶交叉项,所设计的控制器在不破坏任何约束条件的前提下最终达到了渐近跟踪, 并且闭环系统的所有信号最终有界。最后给出数值例子验证了控制器的性能。 关键词:非线性严格反馈系统;非对称障碍李雅普诺夫函数;时变;全状态约束;反步法. 中图分类号: O231.3 Barrier Lyapunov Function-based control for strict-feedback systems with full state constraints WANG Chun-xiao, WU Yu-qiang Department of Engineering, Qufu Normal University, Rizhao, 276862 Abstract: This paper studies an adaptive control design for a class of strict-feedback nonlinear systems with asymmetric time-varying full state constraints. A time-varying asymmetric Barrier Lyapunov Function (ABLF) is employed to ensure time-varying constraints satisfaction. By allowing the barriers to vary with the desired trajectory in time, the initial condition requirements are relaxed. Thus the ABLF can afford greater flexibility in control design. The high order coupling terms caused by backstepping method are canceled through variable substitution. Asymptotic tracking is achieved without violation of any constraints, and all signals in the closed-loop system are ultimately bounded. The performance of the ABLF-based control is illustrated through a numerical example. Key words: nonlinear strict-feedback system; ABLF; time-varying; full state constraints; backstepping. Foundations: The PhD Programs Foundation of Ministry of Education of China under Grant 20123705110002, National Natural Science Foundation (61273091, 61303198 and 61304008),the Project of Taishan Scholar of Shandong Province of China. Author Introduction: Wang Chun-xiao(1979-),female,associate professor,major research direction:nonlinear system control and adaptive control. Correspondence author:Wu Yu-qiang (1962-),male,professor,major research direction: variable structure control, switching control, nonlinear system control. - 1 -
http://www.paper.edu.cn 0 Introduction Constraints are ubiquitous in practical systems [1]. Violation of constraints may cause undesirable performance, degradation, hazards or system damage. Driven by practical require- ments and theoretical challenges, the research of constrained problem has become an attractive research topic [2]–[6]. Model predictive control [7, 8], set invariance notions [9, 10], and reference governors [11, 12] are the main methods to handle constraints. Additionally, barrier Lyapunov Functions (BLFs) have been proposed to guarantee the output and state constraints for vari- ous kinds of nonlinear systems such as Brunovsky form [13], strict-feedback form [14]–[17], and pure-feedback form [18, 19]. In recent years, the study of constrained strict-feedback nonlinear systems have attracted scholars’ attention. BLF was first employed to deal with the tracking control problems for a class of strict-feedback nonlinear systems with an output constraint in [14]. [3, 15] improved the result to strict-feedback nonlinear systems with time-varying output constraint by using a time-varying BLF. Besides the output constraint mentioned in the above works, the state constraints have also been tackled by using BLF. For example, strict-feedback nonlinear system with partial state constraints [20] and the full state constraints [21]–[23] were studied using BLF. However, the existing results are only available for strict-feedback system with static state constraints. One more interesting work is to consider the strict-feedback system with time-varying state constraints. The main difficulties of handling the problem is to deal with the high order coupling terms Motivated by the above observations, in this paper, we employ the time-varying ABLF- based adaptive control to handle the strict-feedback system with asymmetric time-varying full state constraints. The main contributions of this paper are summarized as follows: (1) Time- varying ABLF-based backstepping design is proposed to prevent the violation of the full state constraints and asymptotic output tracking is achieved. (2) An efficient method is given to cancel the high order coupling terms for the first time. (3) The proposed control scheme is able to handle the state constraints that are both time-varying and asymmetric. The rest of the paper is organized as follows. In Section 1, some mathematical preliminaries and statement of the problem are provided. The controller design procedure and the main results are developed in Section 2. A numerical example is given in Section 3 to illustrate the obtained results. Section 5 concludes this paper. - 2 -
http://www.paper.edu.cn 1 Problem Statement Consider the following strict-feedback nonlinear systems with time-varying full state con- straints ˙xi(t) = fi(¯xi(t)) + gi(¯xi(t))xi+1(t); i = 1; 2;··· ; n − 1; ˙xn(t) = fn(x(t)) + gn(x(t))u(t); y(t) = x1(t); (1) where f1(·);··· ; fn(·); g1(·);··· ; gn(·) are known smooth functions, x(t) = (x1(t);··· ; xn(t)) ∈ Rn are the system states and ¯xi = (x1;··· ; xi)T . u(t) ∈ R and y(t) ∈ R are the control input and output respectively. Furthermore, all the states are required to be constrained in a (t) < xi(t) < ¯kci(t);∀t ≥ 0}, where ¯kci(t) : R+ → R time-varying sets as Ωxi := {xi(t) ∈ R; kci (t);∀t ≥ 0. (t) : R+ → R such that ¯kci(t) > kci and kci Remark 1. The state constraints kci (t) < xi(t) < ¯kci(t) considered in this paper are based on the worst-case scenario. The designed constraint functions can be specified according to practical problem’s requirements. The constraints kci (t); ¯kci(t) should be ensured to satisfy (t) < i1(t) < ¯kci(t); i = 1;··· ; n, in which i1 are the virtual stabilizing functions to be kci designed. The control objective is to track a designed trajectory yd(t) while to ensure that the time- varying full state constraints are not violated and all closed-loop signals are bounded. To achieve the control objective, we make the following assumptions on system (1). constant g0 such that 0 < g0 ≤ |gi(·)| for kci this paper, we further assume that gi(·) are all positive. Assumption 1. The functions gi(·)(i = 1; : : : ; n) are known, and there exist a positive (t) < xi(t) < ¯kci(t). Without loss of generality, in Assumption 2. There exist functions ¯Y0 : R+ → R+ and Y 0 : R+ → R+ satisfying (t);∀t ≥ 0, and positive constants Yi; i = 1;··· ; n; such that the d (t)| ≤ Yi, ; ¯dcij i; j = 1;··· ; n, such that ¯kci(t) ≤ ¯Y0(t) < ¯kc1(t) and Y 0(t) > kc1 reference signal yd(t) and its time derivatives satisfy Y 0(t) ≤ yd(t) ≤ ¯Y0(t) and |y(i) i = 1;··· ; n;∀t ≥ 0. Assumption 3. There exist constants ¯Kci; K ci and their derivatives satisfy |k(j) (t) ≥ K ci , dcij (t)| ≤ dcij ;|¯k(j) ci (t)| ≤ ¯dcij. ¯Kci; kci ci Definition 1 [14]. BLF is a continuously differentiable and positive definite scalar func- tion V (x), defined with respect to the system ˙x = f (x) on an open region D containing the origin. It has continuous first-order partial derivatives at every point of D, has the property V (x) → ∞ as x approaches the boundary of D. It satisfies V (x) ≤ b;∀t ≥ 0 along the solution of ˙x = f (x) for x(0) ∈ D and some positive constant b. The following lemmas about BLF are useful for establishing constraint satisfaction and stability analysis. - 3 -
http://www.paper.edu.cn Lemma 1 [14]. For any functions kaj (t); kbj (t), let ¯Zi := {zj(t) ∈ R : −kaj (t) < zj(t) < kbj (t); j = 1; 2; : : : ; i;∀t ≥ 0} ⊂ Ri and N := Rl× ¯Zi ⊂ Rl+i be open sets. Consider the system (2) where (t) := [!(t); ¯zi(t)]T ∈ N , and h : R+×N → Rl+i is piecewise continuous with respect to t and locally Lipschitz about , uniformly in t, on R+×N . Suppose that there exist continuous differentiable and positive definite functions U : Rl → R+ and Vi : ¯Zi → R+ in their respective domains, such that ˙(t) = h(t; (t)); Vj(zj(t)) → ∞ as 1(∥!∥) ≤ U (!) ≤ 2(∥!∥); zj → −kaj or ∑ zj → kbj ; j = 1;··· ; i; where 1 and 2 are class K1 functions. Let V () := to the set ¯Zi. If the inequality holds i j=1 Vj(zj(t)) + U (!) and ¯zi(0) belong ˙V = h ≤ 0; @V @ then ¯zi(t) remain in the open set ¯Zi;∀t ∈ [0;∞]. Lemma 2 [15]. For all |Si| < kbi(t), the following inequality holds log k2 bi (t) k2 (t) − S2 bi i ≤ S2 (t) − S2 i i : k2 bi (3) (4) 2 Time-varying ABLF-based control In this section, we give a time-varying ABLF for system (1) as well as the corresponding stability analysis. The control design is based on backstepping with time-varying ABLFs. For giving the detailed backstepping design, we first give the change of coordinates as follows: S1 = x1 − yd; Si = xi − i1; i = 2;··· ; n; where Si(i = 1;··· ; n) are called the virtual errors, i1(i = 2;··· ; n) are the intermediate functions, which will be designed in Step i. Step 1. Starting with the x1-subsystem, the time derivative of S1 = x1 − yd is ˙S1 = ˙x1 − ˙yd = f1(x1) + g1(x1)x2 − ˙yd = f1(x1) + g1(x1)(S2 + 1) − ˙yd: (5) Since the state constraints are time-varying and asymmetric, we choose an ABLF as follows 1 − q(S1) 2p log( V1 = (t) k2p a1 a1 (t) − S2p k2p 1 ) + q(S1) 2p log( k2p b1 - 4 - (t) k2p b1 (t) − S2p 1 ); (6)
where p is a positive integer satisfying 2p ≥ n, the time-varying barriers are given by http://www.paper.edu.cn (t); { ka1(t) := yd(t) − kc1 kb1(t) := ¯kc1(t) − yd(t); if • > 0; q(•) := if • ≤ 0: 1; 0; (7) Remark 2. The aim of p chosen as 2p ≥ n in (6) is to ensure that the differentiability of i for i = 1;··· ; n − 1. The selection of q(•) in (7) is to ensure that the Lyapunov function in (6) can handle the case of asymmetric time-varying state constraints. Due to Assumptions 2-3, there exist positive constants K b1 ; ¯Kb1, and K a1 ; ¯Ka1, such that ≤ ka1(t) ≤ ¯Ka1; K b1 K a1 (8) Define a set Ωs := {−kai(t) < Si < kbi(t); kai(t) > 0; kbi(t) > 0; i = 1; 2;··· ; n}, where kai(t); kbi(t) are specified later on for i = 2;··· ; n. In the set Ωs, V1 is continuous and differ- entiable, then the time derivative of V1 is given by ≤ kb1(t) ≤ ¯Kb1: ˙V1 = S2p1 1 · ( 1 − q(S1) a1 (t) − S2p k2p 1 + q(S1) (t) − S2p 1 k2p b1 ) · [ ˙S1 + (1 − q(S1)) ˙ka1(t) ka1(t) S1 + q(S1) ˙kb1(t) kb1(t) S1]: (9) In order to simplify the formation, we define Ks1(t) := 1 − q(S1) a1 (t) − S2p k2p 1 + q(S1) (t) − S2p 1 ; k2p b1 and 1 − q(Si) ai (t) − S2p k2p will be used in the following. Then, Ksi(t) := i + q(Si) (t) − S2p i k2p bi ; i = 2;··· ; n; ˙V1 = S2p1 1 Ks1[f1 + g1(S2 + 1) − ˙yd + (1 − q(S1)) ˙ka1(t) ka1(t) S1 + q(S1) ˙kb1(t) kb1(t) S1]: Design stabilising function 1 in the form of 1 = 1 g1 [−(K1 + ¯k1(t))S1 − f1 + ˙yd]; where K1 is a positive design parameter and the time-varying gain is given by √ (1 − q(S1))( ¯k1(t) = ˙ka1 ka1 )2 + q(S1)( ˙kb1 kb1 )2 + : (10) (11) (12) (13) (14) Note that is a positive constant, then it guarantees that the differentiability of 1 and boundedness of the time derivatives of 1 even when ˙ka1 and ˙kb1 are both zero. - 5 -
Substituting (13) into (12), we get ˙V1 ≤ −K1Ks1(t)S2p 1 + Ks1(t)g1S2p1 1 http://www.paper.edu.cn S2: (15) It is noted that the last term of inequality (15) is a high order coupling term which should be canceled in the subsequent step. Step i (i = 2;··· ; n− 1). Consider the xi-subsystem, Si = xi − i1. Take time derivative of Si yields ˙Si = ˙xi − ˙i1 = fi + gixi+1 − ˙i1 = fi + gi(Si+1 + i) − ˙i1: Consider the Lyapunov function 1 − q(Sj) i∑ Vi = [ j=1 2p log( (t) k2p aj aj (t) − S2p k2p j ) + q(Sj) 2p log( k2p bj (t) k2p bj (t) − S2p j )]; (16) (17) (t); kbj (t) := ¯kcj (t) − j1(t). We select parameters Kj1 to where kaj (t) := j1(t) − kcj guarantee that kcj (t) < j1(t) < ¯kcj (t), then kaj (t) > 0 and kbj (t) > 0 can be ensured. Obviously, Vi is positive definite and continuously differentiable in the set Ωs. Then, the time derivative of Vi is given by ˙Vi = ˙Vi1 + S2p1 i Ksi(t)[ ˙Si + (1 − q(Si)) Si + q(Si) Si] ˙kbi(t) kbi(t) ≤ − i1∑ ˙kai(t) kai(t) j + Ksi1(t)gi1S2p1 i1 Si KjKsj (t)S2p j=1 + S2p1 i Ksi(t)[fi + gi(Si+1 + i) − ˙i1 + (1 − q(Si)) ˙kai(t) kai(t) Si + q(Si) ˙kbi(t) kbi(t) Si]: (18) Design stabilising function i as 1 gi i = in which Ki is positive constant and the time-varying gain is given by [−(Ki + ¯ki(t))Si − fi + ˙i1 + i] √ (1 − q)( )2 + q( )2 + ; ¯ki(t) = ˙kai kai ˙kbi kbi (19) (20) The last term of (19) i is used to eliminate the coupling term from the previous step. That is to say the following inequality should be established. i ≤ 0: Adopting variable substitution, we can construct i as follows and i1 Si + Ksi(t)S2p1 Ksi1(t)gi1S2p1 i i = − gi1S2p1 i1 Si[q(Si)(kbi ai1 (1 − q(Si1))(k2p − Si) + (1 − q(Si))(−kai − S2p i1) + q(Si1)(k2p bi1 − Si)] − S2p i1) (21) (22) : - 6 -
Substituting (19) into (18), then we have ˙Vi ≤ − i∑ j=1 KjKsj (t)S2p j + Ksi(t)giS2p1 i http://www.paper.edu.cn Si+1: (23) Remark 3. Note that the high order coupling terms in this paper we hope to be canceled meanwhile to guarantee that Si(i = 1;··· ; n) converges to zero as t → ∞. For this aim, we firstly choose one of virtual errors and reduce its order to be quadratic by a variable substitution. And then, combined with the bound of the chosen virtual error, we obtain the scope of vi(i = 2;··· ; n) for the four different cases. Fortunately, we can find a precise function vi from above scopes for four cases. It can be guaranteed that vi is differentiable. The detailed procedure to get (22) from (21) is provided in Appendix. Step n. As denoted in Step n − 1, Sn = xn − n1, one has ˙Sn = ˙xn − ˙n1 = fn + gnu − ˙n1: Choose the Lyapunov function as 1 − q(Sn) q(Sn) (t) k2p an an(t) − S2p k2p n 2p 2p ) + log( log( Vn = Vn1 + (t) ≤ n1(t) ≤ ¯kcn(t), then kan(t) > 0 and kbn(t) > 0 hold. (25) (t); kbn(t) := ¯kcn(t) − n1. We also select parameters to guarantee where kan(t) := n1 − kcn that kcn In the set Ωs, Vn is continuous and differentiable. So in the same manner of the previous steps, computing the time derivative of Vn, we have ˙Vn = ˙Vn1 + S2p1 n Ksn(t)[fn + gnu − ˙n1 + (1 − q) Sn + q k2p bn Sn] ); (t) k2p bn (t) − S2p n ˙kan(t) kan(t) ˙kbn(t) kbn(t) ≤ − n1∑ (24) (26) (27) KjKsj (t)S2p j + Ksn1(t)gn1S2p1 n1 Sn j=1 + S2p1 n Ksn(t)[fn + gnu − ˙n1 + (1 − q) ˙kan(t) kan(t) Sn + q ˙kbn(t) kbn(t) Sn]: Choose the control law as follows u = − 1 gn [−(Kn + ¯kn(t))Sn − fn + ˙n1 + n]; √ (1 − q)( where Kn is positive constant and ¯kn(t) = _kan kan is used to cancel the coupling term from step n − 1, and )2 + q( )2 + . The last term of (27) _kbn kbn n = − gn1S2p1 n1 Sn[q(Sn)(kbn (1 − q(Sn1))(k2p an1 − Sn) + (1 − q(Sn))(−kan − S2p n1) + q(Sn1)(k2p bn1 − Sn)] n1) − S2p : (28) Substituting the control input (27) into (26), after simple calculation, it can be verified that KjKsj (t)S2p j : (29) ˙Vn ≤ − n∑ j=1 - 7 -
This completes the controller design procedure. http://www.paper.edu.cn Theorem 1. Suppose the investigated system (1) satisfies the Assumptions 1–3. The virtual controllers i; i = 1; 2;··· ; n − 1 in (13) and (19), the actual controller u in (27) are constructed on the sets Ωs. Choosing appropriate positive design parameters Ki; i = 1;··· ; n, given Si(0) ∈ Ωs = {−kai(t) < Si < kbi(t); kai(t) > 0; kbi(t) > 0; i = 1;··· ; n}, the resulting closed-loop system has the following properties: (i) The error signals Si(t); i = 1; 2;··· ; n are bounded by −Dsi (t) ≤ Si(t) ≤ ¯Dsi(t);∀t ≥ 0, √ √ 1 − e2pVn(0)et, ¯Dsi(t) = kbi(t) 2p 1 − e2pVn(0)et with = min{2pKi; (t) = kai(t) 2p where Dsi (i = 1;··· ; n)}. (ii) All the signals in the closed-loop system are bounded. (iii) The full state constraints are not violated. (iv) The origin S = 0 is asymptotically stable. Proof. (i) From (29) and Lemma 2, we have ˙Vn ≤ − n∑ ≤ − n∑ i=1 i=1 ≤ −Vn: Ki[ 1 − q(Si) ai (t) − S2p k2p Ki[(1 − q(Si)) log i + q(Si) (t) − S2p k2p bi k2p ai ai (t) − S2p k2p (t) i i ]S2p i + q(Si) log (t) k2p bi (t) − S2p i ] k2p bi (30) For Si(0) ∈ Ωs, then from Lemma 1 we get Si(t) ∈ Ωs;∀t ≥ 0. Multiplying both sides of (30) by et, then integrating it over [0; t] , it has Vn(t) ≤ Vn(0)e t: In view of (25), it is easy to obtain 1 − q(Si) 2p log( k2p ai − S2p i k2p ai ) + q(Si) 2p log( k2p bi − S2p i k2p bi ) ≤ Vn(0)e t: Taking exponentials on both sides of (32), it yields (1 − q(Si)) k2p ai − S2p + q(Si) t : ≤ e2pVn(0)e k2p bi − S2p i > 0 and k2p i i ai k2p ai Since −kai(t) < Si < kbi(t), we have k2p q(Si) = 1. Multiplying both the sides by k2p following inequality is obtained, k2p bi (t) − S2p √ 1 − e2pVn(0)et: Si(t) ≤ kbi(t) 2p √ Similarly, when Si ≤ 0, q(Si) = 0, we obtain Si ≥ −kai(t) 2p 1 − e2pVn(0)et. √ √ 1 − e2pVn(0)et ≤ Si ≤ kbi(t) 2p concluded as −kai(t) 2p 1 − e2pVn(0)et. (t) − S2p (t) − S2p bi i > 0. When Si > 0, i > 0 and applying manipulations, the bi (34) It could be (31) (32) (33) - 8 -
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