International Journal of Automation and Computing (3), June 24, 38-32 DOI: 7/s633-4-793-6 Tracking Control of a Class of Differential Inclusion Systems via Sliding Mode Technique Lei-Po Liu Zhu-Mu Fu Xiao-Na Song College of Electric and Information Engineering, Henan University of Science and Technology, Luoyang 473, China Abstract: The tracking problem for a class of differential inclusion systems is investigated Using global sliding mode control approach, a tracking control is proposed such that the output of a differential inclusion system tracks the desired trajectory asymptotically An extensive reaching law is proposed to achieve the chattering reduction Finally, an example is given to illustrate the validity of the proposed design Keywords: Tracking, differential inclusion systems, global sliding mode control, chattering reduction, an extensive reaching law Introduction Tracking control is always a very active research area due to its wide applications In the past few decades, various approaches for tracking control have been presented, such as fuzzy control approach [, composite tracking control approach [2, adaptive control approach [3, sliding mode control approach [4, etc Because sliding mode control has attractive features such as fast response, good transient response and insensitivity to variations in system parameters and external disturbances [5, it is a substantial method for the tracking control design of nonlinear systems Generally speaking, the differential inclusion systems are considered a generalization of differential equations, and many practical systems are described by differential inclusion systems, so the study of differential inclusion systems has been paid much attention [2 2 In [2, a necessary and sufficient condition for the stability of polytopic linear differential inclusion systems was derived by bilinear matrix equations In [3, the problem of tracking control of nonlinear uncertain dynamical systems described by differential inclusions was studied In [4, a nonlinear control design method for linear differential inclusion systems was presented by using quadratic Lyapunov functions of their convex hull Using the method in [4, Sun [5 considered the uniformly ultimately bounded tracking control of linear differential inclusions with stochastic disturbance and Huang et al [6 considered the stabilization of linear differential inclusion system with time delay In [7, a frequency-domain approach was proposed to analyze the globally asymptotic stability of differential inclusion systems with discrete and distributed time-delays In [8, 9, the fundamental information about differential inclusions is introduced But the research of tracking control of differential inclusion systems using sliding mode technique is not sufficient In [2, 2, Regular Paper Manuscript received December 29, 22; revised August 2, 23 This work was supported by National Natural Science Foundation of China (Nos 637477 and 62347), fundamental research project (No 4234293) in the Science and Technology Department of Henan province, the science and technology research key project (No 4A43) in the Education Department of Henan province, innovation ability cultivation fund (No 24ZCX5) in Henan University of Science and Technology the authors considered the tracking problem of linear differential inclusion systems This paper applies global sliding mode control to study the tracking control problem for a class of differential inclusion systems The main contributions of this paper lie in the following aspects: ) the system model is a generalized model of those considered in [4, 8, 9; 2) an extensive reaching law is proposed to design a sliding mode controller to make the error system asymptotically stable 2 Problem formulation Consider the following differential inclusion system: ẋ i(t) = x i+(t), i n ẋ n(t) co{[f j(x) + g j(x)(u(t) + w(t))}, j =, 2,, N y(t) = x (t) () where x(t) = [x, x 2,, x n T is the system state, co{ } denotes the convex hull of a set, f j(x) and g j(x) are smooth functions from R n to R for j =,, N, u(t) R is the control input, w(t) is the bounded disturbance, ie, w(t) γ, with a positive constant γ y(t) is the output of the system System () is most often applied in physical systems such as the Duffing-Holmes damped spring system, Van der Pol equation, robot systems and flexible-joint mechanisms [22 This paper assumes that g j(x) >, for all j =, 2,, N The target of this paper is to design a feedback law such that the output y(t) of the system () can track a known reference function r(t) By the conclusion established in convex analysis theory [2, differential inclusion system () is equivalent to the following uncertain system: ẋ i(t) = x i+(t), i n N ẋ n(t) = α j[f j(x) + g j(x)(u(t) + w(t)) y(t) = x (t) where α j are uncertain parameters with the properties that (2)
L P Liu et al / Tracking Control of a Class of Differential Inclusion Systems via Sliding Mode Technique 39 α j and N αj = z (t) z 2(t) Denote z(t) = x (t) r(t) x 2(t) ṙ(t) x n(t) r () (t) z n(t) = y(t) r(t) ẏ(t) ṙ(t) y () (t) r () (t) =, where z(t) is called as the error of tracking z(t) satisfies the following ż i(t) = z i+(t), i n N ż n(t) = α j[f j(x) + g j(x)(u(t) + w(t)) r (n) (t) Then the target of the design is to find a feedback law u(t) such that z(t) as t The following Lemma will be useful in this paper Lemma Consider the following linear system (3) ẋ(t) = Ax(t) + Bu(t) (4) where A is a Hurwitz matrix and u(t) = e λt γ(t) for a positive constant λ and a bounded function γ(t) Then lim x(t) = t Proof Because A is a Hurwitz matrix, there exist positive constants a and b such that e At ae bt And we assume γ(t) q, where q is a constant Solving (4), we obtain thus, x(t) = e At x() + t e A(t τ) Be λτ γ(τ)dτ (5) t x(t) ae bt x() + aq B e b(t τ) λτ dτ = ae bt x() + aq B e bt t, if b = λ ae bt x() + aq B e λt e bt, b λ if b λ (6) where the denotation may have different meanings, but we always assume that they are compatible From (6), obviously we get 3 Main results lim x(t) = (7) t For the nonlinear system (2), the control law design consists of two phases Firstly, an appropriate sliding surface is chosen, so that sliding mode dynamics has desired performance Secondly, a control law is designed, which guarantees that the state of system (2) converges to the sliding surface in a finite time 3 Sliding surface design The sliding surface is designed to ensure the error system stable asymptotically on the sliding surface The sliding surface is considered as s(t) = C(z(t) e µt z()) (8) where C = [c, c 2,, c, and c i(i =, 2,, n ) will be determined, µ > is an appropriate constant, z() is the initial state of the error Let s(t) = Then (8) leads to z n(t) = c iz i + e µt Cz() (9) From (3) and (9), the sliding mode dynamics is obtained as ż i(t) = z i+(t), i n 2 ż (t) = c iz i + e µt () Cz() The matrix description of () is z(t) = A z(t) + Be µt Cz() () where z(t) = [z, z 2,, z T, A =, B = c c 2 c We now choose c i(i =,, n ) such that A is a Hurwitz matrix, then sliding mode dynamics () or () is asymptotically stable by Lemma 32 The controller design Before illustrating the scheme of the controller, a reaching law is introduced in this subsection Lemma 2 When the reaching law is adopted as ṡ(t) = ks εe λt s α sgn(s), k >, ε >, λ, α < (2) where is the Euclidean norm, the state x(t) of system will reach the sliding surface s(t) = in a finite time T, where T = [ k( α) λ ln k( α) λ + s() α, ε( α) if k( α) λ s() α ε( α), if k( α) = λ (3) Proof Pre-multiplying (2) by s T, we have We also have s T ṡ = k s 2 εe λt s α+ (4) s T ṡ = ds T s = d s 2 = s d s 2 dt 2 dt dt (5)
3 International Journal of Automation and Computing (3), June 24 Replacing (4) by (5), we get d s dt = k s εe λt s α (6) Denote z(t) = s α, The time-derivative of z(t) yields dz dt = ( d s α) s α = k( α)z(t) ε( α)e λt dt (7) Integrating (7) from to t, we have z(t)= e k( α)t z() ε( α)e k( α)t e [k( α) λt k( α) λ, if k( α) λ e k( α)t z() ε( α)e k( α)t t, if k( α)=λ (8) Let s(t ) = Then z(t ) = Through a simple calculation, we can obtain [ k( α) λ ln k( α) λ + s() α, ε( α) T = if k( α) λ s() α ε( α), if k( α) = λ (9) Remark The reaching law presented in this paper includes three cases: ) if k =, λ =, < α <, then ṡ(t) = ε s α sgn(s); 2) if λ =, α =, then ṡ(t) = ks εsgn(s); 3) if α =, then ṡ(t) = ks εe λt sgn(s) The reaching law ) is known as power rate reaching law [5, 2) is widely used in many papers, for examples in [7, 3) is presented in [ The term e λt is added to reduce the chattering phenomenon in this paper Remark 2 The right term of the reaching law presented in this paper is continuous in s =, but in [6, the reaching law ṡ(t) = (µ + ηe λt s α )s is not continuous in s = when < α < From the discussion above, the sliding mode controller design is given in the following theorem Theorem For system (), if u(t) is designed as where u(t) = η(x) = [ c iz i+ + µe µt Cz() r (n) (t)+ ks + (η(x) + εe λt s α )sgn(s) ( ) ḡ(x) f(x) + ḡ(x)γ, (2) c iz i+ + µe µt Cz() r (n) (t)+ = min{g (x),, g N (x)} >, ḡ(x) = max{g (x),, g N (x)}, f(x) = max{f (x),, f N (x)}, k >, ε >, λ, > α then the trajectory of the error system (3) converges to the sliding surface s(t) = in a finite time T and remains on it thereafter Proof Consider a Lyapunov function candidate as follows: V (t) = 2 s2 (2) Calculating the time derivative of V (t) with the trajectory of (3), we have { V (t) =s c iz i+ + µe µt Cz() r (n) (t)+ } N α j[f j(x) + g j(x)(u(t) + w(t)) = N s ( c iz i+ + µe µt Cz() Define η(x) = r (n) (t)) N g j(x)w(t) N [ ( ḡ(x) s ks 2 + s N α j[f j(x)+ (η(x) + εe λt s α )s ) c iz i+ + µe µt Cz() r (n) (t)+ f(x)+ḡ(x)γ ( ) ḡ(x) ks 2 (η(x)+εe λt s α )s (22) c iz i+ + µe µt Cz() r (n) (t)+ f(x) + ḡ(x)γ (23) Substituting (23) into (22) yields to V (t) ks 2 εe λt s α+ (24) By Lemma 2 and the sliding mode theory, we conclude that the trajectory of the error system (3) converges to the sliding surface s(t) = in a finite time T and remains in it thereafter Remark 3 In Theorem, we apply the initial condition z() to design the controller (2), the technique is adopted by [6 4 Numerical example In this section, a simulation of a nonlinear system is provided to verify the effectiveness of the method proposed in
L P Liu et al / Tracking Control of a Class of Differential Inclusion Systems via Sliding Mode Technique 3 this paper System model is described by x = x 2 x 2 = α x + α 2x 2 2 + [α + α 2( + x 2 2)(u + w(t)) y(t) = x (t) (25) where f (x) = x, f 2(x) = x 2 2, g (x) =, g 2(x) = + x 2 2, w(t) = 2 sint and α + α 2 =, α, α 2 We choose c = such that the eigenvalue of the matrix A is ξ = The initial value x() = [ T and the reference function is r(t) = sin(t) The sliding surface is designed as s(t) = x (t)+x 2(t) sin(t) cos(t) 8e t The parameters in the controller (2) are chosen as µ = ε =, λ = 2, k = 8, α = 5 Figs and 2 show the trajectories of the output state y(t) and the reference function sin(t) with the different parameters α, α 2 From Figs 3 and 4, we can conclude that the method proposed in this paper reduces the chattering phenomenon So the robustness and stability of system are improved Fig 3 Control input u(t) (α = 9, α 2 = ) Fig 4 Traditional sliding mode control input u(t) adopted by [7 (α = 9, α 2 = ) Fig Output y(t) and the reference function r(t) (α = α 2 = 5) 5 Conclusion The tracking control for a class of differential inclusion systems has been investigated Using global sliding mode control, a novel controller is designed to make the output of differential inclusion systems track the desired trajectory asymptotically Finally, an example illustrates the validity of the proposed method References Fig 2 Output y(t) and the reference function r(t) (α = 9, α 2 = ) [ H F Ho, Y K Wong, A B Rad Robust fuzzy tracking control for robotic manipulators Simulation Modelling Practice and Theory, vol 5, no 7, pp 8 86, 27 [2 Y J Sun Composite tracking control for a class of uncertain nonlinear control systems with uncertain deadzone nonlinearities Chaos, Solitons & Fractals, vol 35, no 2, pp 383 389, 28 [3 X Gong, Z C Hou, C J Zhao, Y Bai, Y T Tian Adaptive backstepping sliding mode trajectory tracking control for a quad-rotor International Journal of Automation and Computing, vol 9, no 5, pp 555 56, 22 [4 H H Tsai, C C Fuh, C N Chang A robust controller for chaotic systems under external excitation Chaos, Solitons & Fractals, vol 4, no 4, pp 627 632, 22
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