Sliding Mode Control of Linear Time-varying Systems Application to Trajectory Tracking Control of Nonlinear Systems Yasuhiko Mutoh and Nao Kogure Department of Engineering and Applied Sciences, Sophia University, 7-1 Kioicho, Chiyoda-ku, Tokyo, Japan Keywords: Sliding Mode Control, Linear Time-varying System, Non-linear System, Tracking Control. Abstract: This paper concerns with the sliding mode controller design method for linear time-varying systems. For this purpose, using the time-varying pole placement technique, the state feedback is designed first so that the time-varying closed loop system is equivalent to the standard linear time invariant system. Then, conventional sliding mode controller design method is applied to this time invariant system to obtain the control input. Finally, using the time-varying transformation matrix, this sliding mode control input is put back to the control input for the original system. In this paper, this controller is applied to the trajectory tracking control problem for nonlinear systems. 1 INTRODUCTION most simple idea might be to approximate the nonlin- ear system along this trajectory using a linear time- This paper concerns with the sliding mode controller varying system. However, since, controller design design for linear time varying systems, and then, we method for linear time-varying system is not neces- apply this control technique to a trajectory tracking sarily simple, it seems that this approach is not com- control of non-linear systems. monly used. In this paper, the above time varying slid- The author proposed the simple design method of ing mode control technique is applied to the trajectory the pole placement controller for linear time varying tracking control problem of non-linear systems. Some systems using the concept of the relative degree of simulation results will be also shown. the system (Mutoh,2011) (Mutoh and Kimura,2011). This pole placement design purpose is to make the time varying closed loop system equivalent to some 2 PRELIMINARIES linear time invariant system that has desired eigenval- ues, by the state feedback. In this paper, we make use of this technique for designing the sliding mode con- In this section, the basic properties of linear time- troller for linear time varying systems. The first step varying systems which we will use later are presented. is to find the state feedback for the linear time varying Consider the following linear time-varying multi- system so that the closed loop system is equivalent to input system. some linear time-invariant standard system. Then, by using the conventional sliding mode controller design x˙(t) = A(t)x(t) + B(t)u(t) (1) method, the sliding mode control input for this linear time invariant system can be obtained (Utkin,1992). Here, x(t) ∈ Rn and u(t) ∈ Rm are the state variable After that, using an equivalent time varying transfor- and the input signal, respectively. A(t) ∈ Rn×n and mation matrix, this control input can be transformed B(t) ∈ Rn×m are time varying coefficient matrices, into the sliding mode control for the original linear which are bounded and smooth functions of t. time varying system. Since, the sliding mode con- The matrix B(t) is written as follows, using col- n troller is designed for the equivalent time invariant umn vectors bi(t) ∈ R (i = 1,··· ,m). system, any type of conventional sliding mode con- troller design method can be applied. B(t) = b1(t) b2(t) ··· bm(t) (2) If we need to control nonlinear systems to follow i n some particular desired trajectory in wide range, the Let bk(t) ∈ R be defined by the following recur- 492 Mutoh Y. and Kogure N.. Sliding Mode Control of Linear Time-varying Systems - Application to Trajectory Tracking Control of Nonlinear Systems. DOI: 10.5220/0005019704920498 In Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2014), pages 492-498 ISBN: 978-989-758-039-0 Copyright c 2014 SCITEPRESS (Science and Technology Publications, Lda.) SlidingModeControlofLinearTime-varyingSystems-ApplicationtoTrajectoryTrackingControlofNonlinearSystems sive equations. vector v(t) ∈ Rm, so that the closed loop system is equivalent to the linear time invariant standard form. b0(t)= b (t) k k (3) Suppose that the system (1) is completely control- bi t A t bi−1 t b˙ i−1 t k( )= ( ) k ( ) − k ( ) lable. Then, if C˜(t) ∈ Rm×n is defined by k = 1,2···m, i = 1,2··· C˜(t)= W(t)R−1(t) (10) Then, the controllability matrix of the system (1) can where be written as follows. W t diag w t ,w t , ,w t 0 0 n−1 n−1 ( )= ( 1( ) 2( ) ··· m( )) (11) U = [b (t)···b (t)|···|b (t)···b (t)] (4) 1×µ c 1 m 1 m wi(t) = [0,··· ,0,λi(t)] ∈ R i (i = 1,··· ,m) Theorem 1. The system (1) is completely control- λi(t) = 0 lable if and only if m ∀ and also if a new outputsignaly ˜(t) ∈ R is defined by rankUc(t)= n t (5) y˜(t)= C˜(t)x(t) (12) If the system (1) is completely controllable, we then, the vector relative degree from u(t) toy ˜(t) is can define the controllability indices, µ1, µ2,···, µm, which satisfy the following equations, µ1,µ2,··· ,µm (Mutoh and Kimura,2011). Lety ˜(t) and C˜(t) be R(t) : nonsingular m y˜1(t) c˜1(t) . ˜ . ∑ µi = n (6) y˜(t)= . , C(t)= . (13) i=1 y˜m(t) c˜m(t) where By differentiatingy ˜(t) successively, we have 0 µ1−1 0 µm−1 R(t)= b1(t)···b1 (t)|···|bm(t)···bm (t) 0 y˜i(t) = c˜i (t)x(t) (7) 1 y˜˙i(t) = c˜i (t)x(t) which is called the truncated controllability matrix. ¨ 2 y˜i(t) = c˜i (t)x(t) In this paper, it is assumed that if the system is com- . pletely controllable, its controllability indices satisfy . µ the inequality, µ1 ≥ µ2 ≥···≥ µm, without loss of y˜( i)(t) = c˜µi (t)x(t)+ c˜µi−1(t)B(t)u(t) generality. i i i µi λ = c˜i (t)x(t)+ i(t)ui(t) Definition 1. Consider the following output equation γ γ for the system (1), + i(i+1)ui+1 ··· + im(t)um(t) i = 1,··· ,m (14) y(t)= C(t)x(t) (8) j Here,c ˜ (t) and γi j(t) are obtained by the following Here, y(t) ∈ Rm is some output signal and C(t) ∈ i recursive equation from C˜(t). Rm×n is a time varying coefficient matrices. Let p be 0 a differential operator. System (1)(8) has the vector c˜i (t)= c˜i(t) j+1 j ˙ j (15) relative degree, r1, r2, ···, rm from u to y, if there ex- c˜i (t)= c˜i (t)A(t)+ c˜i (t) m×n ist some matrix D(t) ∈ R and some nonsingular i = 1,2···m, j = 1,2··· matrix Λ(t) ∈ Rm×m, such that and pr1 γ µi−1 . i j(t)= ci (t)b j(t) (16) .. y(t)= D(t)x(t)+ Λ(t)u(t). (9) Hence, from (14), we have prm pµ1 It should be noted that pri can be replaced by arbitrary .. y˜(t)= D(t)x(t)+ Λ(t)u(t) monic polynomial of p of degree ri. ∇ . pµm (17) 3 STANDARD TIME INVARIANT where, µ1 Λ c˜1 (t) 1(t) SYSTEM µ2 Λ c˜2 (t) 2(t) D t ,Λ t (18) ( )= . ( )= . To design the sliding mode controller for the system . . µm (1), we first design the state feedbackwith a new input c˜m (t) Λm(t) 493 ICINCO2014-11thInternationalConferenceonInformaticsinControl,AutomationandRobotics and where the transformation matrix, T(t), is defined by Λi(t) = [0,··· ,0,λi(t),γi i 1 (t),··· ,γi j(t)] (19) 0 ( + ) c˜1(t) . Thus, by the state feedback . Λ−1 µ1−1 u(t)= (t)(−D(t)x(t)+ v(t)) (20) c˜1 (t) . m T(t)= . (26) with the new input signal v(t) ∈ R , the closed loop . 0 system becomes c˜m(t) . µ1 . p . . µm−1 . y˜(t)= v(t). (21) c˜m (t) . pµm This system has the following state realization. 4 SLIDING MODE CONTROLLER ω˙ (t) = A∗ω(t)+ B∗v(t) DESIGN ∗ A1 0 . = .. ω(t) 4.1 Controller for Linear Time-varying 0 A∗ m Systems ∗ b1 ··· 0 . + . v(t) (22) In this section, the sliding mode controller design . for the linear time varying system (1) is presented. 0 ··· b∗ m For this purpose, we first design the sliding mode where ω(t) ∈ Rn, A∗ ∈ Rn×n, B∗ ∈ Rn×m, and control input v(t) for the linear time invariant sys- tem (22)(23), and then, transform v(t) into the sliding 01 0 mode control input for the original system (1), using . the relation (25)(26). . .. A∗ = ∈ Rµi×µi If we write ω(t) and v(t) as i . . 1 ω 0 0 ... 0 1(t) ω . ω µi (i = 1,...,m) (23) (t) = . , i(t) ∈ R ωm(t) 0 . v1(t) b∗ = . ∈ Rµi . i v t . (27) 0 ( ) = . 1 vm(t) The system (22)(23) is called the linear time invariant (i = 1,··· ,m) standard form. This new state variable ω(t) ∈ Rn is defined by the system (22)(23) is presented as following m sub- systems. y˜1(t) . 0 1 ··· 0 0 . (µ −1) . .. 1 ω˙ i(t) = . ··· . ωi(t)+ . vi(t) y˜1 (t) . 0 ··· 0 1 0 ω(t)= . (24) . 0 ······ 0 1 y˜m(t) (i = 1,··· ,m) (28) . . . Since the system (28) is the standard form, the design (µm−1) y˜m (t) procedure of the ordinary sliding mode controller is very simple stated as follows.