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 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 = 1,2m, 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 = 1,2m, 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)     

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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. First, divide ω (t) into From (14), the original state variable x t and ω t i ( ) ( ) two part. satisfy the relation ωi ω (t) ω t T t x t (25) i(t)= ωµi (29) ( )= ( ) ( ) i (t)

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ωi µi−1 ωµi where (t) ∈ R and i (t) ∈ R. Then, the sliding where surface is defined by 0 c˜i (t) ω ω ωµi Si i(t)= si i(t)+ i (t)= 0 (30) . Ti(t)=  . . (42) where  c˜µi−1(t)  1×µi 1×(µi−1) i Si = [si,1] ∈ R , si ∈ R   (i = 1, ,m). (31) From the above, the design procedure of the slid- ing mode controller for the system (1) is summarized From (28),(29)and (30), the dynamicson the i-th slid- as the following steps. ing surface becomes [Design procedure] 01 0 0 STEP 1. Using the controllability matrix, Uc(t) in  00 1 0  (4), check the controllability of the system (1). If . . . ω˙ i(t)= . . . ωi(t). (32) the system is controllable, calculate the controlla-  . . .    bility indices µ , ,µ and the truncated control-  0 0 1  1 m   lability matrix R(t) in (7).  −si  From the above, if the desired stable characteristic STEP 2. From (10)(11), calculate C˜(t). polynomial of the i-th sliding dynamics is chosen as STEP 3. Using the recursive equation (15), ob- αi µi−1 αi µi−2 αi j γ (p)= p + µi−2 p + + 0 (33) tainc ˜i (t) and, using (16), calculate i j(t) (i = then, the i-th sliding surface is 1, ,m j = i + 1, ,m). Then, using (18)(19), define D(t) and Λ(t). ω αi αi ω Si i(t) = [ 0, , µ 2, 1 ] i(t)= 0. (34) i− STEP 4. Using (26) (or (42)), calculate T(t) (or Since, the i-th subsystem is Ti(t)). ω˙ ∗ω ∗ i(t)= Ai i(t)+ bi vi(t) (35) STEP 5. Define the desired stable characteristic it is well known that the i-th sliding control input vi(t) polynomial (33). Then, define can be defined by ∗ −1 ∗ Si = [si,1] vi(t)= −(Sib ) {SiA ωi(t)+ qi sgn(σi)+ ki fi(σi)} i i s = [ αi , ,αi ] (43) (36) i 0 µi−2 where (i = 1, ,m) σi = Siωi(t) (37) and q > 0 and k > 0 are constant parameters and STEP 6. The sliding mode control input is obtained i i by (40)(41) and (42). fi(σi) is a such that σi fi(σi) > 0. In fact, it is readily shown that, using (36), we have the following Lyapunov function. 4.2 Trajectory Tracking Controller for 1 m m Nonlinear Systems V = ∑ σ2 > 0, V˙ = ∑ σ σ˙ < 0 (38) 2 i i i i=1 i=1 In this paper, sliding mode controller for linear time- Using (23), vi(t) in (36) becomes varying systems is concerned so far. However, in αi αi ω practice, we can hardly find any system that can be vi(t) = −{[0, 0, , µ −2] i(t) i modeled by a linear time-varying system. And, the σ σ . +qi sgn( i)+ ki fi( i)} (39) most of practical systems are nonlinear systems. So, Hence, from (25)(26), the sliding mode control input one of the most important application of the control u(t) for the original system becomes as follows. problem for linear time-varying systems is a control u(t)= Λ−1(t)(−D(t)x(t)+ v(t)) (40) design problem of linear time varying approximate model around some particular trajectory of nonlinear here systems. v1(t) Consider the following non-linear system. . v(t) =  .  . x˙(t)= f (x(t),u(t)) (44)  vm(t)    Here, x(t) ∈ Rn and u(t) ∈ Rm are the state variable v (t) = −{[0 s]T (t)x(t)+ q sgn(σ )+ k f (σ )} i i i i i i i and the input signal. Let x∗(t) and u∗(t) be some σ i = SiTi(t)x(t) particular desired trajectory and the desired input for (i = 1, ,m) (41) x∗(t).

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The problem is to design a controller to track this The sliding mode control input to stabilize the sys- desired trajectory x∗(t) stably around it. This can be tem (50) can be obtained according to the following done by stabilizing this trajectory in the neighborhood steps. of x∗ t and u∗ t . Let ∆x t and ∆u t be defined by ( ) ( ) ( ) ( ) STEP 1. The controllability matrix, Uc(t), ofthesys- ∆x(t)= x(t) − x∗(t) tem (50)-(52) is ∆u(t)= u(t) − u∗(t). 0 0 1 2 Then, we have a linear time-varying approximation Uc(t)=  1 0 0.5sin t − cost ∗ ∗ 1 1 −1 around x (t) and u (t) as follows.  ∆x˙(t)= A(t)∆x(t)+ B(t)∆u(t) (45) 0 ∂ −cost . (53) A t f x∗ t ,u∗ t ( )= ∂x ( ( ) ( )) −1 ∂ ∗ ∗ (46) B(t)= ∂ f (x (t),u (t))  u This implies that the system (50)-(52) is control- Then, using time-varying sliding mode control tech- lable, and the controllability indices are µ1 = 2 nique, error equation can be stabilized around the de- and µ2 = 1. sired trajectory x∗(t) and u∗(t). STEP 2. From STEP 1, the truncated controllability matrix, R(t), becomes 0 1 0 5 NUMERICAL EXAMPLE 1 2 R(t) =  1 2 (sin t − 2cost) 0 (54). 1 −1 1 Consider the following with two in-   put. And from (10)(11), we have ˜ −1 x1(t) = x2(t) C(t) = W(t)R (t) 2 0 1 0 x2(t) = 0.5(1 − x1(t))x2(t) − x1(t)x3(t)+ u1(t) = 2 0 0 1 x3(t) = x1(t) − x3(t)+ u1(t)+ u2(t) (47) −0.5sin2 t + cost 1 0 Let the desired trajectory x∗(t) for this system be  1 00  ∗ 2 x˙1(t) = cost 1 + 0.5sin t − cost −1 1 ∗   x˙2(t) = −sint 1 00 ∗ = 2 x˙3(t) = 1. (48) 1 + 0.5sin t − cost −1 1 Then, the desired input u∗(t) for x∗(t) is obtained as (55) follows. where we choose λ1(t)= λ2(t)= 1. ∗ 2 u1(t) = 0.5sint(1 − cos t) STEP 3. From (15) and (16), we have following k ∗ 2 ∗ c˜ (t) and γi j(t). u2(t) = −cos t + 1 − u1(t) (49) i 1 A linear time-varying approximation of the system c˜1(t) = 0 1 0 (47) around x∗(t) and u∗(t) becomes as follows. 2 c˜1(t) = cost sint − 1 ∆ ∆ 0.5sin2 t −cost d x1(t) x1(t)  ∆x2(t)  = A(t) ∆x2(t)  1 dt c˜2(t) = sint + 2cost + 1 ∆x3(t) ∆x3(t)     1 − cost cost − 1 ∆u1(t) +B(t) (50) γ12(t) = 0 ∆u2(t) And, from these equations, D(t) and Λ(t) are cal- where culated as follows. 0 1 0 cost sint − 1 2 D(t) = A(t) =  cost sint − 1 0.5sin t −cost  sint + 2cost + 1 2cost 0 −1   0.5sin2 t −cost (51) (56) 1 − cost cost − 1 0 0 λ γ Λ 1 12 1 0 B(t) =  1 0 . (52) (t) = λ = 1 1 0 2 0 1  

496 SlidingModeControlofLinearTime-varyingSystems-ApplicationtoTrajectoryTrackingControlofNonlinearSystems

STEP 4. Using (26), the state transformation matrix, 1 T(t), is T (t) 0.8 T (t) = 1 . (57) T2(t) 0.6 2

where σ 0.4 1 0 0 T (t) = . (58) 1 0 1 0 0.2 2 T2(t) = 1 + 0.5sin t − cost −1 1 . 0 (59) 0 2 4 6 8 10 Time[sec] STEP 5. From (33) and the fact that µ1 = 2, µ2 = 1, σ σ2 σ2 Figure 2: Response of = 1 + 2. we choose α1(p)= p + 1. (60) 2 In this case, there is not α (p), because ω2(t) is a 5 scalar. From this, we define u1 u2 S1 = [ 1 1 ], S2 = [ 1 ]. (61) STEP 6. From the above and (40)(41), the sliding mode control input is obtained as follows. 0 Input u 2 u1(t) = (1 − sint cost)∆x1(t) − 0.5sin t∆x2(t) +cost∆x3(t) − sgn(σ1) (62) u2(t) = −(sint + 2cost + 1)∆x1(t) -5 +(cost − 1)∆x2(t) + (1 − cost)∆x3(t) 0 2 4 6 8 10 Time[sec] −sgn(σ ) (63) 2 Figure 3: Response of ∆u(t). where σ1 and σ2 are defined by

σ1 = ∆x1(t)+ ∆x2(t) 2 2 σ2 = (1 + 0.5sin t − cost)∆x1(t) x1 x2 1.5 −∆x2(t)+ ∆x3(t). (64) x3 1 and q1 = q2 = 1, k1 = k2 = 0. 0.5

State x 0

-0.5

-1

-1.5 0 0 2 4 6 8 10 Time[sec]

3 -0.2 Figure 4: Response of x(t).

Δx -0.4

-0.6 Fig.1 shows that the response of ∆x(t) which con- -0.8 verges to the origin. This implies that the state x(t) 0 1 converges to the desired trajectory x∗(t).The value of -0.2 0.5 σ2 σ2 -0.4 1 + 2 is plotted in Fig.2. It decreases monotoni- -0.6 0 cally to 0. Fig.3 and 4 show the control input u(t) and Δx Δx 2 1 state response x(t). According to these graphs, the Figure 1: Response of ∆x(t). time varying sliding mode controller works well for the trajectory tracking control for non-linear systems.

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6 CONCLUSIONS

In this paper, the design procedure of sliding mode controller for linear time-varying system is presented. For this purpose, the time-varying pole placement feedback is used so that the closed loop system is equivalent to some linear time invariant system. Then, the conventional design method of the sliding mode control can be applied to this time invariant system. And, finally by the time-varying transformation ma- trix, this control input is transformed into the sliding mode control input for the original system. It was shown that this controller has a good availability for the trajectory tracking control problem of nonlinear systems.

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