On Euler’s Discretization of Sliding Mode Control Systems with Relative Degree Restriction

On Euler’s Discretization of Sliding Mode Control Systems with Relative Degree Restriction Zbigniew Galias 1 and Xinghuo Yu 2 1 Department of Electrical Engineering, AGH University of Science and Technology, Krak´ow, Poland galias@agh.edu.pl 2 Platform Technologies Institute, RMIT University, Melbourne, VIC 3001, Australia x.yu@rmit.edu.au 1 Introduction Discrete sliding-mode control has been extensively studied to address some basic problems associated with the sliding mode control (SMC) of discrete-time sys- tems that have relatively low switching frequencies (see [2, 3, 4] and references therein). However, the discretization behaviors of continuous-time SMC systems have not been fully explored except some early results in [5, 6, 8]. The Euler dis- cretization method is commonly used for digital implementation and simulation of SMC systems [7]. It is of practical importance to understand the behaviors of Euler’s discretization of SMC systems in order to evaluate ‘deterioration’ of per- formance such as chattering, which is a well known problem in SMC and other switching based control. Furthermore, understanding of the Euler’s discretiza- tion of SMC will help understand discretization effects of other discretization methods such as the zero-order-hold and the Runge-Kutta method. In [7], the first study of Euler’s discretization of the single input equivalent control based SMC systems was carried out and discretization behaviors were fully explored. It was found that only period-2 orbits exist. In this chapter, we perform a complete analysis of discretization behaviors of the most popular SMC systems — equivalent control based (ECB) SMC systems under relative degree restriction (relative degree higher than one) using the Euler’s discretization. First, we review the results for ECB-SMC systems with relative degree one under the Euler’s discretization, including accurate estimates of the bounds of steady states and conditions to guarantee stability [7]. This paves the way to study the ECB-SMC systems with relative degree higher than one. Second, we formulate the ECB-SMC systems with relative degree higher than one in a canonical form that is easy to analyze. Asymptotically stable dynamics is constructed for the higher order sliding mode functions so that the conventional ECB-SMC law for the ECB-SMC systems with relative degree one can be applied. Third, we use the theoretical results for the Euler’s discretization of ECB-SMC systems with relative degree one [7] to analyse the ECB-SMC G. Bartolini et al. (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp. 119–133, 2008. springerlink.com c  Springer-Verlag Berlin Heidelberg 2008 120 Z. Galias and X. Yu systems with arbitrary relative degree (higher than one). Accurate estimates of the bounds of steady states and higher order sliding mode functions are given. We show that there only exist period–2 orbits, similar to the case of the ECB- SMC systems with relative degree one. Finally, the chapter is concluded with some comparisons with the existing results on continuous-time high-order SMC systems [12], where certain commonalities are observed. 2 Euler’s Discretization of Single-Input ECB-SMC Systems 2.1 The Single-Input ECB-SMC Systems Without loss of generality, consider the linear system with a single input u ∈ R in the controllable canonical form ˙x = Ax + bu, (1) where A ∈ R n×n , x, b ∈ R n , and: A = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 010··· 0 001··· 0 . . . . . . . . . . . . . . . 00··· 01 −a 1 −a 2 ··· −a n−1 −a n ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,b= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 . . . 0 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , (2) and the parameters a i (i =1, ,n) are real numbers. The switching function, which is the designated sliding manifold when in the sliding mode, is commonly defined as g(x)=c T x, (3) where c =(c 1 ,c 2 , ,c n ) T ∈ R n and we assume that c T b =0.Therelative degree of g(x) with respect to the control u is one, since ∂ ˙g(x) ∂u =0.Weconsider the popular SMC strategy, the equivalent control of the form: u(x)=−(c T b) −1 c T Ax −α(c T b) −1 sgn(g(x)), (4) where α>0. Since we can always rescale c so that c T b = 1, we assume that c n =1.Notethatg =0isan(n −1)–dimensional dynamics whose characteristic equation is λ n−1 + c n−1 λ n−2 + ···+ c 2 λ + c 1 =0. (5) The coefficients c 1 ,c 2 , ,c n−1 are chosen so that solutions of (5) have negative real parts, i.e. the polynomial (5) is Hurwitz. Substituting (4) into (1) and taking into account that c T b =1yields ˙x = A c x −α sgn(c T x)b, (6) Euler’s Discretization of SMC Systems 121 where A c =(A −bc T A)= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 01 0 ··· 0 00 1 ··· 0 . . . . . . . . . . . . . . . 00··· 01 0 −c 1 ··· −c n−2 −c n−1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ =  0 e T 1 0 B c  , (7) e 1 =(1, 0, ···, 0) T ∈ R n−1 ,and B c = ⎛ ⎜ ⎜ ⎜ ⎝ 01··· 0 . . . . . . . . . . . . 0 ··· 01 −c 1 ··· −c n−2 −c n−1 ⎞ ⎟ ⎟ ⎟ ⎠ . (8) The SMC system is designed in such a way that in the ideal situation, the sliding manifold g = 0 will be reached in finite time. The subsystem y =(x 2 ,x 3 , x n ) T is asymptotically stable because the eigenvalues of B c are zeros of the charac- teristic equation (5) which is Hurwitz. 2.2 Euler’s Discretization We consider the effects of the discrete implementation of the control system (6) by means of the Euler’s numerical integration. The Euler’s discretization with thetimesteph>0 leads to the following discrete-time system: x (k+1) = x (k) + hA c x (k) − αhs k b, (9) where the symbolic sequence s =(s 0 ,s 1 ,s 2 , ) corresponding to the initial condition x (0) =(x (0) 1 ,x (0) 1 , ,x (0) n ) T is defined by s k =sgn(g(x (k) )) = sgn(c 1 x (k) 1 + c 2 x (k) 2 + ···+ c n−1 x (k) n−1 + x (k) n ). Equation (9) is equivalent to x (k+1) i = x (k) i + hx (k) i+1 , for i =1, 2, ,n− 1, (10a) x (k+1) n = x (k) n − h n  i=2 c i−1 x (k) i − αhs k . (10b) From (10) it follows that g(x (k+1) )= n−1  i=1 c i  x (k) i + hx (k) i+1  + x (k) n − h n  i=2 c i−1 x (k) i − αhs k = g(x (k) ) −αhs k . Hence, the formula for the switching function evaluated along the trajectory with the initial point x (0) and the symbolic sequence s =(s 0 ,s 1 ,s 2 , )reads[7] 122 Z. Galias and X. Yu g(x (k) )=g(x (0) ) −αh k−1  j=0 s j . (11) This result says that g(x (k) ) can only differ from g(x (0) ) by a multiple of αh. Using this result it is possible to classify admissible symbolic sequences. The following lemma says each trajectory must cross the sliding manifold at some time and from then on it will cross this manifold in each step [7]. Lemma 1. There exists p>0 such that s p = −s 0 .Ifs 1 = −s 0 then s k = (−1) k s 0 for every k ≥ 0. Proof. Assume that s k = +1 for every k ≥ 0. From (11) it follows that g(x (k) )= g(x (0) ) −αhk > 0 for every k ≥ 0, which is obviously not possible. Hence, there exists p>0 such that s p = −s 0 . Now, we show that s 1 = −s 0 implies that s 2 = −s 1 .From(11)andthe assumptions it follows that s 2 =sgn(g(x (2) )) = sgn(g(x (0) ) −αh(s 0 + s 1 )) = s 0 = −s 1 . It follows that s k+1 = −s k for every k ≥ 0. From the above lemma it follows that all symbolic sequences are eventually period–2. Without loss of generality we can assume that s k =(−1) k+1 for all k ≥ 0. It is always possible to shift the samples in such a way that this assumption is valid. With this assumption the system behavior is described by: x (k+1) =(I + hA c )x (k) +(−1) k αhb. (12) The second iterate of the discretized system defines a time invariant discrete dynamical system: x (2k+2) =(I + hA c ) 2 x (2k) + αh 2 A c b. (13) Remark 1. Using Lemma 1 one can easily calculate bounds for the switching function g in the steady state: |g(x (k) )|≤αh. 2.3 Steady State Behavior Since every symbolic sequence is eventually period–2 it is clear that the period of each periodic orbit must be even. We will show that under certain assumptions each trajectory convergestoaperiod–2orbit. First, let us consider the subsystem y =(x 2 ,x 3 , ,x n ). The dynamics of the y subsystem is described by y (k+1) = y (k) + hB c y (k) +(−1) k αhb  , (14) where b  =(0, ,0, 1) T ∈ R n−1 and B c is defined in (8). In the following, we assume that all eigenvalues of the matrix I + hB c are located within the unit circle, i.e. the discrete system (14) is stable. Euler’s Discretization of SMC Systems 123 Since the symbolic sequence is periodic the trajectory is given by the following recursive formula y (2k+2) =(I + hB c ) 2 y (2k) + αh 2 B c b  . (15) It follows that y (2k) = B k y (0) +(B k−1 + B k−2 + ···B + I)c, (16) where B =(I + hB c ) 2 ,andc = αh 2 B c b  . Note that since all eigenvalues of the matrix I + hB c are located within the unit circle, it follows that the matrix I +0.5hB c has no zero eigenvalues and in consequence it is invertible. Since (5) is Hurwitz, it follows that B c is nonsingular. Thus B −I =(I +hB c ) 2 −I =2hB c +h 2 B 2 c =2hB c (I +0.5hB c )isalsoinvertible and (B k−1 + B k−2 + ···B + I)=(B k − I)(B − I) −1 . It follows that y (2k) =(I + hB c ) 2k y (0) +0.5αh  (I + hB c ) 2k − I  (I +0.5hB c ) −1 b  . (17) Since I + hB c is stable, the trajectory y (2k) converges to the unique period–2 point y  =(x  2 , ,x  n ) T given by y  = −0.5αh (I +0.5hB c ) −1 b  . (18) It can be shown (compare [7]) that x  k = α (−0.5h) n−k+1 /S, for k =2, 3, ,n. (19) where S =det(I +0.5hB c )=1+ n−1  i=1 c i (−0.5h) n−i . Let us define by y  the limit of y (2k+1) when k →∞. It is clear that y  = (I + hB c )y  + αhb, and that the trajectory y (k) converges to the period–2 orbit (y  ,y  ).Theperiodicorbit(y  ,y  ) is symmetric with respect to the origin, i.e. y  = −y  . Indeed, from (14) and (18) it follows that y  + y  =(2I + hB c ) y  + αhb  = −αhb  + αhb  =0. Let us now come back to the full system. Since x (k+1) 1 = x (k) 1 +he T 1 y (k) ,where e 1 =(1, 0, 0) T ∈ R n−1 , it follows that x (2k+2) 1 = x (2k) 1 + he T 1 (2I + hB c )y (2k) and x (2k) 1 = x (0) 1 +2he T 1 (I +0.5hB c )(y (0) + y (2) + ···+ y (2k−2) ). (20) Substituting y (2j) from (17) after some algebraic manipulations we obtain x (2k) 1 = x (0) 1 + e T 1 B −1 c  (I+hB c ) 2k − I   y (0) +0.5αh (I+0.5hB c ) −1 b   . (21) 124 Z. Galias and X. Yu Since (I + hB c ) k → 0, when k →∞it follows that x (2k) 1 converges to x  1 = x (0) 1 − e T 1 B −1 c  y (0) +0.5αh (I +0.5hB c ) −1 b   . (22) Above, we have shown that each trajectory of the system (9) converges to a period–2 orbit defined by (22) and (19). Now, we derive bounds for x  1 .Thepointx  =(x  1 ,x  2 , ,x  n ) T is a period–2 point of the system (9) if its symbolic sequence starts with (s 0 ,s 1 )=(−1, +1), i.e.: g(x  )= n−1  k=1 c k x  k + x  n < 0, (23) g(x  )=g(x  )+αh = n−1  k=1 c k x  k + x  n + αh ≥ 0. (24) where x  is the image of x  after one iteration. These inequalities can be rewritten as − n−1  k=2 c k x  k − x  n − αh ≤ c 1 x  1 < − n−1  k=2 c k x  k − x  n . (25) Since − n−1  k=2 c k x  k − x  n = −x  n  1+ n−1  k=2 c k (−0.5h) n−k  = 0.5αh S  S − c 1 (−0.5h) n−1  =0.5αh  1 − c 1 (−0.5h) n−1 S  , from (25) we obtain conditions for existence of period–2 orbit x  1 ∈ αh 2c 1  −1 − c 1 S  − h 2  n−1 , 1 − c 1 S  − h 2  n−1  . For x  1 there is a shift in the admissible values x  1 = x  1 + hx (0) 2 = x  1 + αh (−0.5h) n−1 /S. It follows that x  1 ∈ αh 2c 1  −1+ c 1 S  − h 2  n−1 , 1+ c 1 S  − h 2  n−1  . Summarizing, we have the following result: Theorem 1. [7] Let us assume that the eigenvalues of B c have negative real parts, and that eigenvalues of I +hB c are located within the unit circle. Then the system trajectory converges to a period–2 orbit whose coordinates are bounded by |x 1 |≤ αh 2c 1  1+ c 1 |S|  h 2  n−1  , |x k |≤ α |S|  h 2  n−k+1 , for 2 ≤ k ≤ n. (26) Euler’s Discretization of SMC Systems 125 Note that if S is close to 0 then the amplitude of period–2 solution is large. If we however choose the time step such that S>0.5, which can always be done by reducing h, we obtain the following bounds: |x 1 |≤αh  1 2c 1 +  h 2  n−1  , |x k |≤2α  h 2  n−k+1 , for 2 ≤ k ≤ n. In the Theorem 1, it is assumed that the system (14) is asymptotically stable, i.e. that all eigenvalues of the matrix I + hB c are located within the unit circle. Now, we show that this condition is satisfied for sufficiently small h. Let us denote by {λ k } n−1 k=1 the eigenvalues of B c . It is clear that hλ k are eigenvalues of hB c . Since all eigenvalues of B c are located in the left half-plane (the continuous subsystem is asymptotically stable) it follows that there exists h  > 0 such that all eigenvalues h  λ k are located within the circle centered at −1 with the radius 1 (the same holds for all h ∈ (0,h  ]). Adding the identity matrix shifts all eigenvalues by 1, and hence for h ∈ (0,h  ] all eigenvalues of I + hB c are located within the unit circle. Therefore, for h ≤ h  the discrete subsystem (14) is asymptotically stable. 3 Euler’s Discretization of ECB-SMC Systems with Relative Degree Restriction 3.1 The Single-Input ECB-SMC Systems with Relative Degree Restriction We now consider the single input linear system with relative degree restriction. The single input systems in the controllable canonical form is the same as in (1) except that the switching function, which is the designated sliding manifold when in the sliding mode, is now defined as g(x)=c T x = c 1 x 1 + c 2 x 2 + ···+ c n−r−1 x n−r−1 + x n−r , (27) where c ∈ R n−r . The relative degree of g(x) with respect to the control u, denoted as r + 1 [12], is defined as ∂g (j) (x) ∂u =0forj =1, ···,r but ∂g (r+1) (x) ∂u =0.The ECB-SMC is then not directly applicable [12, 13]. However, one remedy is to construct a new switching function, s(w)=d T w = d 1 g 1 + d 2 g 2 + ···+ d r g r + g r+1 , (28) where g 1 = g, g 2 =˙g, ,g r+1 = d r g dt r , and w =(g 1 ,g 2 , ···,g r+1 ) T ∈ R r+1 .Itisassumedthats(w) is asymptotically stable, meaning that its characteristic polynomial λ r + d r λ r−1 + ···+ d 2 λ + d 1 = 0 (29) 126 Z. Galias and X. Yu is Hurwitz. In such a way, we can reformulate the system (1) as ˙y = A  y + bu. (30) where y =(v T ,w T ) T ∈ R n with v =(x 1 , ···,x n−r−1 ) T ∈ R n−r−1 .Now, we formulate the state transformation from x to y. Let us denote x  = (x n−r , ,x n ) T ∈ R r+1 . It can be easily seen that  v w  =  I n−r−1 0 P 1 P 2  v x   , (31) where P 1 ∈ R (r+1)×(n−r−1) , P 2 ∈ R (r+1)×(r+1) are given by P 1 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ c 1 c 2 ··· c n−r−1 0 c 1 ··· c n−r−2 . . . . . . . . . . . . 00··· ··· 00··· ··· ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,P 2 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 10··· 00 c n−r−1 1 ··· 00 . . . . . . . . . . . . . . . ··· ··· c n−r−1 10 ··· ··· c n−r−2 c n−r−1 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . We also have A  = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 01··· 000··· 00 00··· 000··· 00 . . . . . . . . . . . . . . . . . . . . . . . . . . . 00··· 100··· 00 −c 1 −c 2 ··· −c n−r−1 10··· 00 00··· 001··· 00 . . . . . . . . . . . . . . . . . . . . . . . . . . . 00··· 000··· 01 −a  1 −a  2 ··· −a  n−r−1 −a  n−r −a  n−r+1 ··· −a  n−1 −a  n ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,b= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 . . . 0 0 0 . . . 0 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . (32) where it can be easily derived that (a  1 ,a  2 , ···,a  n )=(a 1 , ···,a r+1 ,a r+2 − c 1 , ···,a n − c n−r−1 )  I n−r−1 0 P 1 P 2  −1 . (33) We consider the popular ECB-SMC strategy in the new coordinates y.Wefirst derive the equivalent control when the system is in the sliding mode ˙s =0,which leads to ˙s = r  i=1 d i g i+1 +˙g r+1 = r  i=1 d i y n−r+i +˙y n = r  i=1 d i y n−r+i − n  i=1 a  i y i + u. The normal form of ECB-SMC gives u(y)=− r  i=1 d i y n−r+i + n  i=1 a  i y i − α sgn(s(y)), (34) Euler’s Discretization of SMC Systems 127 resulting in ˙ss = −α|s| which is a finite time convergent dynamics [1]. Note that c 1 ,c 2 , ,c n−r−1 are assumed to be coefficients of a Hurwitz polynomial and α>0. Also note that g =0isan(n − r −1)-dimensional dynamics whose characteristic equation is λ n−r−1 + c n−r−1 λ n−r−2 + ···+ c 2 λ + c 1 =0. (35) Substituting (34) into (30) yields ˙x = A  eq x − α sgn(s(y))b, (36) where A  eq = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 010 ··· 0 00··· 00 001 ··· 0 00··· 00 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00··· 01 00··· 00 −c 1 −c 2 ··· −c n−r−2 −c n−r−1 10··· 00 00··· 0001 0 ··· 0 00··· 00 00 1 ··· 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 00··· 00 00··· 01 00··· 00 0 −d 1 ··· −d r−1 −d r ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,b= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 . . . 0 0 0 0 . . . 0 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , A  eq =  B c E 0 A d  . (37) where E ∈ R (n−r−1)×(r+1) has all zero rows except the last row, which is (1, 0, ,0) ∈ R r+1 and B c = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 010 ··· 0 001 ··· 0 . . . . . . . . . . . . . . . 00··· 01 −c 1 −c 2 ··· −c n−r−2 −c n−r−1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ∈ R (n−r−1)×(n−r−1) , (38) A d = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 01 0 ··· 0 00 1 ··· 0 . . . . . . . . . . . . . . . 00··· 01 0 −d 1 ··· −d r−1 −d r ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ∈ R (r+1)×(r+1) . (39) The SMC system is designed in such a way that in the ideal situation, the sliding manifold s = 0 will be reached in finite time. Taking into account that s(y)is an explicit function of w, i.e. s(y)=s(w) we can decompose (36) as 128 Z. Galias and X. Yu ˙v = B c v + Ew, (40a) ˙w = A d w −α sgn(s(w))b, (40b) where b =(0, ···, 0, 1) T ∈ R r+1 . However, if s(w) = 0 is reached, then since the characteristic polynomial of s(w) is Hurwitz, meaning that λ r + d r λ r−1 + ···+ d 2 λ + d 1 =0 has all zeros in the left hand side of the complex plane, then y n−r+1 = g(v)=0 will be reached asymptotically. Since the characteristic polynomial of g(v)=0 is Hurwitz, that is, λ n−r−1 + c n−r−1 λ n−r−2 + ···+ c 2 λ + c 1 =0 has all its zeros in the left hand side of the complex plane, then y 1 = 0 will be reached. Therefore, the whole system is asymptotically stable. It is interesting to see that the characteristic polynomial of A  eq is λ(λ n−r−1 + c n−r−1 λ n−r−2 + ···+ c 2 λ + c 1 )(λ r + d r λ r−1 + ···+ d 2 λ + d 1 )=0. The single zero λ = 0 corresponds to the finite time convergence of s =0which reduces the system dimension by one. It is commonly called the fast sliding motion (versus the slow sliding motion of the other n − 1 dimensions). 3.2 Euler’s Discretization Euler’s discretization of the system (40) with the time step h>0 produces the following discrete-time dynamical system: v (k+1) =(I + hB c )v (k) + h ¯ bw (k) 1 , (41a) w (k+1) =(I + hA d )w (k) − αhs k b, (41b) where s k = s(w (k) )=d 1 w (k) 1 + d 2 w (k) 2 + ··· + d r w (k) r + w (k) r+1 ,and ¯ b = (0, ···, 0, 1) T ∈ R n−r−1 . Let us note that the subsystem (41b) is equivalent to the system (9), being the Euler’s discretization of the single input ECB-SMC system, which was analysed in the previous section. It follows that all symbolic sequences are eventually period–2, and that if we assume that I + hB d where B d = ⎛ ⎜ ⎜ ⎜ ⎝ 01··· 0 . . . . . . . . . . . . 0 ··· 01 −d 1 ··· −d r−1 −d r ⎞ ⎟ ⎟ ⎟ ⎠ (42) has all eigenvalues within the unit circle then every trajectory of the w subsystem converges to a period–2 orbit. Bounds for the steady state of the subsystem w are given by Theorem 1: [...]... structure control systems IEEE Transactions on Industrial Electronics 42, 117–122 (1995) 5 Yu, X., Chen, G.: Discretization behaviors of equivalent control based sliding- mode control systems IEEE Transactions on Automatic Control 48, 1641–1646 (2003) 6 Xia, X.H., Zinober, A.S.I.: Delta-modulated feedback in discretization of sliding mode control Automatica 42, 771–776 (2006) 7 Galias, Z., Yu, X.: Euler’s discretization. .. 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Trajectory with the initial conditions y (0) = (1, 1, 0, 0, 0, 1.01)T converges to a period–2 orbit 4 Conclusions We have shown that the Euler’s discretization with appropriately chosen time step is a good method for discrete implementation of SMC systems We have demonstrated that for the ECB-SMC systems with relative degree restriction, if the discrete system is stable then every trajectory converges... the ZeroOrder-Hold sampling of the multi-input SMC systems References 1 Utkin, V.I.: Sliding Modes in Control Optimization Springer, Berlin (1992) 2 Edwards, C., Spurgeon, S.: Sliding Mode Control: Theory and Applications Taylor and Francis, London (1998) 3 Sabanovic, A., Fridman, L.M., Spurgeon, S.: Variable Structure Systems: From Principles to Implementation IEE, London (2004) 4 Gao, W., Wang, Y.,... Measurement, and Control 122, 793–802 (2000) 12 Fridman, L.M., Levant, A.: Higher order sliding modes In: Sabanovic, A., Fridman, L.M., Spurgeon, S (eds.) 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Journal of Control 76, 924–941 (2003) 14 Wang, B., Yu, X., Chen, G.: Discretization behaviors of sliding mode control systems with matched uncertainties In: Proc WCICA 2006, Dalian, China (2006) 15 Rubio, J.D., Yu, W.: A new discrete-time sliding mode control with time-varying gain and neural identification International Journal of Control 79, 338–348 (2006) ... and higher order sliding mode functions, which allow us to estimate the maximum chattering amplitude when using a given value of the time step It should be noted that there are some commonalities between the ECBSMC systems with relative degree higher than one, discretized by using the Euler method and the results for continuous-time higher-order SMC systems Euler’s Discretization of SMC Systems 133 [12,... system ⎞ ⎛ ⎞ 0 0 0 ⎜0⎟ 0 0⎟ ⎟ ⎜ ⎟ ⎜ ⎟ 0 0⎟ ⎟, b = ⎜0⎟ ⎜0⎟ 1 0⎟ ⎟ ⎜ ⎟ ⎝0⎠ 0 1⎠ 0 0 1 Euler’s Discretization of SMC Systems 131 Let the designated sliding manifold be defined by g(x) = x1 + 2x2 + x3 + x4 = y1 + 2y2 + y3 + y4 , i.e the relative degree of g with respect to the control u is r + 1 = 3 Let the new switching function be defined by s(y) = g1 + g2 + g3 = y4 + y5 + y6 , i.e d = (d1 , d2 ) = (1, 1).. .Euler’s Discretization of SMC Systems |w1 | ≤ αh 2d1 1+ h 2 d1 |Sd | r |wk | ≤ , α |Sd | where h 2 129 r−k+2 , for 2 ≤ k ≤ r + 1, r r+1−i Sd = det (I + 0.5hBd) = 1 + di (−0.5h) i=1 Let us now consider the subsystem (41a) Let us assume that eigenvalues of I + hBc lie within the unit circle Let us denote by w , w the coordinates of the period–2 orbit of the subsystem w to which its trajectory converges . On Euler’s Discretization of Sliding Mode Control Systems with Relative Degree Restriction Zbigniew Galias 1 and Xinghuo Yu 2 1 Department of Electrical Engineering, AGH University of Science. dimension by one. It is commonly called the fast sliding motion (versus the slow sliding motion of the other n − 1 dimensions). 3.2 Euler’s Discretization Euler’s discretization of the system (40) with. Single-Input ECB-SMC Systems with Relative Degree Restriction We now consider the single input linear system with relative degree restriction. The single input systems in the controllable canonical form