Second-Order Sliding Sector for Variable Structure Control

Second-Order Sliding Sector for Variable Structure Control Yaodong Pan 1 and Katsuhisa Furuta 2 1 The 21st COE Century Project Office, Tokyo Denki University, Ishisaka, Hatoyama, Hiki-gun, Saitama 350-0394, Japan pan@ham.coe.dendai.ac.jp 2 Department of Computers and Systems Engineering, Tokyo Denki University, Hiki-gun, Saitama 350-0394, Japan furuta@k.dendai.ac.jp 1 Introduction To design a variable structure (VS) control system with sliding mode, it is neces- sary to determine a sliding surface and a VS control law such that the reduced- order system on the sliding surface is stable and the system converges to the sliding surface in a finite time and stays there since then, i.e. the sliding surface should be a stable invariant subset in the state space[4][3]. There inherently ex- ists chattering phenomena in a VS control system with sliding mode because the invariance can not be kept when it is implemented in a practical control system where the switching frequency is finite. Sliding sectors have been proposed to replace the sliding mode for a chattering free VS controller and for the implementation of the VS controller in discrete- time systems [6][5]. The first sliding sector proposed in [6] is a subset of the state space where the closed-loop system is stable. The VS control law with the sliding sector [6] ensures that the system moves into the sector in a finite time. The robustness of the VS control system with the sliding sector was proved in [7] by using the frequency domain criterion. The PR-sliding sector proposed in [5] for continuous-time systems was defined as a subset of the state space, inside which a norm of state decreases with zero control input. The VS controller with the PR-sliding sector was designed such that the system state moves from the outside to the inside of the sector with a suitable VS control law, the control input is zero inside the sector, and the norm of state keeps decreasing in the state space with specified negativity of its derivative. The VS control system with the PR-sliding sector is quadratically stable and robust stable to bounded parameter uncertainty [5]. The VS control with the PR-sliding sector is called the lazy control because the VS control law is active only outside the PR-sliding sector, which may be useful to some practical control systems. Theoretically the PR-sliding sector with the corresponding VS control law for continuous-time systems is an invariant subset of the state space but with zero G. Bartolini et al. (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp. 97–118, 2008. springerlink.com c  Springer-Verlag Berlin Heidelberg 2008 98 Y. Pan and K. Furuta control input inside the PR-sliding sector, it can not ensure that the system state remains inside the sector forever in a real control system with finite switching frequency. In the most of case there exists a potential possibility for the system state to move out of the sector. The VS control law outside the PR-sliding sector lets the system state move back into the sector as soon as it moves out of the sector. Therefore it is necessary to switch the control input on the boundary of the sector with infinite frequency to guarantee the invariance of the sector. Thus the invariance may be lost because of finite switching frequency when it is implemented in a real control system although it does not affect the stability of the VS control system with the PR-sliding sector. It is proposed to use a hysteresis function to design the VS control law (See Equation (20) of [5]), with which the system may stay inside the sector longer and the invariance is obtained to a certain extent. As an effective method to deal with the chattering problem and to obtain higher sliding accuracy, high-order sliding mode controllers [8][1][2][9] and high- order sliding mode observers [10][11] have been proposed and can be implemented in many mechanical systems[12][13]. The high-order sliding mode with s =˙s = ··· = s (r−1) ensure the sliding mode to be a high-order invariant subset in the state space. Similarly, a second order sliding sector has also been proposed [14] for continuous-time[15] and discrete-time[16] systems to attain the invariant property. The second-order sliding sector is a PR-sliding sector and an invariant subset of the state space. Therefore, it was named an invariant sliding sector in [14]. Inside the invariant sliding sector, a norm of state decreases because it is a PR-sliding sector. And especially inside the sector, the state will not move out of the sector with a suitable control law. The VS controllers with the invariant PR-sliding sector ensure that the system state moves from the outside to the inside of the sector in a finite time, stays inside it forever after being moved into it, and some Lyapunov function keeps decreasing in the state space with a VS control law. The resultant VS control system thus is quadratically stable and without any chattering. Such invariant sliding sector for continuous-time systems remains invariant even if the switching frequency is finite. In [17], Yu and Yu also discussed the existence of an invariant sliding sector for a second order discrete- time system and gave conditions to guarantee the existence. As comparison with those sliding sectors proposed in [6] and [5], the VS control system with the invariant sliding sector is more realizable for practical implementations and may result in a continuous control input. As the objective of this paper, based on the PR-sliding sector proposed in [5], a second-order (i.e. an invariant) PR-sliding sector for continuous-time systems will be designed at first. Then a quadratically stable VS control system with the sector will be proposed, where an internal and an outer sector are introduced to let the VS control law be continuous on the system state and realizable. Finally the proposed VS control algorithm is implemented to an inverted pendulum control system. This chapter is organized as follows. Section 2 presents the PR-sliding sector [5], defines the second-order PR-sliding sector and describes the problem to be Second-Order Sliding Sector for Variable Structure Control 99 considered in this chapter. Section 3 designs the second-order PR-sliding sector based on a normal switch function. Section 4 proposes the VS controller with the second-order PR-sliding sector. Section 5 gives simulation results with the inverted pendulum. 2 Problem Description In this chapter, a linear time-invariant continuous-time single input system with parameter uncertainties and external disturbances, described by the following state equation is taken into consideration. ˙x(t)=Ax(t)+B(u(t)+d(x, t)) (1) where x(t) ∈ R n and u(t) ∈ R 1 are the state and the input vectors, respectively, A and B are constant matrices of appropriate dimensions, pair (A, B) is control- lable, and d(x, t) represents parameter uncertainties and external disturbances. It is assumed that d(x, t) is bounded and satisfies d 2 (x, t) ≤ w(x), ∀x ∈ R n , ∀t ∈ [0, +∞), (2) where w(x)=x T Wxis a quadratic function function on x and W = W T ∈ R n×n is a known positive definite matrix. If the autonomous system ˙x(t)=Ax(t) of the above one in (1) is quadratically stable, then there exists a positive definite symmetric matrix P and a positive semi-definite symmetric matrix R = C T C such that ˙ L(x)=x T (A T P + PA)x ≤−x T Rx, ∀x ∈ R n (3) where P ∈ R n×n , R ∈ R n×n , C ∈ R l×n , l ≥ 1, (C, A) is an observable pair, and L(x) is a Lyapunov function candidate defined as L(x)=||x|| 2 P = x T Px >0, ∀x ∈ R n ,x=0. (4) For an unstable system, the inequality (3) does not hold. It is possible to de- compose the state space into two parts such that one part satisfies the condition ˙ L(x) > −x T Rx for some elements x ∈ R n , and the other part satisfies the con- dition ˙ L(x) ≤−x T Rx for some other elements x ∈ R n . The latter elements form a special subset in which the Lyapunov function candidate L(x) decrease with zero control input. We define a P-norm, denoted by ||x|| P as the square root of the Lyapunov function candidate L(x) in (4), i.e. ||x|| P =  L(x)= √ x T Px, x ∈ R n . (5) Then the P -norm ||x|| P decreases inside this special subset with zero control input as d dt ||x|| 2 P = x T (A T P + PA)x ≤−x T Rx. 100 Y. Pan and K. Furuta Accordingly, we call this special subset a PR-sliding sector because the matrices P and R together with the system parameter A determine the property of this subset. The definition of the PR-Sliding Sector found in [5] is given as follows. Definition 1. The PR-Sliding Sector is defined on the state space R n as S = {x| s 2 (x) ≤ δ 2 (x),x∈ R n }, (6) inside which the P -norm of the system (1) decreases and satisfies d dt ||x|| 2 P = d dt (x T (t)Px(t)) ≤−x T (t)Rx(t), ∀x(t) ∈S where P and R are the matrices as described above, and s(x) and δ 2 (x) are a linear and a quadratic functions, respectively. Such PR-sliding sector is a subset of R n around a hyperplane s(x)=0and is bounded by two surfaces s(x)=±  δ 2 (x). Inside the sector, the P -norm decreases. Therefore a VS control law based on the PR-sliding sector can be designed such that the system moves into the sector where the P -norm decreases. Unfortunately, the PR-sliding sector is not an invariant subset in the state space when it is implemented in a real control system with finite sampling frequency. Similar to the definition of the second-order sliding mode control, i.e., s =˙s =0, one more condition d dt s 2 (x) ≤ d dt δ 2 (x) is included to obtain the invariance of the sector besides the condition s 2 (x) ≤ δ 2 (x) used to define the sliding sector in (6). Definition 2. The PR-sliding sector defined in (6) is said to be a Second-order PR-Sliding Sector denoted by S 2nd = {x| s 2 (x) ≤ δ 2 (x),x∈ R n } (7) whichiscomposedofaninternalsectorS i and an outer sector S o as S i = {x| ξ 2 (x) <s 2 (x) ≤ δ 2 (x),x∈ R n } (8) S o = {x| s 2 (x) ≤ ξ 2 (x),x∈ R n } (9) if 1. The P -norm decreases inside the second-order PR-sliding sector S 2nd with some control input and satisfies d dt ||x|| 2 P = d dt (x T (t)Px(t)) ≤−x T (t)Rx(t), ∀x(t) ∈S 2nd . Second-Order Sliding Sector for Variable Structure Control 101 2. Quadratic functions ξ 2 (x) and δ 2 (x) on x satisfy 0 <ξ 2 (x) <δ 2 (x), ∀x ∈ R n (10) where ξ 2 (x)=x T Ξx, (Ξ = Ξ T > 0,Ξ ∈ R n×n ), (11) δ 2 (x)=x T Δx, (Δ = Δ T > 0,Δ∈ R n×n ). (12) 3. And the following inequality holds inside the outer sector S o . d dt s 2 (x) ≤ d dt δ 2 2 (x), ∀x(t) ∈S o . (13) The internal sector S i and the outer sector S o given above satisfy S i  S o = S 2nd , S i  S o = N where N is the null set in R n . The second-order PR-sliding sector defined above is an invariant subset with the P -norm decreasing inside it because for any system state inside the outer sector, the system will remain inside the outer sector or move into the internal sector, i.e. stay inside the second-order sliding sector, and for any state inside the internal sector, the system may move to the outer sector but will never move out of the sliding sector. Therefore for some time moment, if the system moves into the second-order sliding sector, the system will stay inside the sector since then. The last condition d dt s 2 (x) ≤ d dt δ 2 (x)inthe above definition shows the invariant property of a second-order sliding sector. By decomposing the sector to an internal and an outer sectors, a large control input to ensure the second-order property inside the outer sector can be avoided while the invariance of the sector can be guaranteed. It has been pointed out in [5] that the PR-sliding sector exists with some P and R for any controllable systems and can be designed by using the Riccati equation. In this chapter, the PR-sliding sector and the second-order PR-sliding sector will be designed based on a normal switch function, which is used to realize a VS control system with sliding mode. The control objective with the second- order PR-sliding sector is to let the system move into the sector in a finite time and remain inside the sector since then with some suitable control rule and to stabilize the system quadratically. 3 Second-Order Sliding Sector 3.1 Design of Switch Function A switching function defined as s(x)=Sx(t),S∈ R 1×n (14) should be designed such that the reduced order system on the sliding mode: 102 Y. Pan and K. Furuta  ˙x(t)=Ax(t)+B (u(t)+d(x, t)) s(x)=Sx(t)=0 (15) is stable. Since it is assumed that the pair (A, B) is controllable, there exists a nonsingular matrix F 1 ∈ R n×n that converts the plant of Eq.(1) to the control- lable canonical form: ˙ ¯x(t)= ¯ A¯x(t)+ ¯ B(u(t)+d(x, t)) (16) which can be rewritten as the block form: d dt  ¯x 1 (t) ¯x 2 (t)  =  ¯ A 11 ¯ A 12 ¯ A 21 ¯ A 22  ¯x 1 (t) ¯x 2 (t)  +  ¯ B 1 ¯ B 2  (u(t)+d(x, t)), (17) where ¯x 1 (t) ∈ R n−1 ,¯x 2 (t) ∈ R 1 ,and ¯ A = F −1 1 AF 1 =  ¯ A 11 ¯ A 12 ¯ A 21 ¯ A 22  = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 01 0 . . . . . . . . . 01 0 00··· 0 1 −a 0 −a 1 ··· −a n−2 −a n−1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ¯ B = F −1 1 B =  ¯ B 1 ¯ B 2  = ⎡ ⎢ ⎢ ⎢ ⎣ 0 . . . 0 1 ⎤ ⎥ ⎥ ⎥ ⎦ . Then the switching function given by Eq.(14) can be rewritten as s(x)=Sx(t)= ¯ S¯x(t)= ¯ S 1 ¯x 1 (t)+ ¯ S 2 ¯x 2 (t), (18) where ¯ S := SF 1 =: [ ¯ S 1 ¯ S 2 ], ¯ S 1 ∈ R 1×(n−1) and ¯ S 2 ∈ R 1 . Without loss of generality, it is assumed that ¯ S 2 =1. An equivalent control input guaranteeing ˙s(x)=0isgivenby u eq (t)=−(SB) −1 SAx(t)=−SAx(t), (SB = ¯ S ¯ B =1). (19) With the control input u(t)=u eq (t)+v(t), (20) the system in Eq.(16) can be written as ˙ ¯x(t)= ¯ A eq ¯x(t)+ ¯ B(v(t)+d(x, t)) (21) where the matrix ¯ A eq is determined by ¯ A eq =  ¯ A 11 ¯ A 12 − ¯ S 1 ¯ A 11 − ¯ S 1 ¯ A 12  Second-Order Sliding Sector for Variable Structure Control 103 and v(t) is an alternative control input. Taking a nonsingular transformation ˜x(t)=F 2 ¯x(t) (22) with F 2 =  I n−1 O (n−1)×1 ¯ S 1 1  , the system in Eq.(17) with the control input given by Eq.(20) is transformed into ˙ ˜x(t)= ˜ A eq ˜x(t)+ ˜ B(v(t)+d(x, t)) (23) where the state variable ˜x(t) and the system matrices ˜ A eq and ˜ B can be written as the following block forms: ˜x(t)=  ˜x 1 (t) ˜x 2 (t)  = F 2 ¯x(t)=  ¯x 1 (t) s(x)  , ˜ A eq =  ˜ A 11 ˜ A 12 ˜ A 21 ˜ A 22  = F 2 ¯ A −1 eq F 2 =  ¯ A 11 − ¯ A 12 ¯ S 1 ¯ A 12 O 1×(n−1) 0  , ˜ B =  ˜ B 1 ˜ B 2  = F 2 ¯ B =  O (n−1)×1 1  = ¯ B, I n−1 is the identity matrix of order (n − 1), O (n−1)×1 and O 1×(n−1) are the zero (n −1)-column and row vectors, respectively. It follows from Eq.(23) that ˙ ¯x 1 (t)=( ¯ A 11 − ¯ A 12 ¯ S 1 )¯x 1 (t)+ ¯ A 12 s(x) (24) ˙s(x)=v(t)+d(x, t). (25) Accordingly, the reduced order system of Eq.(15) on the sliding mode s(x)=0 becomes ˙ ¯x 1 (t)=( ¯ A 11 − ¯ A 12 ¯ S 1 )¯x 1 (t). (26) It is obvious that the pair ( ¯ A 11 , ¯ A 12 ) is controllable. Therefore it is easy to design a feedback gain ¯ S 1 such that the above reduced order system with ˜ A 11 = ¯ A 11 − ¯ A 12 ¯ S 1 is stable, for example, by using the pole assignment algorithm or the LQR control method[18][19]. In this way, a switch function s(x)=Sx(t)=[ ¯ S 1 1]F −1 1 x(t) (27) can be designed. 3.2 Design of Sliding Sector With the switch function designed in the last subsection, a PR-sliding sector defined in Eq.(6) can be designed by choosing the switching function s(x)in 104 Y. Pan and K. Furuta Eq.(27) as the linear function s(x) in Eq.(6). In this case, the problem is how to determine the matrices P and R and also the control law to satisfy those conditions for a PR-sliding sector. As the switch function is designed so that the reduced order system in Eq.(26) is stable, there exists a positive definite symmetric matrix ˜ P 11 ∈ R (n−1)×(n−1) such that the following Lyapunov equation holds for some positive definite sym- metric matrix ˜ Q 11 ∈ R (n−1)×(n−1) . − ˜ Q 11 = ˜ A T 11 ˜ P 11 + ˜ P 11 ˜ A 11 (28) Choose positive definite symmetric matrices ˜ P and ˜ R as ˜ P =  ˜ P 11 0 (n−1)×1 0 1×(n−1) h 2  ∈ R n×n (29) ˜ R =  ˜ Q 11 − ˜ P 11 ¯ A 12 − ¯ A T 12 ˜ P 11 h  ∈ R n×n , (30) where h is a large enough positive constant such that the matrix ¯ R is positive definite. Then a PR-sliding sector can be designed as S = {x | s 2 (x) ≤ δ 2 (x, t),x∈ R n } (31) with the linear function s(x) being the switching function given in Eq.(27), the quadratic function δ 2 (x)as δ 2 (x)=x T Δx, (32) and the positive definite symmetric matrices P and R determined by P =(F −1 1 ) T F T 2 ˜ PF 2 F −1 1 (33) R =(F −1 1 ) T F T 2 ˜ RF 2 F −1 1 , (34) where ˜ P and ˜ R are given by Eqs.(29) and (30), respectively, F 1 and F 2 are transformation matrices defined in the last subsection, and Δ is chosen to be Δ = βP (35) with some positive constant β and γ satisfying βP ≥ γW. (36) For the Lyapunov function candidate L(x)=x T (t)Px(t)=˜x T (t) ˜ P ˜x(t), the following holds with above definitions. Second-Order Sliding Sector for Variable Structure Control 105 ˙ L(x)=˜x T (t)( ˜ A T eq ˜ P + ˜ P ˜ A eq )˜x(t)+2˜x T (t) ˜ P ˜ B(v(t)+d(x, t)) =˜x T (t)(  ˜ A T 11 O (n−1)×1 ˜ A T 12 0   ˜ P 11 0 (n−1)×1 0 1×(n−1) h 2  +  ˜ P 11 0 (n−1)×1 0 1×(n−1) h 2   ˜ A 11 ˜ A 12 O 1×(n−1) 0  )˜x(t) +2  ¯x 1 (t) s(x)  T  ˜ P 11 0 (n−1)×1 0 1×(n−1) h 2   O (n−1)×1 1  (v(t)+d(x, t)) =˜x T (t)  ˜ A T 11 ˜ P 11 + ˜ P 11 ˜ A 11 ˜ P 11 ˜ A 12 ˜ A T 12 ˜ P 11 0  ˜x(t)+hs(x)(v(t)+d(x, t)) =  ¯x 1 (t) s(x)  T  − ˜ Q 11 ˜ P 11 ˜ A 12 ˜ A T 12 ˜ P 11 −h   ¯x 1 (t) s(x)  + hs 2 (x)+hs(x)(v(t)+d(x, t)) = −˜x T (t) ˜ R˜x(t)+hs(x)(s(x)+v(t)+d(x, t)) = −x T (t)Rx(t)+hs(x)(s(x)+v(t)+d(x, t)) Therefore inside the PR-sliding sector(i.e., s 2 (x) ≤ δ 2 (x)), if the control law is given by u(t)=u eq + v(t)=u eq − kδ(x) ·sgn (s(x)) , (37) where δ(x)=  δ 2 (x) ,the equivalent control input u eq is given in Eq.(19) and k (k>1+1/ √ γ) is a large enough positive constant parameter, then we have ˙ L(x)=−x T (t)Rx(t)+hs(x)(s(x)+v(t)+d(x, t)) = −x T (t)Rx(t)+hs(x)(s(x) − kδ(x) · sgn (s(x)) + d(x, t)) ≤−x T (t)Rx(t)+|s(x)|(δ(x) −kδ(x)+|d(x, t)|) ≤−x T (t)Rx(t)+|s(x)|(− 1 √ γ δ(x)+ √ x T Wx) = −x T (t)Rx(t)+|s(x)|(− √ β √ γ √ x T Px+ √ x T Wx) ≤−x T (t)Rx(t). The following theorem concludes the above discussion. Theorem 1. The subset designed in Eq.(31) is a PR-sliding sector with corre- sponding parameters discussed above, inside which the P -norm decreases with the control law given by Eq.(37) as d dt L(x)= d dt ||x|| 2 P = d dt (x T (t)Px(t)) ≤−x T (t)Rx(t). 3.3 Second-Order Sliding Sector Based on the PR-sliding sector designed in the last subsection, an internal and an outer sectors for the second-order PR-sliding sector are designed as 106 Y. Pan and K. Furuta S o = {x| αδ 2 (x) <s 2 (x) ≤ δ 2 (x),x∈ R n }, (38) S i = {x| s 2 (x) ≤ αδ 2 (x),x∈ R n } (39) where α is a positive constant satisfying 0 <α<1andαδ 2 (x) is chosen to be the quadraticfunction ξ 2 (x) used in Definition 2 of thesecond-orderPR-slidingsector. Theorem 2. The PR-sliding sector designed in the last subsection with Δ = βP and the internal and outer sectors determined above is a second-order PR-sliding sector as S 2nd = {x| s 2 (x) ≤ δ 2 (x),x∈ R n } (40) with the VS control law of u(t)=u eq (t)+v(t)=u eq (t)+  −k δ 2 (x) s(x) ,x∈S o −k s(x) α ,x∈S i (41) where the equivalent control input u eq (t) is given in Eq.(19), k is a large enough positive constant, and parameters are chosen to satisfy the following relations: k>1+ 1 √ γ (42) β< 2 h (43) R>2h √ α √ γ βP (44) R<(hβα +(2−hβ)(k − 1 √ γ ))P. (45) Proof. First let’s consider the internal sector. In this case s 2 (x) ≤ αδ 2 (x)and the control law is given by u(t)=u eq (t)+v(t),v(t)=−k s(x) α . Thus the derivative of the Lyapunov function candidate L(x) defined by (4) is d dt L(x)== d dt ||x|| 2 P = −x T (t)Rx(t)+hs(x)(s(x)+v(t)+d(x, t)) = −x T (t)Rx(t)+hs(x)(s(x) − k s(x) α + d(x, t)) ≤−x T (t)Rx(t)+h|s(x)|(|s(x)|−k |s(x)| α )+h|s(x)||d(x, t)| ≤−x T (t)Rx(t)+h|s(x)|(|s(x)|−k |s(x)| α )+h √ α √ γ δ 2 (x)) < −x T (t)(R − h √ α √ γ βP)x(t) < − 1 2 x T (t)Rx(t), i.e., the P -norm decreases inside the internal sector. [...]... 2nd-order Sliding Sector Control Fig 7 Motor Angle with Sliding Mode Control Fig 8 Pendulum Angle with Sliding Mode Control Second-Order Sliding Sector for Variable Structure Control 117 Fig 9 Control Input with Sliding Mode Control chattering The control input is continuous Figure 6 shows that the system is in the outside of the second-order P R -sliding sector in the beginning, moves into the outer sector. .. proposed VS controller is shown in Figures 3 – 6 The inverted pendulum can be stabilized at its up position without any Second-Order Sliding Sector for Variable Structure Control Fig 3 Motor Angle with 2nd-order Sliding Sector Control Fig 4 Pendulum Angle with 2nd-order Sliding Sector Control Fig 5 Control Input with 2nd-order Sliding Sector Control 115 116 Y Pan and K Furuta Fig 6 Movement among Sectors... K., Hatakeyama, S.: Invariant Sliding Sector for Variable Structure Control Asian Journal of Control 7, 124–134 (2005) 15 Pan, Y., Furuta, K., Hatakeyama, S.: Invariant Sliding Sector for Variable Structure Control In: Proc 38th IEEE CDC, CDC 1999, Phoenix, US, pp 5152–5157 (1999) 16 Furuta, K., Pan, Y.: Discrete-time Variable Structure Control In: Yu, Xu (eds.) Variable Structure Systems: Towards the... inside the internal sector Therefore the designed P R -sliding sector is a second-order one and the P norm decreases inside the sector 108 Y Pan and K Furuta 4 VS Controller with Second-Order Sliding Sector With the second-order P R -sliding sector designed in Theorem 2, a VS controller should be designed to move the system from the outside to the inside of the second-order P R -sliding sector in a finite... R -sliding sector, the P -norm decreases and the system moves into the sector in a finite time Outside the second-order P R -sliding sector, the following inequality holds: s2 (x) > δ 2 (x) and the control input is given by u(t) = ueq (x) + v(t), v(t) = −ks(x) In this case, the derivative of the Lyapunov function candidate L(x) defined by (4) is Second-Order Sliding Sector for Variable Structure Control. .. Sliding Mode Control Taylor and Francis, London (1998) 118 Y Pan and K Furuta 4 Utkin, V.I.: Sliding modes in control and optimization Springer, Berlin (1992) 5 Furuta, K., Pan, Y.: Sliding Sectors for VS Controller Automatica 36, 211–228 (1999) 6 Furuta, K.: Sliding mode control of a discrete system System & Control Letters 14, 145–152 (1990) 7 Furuta, K., Pan, Y.: VSS Controller Design for Discrete-time... Control Theory and Advanced Technology 10, 669–687 (1994) 8 Levant, A.: Sliding order and sliding accuracy in sliding mode control Int J Contr 66, 1247–1263 (1993) 9 Levant, A.: Principles of 2 -sliding mode design Automatica 43, 576–586 (2007) 10 Fridman, L., Levant, A., Davila, J.: High-Order Sliding- Mode Observer for Linear Systems with Unknown Inputs In: Proc 2006 IEEE Int Workshop on Variable Structure. . .Second-Order Sliding Sector for Variable Structure Control 107 Now let’s consider the outer sector In this case, αδ 2 (x) < s2 (x) ≤ δ 2 (x) and the control law is given by u(t) = ueq (t) + v(t), v(t) = −k δ 2 (x) s(x) Define a quadratic function V (x) as V (x) = s2 (x) − δ 2 (x) (46) Then it follows inside the outer sector that d d d V (x) = (s2 (x) − δ 2 (x))... 21st Century Lecture Notes in Control and Information Science, vol 274, Springer, Berlin (2001) 17 Yu, X., Yu, S.: Discrete Sliding Mode control Design with Invariant Sliding Sectors ASME J Dyn Syst Measur Contr 122, 776–782 (2000) 18 Utkin, V.I., Young, K.K.D.: Methods for Constructing Discontinuity Planes in Multidimensional Variable Structure Systems Automation and Remote Control 39, 1466–1470 (1979)... decreasing Theorem 3 With the internal sector (39), the outer sector (38), and the second-order P R -sliding sector (40) designed in Theorem 2, the following continuous VS control law ⎧ ¯ ⎪ −ks(x) x∈S2nd ⎨ δ 2 (x) (48) u(t) = ueq (t) + v(t), v(t) = −k s(x) x ∈ So ⎪ ⎩ s(x) −k α x ∈ Si ensures that 1 the state moves from the outside to the inside of the second-order P R -sliding sector in a finite time and remains . The second-order sliding sector is a PR -sliding sector and an invariant subset of the state space. Therefore, it was named an invariant sliding sector in [14]. Inside the invariant sliding sector, . ≤−x T (t)Rx(t). 3.3 Second-Order Sliding Sector Based on the PR -sliding sector designed in the last subsection, an internal and an outer sectors for the second-order PR -sliding sector are designed. inside the internal sector. Second-Order Sliding Sector for Variable Structure Control 107 Now let’s consider the outer sector. In this case, αδ 2 (x) <s 2 (x) ≤ δ 2 (x)and the control law is given