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ISA Transactions 51 (2012) 65–73
Contents lists available at SciVerse ScienceDirect
ISA Transactions
journal homepage: www...
66 S. Balochian, A.K. Sedigh / ISA Transactions 51 (2012) 65–73
and engineering systems such as electrochemistry and diffu...
S. Balochian, A.K. Sedigh / ISA Transactions 51 (2012) 65–73 67
Fig. 1. Stability region of LTI-FOS with order 1 < q < 2.
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68 S. Balochian, A.K. Sedigh / ISA Transactions 51 (2012) 65–73
Fig. 2. Switching between two controllers.
where Aic = Ai ...
S. Balochian, A.K. Sedigh / ISA Transactions 51 (2012) 65–73 69
may not hold especially when the subsystem is unstable. It...
70 S. Balochian, A.K. Sedigh / ISA Transactions 51 (2012) 65–73
(a) Continuous state X(t). (b) Switching function σ(t) bet...
S. Balochian, A.K. Sedigh / ISA Transactions 51 (2012) 65–73 71
(a) Continuous state X(t). (b) Switching function σ(t) bet...
72 S. Balochian, A.K. Sedigh / ISA Transactions 51 (2012) 65–73
(a) Continuous state X(t). (b) Switching function σ(t) bet...
S. Balochian, A.K. Sedigh / ISA Transactions 51 (2012) 65–73 73
[19] Pommier V, Sabatier J, Lanusse P, Oustaloup A. Crone ...
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Sufficient condition for stabilization of linear time invariant fractional order switched systems and variable structure control stabilizers

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This paper presents the stabilization problem of a linear time invariant fractional order (LTI-FO) switched system with order 1<q><21><q><2><q><21><q><2 based on the convex analysis and linear matrix inequality (LMI) is presented and proved. Then a single Lyapunov function, whose derivative is negative, is constructed based on the extremum seeking method. A sliding sector is designed for each subsystem of the LTI-FO switched system so that each state in the state space is inside at least one sliding sector with its corresponding subsystem, where the Lyapunov function found by the extremum seeking control is decreasing. Finally, a switching control law is designed to switch the LTI-FO switched system among subsystems to ensure the decrease of the Lyapunov function in the state space. Simulation results are given to show the effectiveness of the proposed VS controller.

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Sufficient condition for stabilization of linear time invariant fractional order switched systems and variable structure control stabilizers

  1. 1. ISA Transactions 51 (2012) 65–73 Contents lists available at SciVerse ScienceDirect ISA Transactions journal homepage: www.elsevier.com/locate/isatrans Sufficient condition for stabilization of linear time invariant fractional order switched systems and variable structure control stabilizers Saeed Balochiana,∗ , Ali Khaki Sedighb a Department of Electrical Engineering, Science And Research Branch, Islamic Azad University, Tehran, Iran b Department of Electrical and Computer Engineering, K.N.Toosi University of Technology, Tehran, Iran a r t i c l e i n f o Article history: Received 17 January 2011 Received in revised form 16 July 2011 Accepted 29 July 2011 Available online 14 September 2011 Keywords: Fractional order switched system Linear matrix inequality (LMI) Variable structure control a b s t r a c t This paper presents the stabilization problem of a linear time invariant fractional order (LTI-FO) switched system with order 1 < q < 2 by a single Lyapunov function whose derivative is negative and bounded by a quadratic function within the activation regions of each subsystem. The switching law is extracted based on the variable structure control with a sliding sector. First, a sufficient condition for the stability of an LTI-FO switched system with order 1 < q < 2 based on the convex analysis and linear matrix inequality (LMI) is presented and proved. Then a single Lyapunov function, whose derivative is negative, is constructed based on the extremum seeking method. A sliding sector is designed for each subsystem of the LTI-FO switched system so that each state in the state space is inside at least one sliding sector with its corresponding subsystem, where the Lyapunov function found by the extremum seeking control is decreasing. Finally, a switching control law is designed to switch the LTI-FO switched system among subsystems to ensure the decrease of the Lyapunov function in the state space. Simulation results are given to show the effectiveness of the proposed VS controller. © 2011 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction Integer order switched systems have been the subject of inter- est for the past decades, for their wide application areas. Switched systems are a class of hybrid systems consisting of several sub- systems and a switching rule indicating the active subsystem at each instant of time. Many real-world processes and systems can be modeled as switched systems [1]. The switching signals in sys- tems or control can be classified into autonomous and controlled ones, which result from the system itself and the designers’ inter- vention, respectively [2,3]. Autonomous switched systems, arising from nonlinear systems, large-scale uncertain systems, parameter- varying systems, etc., have been investigated by probing the in- ternal switching features in such systems [4,5]. There are several constructive propositions in the literature to stabilize classes of lin- ear switched systems without input signals; see for instance [5]. Three basic problems in the stability analysis of switched systems are: (i) finding necessary and/or sufficient conditions for stabil- ity/stabilizability under arbitrary switching [5,6], (ii) identifying the limited but useful class of stabilizing switching laws [7,8], and (iii) constructing a stabilizing switching law [9]. There are many ∗ Corresponding author. E-mail addresses: saeed.balochian@gmail.com (S. Balochian), sedigh@kntu.ac.ir (A.K. Sedigh). existing works on these problems for the case where the switched systems are composed of continuous-time subsystems. The study of switching systems is closely related to some inves- tigations on differential inclusions. By using techniques from the convex non-smooth analysis, a series of very powerful results can be deduced. For an overview of the most important ideas related to differential inclusions and stability problems related to switching systems, refer to [10–12]. Also, the controller design with guaran- teed stability and performance in the integer order switching sys- tems is the main topic of many papers; see for example [13]. Also, one of the problems is the stabilization of a switched system us- ing a single Lyapunov function whose derivative is negative and bounded by a quadratic function within the activation regions of each subsystem, and may eventually be positive outside these re- gions. In the field of switched systems, this stability property is usually defined as the quadratic stability [1]. The method proposed in [8] points out that the existence of a stable convex combination of the linear subsystem Ai implies the possibility of quadratic stabi- lization of the origin through a suitable switching rule. The author of [14] has demonstrated that this condition is not only sufficient but also necessary for quadratic stability in the case of two subsys- tems. In order to avoid the occurrence of fast switching, a hybrid switching strategy has been proposed [1,5]. On the other hand, the theory of fractional calculus has a long and prominent history. In fact, one may trace it back to the very origins of differential calculus itself. The fractional order calculus plays an important role in many physical and chemical processes 0019-0578/$ – see front matter © 2011 ISA. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.isatra.2011.07.010
  2. 2. 66 S. Balochian, A.K. Sedigh / ISA Transactions 51 (2012) 65–73 and engineering systems such as electrochemistry and diffusion waves, electromagnetic waves, fractal electrical networks, elec- trical machines, nanotechnology, viscoelastic material and sys- tems, quantum evolution of complex systems, heat conduction, robotics and so on [15–17] and references therein. These sys- tems exhibit hereditary properties and long memory transients. Furthermore, fractional order controllers have recently gained an increasing attention such as optimal control [18], CRONE con- troller [19], TID controller [20], fractional PID controller [21], slid- ing mode controller and VSC controller. Consequently, the study of fractional-order switching system instead of integer order seems indispensable. A Variable Structure Control (VSC) system as a powerful control approach is proposed by Utkin. This method changes the structure of the system by switching at precisely defined states to another member of a set of possible continuous functions of the state and its strategy is based on the concept of an attractive manifold [22]. Numerous practical applications of VSC have been reported in the literature. VSC systems provide the framework to define the appropriate control laws and the switching structure. Lyapunov’s direct method is often used to design control surfaces which guide the system to a given goal. The distinguishing feature of VSC is sliding motion, which occurs when the system state repeatedly crosses certain subspaces, or sliding hyperplanes, in the state space. The sliding sector was proposed to replace the sliding mode for a chattering free VS control [23]. It has been shown for any system that there exists a subset in the state space, inside which some norm of the system decreases without any control input although sometimes the subset has only one element, i.e., the origin. The subset is named a sliding sector, which can be design using a Riccati equation [24,25]. An extremum seeking control algorithm has been used to find the predetermined norm of the state, i.e. a Lyapunov function as in [24]. Variable structure control that defined the sliding surface for linear time invariant fractional order systems (LTI-FOS) was introduced in [26] and further developed in [27]. Variable structure control systems for LTI-FOS with time delay using specification of sliding surface, without limitations on the control inputs is considered in [28]. Authors in [29] have considered the design of sliding control for LTI-FOS with time delay in the control input. In [30] the stabilization of a particular class of multi-input linear systems of fractional order differential inclusions with state delay using variable structure control is considered. Also, [31] has proposed a controller based on active sliding mode theory to synchronize chaotic fractional order systems in a master–slave structure. In [32] Sliding mode control approaches are developed to stabilize a class of linear uncertain fractional-order dynamics. All the above mentioned papers are focused on LTI-FOS with fractional order 0 < q < 1. In this paper, a sufficient condition for stability of LTI-FO switched systems with order 1 < q < 2 is proved, and a switch- ing law for continuous time LTI-FO switched systems is derived. For extracting stabilizer switching law, first, the existence of a sys- tem with integer order derivatives which has stability properties equivalent to the fractional system with 1 < q < 2 is presented. Then the extremum seeking control algorithm is used to find a Lya- punov function for the LTI-FO switched system. In general, a sliding sector is defined as a subset of the pseudo-state space inside which some norm of state decreases. Finally, a switching law based on the sliding sector is derived. The paper is organized as follows. In Section 2, a review of frac- tional calculus and two important type of fractional order switch- ing systems are presented. In Section 3, the problem formulation is given. In Section 4, we consider sufficient condition of stability of the LTI-FO switched system and the Lyapunov function for LTI- FO switched system based on an equivalent integer order system from the stability point of view. In Section 5, we present the condi- tions for stabilization of the LTI-FO switched system based on the extremum seeking method for computation of Lyapunov function and the sliding sector is defined. Also, the VS control law i.e. switch- ing law based on the definition of the sliding sector is discussed. Finally, the results are illustrated using three examples. 2. Preliminaries Given a family of linear fractional systems 0D q t X(t) = AiX(t) i = 1, 2, . . . , N. (1) The following two types of switching systems can be defined. Type I switching systems: Switching logic among the systems is unknown. At each time instant, it is only know that 0D q t X(t) ∈ {AiX(t) : i = 1, 2, . . . , N} i = 1, 2, . . . , N. (2) To analyze this linear differential inclusion (LDI) we assume that the switch is arbitrary. Type II switching systems: The switch is orchestrated by a con- troller/supervisor that can choose one of the systems on a time scheduling basis, measurement of the states or a certain output. It is assumed that the state X is available for measurement. For this case, the switching strategy can be optimized for the best perfor- mances. The system can be written as 0D q t X(t) = Aσ(x)X(t), 1 < q < 2 (3) where σ(x) = i for X ∈ Ωi and ∪ N i=1 Ωi = Rn . The design problem boils down to the construction of the sets Ωi with a well designed switching law. To differentiate the above two switching types, we simply call the system (2) the fractional order LDI and the system (3) the switched system. Given n−1 < q < n Riemann–Liouville definition of q-th order fractional derivative operator 0D q t is given by 0D q t f (t) = 1 (n − q)  d dt n ∫ t 0 f (τ) (t − τ)q−n+1 dτ where (.) is the Gamma function generalizing factorial for non- integer arguments (q) = ∫ ∞ 0 e−t tq−1 dt. Other definitions of fractional derivative are: Grünwald–Letnikov definition: 0D q t f (t) = lim h→0 1 hq [ t h ]− j=0 (−1)j  q j  f (t − jh) where [.] is a flooring operator. Caputo definition: 0D q t f (t) =    1 (m − q) ∫ t 0 f (m) (τ) (t − τ)q+1−m dτ, m − 1 < q < m dm dtm f (t), q = m. For more details refer to [33]. A fractional order linear time invariant (FO-LTI) system can be represented in the following state-space form:  0D q t X(t) = AX(t) + Bu(t) y(t) = CX(t) where X(t) ∈ Rn , u(t) ∈ Rr , y(t) ∈ Rp are states and input and output vectors of the system and A ∈ Rn×n , B ∈ Rn×r , C ∈ Rp×n ,
  3. 3. S. Balochian, A.K. Sedigh / ISA Transactions 51 (2012) 65–73 67 Fig. 1. Stability region of LTI-FOS with order 1 < q < 2. and q is the fractional commensurate order. It has been shown that the system Dq X(t) = AX(t) is asymptotically stable if the following condition is satisfied [34] |arg(eig(A))| > qπ 2 , where 0 < q < 2, and eig(A) are eigenvalues of the matrix A. The stable and unstable regions for 1 < q < 2 are shown in Fig. 1. 3. Problem description Consider a LTI-FO switched system described by the pseudo- state space equation as follows 0D q t X(t) = Aσ X(t) + BUσ (t), 1 < q < 2 (4) where X(t) ∈ Rn is the continuous state variable, σ denote a switching signal taking values as σ = 1, 2, . . . , N and a finite set of matrices  A := {Aσ : σ = 1, 2, . . . , N} is given, and Uσ (t) = Kσ X(t) is input signal, where Kσ ∈ R1×n is state feedback gain, and 0D q t X(t) is the fractional derivative of order q, 1 < q < 2, of X(t) relative to time. This paper presents sufficient condition for stabilization of LTI-FO switched systems and extracts stabilizer switching control based on the sliding sector. 4. Sufficient condition for the stability of an LTI-FO switched system The main objective of this section is to construct a continuous Lyapunov function whose derivative along any state trajectory is negative for each subsystem, within some regions in the state space. Moreover, these regions must cover the entire state space. A sufficient condition for stabilization in the case of two subsystems and a sufficient condition in the general case will be presented. The following theorem has been proved in LMI studies for LTI-FO system based on the mapping in Ω plane. Theorem 1 ([35]). The system Dq X(t) = AX(t), where 1 < q < 2, is stable if and only if there is a positive definite Hermitian matrix P = P∗ > 0 such that βPA + β∗ AT P < 0 (5) where β = η + jζ, and η, ζ are defined from tan(π/2 − θ) = η/ζ with θ = (q − 1)π/2. Theorem 2. Given the switched system (4) with N = 2, the point x = 0 is a stabilized switched equilibrium if there exists α ∈ (0, 1) such that Ac = αA1c + (1 − α)A2c (6)  arg λj (Ac )   > qπ 2 , (j = 1, 2, . . . , n) (7) where A1c = A1 + BK1 and A2c = A2 + BK2. Proof (Sufficiency). If the convex combination A is stable, there exist two positive definite symmetric matrices P, Q such that according to Theorem 1 βPAc + β∗ AT c P = −Q . (8) Using (6), we can rewrite (8) as αXT  βPA1c + β∗ AT 1c P  X + (1 − α)XT ×  βPA2c + β∗ AT 2c P  X = −XT QX. (9) Let λmin be the smallest (positive real) eigenvalue of Q . Given 0 < ε ≤ λmin, we have −XT QX ≤ −εXT X, so that (9) can be rewritten as αXT  βPA1c + β∗ AT 1c P  X + (1 − α)XT ×  βPA2c + β∗ AT 2c P  X ≤ −εXT X (10) or equivalently α  XT  βPA1c + β∗ AT 1c P  X + εXT X  + (1 − α) ×  XT  βPA2c + β∗ AT 2c P  X + εXT X  ≤ 0. (11) This means that for every nonzero X we have that either XT  βPA1c + β∗ AT 1c P  X + εXT X ≤ 0 or XT  βPA2c + β∗ AT 2c P  X + εXT X ≤ 0 or equivalently we have that either XT  βPA1c + β∗ AT 1c P  X ≤ −εXT X (12) or XT  βPA2c + β∗ AT 2c P  X ≤ −εXT X. (13) Now, define the two regions Ωi =  XT  βPAic + β∗ AT ic P  X ≤ −εXT X  , i ∈ {1, 2} . (14) There are two closed regions which overlap and cover Rn {0}. It is easy to show that any strategy where the system ∑ i is active in region Ωi assures stability, using Lyapunov function V(x) = XT PX (with P given by (8)). In fact, within the region Ωi ˙V(x) = ˙Vi(x) = XT  βPAic + β∗ AT ic P  X ≤ −εXT X. While at the switching points (which are interior to the region Ω1 ∩ Ω2) ˙V(x) = sup γ ∈[0,1]  γ ˙V1(x) + (1 − γ ) ˙V2(x)  ≤ max i=1,2  ˙Vi(x)  ≤ −εXT X. When there are more than two subsystems, it is possible to search for a pair of subsystems satisfying (6) and (7). Moreover, Theorem 2 can be generalized to the case of N subsystems as a sufficient condition only. Theorem 3. Given the switched system (4), if there exist αi ∈ (0, 1), i = 1, . . . , N such that N− i=1 αi = 1 (15) Ac = N− i=1 αiAic , (16)  arg λj (Ac )   > qπ 2 , (j = 1, 2, . . . , n)
  4. 4. 68 S. Balochian, A.K. Sedigh / ISA Transactions 51 (2012) 65–73 Fig. 2. Switching between two controllers. where Aic = Ai +BKi and arg λj (Ac ) is jth eigenvalue argument of the matrix Ac , then the point x = 0 is a stabilized switched equilibrium. Proof. The proof with minor modifications in the proof of suffi- ciency for Theorem 2 is straightforward. Remark 1. Modern controller design schemes may yield two (or more) controllers for a certain plant, together with a decision maker module that switches between the two controllers accord- ing to some criteria (see e.g., [1,2]). The resulting configuration, de- picted in Fig. 2, can be described mathematically using a switched system, and its stability analysis is clearly of great importance. According to result of the Theorems 2 and 3, this problem can be analyzed. In this case, the switched system is defined by A = A1 = A2, Ui = KiX(t), A1c = A + BK1, A2c = A + BK2. Also, the following lemma can be deducted from Theorem 1. Lemma 1. The system 0D q t X(t) = AX(t) + BU(t), where 1 < q < 2, and U(t) = K(t)X(t) is stable if and only if the following integer- order system is stable: ˙X(t) = (βAcl)X(t) (17) where Acl = A + BK, β = η + jζ, and η, ζ are defined from tan(π/2 − θ) = η/ζ with θ = (q − 1)π/2. Proof. Assume that system (17) is stable. Considering the Lyapunov function below, V(t) = ‖X‖2 p = XH (t)PX(t) > 0, ∀x ∈ Rn , x ̸= 0 (18) where P is a symmetric positive definite matrix, and according to system (17), we have ˙V(t) = XT (t)  βPAcl + β∗ AT clP  X(t). Since system (17) is stable, we will have ˙V(t) < 0. Therefore, Eq. (5) holds. The proof for the reverse case is obvious. Lemma 1 and Theorem 3 shows the relationship of the LMI inequal- ity in Eq. (5) with an integer-order linear system which ensures the stability of the LTI-FO switched system under consideration in Eq. (4). Therefore, system (17) is a shadow (equivalent) system of LTI-FO system from the stability point of view. Therefore, the sufficient condition for stability of LTI-FO switched system in Eq. (4) is that, according to Theorem 3 and Lemma 1, the following relationship holds after choosing the Lyapunov function: ˙V(t) = XT (t)  βPAc + β∗ AT c P  X(t) < −XT RX, ∀X ∈ Rn (19) where Ac is given in Eq. (16), P is a symmetric positive definite matrix, and R is a symmetric positive semi-definite matrix. Remark 2. It should be noted that even if subsystems of switched system Eq. (4) are individually stable, LTI-FO switched system can be unstable under some switching law. Therefore, in the following section, stabilizer switching law based on Theorems 2 and 3 and Lemma 1 is presented. 5. Stabilizer switching law 5.1. Lyapunov function found by extremum seeking algorithm In this section, we propose a method based on the extremum seeking algorithm for finding the Lyapunov function Eq. (18) and its derivative in Eq. (19). The extremum seeking algorithm has been used in [23,36,37] for determination of the sliding sectors in systems with integer derivatives. Here we need to make some fundamental modifications to this method, both in objective and implementation. Define a cost function J as J = Min 1≤j≤n  arg λj(Ac )   − qπ 2 (20) where Ac is the matrix defined in Eq. (16). By maximizing the cost function (20) subject to the convex combination in Eq. (16), if the value of the maximized objective function is positive, we can guarantee the existence of the stabilizing control for system (4) using a convex combination of the subsystems, and consequently, we can guarantee the presence of α∗ i and define its value. This is because the convex combination ensures that system states converge by the obtained switching between subsystems. It is assumed that there is a set of points α∗ 1 , . . . , α∗ N that maximizes the cost function J with constraints αi to the maximum value J∗ (α∗ 1 , α∗ 2 , . . . , α∗ N ), i.e. J(α1, α2, . . . , αN ) ≤ J∗ (α∗ 1 , α∗ 2 , . . . , α∗ N ) ∀αi ∈ (0, 1),  i = 1, 2, . . . , N; N− i=1 αi = 1  (21) J∗ (α∗ 1 , α∗ 2 , . . . , α∗ N ) = Min 1≤j≤n  arg λj  Aeq   − qπ 2 ≥ 0 (22) where Aeq is determined by Aeq = N− i=1 α∗ i Aic . Note. If the maximum value of the cost function (20) subject to constraints in Theorem 3 i.e. αi ∈ (0, 1), i = 1, . . . , N and ∑N i=1 αi = 1 turns out to be positive, it means that, we can place the eigenvalues of the LTI-FO switched system in the stable area by switching between subsystems. Now, consider a linear time-invariant continuous-time system described by the following state equation 0D q t X(t) = AeqX(t) (23) where X(t) ∈ Rn is the stable variable and Aeq is the system matrix determined by the extremum seeking method algorithm for eigenvalues argument. It is clear that the system in (23) is stable as all eigenvalues of Aeq are in the stable region of the complex plane. Thus there exists a positive definite symmetric matrix P and a positive semi-definite symmetric matrix R according to Lemma 1 such that ˙V(t) = XT (t)  βPAeq + β∗ AT eqP  X(t) < −XT RX, ∀X ∈ Rn (24) where P ∈ Rn×n , R ∈ Rn×n , and V(t) is a candidate Lyapunov func- tion defined as Eq. (18) which will be used to design a sliding sector in the next section. 5.2. Sliding sector for LTI-FO switched systems For each subsystem 0D q t x(t) = Aσc x(t) (25) where Aσc = Aσ + BKσ . The inequality ˙V(t) = XT (t)  βPAσc + β∗ AT σc P  X(t) < −XT RX, ∀X ∈ Rn
  5. 5. S. Balochian, A.K. Sedigh / ISA Transactions 51 (2012) 65–73 69 may not hold especially when the subsystem is unstable. It is possible to decompose the state space for each subsystem into two parts such that one part satisfies the condition ˙V(t) = XT (t)  βPAσc + β∗ AT σc P  X(t) > −XT RX for some element X ∈ Rn , and the other part satisfies the condition ˙V(t) = XT (t)  βPAσc + β∗ AT σc P  X(t) ≤ −XT RX for some other element X ∈ Rn . Definition. The sliding sector defined in the state space Rn for the system given by Eq. (4) is as Sσ =  X|XT (t)  βPAσc + β∗ AT σc P  X(t) ≤ −XT RX, X ∈ Rn  (26) where P and R are matrices used in the Lyapunov function in Eqs. (18) and (24). Inside the mentioned sliding sector the norm of the LTI-FO switched system decreases. The presence of such a sliding sector for systems with integer derivatives is proved in [23], and based on Lemma 1, this result is also valid for LTI-FO switched systems. 5.3. Switching law Based on the sliding sectors defined in the last section, a switching law is designed in the following theorem. Theorem 4. The variable structure control to stabilize the switched system given by Eq. (4) is u(t) = σ(t) (27) where σ(t) is a switching function of t taking values from a finite set∑ := {σ : σ = 1, 2, . . . , N} and specified by the VS control rule i.e. σ(t) = j, j ∈ {1, 2 . . . N}. If X(t) ∈ Sj and ˙V(t) = XT (t)  βPAjc + β∗ AT jc P  X(t) ≤ XT (t)  βPAic + β∗ AT ic P  X(t), ∀i = 1, 2, . . . , N. (28) In which Sj is the sliding sector defined in Eq. (26). Proof. Consider the Lyapunov function defined in (18), i.e. V(t) = ‖X‖2 p = X(t)PX(t), ∀X ∈ Rn , X ̸= 0. Its derivative for the system in (23) is XT (t)  βPAeq + β∗ AT eqP  X(t) ≤ −XT RX, ∀X ∈ Rn . As we have ˙V(t) = XT (t)  βPAeq + β∗ AT eqP  X(t) + XT RX = N− σ=1 λσ  XT (t)  βPAσc + β∗ AT σc P  X(t) + XT RX  ≤ 0, ∀X ∈ Rn there exists a σ = j such that ˙V(t) = XT (t)  βPAjc + β∗ AT jc P  X(t) ≤ −XT RX. Therefore, we choose the discrete state σ to be equal to j, then the derivative function of the Lyapunov function in (18) for the LTI-FO switched system in (4) satisfies the following inequality: ˙V(t) = XT (t)  βPAjc + β∗ AT jc P  X(t) ≤ −XT RX, ∀X ∈ Rn which means the proposed control law in (27) stabilizes the LTI-FO switched system in (4). Remark 3. According to the VS control proposed in Theorem 4, the current subsystem Aσc should be chosen as Ajc . If there is only Fig. 3. A trajectory X(t) of the switched system (sampling interval, h = 0.02 s). one sliding sector for the current state X(t), then the derivative of the Lyapunov function in Eq. (18) for the system (4) satisfies the following inequality: ˙V(t) = XT (t)  βPAjc + β∗ AT jc P  X(t) < −XT (t)RX(t), ∀X ∈ Rn . If there is more than one sliding sector for the current state, the condition in (28) given in the theorem ensure that the Lyapunov function (18) decreases with quickest speed over all possible switching laws. Remark 4. Theorem 4 gives a VS control law for a continuous-time system. With the VS controller, the system relation (4) is stable while there is no need for a sliding mode, sliding boundary layer, or sliding sector. The VS control law switches the control input among the subsystems such that a Lyapunov function continues to decrease inside a sliding sector associated with the control law. 6. Simulation results In this section, Riemann–Liouville definition of the fractional derivative is used. Example 1. Consider the system given by Eq. (4) with q = 1.05, and A1 = [ 0 1 −2 −1 ] , A2 = [ 0 1 −10 −1 ] , B = [ 0 1 ] , U1 = 0, U2 = 0 where eigenvalues of Ai are λ(A1) = {−0.5 ± j1.3229} ⇒ arg λi(A1) = {110.7048°, 249.2952°} λ(A2) = {−0.5 ± j3.1225} ⇒ arg λi(A1) = {99.0974°, −99.0974°} . Subsystems are stable. According to Remark 2, Fig. 3 depicts the trajectory of this system corresponding to the state-dependent switching law σ(t) =  1 x1(t)x2(t) > 0 2 otherwise. With the initial condition  0D−0.95 t x1(t)|t=0 0D0.05 t x1(t)|t=0 0D−0.95 t x2(t)|t=0 0D0.05 t x2(t)|t=0 T = [1 2 1 −2]T . Therefore, the system is unstable and X(t) does not converge to the origin.
  6. 6. 70 S. Balochian, A.K. Sedigh / ISA Transactions 51 (2012) 65–73 (a) Continuous state X(t). (b) Switching function σ(t) between subsystems. Fig. 4. VSC for LTI-FO switched system (sampling interval, h = 0.02 s). Proposed VS control in this paper can be used for stabilization of the above LTI-FO switched system. Therefore, with the extremum seeking method, the positive coefficients of the convex combina- tion are found as α∗ 1 = 1, α∗ 2 = 0 which gives Aeq = N− i=1 α∗ i Ai = α∗ 1 A1 + α∗ 2 A2 = A1 = [ 0 1 −2 −1 ] . Choose the positive definite matrix as the identity matrix, i.e. R = I2 = [ 1 0 0 1 ] . This gives P = [ 28.5146 3.5458 3.5458 11.5036 ] . The simulation result with the proposed VS controller, the above parameters and the initial condition [0D−0.95 t x1(t)|t=0 0D0.05 t x1(t)|t=0 0D−0.95 t x2(t)|t=0 0D0.05 t x2(t)|t=0]T = [1 2 1 − 2]T are as shown in Fig. 4. The notation for initial conditions is used as [38]. Example 2. In this example the results of the paper are used to eliminate chaotic oscillations in the following fractional order Chen system [39,40].    D1.1 t x1(t) = 40(x2 − x1), D1.1 t x2(t) = −12x1 − x1x3 + 28x2 D1.1 t x3(t) = x1x2 − 3x3 (29) Kheirizad et al. [41] shows that chaotic oscillations are generated by the above fractional order system. This system has three unsta- ble fixed points X1eq = (0, 0, 0) , X2eq = (6.9282, 6.9282, 16) , X3eq = (−6.9282, −6.9282, 16) [40]. Switched system are now used for eliminating chaotic oscillations. Linearization of the fractional order Chen system (29) in its equilibrium points, gives the Jacobian matrix of the system evaluated at equilibrium point Xeq = (x1eq, x2eq, x3eq) as: J =  −40 40 0 −12 − x3eq 28 −x1eq x2eq x1eq −3  . Now,consider the system (4) with q = 1.1, and A1 =  −40 40 0 −12 28 0 0 0 −3  , A2 =  −40 40 0 −28 28 −6.9282 6.9282 6.9282 −3  , A3 =  −40 40 0 −28 28 6.9282 −6.9282 −6.9282 −3  ; B = 0, U1 = 0, U2 = 0, U3 = 0 where A1, A2, A3, B, U1, U2, U3 are extracted from linearization of the fractional order Chen system Eq. (29) in the equilibrium points. It is desired to stabilize the above system by switching between the subsystems given by A1, A2, A3. Note that, eigenvalues of Ai are λ(A1) = {−32, 20, −3} ⇒ arg(λ) = {180°, 0°, 180°} λ(A2) = {2.6152 ± j13.5268, −20.2304} ⇒ arg(λ) = {79.0577°, −79.0577°, 180°} λ(A3) = {2.6152 ± j13.5268, −20.2304} ⇒ arg(λ) = {79.0577°, −79.0577°, 180°} with the extremum seeking method, the positive coefficients of the convex combination are found as α∗ 1 = 0.0035, α∗ 2 = 0.5402, α∗ 3 = 0.4562, which gives Aeq = N− i=1 α∗ i Ai = α∗ 1 A1 + α∗ 2 A2 + α∗ 3 A3 =  −40 40 0 −27.9434 28 −0.5821 0.5821 0.5821 −3  and the respective eigenvalues are λ(Aeq) = {−12.3803, −1.036, −1.5837} ⇒ arg(λ) = {180°, 180°, 180°} . Choose the positive definite matrix as the identity matrix, i.e. R = I3 =  1 0 0 0 1 0 0 0 1  . This gives P =  33.2776 −39.178 4.9127 −39.1780 48.5821 −6.072 4.9127 −6.072 2.1658  .
  7. 7. S. Balochian, A.K. Sedigh / ISA Transactions 51 (2012) 65–73 71 (a) Continuous state X(t). (b) Switching function σ(t) between subsystems. Fig. 5. VSC for LTI-FO switched system (sampling interval, h = 0.01 s). (a) State X(t). (b) Switching function σ(t) between subsystems. Fig. 6. VSC for LTI-FO switched system (sampling interval, h = 0.02 s). The simulation result with the proposed VS controller, the above parameters and the following initial condition are shown in Fig. 5.  0D−0.9 t x1(t)|t=0 0D0.1 t x1(t)|t=0 0D−0.9 t x2(t)|t=0 0D0.1 t x2(t)|t=0 0D−0.9 t x3(t)|t=0 0D0.1 t x3(t)|t=0 T = [−10 0 − 4 1 10 − 2]T . For the proposed VS control law, there is no requirement for any sliding mode, the switching frequency can be very slow whilst the VS control system is still stable. Fig. 6 shows the simulation result with the sampling interval being h = 0.02 s. Example 3. Consider the system given by the Eq. (4) with q = 1.5, A1 = [ 0 1 1 1 ] , A2 = [ 0 2 2 2 ] , B = [ 0 1 ] , where eigenvalues of A are λ(A1) = {−0.618, 1.618} ⇒ arg λi(A) = {180, 0} λ(A2) = {−1.2361, 3.2361} ⇒ arg λi(A) = {180, 0} . System is unstable. Two state feedback controllers U1 = [0 −4] X(t), U2 = [−4 0]X(t) are available that none of them can indi- vidually stabilize the system λ(A1c ) = λ(A1 + BU1) = {−3.3028, 0.3028} λ(A2c ) = λ(A2 + BU2) = {1 ± j1.7321} . With the extremum seeking method, the positive coefficients of the convex combination are found as α∗ 1 = 0.6154, α∗ 2 = 0.3846, which gives Aeq = N− i=1 α∗ i Aic = α∗ 1 A1c + α∗ 2 A2c = [ 0 1.3846 −0.1539 −1.0768 ] and the respective eigenvalues are λ(Aeq) = {−0.2614, −0.8154} ⇒ arg(λ) = {180°, 180°} . Choose the positive definite matrix as the identity matrix, i.e. R = I2 = [ 1 0 0 1 ] . This gives P = [ 6.5933 8.1911 8.1911 16.5047 ] . The simulation result with the proposed VS controller, the above parameters and the initial condition [0D−0.5 t x1(t)|t=0 0D0.5 t x1 (t)|t=0 0D−0.5 t x2(t)|t=0 0D0.5 t x2(t)|t=0]T = [1 10 −5 1]T are as shown in Fig. 7. For the proposed VS control law, there is no requirement for any sliding mode, the switching frequency can be very slow whilst the
  8. 8. 72 S. Balochian, A.K. Sedigh / ISA Transactions 51 (2012) 65–73 (a) Continuous state X(t). (b) Switching function σ(t) between subsystems. Fig. 7. VSC for LTI-FO switched system (sampling interval, h = 0.05 s). (a) State X(t). (b) Switching function σ(t) between subsystems. Fig. 8. VSC for LTI-FO switched system (sampling interval, h = 0.2 s). VS control system is still stable. Fig. 8 shows the simulation result with the sampling interval being h = 0.2 s. 7. Conclusion In this paper, sufficient condition for stability of the LTI-FO switched systems is proved. Then based on the proved sufficient condition, a Lyapunov function is found using the extremum seeking method. Also the definition of sliding sector in LTI-FO system is presented. Finally, a stabilizer switching law based on sliding sector is designed for the switched system. Simulation results are used to show the main points of the paper. As is shown in the simulation results, the level of chattering can be high because the system is switched frequently to converge at the fastest speed. To decrease the switching frequency, the controller may be modified such that the control system law is switched only if the system diverges with the current control. References [1] Liberzon D. Switching in systems and control. Boston (MA): Birkhäuser; 2003. [2] Morse AS. Control using logic-based switching. Heidelberg (Germany): Springer; 1997. [3] Persis D, Santis RD, Morse AS. Switched nonlinear systems with state- dependent dwell time. Systems Control Lett 2003;50(4):291–302. [4] Sanfelice R, Goebel R, Teel A. Invariance principles for hybrid systems with connections to detectability and asymptotic stability. IEEE Trans Autom Control 2007;52(12):2282–97. [5] Liberzon D, Morse AS. Basic problems in stability and design of switched systems. IEEE Control Syst Mag 1999;19:59–70. [6] Zhai G. Stability and L2 gain analysis of switched symmetric systems. In: Liu D, Antsaklis PJ, editors. Stability and control of dynamical systems with applications. Boston: Birkhäuser; 2003. [7] Zhai G, Yasuda K. Stability analysis for a class of switched systems. Trans Soc Instrum Control Eng 2000;36:409–15. [8] Wicks MA, Peleties P, DeCarlo RA. Construction of piecewise Lyapunov functions for stabilizing switched systems. In: Proceedings of the 33rd IEEE conference on decision and control. 1994. p. 3492–7. [9] Wicks MA, Peleties P, DeCarlo RA. Switched controller synthesis for the quadratic stabilization of a pair of unstable linear systems. Eur J Control 1998; 4(2):140–7. [10] Blanchini F. Set invariance in control. Automatica 1999;35:1747–67. [11] Smirnov GV. Introduction to the theory of differential inclusions. Graduate studies in mathematics, vol. 41. Providence (RI): American Mathematical Society; 2002. [12] Hu Tingshu, Ma Liqiang, Lin Zongli. Stabilization of switched systems via composite quadratic functions. IEEE Trans Automat Control 2008;53(11): 2571–85. [13] Bemporad A, Morari M. Control of systems integrating logic, dynamics, and constraints. Automatica 1999;35(3):407–27. [14] Feron E. Quadratic stabilizability of switched systems via state and output feedback. Technical report CICS-P-468. MIT. 1996. [15] Baleanu D, Güvenç ZB, Machado JAT. New trends in nanotechnology and fractional calculus applications. Springer; 2010. [16] Sabatier J, Agrawal OP, Machado JAT. Advances in fractional calculus, theoretical developments and applications in physics and engineering. London: Springer; 2007. [17] Ortigueira MD. An introduction to the fractional continuous-time linear systems: the 21st century systems. IEEE Circuits Syst Mag 2008;8(3):19–26. [18] Agrawal OP. A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dynam 2004;38:323–37.
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