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- 1. ISA Transactions 51 (2012) 105–110 Contents lists available at SciVerse ScienceDirect ISA Transactions journal homepage: www.elsevier.com/locate/isatrans Simulation and stability analysis of neural network based control scheme for switched linear systems H.P. Singh∗ , N. Sukavanam Department of Mathematics, Indian Institute of Technology Roorkee (IITR), Roorkee-247667, Uttarakhand, India a r t i c l e i n f o Article history: Received 7 April 2011 Received in revised form 18 July 2011 Accepted 18 August 2011 Available online 18 September 2011 Keywords: Switched systems Stability analysis Feedforward neural network Lyapunov function Uniformly ultimate boundedness Uncertainty a b s t r a c t This paper proposes a new adaptive neural network based control scheme for switched linear systems with parametric uncertainty and external disturbance. A key feature of this scheme is that the prior information of the possible upper bound of the uncertainty is not required. A feedforward neural network is employed to learn this upper bound. The adaptive learning algorithm is derived from Lyapunov stability analysis so that the system response under arbitrary switching laws is guaranteed uniformly ultimately bounded. A comparative simulation study with robust controller given in [Zhang L, Lu Y, Chen Y, Mastorakis NE. Robust uniformly ultimate boundedness control for uncertain switched linear systems. Computers and Mathematics with Applications 2008; 56: 1709–14] is presented. © 2011 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction A switched system is an important class of hybrid dynamical system which consists of a collection of subsystems and a switching law that decides which of the subsystem is active at each moment. There are many applications of switched systems in control of mechanical systems, process control, power systems, traffic management, and many other practical control fields. The motivation for studying switched systems is from the fact that many practical systems are inherently multi-models. In last two decades, there have been numerous control design methods and stability analysis available in the literature for switched systems [1–7]. A main problem which is always inherent in all dynamical systems is the presence of uncertainties and external disturbances. Even for switched systems this problem has been an active area of research for many years. In [8], a robust H∞ control scheme is proposed for switched linear systems with norm- bounded time-varying uncertainties by using a multiple Lyapunov function method. In [9], an adaptive robust stabilizing controller is designed for a class of uncertain switched linear systems, in which uncertainties satisfying the matching condition and switching among subsystems are determined by using a nominal system. ∗ Corresponding author. Tel.: +91 8859866804; fax: +91 1332 273560. E-mail addresses: harendramaths@gmail.com, hpsmaths@gmail.com (H.P. Singh), nsukvfma@iitr.ernet.in (N. Sukavanam). A new class of uncertain switched fuzzy systems is proposed in [10]. In [11], an algorithm for robust adaptive control for parametric-strict output feedback switched systems is developed. This control scheme guarantees system stability for bounded disturbances and parameters without requiring a priori knowledge on such parameters or disturbances. In [12], L2-gain analysis and control synthesis problem is designed for a class of uncertain switched linear systems subject to actuator saturation and external disturbances. The problem of optimal switching for a class of switched systems with parameter uncertainty is proposed in [13]. The conditions for robust stabilization of this class of switched systems with parameter uncertainty are presented based on a multi-Lyapunov function technique and a Linear matrix inequality technique. In [14], a continuous state feedback control scheme for uncertain switched linear systems is proposed in which external disturbance is not considered and the exact knowledge of the possible upper bound of uncertainty is known a priori. However in practical applications the calculation of this bound is difficult and time consuming. Therefor further improvement is needed for this type of control scheme. Artificial neural network (ANN) can be defined as nonlinear systems consisting of a number of interconnected processing units or neurons. Due to their versatile features such as learning capability, nonlinear mapping and parallel processing, ANN is successfully applicable in real life problems arising in the areas such as dynamics of economy, system control, robotics, etc. The limitations of ANN come from its architecture and the accuracy of the results vary depending on network quality. A European study 0019-0578/$ – see front matter © 2011 ISA. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.isatra.2011.08.004
- 2. 106 H.P. Singh, N. Sukavanam / ISA Transactions 51 (2012) 105–110 on the Stimulation Initiative for European Neural Applications (SIENA project) indicates ANN is used in 39% of the production or manufacturing sectors, with 35% of the usage related to control, monitoring, modeling and optimization [15]. In [16], the possibility to apply neural network modeling for simulation and prediction of the EU-27 global index of production and domestic output price index behavior is investigated. The most useful property of neural network in control is their ability to approximate arbitrary linear or nonlinear functions through learning. Due to this property neural networks have proven to be a suitable tool for controlling complex nonlinear dynamical systems. The basic idea behind neural network based control is to learn unknown nonlinear dynamics and compensate for uncertainties existing in the dynamic model. There are many NN based control schemes available for a class of switched systems [17–19]. But the case of uncertainty bound estimation using neural network is not found in the literature for switched systems. In this paper, we propose a new adaptive neural network based control scheme for switched linear systems with parametric uncertainties and external disturbances. The proposed control scheme has the following salient features: (i) prior information of the upper bound of the uncertainty is not required, (ii) the feedforward neural network is able to learn the upper bound of uncertainty, and (iii) uniformly ultimately bounded (UUB) stability of system re- sponse and NN weight error are guaranteed under arbitrary switching laws. This paper is organized as follows. Some basics of stability analysis are given in Section 2. In Section 3, a review of feedforward neural network and a new robust adaptive controller is proposed. A common Lyapunov function approach is used to show that system response and NN weight error are all UUB. A simulation example is provided in Section 4 to illustrate the effectiveness of the proposed control scheme. Section 5 gives conclusions. 2. Preliminaries Consider a switched linear system represented by a differential equation of the form ˙x(t) = Aσ(t)(ω)x(t) + Bσ(t)(ω)u(t), (1) σ(t) : R+ −→ S = {1, . . . , N} where t ≥ 0, x = (x1, x2, . . . , xn) ∈ Rn denotes the state vector of the system. u(t) ∈ Rm is the control input and R+ denotes the set of nonnegative real numbers. σ(t) is a piecewise constant function called a switching law, which indicates the active subsystem at each instant. Ai(ω), Bi(ω), i = 1, . . . , N are matrices whose elements are continuous functions of a time-varying vector ω on a compact set Ω ⊂ Rq . When switched system (1) has parametric uncertainty and external disturbance then it can be written in the following form ˙x(t) = ( ¯Ai + Ai(ω))x(t) + ( ¯Bi + Bi(ω))u(t) + d(t), (2) where d(t) is the bounded external disturbance, ¯Ai, i = 1, . . . , N are commuting Hurwitz matrices and Ai(ω), Bi(ω) are uncertainty terms which satisfy the following conditions [20,21] Ai(ω) = ¯BDi(ω) (3) Bi(ω) = ¯BEi(ω) (4) I + 1 2 (Ei(ω) + ET i (ω)) ≥ δI (5) where δ is a positive constant, Di : Ω −→ Rm×n and Ei : Ω −→ Rm×m are continuous matrix functions. Now consider the nominal case with u(t) = 0 and d(t) = 0 ˙x(t) = Aσ(t)(ω)x(t). (6) Then stability conditions for system (6) are given by the following theorems [22]. Theorem 2.1. If {Ai : i = 1, . . . , N} is a set of commuting Hurwitz matrices, then the switched linear system (6) is globally uniform asymptotic stable for any arbitrary switching sequence between Ai. Theorem 2.2. For a given symmetric positive definite matrix Q , let P1, P2, . . . , PN > 0 be the unique symmetric positive definite solutions of the following equations AT 1 P1 + P1A1 = −Q , (7) AT i Pi + PiAi = −Pi−1, (8) with the condition of Theorem 2.1, then the function L = xT PN x is a common Lyapunov function for the switched linear system (6). Because of the parametric uncertainty and external disturbance in switched linear system (2), we cannot derive the state response x(t) which exactly converges to the equilibrium point. Therefore it is only reasonable to expect that the system response converges into a neighborhood of the equilibrium point and remains within it thereafter, which is the so-called uniformly ultimate boundedness. In [23], the following uniformly ultimate boundedness is proposed for dynamical systems. Definition 2.1. The solution of a dynamical system is uniformly ultimately bounded (UUB) if there exists a compact set S ⊂ Rn such that for all x(t0) = x0 ∈ S, there exists a λ > 0 and a number T(λ, x0) such that ‖x(t)‖ < λ for all t ≥ t0 + T. Theorem 2.3. If there exists a function L with continuous partial derivatives such that for x in a compact set S ⊂ Rn , L is positive definite and ˙L < 0 for ‖x‖ > R, R > 0 such that the ball of radius R is contained in S, then the system is UUB and the norm of the state is bounded to within some neighborhood of R. 3. Neural network based control design In this section, our aim is to design a control input u(t) in such a way that the switched linear system (2) is uniformly ultimately bounded (UUB). The following robust control scheme u(t) = − w ‖w‖ ρ, if ‖ω‖ > ϵ − w ϵ ρ, if ‖ω‖ ≤ ϵ (9) is proposed in [14], where w = ¯BT PN ρx, ϵ > 0 is any constant and ρ denotes the possible upper bound of the parametric uncertainty defined as ρ = 1 δ max i max ωϵΩ ‖Di(ω)‖ ‖x‖ (10) which is assumed to be known. However, there are some potential difficulties associated with the robust controller (9). (i) Implementation requires a precise knowledge of the uncertainty bound, (ii) if an uncertain switched system has many subsystems, the computation of the uncertainty bound will be a complex and time consuming task, and (iii) if a new switched system comes in, the whole procedure has to start again. Therefore, we introduce a new adaptive control approach to avoid the requirement of a prior knowledge of the upper bound ρ in the expression (10) by using a feedforward neural network (FFNN). Mathematically, a two-layer feedforward neural network (Fig. 1) with n input units, m output units and L units in the hidden layer is given as
- 3. H.P. Singh, N. Sukavanam / ISA Transactions 51 (2012) 105–110 107 zl = L− j=1 wljσ n− k=1 vjkxk + θvj + θwl l = 1, 2, . . . , m (11) where σ(.) are the activation functions of the neurons of the hidden-layer [23]. The inputs-to-hidden-layer interconnection weights are denoted by vjk and the hidden-layer-to-outputs interconnection weights by wlj. The bias weights are denoted by θvj, θwl. There are many classes of activation functions e.g. sigmoid, hyperbolic tangent and Gaussian. The usual choice is the sigmoid activation function, defined as σ(x) = 1 1 + e−x . (12) By collecting all the NN weights vjk and wlj into matrices of weights VT and WT , we can write the NN equation in terms of vectors as z = WT σ(VT x) (13) with the vector of activation functions defined by σ(y) = [σ(y1) · · · σ(yn)]T for a vector y ∈ Rn . Without losing generality first-layer weights are fixed and only the second-layer weights are tuned, then the NN has only one layer of tunable weights. Then according to [23–26], ρ can be written in neural network form as ρ = WT φ(x) + ρ and an estimate of ρ can be given by ˆρ = ˆWT φ(x) where ρ denotes NN approximation error, φ(.) is a basis matrix and ˆW is the estimation of the weight matrix W provided by some on-line weight tuning algorithms. For further analysis, the following assumptions are selected as in [24,26,27]. Assumption 3.1. Given an arbitrary small positive constant ξ and a continuous function ρ on a compact set Ω, there exist an optimal vector W∗ such that ‖ ρ‖ = ‖W∗T φ(x) − ρ‖ < ξ. Assumption 3.2. The norm of parametric uncertainty µ defined as µ = Di(ω)x and its upper bound satisfy the following relationship ρ − ‖µ‖ > ξ. For the design of a neural network based control input u(t) and the weight vector ˆW, we have the following theorem. Theorem 3.1. Consider the switched linear systems (2). If u(t) in (2) is designed as − ˆρ2 w ˆρ‖w‖ + ϵ , (14) Singh et al. [24], where ˙ϵ = −γ ϵ, ϵ(0) > 0, γ is a positive constant, w = ¯BT PN x and the adaptive neural network weight law is given as ˙ˆW = F‖w‖(φ(x) − κ ˆW), with a positive definite diagonal gain matrix F, and a positive constant κ, then the system response x and ˜W are uniformly ultimately bounded (UUB), and x can be made small by a suitable choice of F and κ. Proof. Consider the following Lyapunov function candidate L = 1 2 xT PN x + 1 2 tr( ˜WT F−1 ˜W) + k−1 ϵ (15) where ˜W = W∗ − ˆW, ˙˜W = − ˙ˆW. Then, the time derivative of the Lyapunov function is given as ˙L = 1 2 ˙xT PN x + 1 2 xT PN ˙x + tr( ˜WT F−1 ˙˜W) + k−1 ˙ϵ ˙L = 1 2 [xT ( ¯Ai T + AT i ) + uT (¯BT + BT i )]PN x + 1 2 xT PN [( ¯Ai + Ai)x + (¯B + Bi)u] + 1 2 dT PN x + 1 2 xT PN d + tr( ˜WT F−1 ˙˜W) + k−1 ˙ϵ. Now using condition (3) and (4), we get ˙L = 1 2 [xT ( ¯Ai T PN + PN ¯Ai) + (¯BDi)T PN + PN (¯BDi)]x + 1 2 [uT (¯BT + Ei T ¯BT )PN x + xT PN (¯B + ¯BEi)u] + 1 2 dT PN x + 1 2 xT PN d + tr( ˜WT F−1 ˙˜W) + k−1 ˙ϵ ˙L = 1 2 xT ( ¯Ai T PN + PN ¯Ai)x + 1 2 xT [(¯BDi)T PN + PN (¯BDi)]x − ˆρ2 wT 2(ˆρ‖w‖ + ϵ) (I + Ei)T ¯BT PN x − ˆρ2 2(ˆρ‖w‖ + ϵ) xT PN ¯B(I + Ei)w + 1 2 dT PN x + 1 2 xT PN d + tr( ˜WT F−1 ˙˜W) + k−1 ˙ϵ using condition (5) with δ = 1 and condition (8) with ¯Ai T PN + PN ¯Ai = −Ri, where Ri is a positive definite symmetric matrix, then we have ˙L ≤ − 1 2 λmin(Ri)‖x‖2 − ˆρ2 ‖w‖2 ˆρ‖w‖ + ϵ + ‖w‖(‖µ‖ − ρ) + ‖w‖(W∗T φ(x) + ρ) + tr( ˜WT F−1 ˙˜W) + η‖PN ‖ ‖x‖ − ϵ where η is the disturbance bound. ˙L ≤ − 1 2 λmin(Ri)‖x‖2 − ˆρ2 ‖w‖2 ˆρ‖w‖ + ϵ + ‖w‖(−ξ + ρ) + ‖w‖W∗T φ(x) + tr( ˜WT F−1 ˙˜W) + η‖PN ‖ ‖x‖ − ϵ ˙L ≤ − 1 2 λmin(Ri)‖x‖2 − ˆρ2 ‖w‖2 ˆρ‖w‖ + ϵ + ˆρ‖w‖ + tr(‖w‖W∗T φ(x) − ‖w‖W∗T φ(x)) + κ‖w‖tr ˜WT (W∗ − ˜W) + η‖PN ‖ ‖x‖ − ϵ ˙L ≤ − 1 2 λmin(Ri)‖x‖2 + κ‖w‖tr ˜WT (W∗ − ˜W) + η‖PN ‖ ‖x‖. Let the optimal weight W∗ is bounded by ‖W∗ ‖F < Wmax on a compact set Ω, where Wmax > 0. Then the following identity is used in the derivation of the above inequality tr ˜WT (W∗ − ˜W) = ⟨ ˜W, W∗ ⟩F − ‖ ˜W‖2 F ≤ ‖ ˜W‖F ‖W∗ ‖F − ‖ ˜W‖2 F . Then ˙L ≤ − 1 2 λmin(Ri)‖x‖2 + κ‖¯BT PN x‖ ‖ ˜W‖F (‖W∗ ‖F − ‖ ˜W‖F ) + η‖PN ‖ ‖x‖ ˙L ≤ −‖x‖ [ 1 2 λmin(Ri)‖x‖ + κ‖¯B‖ ‖PN ‖ ‖ ˜W‖F (‖ ˜W‖F − ‖W∗ ‖F ) − η‖PN ‖ ] .
- 4. 108 H.P. Singh, N. Sukavanam / ISA Transactions 51 (2012) 105–110 Fig. 1. Feedforward neural network. Thus ˙L is negative as long as the term in brackets is positive. Completing square terms, it is obtained that 1 2 λmin(Ri)‖x‖ + κ‖¯B‖ ‖PN ‖ ‖ ˜W‖F (‖ ˜W‖F − ‖W∗ ‖F ) − η‖PN ‖ = 1 2 λmin(Ri)‖x‖ + κ‖¯B‖ ‖PN ‖ ‖ ˜W‖F − Wmax 2 2 − κ 4 ‖¯B‖ ‖PN ‖W2 max − η‖PN ‖. Then RHS is positive if ‖x‖ > κ‖¯B‖ ‖PN ‖W2 max + 4η‖PN ‖ 2λmin(Ri) (16) or ‖ ˜W‖F > Wmax 2 + η ‖¯B‖κ + W2 max 4 . (17) Thus ˙L is negative outside a compact set. According to the Theorem 2.3, x and ˜W are UUB. Moreover, x can be made small by a suitable choice of F and κ. 4. Simulation results In this section, we consider a numerical example to demon- strate the utility of the proposed control scheme for the switched linear system (2) with the following two subsystems similar to [14] A1(ω) = [ 0 1 −6 + ω2(t) −3 + ω1(t) ] , B1(ω) = [ 0 1.4387 + ω3(t) ] , A2(ω) = [ 0 1 −6.225 + ω2(t) −3 + ω1(t) ] , B2(ω) = [ 0 0.5613 + ω3(t) ] , where ω1(t) = 0.5 cos(t), ω2(t) = sin(t), ω3(t) = 0.25 sin(t). ¯Ai = [ 0 1 −6 −3 ] , ¯B = [ 0 1 ] , d(t) = [ 0.1 sin(2t) 0.1 cos(2t) ] , D1(w) = [w2(t) w1(t)], D2(w) = [−0.225 + w2(t) w1(t)], Fig. 2. Response of state variable x1 with control design (9). Fig. 3. Response of state variable x2 with control design (9). E1(w) = 0.4387 + w3(t), E2(w) = −0.4387 + w3(t), A1(ω) = [ 0 0 ω2(t) ω1(t) ] , B1(ω) = [ 0 0.4387 + ω3(t) ] , A2(ω) = [ 0 0 −0.225 + ω2(t) ω1(t) ] , B2(ω) = [ 0 −0.4387 + ω3(t) ] which satisfies the condition (3) and (4). We choose a positive definite matrix Q as Q = [ 36 3 3 24 ] (18) then the common Lyapunov solution matrix PN = [ 17 3 3 5 ]
- 5. H.P. Singh, N. Sukavanam / ISA Transactions 51 (2012) 105–110 109 Fig. 4. Response of state variable x1 with control design (14) for F = I10 and κ = 1. Fig. 5. Response of state variable x2 with control design (14) for F = I10 and κ = 1. can be easily obtained using the Lyapunov equations (7) and (8). ρ = 1.15‖x‖ is selected as in [14] for comparison between the robust controller (9) and our proposed controller (14). The architecture of the FFNN is composed of 2 input units, 10 sigmoid units and a single output unit. The controller parameters F, κ and γ in the proposed scheme are selected as I10, 0.01 and 0.01, respectively. For simulation, we take a particular switched law σ(t) defined as σ(t) = 1 for t < 30 and σ(t) = 2 for 30 ≤ t ≤ 70. The simulation of the whole system is shown for 70 s. The effectiveness of the FFNN to learn the upper bound of the switched linear system uncertainty for producing the desired system response is shown in the following figs. Response of state variables x1 and x2 are shown in Figs. 2 and 3, respectively with known uncertainty bound (i.e. with robust control design (9)). Figs. 4–5, Figs. 6–7 and Figs. 8–9 depict the response of the state variable x with neural network based control design (14) for (F = I10, κ = 1), (F = 0.1I10, κ = 0.01) and (F = I10, κ = 0.01), respectively. Fig. 6. Response of state variable x1 with control design (14) for F = 0.1I10 and κ = 0.01. Fig. 7. Response of state variable x2 with control design (14) for F = 0.1I10 and κ = 0.01. From Figs. 4–5 and Figs. 6–7, we see that the UUB region of the state x is increasing for (F = I10, κ = 1) and (F = 0.1I10, κ = 0.01). In Figs. 8 and 9, the UUB region of the state x is small for F = I10, κ = 0.01 in comparison to Figs. 2–7. In Figs. 4–9, for the fixed γ , we observe that the state x is improved (decreased) by increasing F and decreasing κ. Overall, we can say that the FFNN based controller is quite effective without prior knowledge of the upper bound of uncertainty, in comparison to the controller using known upper bound of uncertainty as in [14]. 5. Conclusion In this paper, we investigated the uniformly ultimately bound- edness of a switched linear system with parametric uncertainty and external disturbance under arbitrary switched signals using a Lyapunov function generated by the weighting matrices. A feed- forward neural network is employed to learn the upper bound of
- 6. 110 H.P. Singh, N. Sukavanam / ISA Transactions 51 (2012) 105–110 Fig. 8. Response of state variable x1 with control design (14) for F = I10 and κ = 0.01. Fig. 9. Response of state variable x2 with control design (14) for F = I10 and κ = 0.01. uncertainty. The NN weights may be simply initialized to zero or randomized. Finally, the numerical simulation is carried out for two subsystems. The simulation results show that the feedforward neural network with the on-line updating law can compensate the switched linear system efficiently. In our proposed controller (14) any prior knowledge of the uncertainty bound is not required. This is the sharp contrast to the control approach in [14], which has some potential difficulties such as (i) implementation requires a prior knowledge of the uncertainty bound, (ii) if an uncertain switched system has many subsystems, the computation of the uncertainty bound will be a complex and time consuming task, and (iii) if a new switched system comes in, the whole procedure has to start again. Acknowledgment This work is financially supported by the Council of Scientific and Industrial Research (CSIR), New Delhi, India and Department of Science and Technology (DST), Government of India through grant No. DST-347-MTD. References [1] Liberzon D, Morse AS. Basic problems in stability and design of switched systems. IEEE Control Systems Magazine 1999;19:59–70. [2] Asarin E, Bournez O, Dang T, Maler O, Pnueli A. Effective synthesis of switching controllers for linear systems. Proceedings of the IEEE 2000;88:1011–25. [3] Hespanha JP, Morse AS. 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