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Multiple time-scale dynamic systems also known as singularly perturbed systems normally occur due to the presence of small “parasitic” parameters, typically small time constants, masses, etc. In state space, such systems are commonly modelled using the mathematical framework of singular perturbations, with a small parameter, say ε, determining the degree of separation between the “slow” and “fast” modes of the system. However, it is necessary to note that it is possible to solve the singularly perturbed systems without separating between slow and fast mode subsystems. But the requirement is that the “parasitic” parameters must be large enough. In the case of having very small “parasitic” parameters which normally occur in the description of various physical phenomena, a popular approach adopted to handle these systems is based on the so-called reduction technique. According to this technique the fast variables are replaced by their steady states obtained with “frozen” slow variables and controls, and the slow dynamics is approximated by the corresponding reduced order system. This time-scale is asymptotic, that is, exact in the limit, as the ratio of the speeds of the slow versus the fast dynamics tends to zero.
In the last few years, the research on singularly perturbed systems in the H∞ sense has been highly recognized in control area due to the great practical importance. H∞ optimal control of singularly perturbed linear systems under either perfect state measurements or imperfect state measurements has been investigated via differential game theoretic approach. Although many researchers have studied the H∞ control design of linear singularly perturbed systems for many years, the H∞ control design of nonlinear singularly perturbed systems remains as an open research area. This is due to, in general, nonlinear singularly perturbed systems can not be decomposed into slow and fast subsystems.
Recently, a great amount of effort has been made on the design of fuzzy H∞ for a class of nonlinear systems which can be represented by a Takagi-Sugeno (TS) fuzzy model; see (Nguang & Shi, 2001; Han & Feng, 1998; Chen et al., 2000; Tanaka et al., 1996). Recent studies (Nguang & Shi, 2001; Han & Feng, 1998; Chen et al., 2000; Tanaka et al., 1996; Wang et al., 1996) show that a fuzzy model can be used to approximate global behaviors of a highly complex nonlinear system. In this fuzzy model, local dynamics in different state space regions are represented by local linear systems. The overall model of the system is obtained by “blending” of these linear models through nonlinear fuzzy membership functions. Unlike conventional modelling which uses a single model to describe the global behavior of a system, fuzzy modelling is essentially a multi-model approach in which simple submodels (linear models) are combined to describe the global behavior of the system. Employing the existing fuzzy results (Nguang & Shi, 2001; Han & Feng, 1998; Chen et al., 2000; Tanaka et al., 1996; Wang et al., 1996) on the singularly perturbed system, one ends up with a family of ill-conditioned linear matrix inequalities resulting from the interaction of slow and fast dynamic modes. In general, ill-conditioned linear matrix inequalities are very difficult to solve.
What we intend to do in this chapter is to design a robust H∞ fuzzy controller for a class of uncertain nonlinear two time-scale dynamic systems with Markovian jumps. First, we approximate this class of uncertain nonlinear two time-scale dynamic systems with Markovian jumps by a Takagi-Sugeno fuzzy model with Markovian jumps. Then based on an LMI approach, we develop a technique for designing a robust H∞ fuzzy controller such that the L2-gain of the mapping from the exogenous input noise to the regulated output is less than a prescribed value. To alleviate the ill-conditioned linear matrix inequalities resulting from the interaction of slow and fast dynamic modes, these ill-conditioned LMIs are decomposed into ε-independent LMIs and ε-dependent LMIs. The ε-independent LMIs are not ill-conditioned and the ε-dependent LMIs tend to zero when ε approaches to zero. If ε is sufficiently small, the original ill-conditioned LMIs are solvable if and only if the ε-independent LMIs are solvable. The proposed approach does not involve the separation of states into slow and fast ones, and it can be applied not only to standard, but also to nonstandard singularly perturbed systems.
This chapter is organized as follows. In Section 2, system descriptions and definition are presented. In Section 3, based on an LMI approach, we develop a technique for designing a robust H∞ fuzzy controller such that the L2-gain of the mapping from the exogenous input noise to the regulated output is less than a prescribed value for the system described in Section 2. The validity of this approach is demonstrated by an example from a literature in Section 4. Finally, conclusions are given in Section 5.
The class of uncertain nonlinear two time-scale dynamic systems with Markovian jumps under consideration is described by the following TS fuzzy model with Markovian jumps:
where Eε=[I00εI]ε0 is the singular perturbation parameter, υ(t)=[υ1(t)⋯υϑ(t)]is the premise variable that may depend on states in many cases, μi(υ(t))denote the normalized time varying fuzzy weighting functions for each rule, ϑ is the number of fuzzy sets, x(t)∈ℜnis the state vectoru(t)∈ℜmis the input, w(t)∈ℜpis the disturbance which belongs to L2[0,∞), z(t)∈ℜsis the controlled output, the matrix functions Ai(η(t))B1i(η(t))B2i(η(t))C1i(η(t))D12i(η(t))ΔAi(η(t))ΔB1i(η(t))ΔB2i(η(t))ΔC1i(η(t))and ΔD12i(η(t)) are of appropriate dimensions. {(η(t))}is a continuous-time discrete-state Markov process taking values in a finite set S = {1,2, , s} with transition probability matrix PrΔ__{Pιk(t)} given by
where Δ > 0, andlimΔ→0O(Δ)Δ=0. Here λιk≥0 is the transition rate from mode ι (system operating mode) to modek(ι≠k), and
λιι=−∑k=1,k≠ιsλιkE3
For the convenience of notations, we let μiΔ__μi(υ(t))η=η(t) and any matrix M(μι)Δ__M(μη=ι). The matrix functions ΔAi(η)ΔB1i(η)ΔB2i(η)ΔC1i(η)and ΔD12i(η) re-present the time-varying uncertainties in the system and satisfy the following assumption.
where Hji(η)j=1,2,⋯,5 are known matrices which characterize the structure of the uncertainties. Furthermore, there exists a positive function ρ(η) such that the following inequality holds:
‖F(x(t)ηt)‖≤ρ(η)E4
We recall the following definition.
Definition 1: Suppose γ is a given positive number. A system of the form (1) is said to have the L2-gain less than or equal to γ if
Ε[∫0Tf{zT(t)z(t)−γ2wT(t)w(t)}dt]≤0x(0)=0E5
where Ε[]stands for the mathematical expectation, for all Tf and all w(t) ∈ L2 [0, Tf].
Note that for the symmetric block matrices, we use (*) as an ellipsis for terms that are induced by symmetry.
This section provides the LMI-based solutions to the problem of designing a robust H∞ fuzzy controller that guarantees the L2-gain of the mapping from the exogenous input noise to the regulated output to be less than some prescribed value.
First, we consider the following H∞ fuzzy state feedback which is inferred as the weighted average of the local models of the form:
u(t)=∑j=1rμjKj(l)x(t)E6
Then, we describe the problem under our study as follows.
Problem Formulation: Given a prescribed H∞ performance γ > 0, design a robust H∞ fuzzy state-feedback controller of the form (6) such that the inequality (5) holds.
Before presenting our first main result, the following lemma is needed.
Lemma 1: Consider the system (1). Given a prescribed H∞ performance γ > 0, for ι = 1, 2, , s, if there exist matricesPε(ι)=PεT(ι), positive constants δ(ι) and matrices Yj(ι), j = 1, 2, , r such that the following ε-dependent linear matrix inequalities hold:
Consider a Lyapunov functional candidate as follows:
V(x(t)ιε)=γxT(t)Qε(ι)x(t)∀ι∈SE22
Note that Qε(ι) is constant for eachι. For this choice, we have V(0, ι0, ε) = 0 and V(x(t),ι,ε)→∞ only when ||x(t)|| → ∞.
Now let consider the weak infinitesimal operatorΔ˜of the joint process {(x(t), ı, ε), t ≥ 0}, which is the stochastic analog of the deterministic derivative. {(x(t), ı, ε), t ≥ 0} is a Markov process with infinitesimal operator given by (Souza & Fragoso, 1993),
Hence, the inequality (5) holds. This completes the proof of Lemma 1.
Remark 1: The linear matrix inequalities given in Lemma 1 becomes ill-conditioned when ε is sufficiently small, which is always the case for the uncertain nonlinear two time-scale dynamic systems. In general, these ill-conditioned linear matrix inequalities are very difficult to solve. Thus, to alleviate these ill-conditioned linear matrix inequalities, we have the following theorem which does not depend on ε.
Now we are in the position to present our first result.
Theorem 1: Consider the system (1). Given a prescribed H∞ performance γ > 0, for ι = 1, 2, , s, if there exist matrices P(ι), positive constants δ(ι) and matrices Yj(ι), j = 1, 2, , r such that the following ε-independent linear matrix inequalities hold:
with Zε(ι)=(λι1P¯^(ι)⋯λι(ι−1)P¯^(ι)λι(ι+1)P¯^(ι)⋯λι1P¯˜(ι))Pε(ι)=diag{P¯^(1)⋯P¯^(ι−1)P¯^(ι+1)⋯P¯^(s)}P¯^(ι)=P˜(ι)+P˜T(ι)2 andYεj(ι)=Kj(ι)Mε−1(ι). Note that the ε- dependent linear matrix tends to zero when ε approaches zero.
Employing (38)-(40) and knowing the fact that for any given negative definite matrix W, there exists an ε > 0 such that W + εI < 0, one can show that there exists a sufficiently smallε^0 such that for ε∈(0,ε^] (8) and (9) hold. Since (7)-(9) hold, using Lemma 1, the inequality (5) holds forε∈(0,ε^].
Consider a modified series dc motor model based on (Mehta & Chiasson, 1998) as shown in Figure 1 which is governed by the following difference equations:
whereω˜(t)=ω(t)−ωref(t) is the deviation of the actual angular velocity from the desired angular velocity,i˜(t)=i(t)−iref(t) is the deviation of the actual current from the desired current, V˜(t)=V(t)−Vref(t)is the deviation of the actual input voltage from the desired input voltage, J is the moment of inertia, Km is the torque/back emf constant, D is the viscous friction coefficient, andRa, Rf, Laand Lf are the armature resistance, the field winding resistance, the armature inductance and the field winding inductance, respectively, withRΔ__Rf+RaandLΔ__Lf+La. Note that in a typical series-connected dc motor, the condition LfLa holds. When one obtains a series connected dc motor,i(t)=ia(t)=if(t) we haveLfLa. Now let us assume that |ΔJ| ≤ 0.1J.
where x(t)=[x1T(t)x2T(t)]T is the state variables, w(t)=[w1T(t)w2T(t)]Tis the disturbance input, u(t) is the controlled input and z(t) is the controlled output.
The control objective is to control the state variable x2(t) for the range x2(t) ∈ [N1 N2]. For the sake of simplicity, we will use as few rules as possible.
Note that Figure 2 shows the plot of the membership function represented by
M1(x2(t))=−x2(t)+N2N2−N1 and M2(x2(t))=x2(t)+N2N2−N1
Knowing that x2(t) ∈ [N1 N2], the nonlinear system
(59) can be approximated by the following TS fuzzy model
H11(ι)=[−0.05J(ι)0.05J(ι)N100] and H12(ι)=[−0.05J(ι)0.05J(ι)N200]
In this simulation, we select N1 = −3 and N2 = 3. Using the LMI optimization algorithm and Theorem1 with ε = 0.005, γ = 1 and δ(1) = δ(2) = δ(3) = 1, we obtain the results given in Figure 3, Figure 4 and Figure 5
Remark 2: Employing results given in (Nguang & Shi, 2001; Han & Feng, 1998; Chen et al., 2000; Tanaka et al., 1996; Wang et al., 1996) and Matlab LMI solver [28], it is easy to realize that when ε< 0.005 for the state-feedback control design, LMIs become ill-conditioned and Matlab LMI solver yields an error message, “Rank Deficient”. However, the state-feedback fuzzy controller proposed in this paper guarantee that the inequality (5) holds for the system (59). Figure 3 shows the result of the changing between modes during the simulation with the initial mode at mode 1 and ε = 0.005. The disturbance input signal, w(t), which was used during simulation is given in Figure 4. The ratio of the regulated output energy to the disturbance input noise energy obtained by using the H∞ fuzzy controller is depicted in Figure 5. The ratio of the regulated output energy to the disturbance input noise energy tends to a constant value which is about 0.0094. Soγ=0.0094=0.0970 which is less than the pres-cribed value 1. Finally, Table 2 shows the performance index, γ, for different values of ε.
The performance index γ
ε
State-feedback control design
0.005
0.0970
0.10
0.4796
0.30
0.8660
0.40
0.9945
0.41
> 1
Table 2.
The performance index γ for different values of ε.
This chapter has investigated the problem of designing a robust H∞ controller for a class of uncertain nonlinear two time-scale dynamic systems with Markovian jumps that guarantees the L2- gain from an exogenous input to a regulated output to be less or equal to a prescribed value. First, we approximate this class of uncertain nonlinear two time-scale dynamic systems with Markovian jumps by a class of uncertain Takagi-Sugeno fuzzy models with Markovian jumps. Then, based on an LMI approach, sufficient conditions for the uncertain nonlinear two time-scale dynamic systems with Markovian jumps to have an H∞ performance are derived. The proposed approach does not involve the separation of states into slow and fast ones and it can be applied not only to standard, but also to nonstandard uncertain nonlinear two time-scale dynamic systems. An illustrative example is used to illustrate the effectiveness of the proposed design techniques.
References
1.AssawinchaichoteW.NguangS. K.2002 Fuzzy control design for singularly perturbed nonlinear systems: An LMI approach, ICAIET, 146151 , Kota Kinabalu, Malaysia.
2.AssawinchaichoteW.NguangS. K.2002 Fuzzy observer-based controller design for singularly perturbed nonlinear systems: An LMI approach, Proc. IEEE Conf. Decision and Contr., 21652170 , Las Vegas.
3.BenjellounK.BoukasE. K.CostaO. L. V.1997 H∞ control for linear time delay with Markovian jumping parameters, J. of Opt. Theory and App., 105 page (73-95).
4.BoukasE. K.LiuZ. K.2001 Suboptimal design of regulators for jump linear system with timemultiplied quadratic cost, IEEE Tran. Automat. Contr., 46 page (944-949).
5.BoukasE. K.YangH.1999 Exponential stabilizability of stochastic systems with Markovian jump parameters, Automatica, 35 page (1437-1441).
6.BoukasE. K.LiuZ. K.LiuG. X.2001 Delay dependent robust stability and H∞ control of jump linear systems with time-delay,” Int. J. of Contr.
7.BoydS.El GhaouiL.FeronE.BalakrishnanV.1994 Linear Matrix Inequalities in Systems and Control Theory, 15 SIAM, Philadelphia.
8.ChenB. S.TsengC. S.HeY. Y.2000 Mixed H∈ /H∞ fuzzy output feedback control design for nonlinear dynamic systems: An LMI approach, IEEE Trans. Fuzzy Syst., 8 page (249-265).
9.DraganV.ShiP.BoukasE. K.1999 Control of singularly perturbed system with Markovian jump parameters: An H∞ approach, Automatica, 35 page (1369-1378),.
11.FariasD. P.GeromelJ. C.ValJ. B. R.CostaO. L. V.2000 Output feedback control of Markov jump linear systems in continuous-time, IEEE Trans. Automat. Contr., 45 page (944-949).
12.FengX.LoparoK. A.JiY.ChizeckH. J.1992 Stochastic stability properties of jump linear system, IEEE Tran. Automat. Contr., 37 page (38-53).
13.FridmanE.2001 State-feedback H∞ control of nonlinear singularly perturbed systems, Int. J. Robust Nonlinear Contr., 6 page (25-45).
14.GahinetP.NemirovskiA.LaubA. J.ChilaliM.1995 LMI Control Toolbox- For Use with MATLAB, The MathWorks, Inc., MA.
15.HanZ. X.FengG.1998 State-feedback H∞ controller design of fuzzy dynamic system using LMI techniques, Fuzzy-IEEE’98, page (538-544).
16.IkedaK. T.WangH. O.1996 Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: Quadratic stability, H∞ control theory, and linear matrix inequality, IEEE Trans. Fuzzy. Syst., 4 page (1-13).
17.KokotovicP. V.KhalilH. K.O’ReillyJ.1986 Singular Perturbation Methods in Control: Analysis and Design, Academic Press, London.
18.KushnerH. J.1967 Stochastic Stability and Control, Academic Press, New York.
19.MehtaS.ChiassonJ.1998 Nonlinear control of a series dc motor: Theory and experiment, IEEE Trans. Ind. Electron., 45 page (134-141).
20.NguangS. K.ShiP.2001 H∞ fuzzy output feedback control design for nonlinear systems: An LMI approach, Proc. IEEE Conf. Decision and Contr., pages (4352-4357).
21.NguangS. K.ShiP.2003 H∞ fuzzy output feedback control design for nonlinear systems: An LMI approach, IEEE Trans. Fuzzy Syst., 11 page (331-340).
23.PanZ.BasarT.1993 H∞-optimal control for singularly perturbed systems Part I: Perfect state measurements, Automatica, 29 page (401-423).
24.PanZ.BasarT.1994 H∞-optimal control for singularly perturbed systems Part II: Imperfect state measurements, IEEE Trans. Automat. Contr., 39 page(280-299).
25.RamiM. A.El GhaouiL.1995 H∞ statefeedback control of jump linear systems, Proc.Conf. Decision and Contr., page (951-952),
26.ShiP.BoukasE. K.1997 H∞ control for Markovian jumping linear system with parametric uncertainty, J. of Opt. Theory and App., 95 page numbers (75-99),.
27.ShiP.DraganV.1999 Asymptotic H∞ control of singularly perturbed system with parametric uncertainties, IEEE Trans. Automat. Contr., 44 page (1738-1742).
28.SouzaC. E.FragosoM. D.1993 H∞ control for linear systems with Markovian jumping parameters, Control-Theory and Advanced Tech., 9 page (457-466).
29.WangH.TanakaO. K.GriffinM. F.1996 An approach to fuzzy control of nonlinear systems: Stability and design issues, IEEE Trans. Fuzzy Syst., 41 page (14-23).
30.WonhamW. M.1968 On a matrix Riccati equation of stochastic control, SIAM J. Contr., 6 page (681-697).
Written By
Wudhichai Assawinchaichote, Sing Kiong Nguang and Non-members