System Terminology.
1. Introduction
Markovian jump systems consists of two components; i.e., the state (differential equation) and the mode (Markov process). The Markovian jump system changes abruptly from one mode to another mode caused by some phenomenon such as environmental disturbances, changing subsystem interconnections and fast variations in the operating point of the system plant, etc. The switching between modes is governed by a Markov process with the discrete and finite state space. Over the past few decades, the Markovian jump systems have been extensively studied by many researchers; see (Kushner, 1967; Dynkin, 1965; Wonham, 1968; Feng et al., 1992; Souza & Fragoso, 1993; Boukas & Liu, 2001 ; Boukas & Yang, 1999; Rami & Ghaoui, 1995; Shi & Boukas, 1997; Benjelloun et al., 1997; Boukas et al., 2001 ; Dragan et al., 1999). This is due to the fact that jumping systems have been a subject of the great practical importance. For the past three decades, singularly perturbed systems have been intensively studied by many researchers; see (Dragan et al., 1999; Pan & Basar, 1993; Pan & Basar, 1994; Fridman, 2001; Shi & Dragan, 1999; Kokotovic et al., 1986).
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
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
2. System Descriptions and Definitions
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
where Δ
For the convenience of notations, we let
and
where
We recall the following definition.
where
Note that for the symmetric block matrices, we use (*) as an ellipsis for terms that are induced by symmetry.
3. Robust H∞ Fuzzy Control Design
This section provides the LMI-based solutions to the problem of designing a robust H∞ fuzzy controller that guarantees the
First, we consider the following
Then, we describe the problem under our study as follows.
Before presenting our first main result, the following lemma is needed.
where
with
then the inequality (5) holds. Furthermore, a suitable choice of the fuzzy controller is
where
or in a more compact form
where
Consider a Lyapunov functional candidate as follows:
Note that
Now let consider the weak infinitesimal operator
Adding and subtracting
Now let us consider the following terms:
and
where
where
Substituting (27) into (24), we have
where
Using the fact
we can rewrite (28) as follows:
where
Pre and post multiplying (30) by
with
Note that (31) is the Schur complement of Ψ
Following from (29), (32) and (33), we know that
Applying the operator
From the Dynkin’s formula [2], it follows that
Substitute (36) into (35) yields
Using (34) and the fact that we have
Hence, the inequality (5) holds. This completes the proof of Lemma 1.
Now we are in the position to present our first result.
where
with
then there exists a sufficiently small
where
with
with
Substituting (50) and (52) into (51), we have
Clearly,
Using the matrix inversion lemma, we learn that
where
where the
with
Employing (38)-(40) and knowing the fact that for any given negative definite matrix
4. Illustrative Example
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:
where
![](http://cdnintech.com/media/chapter/8721/1512345123/media/image136.png)
Figure 1.
A modified series dc motor equivalent circuit.
Giving
where ε = L represents a small parasitic parameter. Assume that, the system is aggregated into 3 modes as shown in Table 1:
Mode | Moment of Inertia | J ( ) ± ∆ J ( )(kg· m 2 ) |
1 | Small | 0.0005 ± 10% |
2 | Normal | 0.005 ± 10% |
3 | Large | 0.05 ± 10% |
Table 1.
The transition probability matrix that relates the three operation modes is given as follows:
The parameters for the system are given as R = 10 Ω, Lf = 0.005 H, D = 0.05 N m/rad/s and Km = 1 N m/A. Substituting the parameters into (58), we get
where
The control objective is to control the state variable
Note that Figure 2 shows the plot of the membership function represented by
Knowing that
![](http://cdnintech.com/media/chapter/8721/1512345123/media/image147.png)
Figure 2.
Membership functions for the two fuzzy set.
(59) can be approximated by the following TS fuzzy model
where μi is the normalized time-varying fuzzy weighting functions for each rule,
and
with ||
In this simulation, we select
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
![](http://cdnintech.com/media/chapter/8721/1512345123/media/image156.png)
Figure 3.
The result of the changing between modes during the simulation with the initial mode at mode.
![](http://cdnintech.com/media/chapter/8721/1512345123/media/image157.png)
Figure 4.
The disturbance input,
![](http://cdnintech.com/media/chapter/8721/1512345123/media/image158.png)
Figure 5.
The ratio of the regulated output energy to the disturbance noise energy,
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.
5. Conclusion
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
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