2.3 Summary
3.1.2 Constructing smooth TS fuzzy models
Generally, there are two major approaches to construct a smooth TS fuzzy model: non-linear identification using experimental input-output data and derivation (non-linearization) from given nonlinear system equations. Synthesizing a fuzzy model based on input-output data was introduced by Takagi and Sugeno [27] and later elaborated by Kang [82, 83]. An exhaustive literature exists on different methods for TS fuzzy identification [84]. Identi-fying a system using TS fuzzy approach normally involves two major steps of structure identification and parameter identification. In control problems, the TS identification ap-proach is found advantageous for the plants whose direct mathematical models (or part of their model) are difficult to obtain [84]. In this respect, automatic identification to facilitate trial-and-error identification procedure of modeling is even suggested (see [85]
and the references therein).
The second approach of derivation from a given nonlinear system is suitable for me-chanical or electrical systems whose physical models are readily available. Thereby, this section intends to give an outline in constructing smooth TS fuzzy models from a given smooth dynamical system (3.5) and not to discourse the first approach. Regardless, inter-ested readers can refer to the afore-mentioned references for more details on the subject.
The second approach of TS fuzzy modeling can, in turn, be obtained through the two ideas of sector nonlinearity and off-equilibrium linearization [28, 86]. The former is based on building local fuzzy sub-systems through linearizing a nonlinear system by its Taylor expansion for a selection of points far from the system’s equilibrium point. The latter is about finding an accurate representation of the original mathematical function by a TS fuzzy model over a domain of interest.
3.1.2.1 Off-equilibrium linearization
Using the off-equilibrium linearization approach, fuzzy sub-systems of a smooth TS fuzzy model are constructed via expanding the 1st-order Taylor series in a chosen set of lin-earization points θj excluding any equilibrium point of the original system. The whole dynamics can then be formed by allowing the rule base to define the degree of validity (or activation degree) of the resulting linear (or affine) fuzzy sub-systems. As a first step, one has to determine which variables in x, u and ρ, i.e. the premise variables θ, participate as nonlinear term in (3.5). The consequent local sub-systems are then obtained by derivation of the functions f and g of a smooth system (1.2) with respect to the vector of x, u and ρ:
Aj= ∂f a partition of the universe of discourse are carefully chosen to cover the approximation area. With the above matrices and the corresponding membership functions, a smooth
Chapter 3: Takagi-Sugeno fuzzy modeling 42
TS fuzzy model can be practically constructed. If an affine TS fuzzy model is to be obtained, fixed terms can be further derived as aj = f (xj, uj, ρj) − Ajxj − Bjuj and cj = g(xj, uj, ρj) − Cjxj.
The idea behind this approach is inspired by classical gain scheduling methods where a nonlinear system is approximated with several linear systems obtained through lineariza-tion at the equilibrium manifold of the system (3.4) [87]. If the approximalineariza-tion area needs to be expanded around the equilibrium manifold, fixed affine terms like aj and cj should be also taken into account [88].
Example 3.1: Consider the following smooth nonlinear system originally adopted from
[88]: (
˙x1= x2,
˙x2= x21+ x22, (3.8)
where the input function u is intentionally dropped for modeling purposes. The goal is to derive a TS fuzzy model to represent the nonlinear system (3.8). The model is constructed using the off-equilibrium linearization approach over the chosen set of linearization points x1 ∈ {0.5, 1.5, 2.5, 3.5} and x2 ∈ {−1, 0, 1, 4}, with selected membership functions illus-trated in Fig. 3.1. Checking the approximation accuracy of the resulting model, Fig. 3.2 shows the generated surface of the original nonlinear function ˙x2 in (3.8) compared with that of the obtained TS fuzzy model. Although the approximation accuracy of the TS fuzzy model is acceptable over the chosen linearization points, beyond the range of these points, the accuracy is gradually becoming poorer. To increase the accuracy of modeling using this approach, more linearization points may be added over a wider range or dif-ferent forms of membership functions should be selected, i.e. sigmoid or gaussian instead of trapezoidal ones used in Fig. 3.1. For a very complex (smooth) nonlinear systems, the selection of the set of linearization points and respective membership functions can be almost infinitely large.
)
1(t z ))
( (1
1z t
M M2(z1(t)) M3(z1(t)) M4(z1(t))
)
2(t z ))
( (1
1z t
N N2(z1(t)) N3(z1(t)) N4(z1(t))
Figure 3.1: Membership functions for the off-equilibrium linearization method. Variables z1and z2represent the equivalent fuzzy variables for system states x1and x2, respectively.
43 3.1 Models for smooth dynamical systems
−2 0
2 4
6
−2 0 2 4 6
−20 0 20 40 60 80
x
1x
2˙x
2(a)
−2 0
2 4
6
−2 0 2 4 6
−20
−15
−10
−5 0
x
1x
2˙x
2(b)
Figure 3.2: The surface function of (a) the original system (3.8) (white mesh) and it’s TS fuzzy model (grey mesh underneath), (b) the error between the original system and its TS fuzzy model.
In this respect, few methods have been also suggested in the literature to automate the problem of selecting an appropriate set of linearization points and the corresponding membership functions by employing classical optimization algorithms (see for instance [89]) or AI optimization methods like genetic algorithms [90,91]. Both types of algorithms have been shown to be disadvantageous specifically when the number of parameters grow within the algorithm or the dimension of system states are increased.
Chapter 3: Takagi-Sugeno fuzzy modeling 44
3.1.2.2 Sector nonlinearity
This approach, initially suggested by Kawamoto et al. [92] and later generalized by Tanaka et al. [28, 93], is about exactly representing a smooth nonlinear system (3.4) by a TS fuzzy model in some specific sector boundaries instead of approximating over a chosen set of linearization points. Therefore, the target of modeling is to find an exact TS fuzzy representation of an original nonlinear function ˙x = f (x(t)), f (0) = 0 in a bounded sector [α1 α2], or so-called global sector, such that ˙x = f (x(t)) ∈ [α1 α2]x(t). However, since constructing a TS fuzzy model over global sector is usually found difficult, system states can be further bounded to local sectors as x(t) ∈ [−d, d] to achieve the highest approximation accuracy possible [28].
The rationale behind this method can be expressed by showing a nonlinear system in matrix form as
where any nonlinear term in the matrix An×n describes a fuzzy variable aij = zi, and zi is bounded as aij ∈ [minxaij(x) maxxaij(x)] for x belongs to the universe of dis-course. Therefore each fuzzy variable zi can be formulated as the convex combination of membership functions
zi(x) = M (zi(x)) · min
x aij(x) + (1 − M (zi(x))) · max
x aij(x).
The sector boundaries of a fuzzy variable ziare determined as maxzi∈Mziand minzi∈Mzi, which in turn, are substituted by the nonlinear terms in (3.9) to form the fuzzy sub-system matrices of a smooth TS fuzzy model.
Example 3.2: Assume the same nonlinear system in Example 3.1. The original system (3.8) is again intended to be represented by a smooth TS fuzzy model using the sector nonlinearity approach over the local sector boundary of x1∈ [0.5, 3.5] and x2∈ [−1, 4] with the derived membership functions as shown in Fig. 3.3. For the sake of brevity, detailed steps of constructing this model are omitted here since it is treated in the article [94]
written by the author. A full description of the general procedure can also be found in [28]. The surface of the original nonlinear function ˙x2 in (3.8) along with that of its TS fuzzy approximation are depicted in Fig. 3.4. As noticed, this approach is more powerful than the former one as it can exactly represent the system dynamics in the sector boundaries. However, a disadvantage appears in control applications where (should we consider the control input function u(t)), the resulting fuzzy sub-systems may turn out to be not controllable or observable using Parallel Distributed Control (PDC) design or the other fuzzy state feedback control approaches [28, 86]. This is mainly because the relationship between system states and control inputs may not be explicitly incorporated
45 3.1 Models for smooth dynamical systems
Figure 3.3: Membership functions for the sector nonlinearity method.
−1
Figure 3.4: The surface function of (a) the original system (3.8) (white mesh) and it’s TS fuzzy model (grey mesh underneath), (b) the error between the original system and its TS fuzzy model.
Chapter 3: Takagi-Sugeno fuzzy modeling 46