7. Extensions
7.1 Fuzzy Logic Systems
While fuzzy control systems have quickly gained acceptance in industry, a large part of academia is still approaching works on fuzzy control with suspicion. One reason for this is the evident lack of systematic methods for analysis and design of fuzzy control systems. In this section, we will
7.1 Fuzzy Logic Systems using LMI computations, see[156, 135]. Unfortunately, most of these re-sults are derived by embedding the fuzzy system (7.2) into the class of linear time-varying systems
˙x XL
i1
λi(t)Aix (7.4)
where the weightsλi(t) may vary arbitrarily with time while satisfying 0≤λi(t) ≤1,P
iλi(t) 1. In other words, the stability conditions guar-antee stability for the system(7.4)from which stability of(7.2)follows. By this embedding, the structural information structural information about the system encoded in the rule premises is disregarded. As we have seen in Chapter 4, this structural information is crucial in many cases.
It is natural to try to extend the piecewise quadratic analysis to fuzzy systems. In this way, fuzzy controllers can be analyzed efficiently using more powerful Lyapunov functions and structural information can be ac-counted for. This also allow us to treat affine Takagi-Sugeno systems that have an additional offset term in the consequent dynamics. This was the model structure that what was originally proposed in [134]. The affine Takagi-Sugeno systems are described by rules on the form
Ri: IF x1is Fi,1AND . . . AND xn is Fi,n
THEN ˙xAix+ai, i1, . . . ,L and the inferred dynamics can be written as
˙x XL
i1
µi(x)⋅{Aix+ai}, (7.5)
with µi(x) ≥ 0 and P
iµi(x) 1. Many applications use affine Takagi-Sugeno systems, see e.g.[132, 7]. As shown in [37], the function approx-imation capabilities of the Takagi-Sugeno system are also substantially improved when offset terms are allowed.
Takagi-Sugeno Fuzzy Systems – A Piecewise Linear Perspective In order to clarify the link between fuzzy systems and the piecewise linear systems considered in this thesis, it is fruitful to consider fuzzy systems as a particular instance of operating regime based models [56, 97]. Op-erating regime based modeling is a common name for techniques where a globally valid model of the system dynamics is obtained by combining simple local models, each valid within a certain operating regime. In this
Chapter 7. Extensions
context, the special feature of fuzzy systems is that prior knowledge of op-erating regimes and locally valid dynamics is encoded using fuzzy rules.
Each rule antecedent defines an operating regime and the associated rule consequent specifies the local model valid within this region:
Ri: IF x1is Fi,1AND . . . AND xn is Fi,n
| {z }
operating regime specification
THEN ˙x|A{zix+a}i local dynamics
, i1, . . . ,L
Comparing with the mathematical description(7.5), we see that in regions where µi(x) 1 for some i, all other normalized membership functions evaluate to zero and the dynamics of the system is given by ˙xAix+ai. We will call such a region the operating regime of model i. Between oper-ating regimes there are regions where 0<µi(x) <1. In these regions, the system dynamics is given by a convex combination of several affine sys-tems. We will call these regions interpolation regimes. As we will see next, the partitioning into operating and interpolation regimes is often polyhe-dral. This will allow us to view fuzzy systems as pwLDIs and directly apply the associated analysis techniques to fuzzy systems.
The Geometry of Fuzzy Partitions
There is more structure in fuzzy system partitions than what is directly visible in the formulation(7.5). Consider for simplicity the case when the model scheduling is governed by one variable, w C x. In this case, the rules are on the form
Ri: IF w is Fi THEN ˙xAix+ai i1, . . . ,L. (7.6) The operating regimes where µi(w) 1 induce intervals in the schedul-ing space w C x, and polyhedral cells in the state space. An example of membership functions and the associated partitioning is shown in Fig-ure 7.1. Note that if the scheduling intervals{wtµi(w) 1}are connected sets, the induced regimes are convex polyhedral sets. This is for example the case for the membership functions in Figure 7.1.
A similar structure in the induced partition can be found when rules are formed using the AND connective in a higher dimensional scheduling space. The rules then take the form(7.1), and the normalized membership functions are obtained as in(7.3). Sinceµl(x) 1 in operating regime l, we must haveµl,m(xm) 1 for m1, . . . ,n. Operating regime l is thus given by the intersection of the cells that are induced by the fuzzy sets
7.1 Fuzzy Logic Systems
−50 −2 −1 1 2 5
0.5 1
wx1
µ1(⋅) µ2(⋅) µ3(⋅)
−5
0
5 −5 0
5 0
0.5 1
wx1
x2
Figure 7.1 The normalized membership functions(left)and the resulting parti-tion of the state space into operating and interpolaparti-tion regimes(right).
used in each propositional variable, Xl \
m
{x :µl,m(xm) 1}.
The partition resulting from rules formed with the AND connective can be seen as a composition of several simple partitions, as illustrated in Figure 7.2. Moreover, the induced operating and interpolation regimes are then convex polyhedral sets that can be obtained directly from the membership functions of the simple propositions.
Fuzzy Systems as Piecewise Linear Differential Inclusions
The discussion above establishes that affine Takagi-Sugeno systems can be viewed as pwLDIs as introduced in Chapter 2. The fuzzy rules(7.1) induce a partitioning of the state space into a number of convex polyhedral cells{Xi}i∈I. The cells act either as operating regimes or as interpolation regimes. In each region, the dynamics is given by a convex combination of affine systems
˙x X
k∈K(i)
µk(x){Akx+ak}, x∈Xi
with 0≤µk(x) ≤1,P
k∈K(i)µk(x) 1. Here, we have introduced the index set K(i) to specify what system matrices are used in the interpolation within cell Xi, i.e.,
K(i) {ktµk(x) >0 for x∈Xi}
For operating regimes, the set K(i)contains one single element. By disre-garding the state dependence of the membership functions, we can embed
Chapter 7. Extensions
−6 0
6
−6 0 6 0
1 −6
0 6
−6 0 6 0
1 −6
0 6
−6 0 6 0 1
Figure 7.2 The fuzzy partition with scheduling in two variables(bottom)can be derived from the intersection of simple partitions induced by each propositional variable(top and center).
these fuzzy systems into the class of pwLDIs,
˙¯x co
k∈K(i)
¯ Ak¯x
x∈Xi (7.7)
with 0≤µk(t) ≤1,P
k∈K(i)µk(t) 1. Now, Theorem 4.2 applies directly and gives a novel procedure for studying stability of fuzzy systems.
An Example
In order to demonstrate the feasibility of the approach to problems of more realistic size, this section presents a piecewise quadratic stability analysis of a 25-region fuzzy system. The system dynamics is given by the nine rules in Table 7.1. The membership functions of the fuzzy propositions “xi
is Fl,i” are trapezoidal and shown in Figure 7.1. The rules partition the state space into the operating regimes and interpolation regimes shown in Figure 7.3.
Note that all regions off the origin have bias terms, and that the system matrices associated to some of the operating regions are non-Hurwitz (these operating regimes, given by the two last rules of the rule base, are lightly shaded in Figure 7.3). Hence, the standard conditions for quadratic stability can not be applied. As shown in Figure 7.4, simulations reveal a highly nonlinear behavior but suggest that the system is stable.
7.2 Hybrid Systems