Model uncertainties in selecting fuzzy sub-vector fields

The Lyapunov framework proposed in this chapter can be essentially employed for the robustness analysis of existing uncertainties in different fuzzy sub-vector fields and the corresponding subsystems of a non-smooth TS fuzzy system. These uncertainties can be traced between two dichotomies: uncertainties as to which fuzzy state combinations are possible in a fuzzy system, which, in turn, result in uncertain selection of the differ-ent sub-vector fields, and uncertainties in the values of the sub-vector fields at differdiffer-ent fuzzy states. These types of uncertainties may also arise from uncertainties in parameter identification or unmodeled dynamics in the original mathematical model of the physical system. The second type of uncertainties is the out of the scope of this discussion but there is literature covering this aspect (see for instance [178–181]). When not all of the fuzzy state combinations are realizable in a non-smooth TS fuzzy system, this may lead to uncertainties in selecting the possible sub-vector fields at certain continuous states. This is a major attribute of TS fuzzy systems, which are the convex combination of the weighted affine subsystems and sub-vector fields, or any system of the form of a weighed sum of affine subsystems. The problem arises from the fact that the associated weight of a fuzzy sub-vector filed (or in our case support [w]⁰ as defined in (3.24)) may be quite uncertain as a result of the use of a given identification method.

The conditions in LMI stabilization problems are confined to the nominal regions where the fuzzy states are possible. Nevertheless, as highlighted above, the formulation in the second, third, and fourth stability conditions provides the possibility for the flexible regions to be larger than the nominal specified ones. This can be implemented by the substitution of regions by quadratic inequalities, although this may lead to a conservative formulation as pointed out in Sections 4.3.1 and 4.5. The larger regions allow, for instance, the fuzzy sub-vector fields Fm1 and Fm3 to switch to Fm2 anywhere in the fuzzy state space, even though the Lyapunov function is originally bound to be reduced for Fm₂ in the region

|x1| < x0. Uncertainties arising from the unknown location of fuzzy sub-vector fields can also be formulated in the LMI stabilization problems by confining the stability conditions to the uncertain regions, in the same format highlighted above.

It is already known that in a non-smooth TS fuzzy model as proposed by Definition

129 4.7 Summary

3.1, a fuzzy sub-vector field is the convex hull of affine sub-systems. There might be the case when a local Lyapunov function can be found to verify the stability of all extreme members of the convex hull for different fuzzy sub-vector fields. In this case, the Lyapunov (energy) function can be reduced for such extreme members (sub-systems). However, this might not result in verifying the stability of a non-smooth TS fuzzy system due to two possible scenarios.

• First, when local Lyapunov functions allocated to regions sharing the same boundary, are not equal at some boundary states, the stability is verifiable, if and only if, the solution trajectories evolve in a direction such that the non-smooth Lyapunov function reduces at these states. Therefore, all extreme members of the convex hull must evolve in the same direction.

• Second, if local Lyapunov functions are equal at some boundary states, solution tra-jectories can evolve bidirectionally. However, verifying stability when sliding motion occurs at the region boundaries will be problematic.

According to our assumptions, all sliding motions (in case of their occurrence) should be substituted by thier equivalent dynamics in a TS fuzzy model, cf. Section 3.2.4).

Nevertheless, if two sub-vector fields belonging to different fuzzy regions are not equal at the region boundaries there is still a possibility that another sliding motion occurs, which is not considered in the dynamics of TS fuzzy model before verifying the stability. Therefore, the stability of a non-smooth TS fuzzy system depends on the event of the afore-mentioned possible scenarios. If these two scenarios occur, stability cannot be verified. Indeed, in the event of sliding motion, one of the local Lyapunov functions must also be the Lyapunov function defined for the sliding dynamics.

4.7 Summary

Based on the proposed non-smooth TS fuzzy model structure of Definition 3.1, a stability theory is developed in the Lyapunov sense to asses structural stability of the most im-portant classes of non-smooth dynamical system showing different degrees of smoothness.

It’s been asserted that the existing theory for stability analysis of TS fuzzy systems is fundamentally restricted to the classical notion of stability (stability of equilibria). The proposed stability theorems are formulated as linear matrix inequalities (LMIs) to be solved numerically by efficient interior-point methods. Solving LMI stability conditions as a feasibility problem can analytically predict the edge of discontinuity-induced bifurcation (DIBs), the bifurcation phenomena unique to non-smooth systems, through investigat-ing the stability of periodic solutions. The stability theorems presented in this chapter can be fairly generalized to expand the application of model-based TS fuzzy approach to non-smooth dynamical systems by suggesting a Lyapunov framework for bifurcation analysis and ensuing chaotic dynamics. A number of relevant but archetypal examples were included to support this assertion.

Using the non-smooth Lyapunov function approach is fundamental to the development of stabilization theorems for studying Filippov-type systems (sliding and non-sliding) and

Chapter 4: Stability Analysis 130

impacting systems, as presented in Sections 4.3 and 4.4. Lyapunov function candidates are defined as non-smooth functions based on the switching manifold, and further piece-wise smooth in time, based on detached but flexible regions of fuzzy state space. In this manner, the stability conditions are confined in different local regions instead of the entire state space. The regions are then expressed as quadratic inequalities. By adopting the S-procedure technique, the confined stability conditions are substituted by unconfined conditions. A few methods were proposed to substitute fuzzy state-space region partitions with quadratic inequalities. The chain of substitutions will result in LMI problems formulated as numerous quadratic functions with unknown parameters in a non-conservative manner. Therefore, all the stability conditions are fully recast on LMIs.

The flexibility of partitioning the fuzzy state space also plays an important role in LMI formulation of the proposed stability theorems. This flexibility allows identical local Lyapunov functions to be searched for in different fuzzy sub-vector fields or multiple local Lyapunov functions to be searched for in single fuzzy sub-vector field (and the associated discrete state). Therefore, the conservative formulation of bilinear matrix inequalities (BMI), which normally results from the piece-wise structure of quadratic Lyapunov func-tions, can be avoided. Moreover, region partitions can be constructed arbitrarily close to the switching manifold. It was shown how this approach can relax the conservative LMI formulations in finding a feasible solution, specially in the event of a grazing bifurcation in an impacting system and a sliding-grazing bifurcation in a sliding Filippov-type system.

Model uncertainties and inaccuracies are inevitable in any mathematical modeling attempt when trying to mimic reality. This remains true for the non-smooth TS fuzzy modeling proposed in this thesis. However, this uncertainty can be dealt with very well by the essential discontinuities and uncertainties affiliated with non-smooth dynamical systems. Robustness analysis takes advantage of the proposed Lyapunov structure when uncertainties occur in modeling the discrete states governed by switch sets and in selecting the fuzzy sub-vector fields to achieve robust control strategies and guarantee the stability (of periodic solutions in the sense of Lyapunov), which will be presented in the next chapter.

Chapter 5

Controller design for DC-DC converters

Chaos often breeds life, when order breeds habit . . . Henry Adams (1858-1918)

This chapter proposes Lyapunov-based control strategies to suppress nonlinear phe-nomena and unwanted chaos in electrical Filippov-type system like DC-DC electronic converters. The strategies are applicable to the non-smooth TS fuzzy modeling methods proposed in Chapter 3 and are mainly based on the structural stability theorems discussed in Chapter 4.

5.1 General motive

Consider a simple problem of controlling a swinging pendulum to its upright position.

Although the original system is not a non-smooth system, a switching control strategy can be devised to switch between one controller pushing the pendulum up, and another controller maintaining it in its upright position. This switching control strategy can con-vert the close-loop system in its entirety to a piece-wise smooth system. By doing so, the fundamental problem will be narrowed down to when the switching events should take place to stabilize the pendulum at its upright position. Generally speaking, a controller can be designed in such a way that an algorithm generates switching events, based on some stability criterion, to activate locally designed controllers [182]. A common approach in studying a large class of nonlinear systems is linearization around some local points to break the system down into a number of linear sub-systems (see Definition 2.4 and Theo-rem 2.1 in Section 2.2.4). In a similar fashion, a nonlinear controller can be composed of a number of linear controllers that can only operate in certain pre-specified areas of state space. At the same time, a scheduling variable can be used to determine which operating

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Chapter 5: Controller design for DC-DC converters 132

region the system is currently active and to enable the appropriate linear sub-controller.

These type of controllers are known as gain-schedulers [183, 184], and are widely-used in industry. However, design of linear controllers and switching algorithms in the control industry are mainly based on engineering experience combined with lengthy simulations.

Intrinsically, fuzzy logic imitates human reasoning through approximate information and underlying uncertainty to generate decisions. Non-smooth TS fuzzy models in the sense of Definition 3.1, can be ultimately viewed as an aggregation of certain local affine models.

Thus, the complex control task can be distributed into several local task components. In summation, it is sensible to establish a controller design framework such as fuzzy model-based switching/gain-scheduling or fuzzy model-model-based adaptive control scheme.

In our design, the location of switching events in state space is crucial in terms of curbing the chaotic behavior of a non-smooth model. The design objective for a closed-loop TS fuzzy system, which is normally composed of a number of local controllers in different fuzzy operating regions, is to locate a specific fuzzy sub-vector field or a set of sub-systems such that the closed-loop system, according to the stability conditions proposed in Chapter 4, becomes structurally stable. Nevertheless, practically different choices of fuzzy sub-vector fields (and the corresponding sub-systems) are available to ensure close-loop stability. Therefore additional performance criteria must be imposed to satisfy the selection of the right fuzzy sub-vector fields, driving the close-loop system to a stable periodic solution.