Comments and References

Chapter 4. Lyapunov Stability

we obtain a system which is exponentially unstable. This system can not be globally stabilized using a bounded feedback. Using the approach de-scribed above we can still obtain an estimate of the region of attraction, as shown in dashed line in Figure 4.22. For comparison, the exact region of attraction obtained by simulation is shown in full line in the same figure.

−20 −10 0 10 20

−5 0 5

Figure 4.22 Limited region of attraction estimated by piecewise quadratic Lya-punov function(^dashed), and exact region of attraction(^full)^.

Note that a similar approach could be used to show convergence to a set.

By restricting the analysis to the region outside some ball

B

^r
{^xt^xTx≤ r²}, convergence be established to the largest level set of the computed Lyapunov function that contains

B

^r. Such a result could be combined with the local instability analysis to do limit cycle computations.

4.13 Comments and References convex optimization are two important ingredients that allow complex systems to be analyzed using efficient numerical computations. Around the same time that these results were published[⁶³], similar ideas were reported for the analysis of hybrid systems in[¹⁰⁹]. An interesting exten-sion to systems with multiple equilibrium points was given in[⁴⁴]^. Strict vs. Nonstrict Inequalities

As the main benefit of the stability conditions presented in this thesis is the possibility for numerical verification, our philosophy has been to derive stability conditions that can be implemented in software as they stand. This has led to two peculiars in our formulation of the stability conditions.

The first feature is the separation of cells that contain the origin from cells that do not. From a theoretical point-of-view it may appear unnec-essary to eliminate affine terms in regions that contain the origin. An alternative approach would be to use non-strict inequalities

A¯^T_i P¯i+^P¯iA¯i+^E¯i^TUiE¯i≤

−εI 0

0 0



∀i∈^I.

A formulation of this kind could be useful for analysis of systems with multiple equilibria[⁴⁴]. However, at the writing of this thesis neither[⁴²] which was used for all computations in this thesis, nor the freely available semidefinite programming environment[¹⁵¹]supported non-strict LMIs.

Another feature of the stability conditions is that they use a param-eterization that enforces continuity on the Lyapunov function candidate.

An alternative approach would simply be to have different matrices ¯Pi in each region, and then enforce continuity via constraint equations of the type(^4.13). This is the approach used in[^{108, 44}]. There are many factors that contributed to our use of constraint matrices. Not every optimization environment supports the mixture of linear equations and LMI conditions.

It may also be numerically sensitive to introduce a large number of super-fluous parameters, and then trust the optimization software to eliminate redundant constraints. Moreover, the approach with constraint matrices is natural for many classes of partitions(as will be shown in Chapter 8)^, and can be used with ease also for complicated partitions.

Piecewise Linear Lyapunov Functions

While efficient software for LMI optimization have not appeared until quite recently[¹⁰⁰], the simplex method for solving linear programming problems is more than 50 years old[³⁵]. Consequently, researchers have for a long time been aware of the benefits of deriving results that can be verified via linear programming.

Chapter 4. Lyapunov Stability

Computer algorithms for construction of piecewise linear Lyapunov functions have been reported in, among others,[24, 25, 95, 16, 96, 102]^. The focus has been on polytopic Lyapunov functions and uncertain linear systems. An important exception is the work [¹⁰²]that considers poly-topic Lyapunov functions for piecewise linear systems. Highly related to our approach is the stability analysis proposed in [⁸⁰], in which piece-wise linear Lyapunov functions(that may have affine terms, and are not necessarily polytopic)were constructed using so-called facet functions.

Polytopic vs. Ellipsoidal Cell Boundings

Ellipsoidal cell boundings have also been used in the references[^{108, 44}]^. In [⁴⁴], ellipsoidal cell boundings were used to allow the use of the S-procedure in control design based on quadratic Lyapunov functions. In [¹⁰⁸], ellipsoidal cell boundings were used to assure robustness to un-certainties in regional descriptions of hybrid systems. As shown in this chapter, computational efficiency may be another reason for trying ellip-soidal cell boundings before using the full power of polyhedral relaxation terms.

Numerical Lyapunov Function Construction

Most analysis methods for dynamical systems are somehow related to Lya-punov functions, and a LyaLya-punov function appears more or less explicitly in most analysis conditions. Dissipativity analysis[^{149, 47}], absolute sta-bility[^{152, 113}], and analysis based on integral quadratic constraints[⁹⁴] can all be viewed as methods for Lyapunov function construction.

5

Dissipativity Analysis

A fundamental idea in systems and control is to view complex systems as the interconnection of simpler subsystems. Such a perspective is of-ten helpful in bringing insight into and understanding about a dynamic system. Viewing a system as the interconnection of its components, it is natural to ask whether the analysis of a complex system can be based on the (hopefully simpler) analysis of its components. This is the idea behind input-output analysis, which has been a very successful tool in system theory. Roughly speaking, the idea is to replace detailed models of system components by relationships between their input and output energies, and then derive results for interconnections of such models. The most well-known results may be the small-gain and the passivity theo-rems. Both allow stability of a feedback interconnection to be verified from the analysis of its components. Hence, by establishing

L

²-gain or passivity properties of piecewise linear systems, we can hope to establish stability of interconnections by invoking small gain and passivity theorems. This chapter will provide such tools.

This approach is useful for robustness analysis and it also allows us to use different tools for analyzing different components. For example, physical insight may help us to establish passivity of one subsystem and piecewise linear techniques can allow us to prove strict passivity of an-other subsystem. This allows us to analyze systems that combine piece-wise linear systems with components that can not be or are not efficiently described by piecewise linear techniques. Time delays is one example of such components.

As we have seen in Chapter 2, several important interconnections of piecewise linear systems are themselves piecewise linear. Although this route will not be explored in depth here, we note that these tools will also allow us to choose whether to analyze a interconnected system in one step, or to first analyze the subsystems and then use small-gain and passivity results. This will enable us to trade-off complexity in the computations

Chapter 5. Dissipativity Analysis

In document Piecewise Linear Control Systems Johansson, Mikael (Page 99-103)