Summary of Contributions set was derived. This has a direct application to estimation of stability regions, but may also be useful in limit cycle computations.
Dissipativity Analysis using Piecewise Quadratic Storage Functions The Lyapunov functions were applied to dissipativity analysis. We gave an approach for dissipativity analysis based on piecewise quadratic storage functions. By use of upper and lower bounds on the estimated energy quantities, it was shown how partition refinements could give improved estimates. The dissipativity analysis also opens up possibilities to combine the piecewise linear analysis with analysis done by other methods. The ideas were illustrated on small gain analysis of a piecewise linear system with dynamic uncertainty.
Piecewise Linear-Quadratic Optimal Control
The piecewise quadratic functions were also used in solving optimal con-trol problems via convex optimization. In this way, feedback concon-trol laws were obtained using very simple methods. The idea was only taken a small step in this thesis, and further developments are necessary in or-der to make this approach a useful design methodology.
Fuzzy Systems, Hybrid Systems and Smooth Nonlinear Systems The basic results were extended in several useful ways, providing new analysis techniques for fuzzy systems, hybrid systems and smooth non-linear systems.
The first extension was to fuzzy systems. An important class of fuzzy systems are close to being piecewise linear, and we showed how these sys-tems could be described by piecewise linear differential inclusions. Given a system model in terms of fuzzy logic-based rules, a corresponding un-certain piecewise linear model can be derived using a simple procedure.
This allowed the piecewise quadratic Lyapunov function computations to be tailored to fuzzy systems, hence providing a novel and powerful toolset for analysis of such systems.
A second extension was to hybrid systems. Hybrid systems have at-tracted a large interest in the control community over the last few years, and many intriguing questions remain to be answered. We showed how a class of hybrid systems could be analyzed via convex optimization. The analysis uses Lyapunov functions that have a discontinuous dependence on the discrete state.
The third extension was to smooth nonlinear systems. Piecewise linear systems can approximate smooth systems to arbitrary precision. The same statement holds true for the piecewise linear and piecewise quadratic Lyapunov functions suggested in this thesis. It is therefore natural to
Chapter 9. Concluding Remarks
try to use the piecewise linear approach to analysis of smooth systems.
We showed approximation errors could be taken into account explicitly, providing formal results for smooth systems based on piecewise linear analysis.
Partition refinements play an important role throughout this thesis.
They increase the accuracy in piecewise linear approximations and im-prove the flexibility of the Lyapunov function candidates. A novel approach for automated partition refinements was proposed, based on linear pro-gramming duality.
Computational Aspects and Toolbox Implementation
Computational issues were also treated. Explicit formulas for covering el-lipsoids of minimal volume were given for simplex and hyper-rectangular cells. This makes it easy and computationally efficient to use ellipsoidal cell boundings in the analysis, and has also a theoretical interest in itself.
We provided some insight in the S-procedure, proving that for some par-tition types, the polytopic S-procedure is always less conservative than the quadratic procedure. We also proved non-conservatism of the S-procedure for simplex partitions in dimensions up to 3.
Finally, we presented a Matlab toolbox that makes it easy to describe, simulate and analyze piecewise linear systems. It gives easy access to many of the results from the thesis and provides a simulation engine for efficient simulation of piecewise linear systems with sliding modes.
Open Problems and Ideas for Future Research
Numerical analysis and design of control systems is only in its infancy.
The progress in hardware and software opens many possibilities. In this thesis, we have shown how convex optimization can be used to analysis of nonlinear dynamical systems. The results have direct applications, and can be extended in many promising ways. In this section, we point out some open problems and give ideas for future research.
The matrix representation for piecewise linear systems was derived to be convenient in computations. A drawback with this representation is that it requires a large amount of memory for its representation. As discussed in Chapter 2, model representations that are much more mem-ory efficient can be derived for piecewise linear systems with continuous vector fields. The implicit piecewise linear representation was shown to be closely related to our model. Is it possible to derive LMI conditions for piecewise quadratic stability that makes efficient use of this model?
Partition refinements are very useful to increase the flexibility of the Lyapunov function candidate. We gave a simple procedure for automated
Open Problems and Ideas for Future Research partition refinements based on linear programming duality. The develop-ments were based on a large portion of heuristics, and many issues are left open. Is this partition refinement strategy the best possible? Can con-vergence be proven in a formal way? The extension to the LMI case would also be very interesting.
It would be very useful to be able to analyze piecewise linear sys-tems that are interconnected with nonlinearities described by integral quadratic constraints. This would, for example, allow less conservative analysis of piecewise linear systems with time delays. It appears to be fruitful to consider an approach along the lines of the Lyapunov tech-nique presented in[70], Section 1.7.
The idea of solving optimal control problems via convex optimization and the Hamilton-Jacobi-Bellman inequality is very promising. Advances in hardware and software now allows design of feedback laws based on optimality considerations using convex optimization. This theme has only be touched upon in this thesis. Several interesting extensions along these lines have been given in[118].
There is currently a large interest in fuzzy control and hybrid control systems. We have extended the piecewise quadratic Lyapunov functions to apply also to these cases. The piecewise linear techniques Lyapunov functions could be extended similarly, resulting in potentially very useful results.
Many system theoretic issues related to well-posedness of solutions, observability and controllability are left open. These are important com-ponents of a more complete theory for piecewise linear dynamical systems.
Finally, the majority of the problems treated in this thesis are con-cerned with properties of equilibria. Tracking problems have not been considered. Extensions to tracking problems would be of great practical relevance.
Chapter 9. Concluding Remarks
A
Proofs
A.1 Proofs for Chapter 2
Proof of Proposition 2.1
We consider two polyhedral piecewise linear systems
Σ1:
(˙xAix+ai+Biu1
y1Cix+ci+Diu1
x∈Xi
Σ2:
(˙zAjz+aj+Bju2
y2Cjz+cj+Dju2 z∈Zj
where Xi and Zj denote polyhedral sets, represented as Xi {xtGix+gi0}
Zj {ztGjz+gj 0}
Throughout, this proof we will let x∈Xi and z∈Zj. This is a polyhedral constraint in x and z, which can be represented as
Xeij
x z
t G
i 0 gi
0 Gj gj
x z 1
: eGij
x z 1
0
The partition of the interconnected system is made up from cells derived for all i,j such that x∈Xi, z∈Zj, and has thus ij cells.