Abstract
Application of the Lyapunov method to 2D system stability and performanceanalysis yields algebraic systems that can be interpreted as eithersum-of-squares problems for nontrivial matrix polynomials, or parameterizedlinear matrix inequalities that need to be satisfied for certain ranges ofparameter values. In this paper we show that dualizing core inequalities inthe latter forms allows converting these systems to conventionaloptimization problems on sets described by polynomial matrix inequalities.Methods for solving these problems include moment-based methods or the“atomic optimization” method proposed earlier by the author. As aresult, we obtain necessary conditions for 2D system stability and lowerbounds on system performance. In particular, we demonstrate respectiveresults for discrete-discrete system stability and mixedcontinuous-discrete system $\mathcal{H}_\infty$ performance. A numerical example isprovided.
Highlights
The formalism of multidimensional systems, in particular, their most well-studied case—n = 2 with both independent variables having some variation of temporal semantics—has been receiving much interest in recent years
The current paper addresses these two types of problems applied to certain representations of 2D systems
As with the more usual kinds of dynamic systems, there are various approaches to 2D systems’ stability analysis. They are generally based on characteristic polynomials/multinomials, or specialized Lyapunov functions. The latter often allows using techniques based on linear matrix inequalities (LMIs) and convex optimization
Summary
This can be achieved by partial dualization Applying this transformation to the basic Lyapunov inequality (see [3, 11] for more details with an accent on continuous systems), we can show that infeasibility of the latter is equivalent to feasibility of the dual-like form F (ejω)Z(ω)F (ejω)H − Z(ω) ≥ 0 (with Z(ω) = Z(ω)H ≥ 0 as the new unknown matrix). The part that generally becomes harder in this case is obtaining bounds on variables of intermediate 1D descent subproblems (formed by current approximations and descent directions), as the respective equations will no longer be reducible to reasonably simple eigenvalue problems Another way to approach (7) is to force it into the polynomial form by replacing ejω, ω ∈ [0; 2π], with an appropriate parametrization of the unit circle.
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