6. Optimization of Integral Quadratic Constraints
Ulf T. Jonsson, Anders Rantzer · Society for Industrial and Applied Mathematics eBooks · 1999
6.1 Introduction Computational issues in stability and performance analysis have been studied for a long time. In particular, it was shown in a classical paper by Zames [444] that a stable linear system G in negative feedback interconnection with a slope restricted nonlinearity is stable if there exists a “multiplier” M such that ℜ (M (jω) (G (jω) +1) )>0 ∀ω ∈ [0,∞] . The multiplier can be selected from a specific class, and Zames noted that the main difficulty in applying this stability condition was to find a suitable multiplier from the class. Several more sophisticated multiplier-based stability results were developed during the period 1965–1975. For example, a criterion for slope restricted nonlinearities was derived in [445] and a criterion for slowly time-varying parameters in [385]. See also the books [98, 419]. All these results had their main limitation in the computability of the multipliers. The difficulty was mainly due to the lack of powerful computers and suitable numerical software. Since the early 1980s a lot of work has been focused on robustness analysis of uncertain systems. In particular, multiplier-based methods for diagonally perturbed linear time-invariant (LTI) systems received attention through the work in [102, 350]. The theory was later extended to treat also parametric uncertainty [131, 441]. Since both the system and the uncertainty are time invariant, it is possible to do the analysis frequency by frequency; see [31, 314, 443]. This is not the case for systems that contain time-varying or nonlinear components. The development and implementation of efficient algorithms for solution of convex optimization problems involving linear matrix inequalities (LMIs) has greatly increased the possibilities for efficient computations in system analysis. For references, see [64, 119, 153, 296]. Multiplier computation in analysis of systems with various forms of dynamic and parametric uncertainty was treated in [28, 29, 191, 262].