The BMI formulation of the u]Km-synthesis problem also naturally leads one to a ge<meG'ic perspective for the problem that reflects the structure of a fundamental paradigm in robust control synthesis and analysis based on the topological separation of plant and unceg raphs. The fundamental importance of the BMI structure in robust synthesis is partia1lY reflected in the fact that a wide array of controller robust synthesis problems may be formulated asBMIs.
This paper summarizes and extends the conference papers.!" The main contributions of this paper are as follows: Firstly, the I-liKm-synthesis problem for mixed parametric and linear time invariant uncertainty is shown to be reducible to a BMI problem. This suggests new, rigorous approaches for robust control synthesis, in which the search for the stability multipliers and the controller is conducted jointly. The BMI approach improves on the D, G-K-iterationlM-K.iteration approaches for I-liKm-synthesis, and is applicable to a wide array of other controller synthesis problems. Secondly, this paper discusses a fundamental underlying paradigm for robust control synthesis, that of topological separation, that explains why the BMI problem inevitably holds such a central place in robust control synthesis theory. Finally, for the sake of completeness. a full exposition of the so-called 'positivity' approach to I-liKm-synthesis is included.
This paper is organized as follows: firstly, a brief general discussion of the basic structure of the robust synthesis problem based on topological separation arguments is presented in Section 2. The equivalence between positive real and the bounded real frameworks which leads to the multiplier approach to IIIKm-synthesis is then outlined in Section 3. The main results with regard to I-liKm-synthesis via BMIs are given in Section 4. The BMI formulation and its implications are then discussed at some length in Section 5.
Notation is standard, and will be defined as the need arises. Much of the notation and terminology follows that of References 7 -11. In particular, R denotes the set of real numbers, C the set of complex numbers, Co. the extended JW axis and C+ the closed right half complex plane. The space of square integrable functions [0, oo)~RP is denoted by L~,(o.-)J' and L~.,(o._)J denotes the space of functions [0, 00 ) ~R P square integrable over finite intervals. A causal operator from Lt.(o._Β») to L~e,(o._)J is said to be stable if it has bounded induced L~.(o._)) gain. The set of linear time invariant (LTI) stable causal operators L~,(o._Β»)~L~e,(o,_)J is denoted by :Β£lft P , and :Β£lftJ denotes the elements of ~fJ;tP which are memoryless. Refer to Table 1 for a further selection of the notation that will be used.
2. BACKGROUND: TOPOLOGICAL SEPARATION
This section gives a brief outline of the background of the I-liKm-synthesis problem, and suggests why the LMI and BMI problems are of fundamental interest to robust analysis and synthesis, respectively. Much of robust control analysis is based on the absolute stability results of References 12-14. In particular, one may note that Reference 7, Chapter 2, building on the framework of Reference 14, gives an interpretation of absolute stability criteria in terms of the requirement for a topological separation between the inverse graph of the plant uncertainty and the graph of the nominal plant. See also References 15 and 16. Consider the interconnection , A of Figure 1, where I:i and T are p-input p-output causal, stable operators from L~.,(o,_Β») to L~.,[o._))). The operator T represents the nominal plant (or the nominal closed loop, in robust controller synthesis), and the operator I:i represents an element of the plant uncertainty set. The interconnection j A is said to be stable if and only if the induced L~.{O.\Β»1 gain from (r, d) to (u, y) in the interconnection j B is bounded. For any function 'V(u,y)E~~. 'V'rE [0,00) (6) A recent reinterpretation of this viewpoint is given in Reference 15. Special cases include d r (u, Y) = lI u r Il 2 -II Yr Il 2 and dr(u,y)-(ur.Yr), the cases where the set of possible d'S is