U-Parameter Design: Feedback System Design with Guaranteed Robust Stability

We propose here an extension of the Q-parameter design technique developed by Youla, Jabr and Bongiorno [1], and Desoer, Liu, Murray, and Saeks [2] for the design of nominally stable closed-loop systems. The basic idea behind the Q-parameter approach is the characterization of all compensators, which guarantee closed-loop stability for a nominal plant, in terms of a compensator parameter Q(s), where Q(s) is an arbitrary stable function. Here stable is taken to be BIBO stable, i.e., Q(s) is proper and all its poles have negative real parts. This free parameter function, Q(s), is then used to design a nominally optimal system, or to meet other design objectives. Boyd et al. [3] exploit this parameterization to develop computer-aided design software which permits the minimization of a performance measure, while simultaneously meeting a set of additional deisgn specifications. The key point is that with Q-parameterization many design specifications reduce to convex constraints on the Q(s) function space, and by further parameterization of Q(s), the optimal design problem can be reduced to a convex programming in a finite dimensional space. However, the Q-parameterization approach only guarantees stability of a fixed “nominal” plant. No robustness of the solution is assured for plant uncertainties. To provide for the design of closed-loop systems which are guaranteed to be robustly stable, we propose a “U-parameterization” of the compensator in terms of an arbitrary strongly bounded real (SBR) function, U(s).