The Simultaneous Optimization Problem for Sensitivity and Gain Margin

In this note, the combined sensitivity and gain margin problem for SlSO linear systems is formulated and solved using a complex function interpolation technique. i t is proved that this problem always has a real rational solution provided it is solvable in the complex irrational sense. The sensitivity minimization problem subject to a gain margin constraint and its dual problem are also considered. In addition, the range of the gain margin constraint is given subject to which the optimal constrained sensitivity is identical with the optimal unconstrained sensitivity. Finally, it is shown that, not unexpectedly, the gain margin maximization conflicts with the sensitivity minimization for a nonminimum phase plant. In control systems design, one of the most important objectives concerns robustness of feedback systems to uncertainty in plants and to disturbance inputs. As robustness measures, sensitivity (and particularly the maximal magnitude of a sensitivity function) and gain margin depict different aspects of this robustness and have played a key role in the classical design and theory of feedback systems. The former quantifies output disturbance rejection and sensitivity to small additive variations, while the latter quantifies sensitivity to real multiplicative gain variation [I], [2]. Obviously, it is generally desirable to design a controller to have as small as possible sensitivity and as large as possible gain margin. In view of this, two kinds of problems arise associated with sensitivity and gain margin, respectively. The sensitivity problem is to find a proper compensator such that the closed-loop sensitivity is less than some tolerance value, and sensitivity minimization involves finding a proper compensator such that the closed-loop sensitivity equals or is arbitrarily close to the minimal attainable sensitivity. Similarly, the gain margin problem is I I Fig. I. Feedback system. achieve the maximal attainable gain margin. Up to now, all of the above problems have been discussed and solved individually by many authors, see, e.g., [3]-[q and [lo]. However, relationships between the two kinds of problems have not yet been developed. In fact, one should care about sensitivity when one maximizes gain margin, and in the same way one should care about gain margin when one minimizes sensitivity. The purpose of the note is to relate sensitivity to gain margin from a design point of view and to reveal tradeoffs between these two quantities. More specifically, the basic problem studied in this note is to optimize simultaneously the closed-loop sensitivity and gain margin via a proper compensator given a tolerance on sensitivity and a tolerance on the gain margin. Also, some related problems are considered. The basic tool used to tackle these issues was presented by Khargonekar and Tannenbaum [4]. The remainder of this note is outlined as follows. In the next section we briefly review the approach in [4] and its existing application, and prove that the approach can be applied to a more complicated class of control problems without causing the problem of irrationality or complexity of solutions. Section I11 is devoted to the combined sensitivity and gain margin problem. Section IV discusses the sensitivity minimization problem subject to a gain margin constraint, its dual constrained gain margin maximization problem, and how sensitivity minimization and gain margin maximization conflict with each other. The note concludes with Section V. Some proofs appear in the Appendix. All results are for single-input single-output (SISO), linear time-invariant (LTI) plants. Let P ( s ) be a scalar linear time-invariant (LTI) nominal plant with closed right half plane (RHP) zeros zl , Z: ,. . . , z,,, (x possibly included) and closed RHP poles pl , pz , . . . , p, . The closed-loop configuration is shown in Fig. 1 , where C(s) is an LTI proper stabilizing compensator. As usual, we define the sensitivity function