IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 7, JULY 1998 1003 Robust Controller Synthesis via Shifted Parameter-Dependent Quadratic Cost Bounds Vikram Kapila, Wassim M. Haddad, Richard S. Erwin, and Dennis S. Bernstein Abstract—Parameterized Lyapunov bounds and shifted quadratic guar- anteed cost bounds are merged to develop shifted parameter-dependent quadratic cost bounds for robust stability and robust performance. Robust fixed-order (i.e., full- and reduced-order) controllers are devel- oped based on new shifted parameter-dependent bounding functions. A numerical example is presented to demonstrate the effectiveness of the proposed approach. Index Terms—Fixed-structure controllers, real parameter uncertainty, shifted parameter-dependent bounding functions. NOMENCLATURE Real numbers, real matrices, tr Transpose, inverse, trace, expectation. identity matrix, zero matrix. IN symmetric, nonnegative-definite, positive-definite matrices. IN I. INTRODUCTION One of the principal objectives of robust control theory is to synthesize feedback controllers with a priori guarantees of robust stability and performance. In structured singular value synthesis [3], [9] these guarantees are achieved by means of bounds involving frequency-dependent scales and multipliers which account for the structure of the uncertainty as well as its real or complex nature. An alternative robustness approach involves bounding the effect of real or complex uncertain parameters on the H performance of the closed- loop system [6], [11]. These guaranteed cost bounds take the form of modifications to the usual Lyapunov equation to provide bounds for robust stability and performance [1], [4]–[6]. A diverse collection of guaranteed cost bounds have been devel- oped. Bounded-real-type guaranteed cost bounds were developed in [8] and [10], while positive-real-type bounds are discussed in [4]. More recently, parameter-dependent Popov guaranteed cost bounds [6] have provided links with frequency-dependent scales and mul- tipliers while providing reliable bounds for the peak real structured singular value [6], [11]. Finally, the introduction of shift terms in [12] has been shown to reduce the conservatism of guaranteed cost bounds for structured real uncertainty without requiring frequency-dependent scales and multipliers. Manuscript received October 1, 1996. This work was supported in part by the National Science Foundation under Grant ECS-9496249 and the Air Force Office of Scientific Research under Grants F49620-95-1-0019 and F49620-96-1-0125. V. Kapila is with the Department of Mechanical, Aerospace, and Manufacturing Engineering, Polytechnic University, Brooklyn, NY 11201 USA. W. M. Haddad is with the School of Aerospace Engineering, Geor- gia Institute of Technology, Atlanta, GA 30332-0150 USA (e-mail: wm.haddad@aerospace.gatech.edu). R. S. Erwin and D. S. Bernstein are with the Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109-2118 USA. Publisher Item Identifier S 0018-9286(98)04638-8. It can easily be seen that parameter-independent guaranteed cost bounds provide the means for obtaining solutions to the quadratic stability linear matrix inequality (LMI) for all admissible uncertainty . The solution to this LMI then provides a bound for the worst case H cost. It was shown in [12] that the inclusion of the shift terms in both the bounded-real and positive- real guaranteed cost bounds can reduce the conservatism of these bounds. Since the Popov guaranteed cost bound [6] also entails less conservatism than classical bounded-real and positive-real guaranteed cost bounds, the objective of this paper is to combine features of both the Popov bound and shifted quadratic bounds. The bound we construct in this paper is the most general of its kind developed thus far, encompassing the Popov, positive- real, and shifted positive-real bounds as special cases. The benefits of this generalization are demonstrated by a numerical example involving robust controller synthesis. Specifically, our numerical results show that the combination of both the shift terms and the parameter-dependent terms provides reduced conservatism and improved robustness/performance tradeoffs as compared to either the Popov bound [6], [11] or the shifted positive-real bound [12] separately. The contents of the paper are as follows. In Section II, we state the robust fixed-order dynamic compensation problem. In Section III, we restate a key theorem from [6] to provide sufficient conditions for robust stability and performance. In Section IV, we develop a novel shifted parameter-dependent bounding function for robust stability and performance. In Section V, we provide constructive sufficient conditions for robust stability and performance via fixed-order (i.e., full- and reduced-order) dynamic compensation. Section VI provides a numerical example to demonstrate the effectiveness of the newly developed bounds for robust controller synthesis. Finally, Section VII gives conclusions. II. ROBUST FIXED-ORDER DYNAMIC COMPENSATION In this section, we introduce the robust stability and performance problem. This problem involves a set of constant uncertain perturbations of the nominal system matrix . The objective of the problem is to determine a fixed-order strictly proper dynamic compensator that stabilizes the plant for all variations in and minimizes the worst case H performance of the closed-loop system. In this and the following section, no explicit structure is assumed for the elements of . In Section IV, the structure of will be specified. A. Robust Dynamic Compensation Problem Given the th-order stabilizable and detectable plant (1) (2) where denotes a unit-intensity white noise signal, determine an th-order dynamic compensator (3) (4) that satisfies the following criteria: 1) the closed-loop system (1)–(4) is asymptotically stable for all ; 0018–9286/98$10.00  1998 IEEE