IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 1, JANUARY 2005 41 Generalized KYP Lemma: Unified Frequency Domain Inequalities With Design Applications Tetsuya Iwasaki, Senior Member, IEEE, and Shinji Hara, Senior Member, IEEE Abstract—The celebrated Kalman–Yakuboviˇ c–Popov (KYP) lemma establishes the equivalence between a frequency domain inequality (FDI) and a linear matrix inequality, and has played one of the most fundamental roles in systems and control theory. This paper first develops a necessary and sufficient condition for an -procedure to be lossless, and uses the result to generalize the KYP lemma in two aspects—the frequency range and the class of systems—and to unify various existing versions by a single theorem. In particular, our result covers FDIs in finite frequency intervals for both continuous/discrete-time settings as opposed to the standard infinite frequency range. The class of systems for which FDIs are considered is no longer constrained to be proper, and nonproper transfer functions including polynomials can also be treated. We study implications of this generalization, and develop a proper interface between the basic result and various engineering applications. Specifically, it is shown that our result allows us to solve a certain class of system design problems with multiple specifications on the gain/phase properties in several frequency ranges. The method is illustrated by numerical design examples of digital filters and proportional-integral-derivative controllers. Index Terms—Control design, digital filter, frequency domain inequality, Kalman–Yakuboviˇ c–Popov (KYP) lemma, linear ma- trix inequality (LMI). I. INTRODUCTION O NE OF THE most fundamental results in the field of dynamical systems analysis, feedback control, and signal processing, is the Kalman–Yakuboviˇ c–Popov (KYP) lemma [1]–[3]. Various properties of dynamical systems can be char- acterized by a set of inequality constraints in the frequency domain. The KYP lemma establishes equivalence between such frequency domain inequality (FDI) for a transfer function and a linear matrix inequality (LMI) for its state space realization. The basic roles of the KYP lemma are two fold: it provides 1) insights into analytical approaches to systems theory, and 2) a framework for numerical approaches to systems analysis and synthesis. Manuscript received September 4, 2003; revised May 25, 2004 and September 11, 2004. Recommended by Associate Editor Y. Ohta. This work was supported in part by the National Science Foundation under Grant 0237708, by The Ministry of Education, Science, Sport, and Culture, Japan, under Grant 14550439, by CREST of the Japan Science and Technology Agency (JST), and by the 21st Century COE Program on Information Science and Technology Strategic Core. T. Iwasaki is with the Department of Mechanical and Aerospace Engi- neering, University of Virginia, Charlottesville, VA 22904-4746 USA (e-mail: iwasaki@virginia.edu). S. Hara is with the Department of Information Physics and Computing, Grad- uate School of Information Science and Engineering, The University of Tokyo, Tokyo 113-8656, Japan (e-mail: Shinji_Hara@ipc.i.u-tokyo.ac.jp). Digital Object Identifier 10.1109/TAC.2004.840475 The KYP lemma [3] states that, given matrices , , and a Hermitian matrix , the FDI (1) holds for all if and only if the LMI (2) admits a Hermitian solution . Thus, the infinitely many in- equalities (1) parametrized by can be checked by solving the finite-dimensional convex feasibility problem (2). Appropriate choices of in (1) allows us to represent various system prop- erties including positive-realness and bounded-realness. While the KYP lemma has been a major machinery for de- veloping systems theory, it is not completely compatible with practical requirements. In particular, each design specification is often given not for the entire frequency range but rather for a certain frequency range of relevance. For instance, a closed-loop shaping control design typically requires small sensitivity in a low frequency range and small complementary sensitivity in a high frequency range. Thus a set of specifications would gen- erally consists of different requirements in various frequency ranges. On the other hand, the standard KYP lemma treats FDIs for the entire frequency range only. The current state of the art for fixing the incompatibility is to introduce the so-called weighting functions. A low/band/high- pass filter would be added to the system in series as a weight that emphasizes a particular frequency range and then the de- sign parameters are chosen such that the weighted system norm is small. The weighting method has proven useful in practice, but there remains some room for improvement. First, the addi- tional weights tend to increase the system complexity (e.g., con- troller order), and the amount of increased complexity is posi- tively correlated with the complexity of the weights. Second, the process of selecting appropriate weights can be time-con- suming, especially when the designer has to shoot for a good tradeoff between the complexity of the weights and the accu- racy in capturing desired specifications. We remark that some of these deficiencies may be addressed by recent developments for new characterizations of disturbance signals [4], [5]. An alternative approach is to grid the frequency axis. In this case, infinitely many FDIs are approximated by a finite number of FDIs at selected frequency points. This approach has a prac- tical significance especially when the system is well damped and the frequency response (after the design) is expected to be “smooth” (i.e., no sharp peaks). The resulting computational 0018-9286/$20.00 © 2005 IEEE