The Kalman-Yakubovich-Popov (KYP) lemma is a useful tool in control and signal processing that allows an important family of computationally intractable semi-infinite programs in the entire frequency range to be characterized by computationally tractable semidefinite programs. The first part of this thesis presents a new variation of the frequency selective Kalman-Yakubovich-Popov (FS-KYP) lemma for single input single output systems, which generalizes the conventional KYP lemma on given frequency intervals. Based on the transfer function representation of single input single output systems, the proposed FS-KYP lemma provides a unified framework to convert an important family of semi-infinite programs with generic frequency selective constraints that arise from a variety of analysis and synthesis problems for infinite impulse response systems into semidefinite programs. In contrast to existing variations of the FS-KYP lemma, which invariably involves Lyapunov variables of large dimensions, the proposed FS-KYP lemma is free from Lyapunov variables. As a consequence, the proposed semidefinite programs require a minimal number of additional variables, thus can be efficiently solved by general purpose semidefinite programming solvers on a standard personal computer. The second part of this thesis studies several applications of the FS-KYP lemma to control and signal processing. Firstly, we investigate the beam pattern synthesis of an antenna array with bounded sidelobe and mainlobe levels. It is shown that the pattern synthesis problem can be posed as a convex semi-infinite program that is turned into an semidefinite program via the proposed FS-KYP lemma. The attractive feature of the proposed method is that our semidefinite program uses only a minimal number of auxiliary variables. This subsequently enables the design of patterns for arrays with several hundred elements to be achieved on a standard personal computer using existing SDP solvers. Secondly, we develop an efficient method to design several types of digital and analog infinite impulse response filters and filter banks via the new FS-KYP lemma. The proposed method is more flexible than analytical methods in the sense that it allows direct control of more design parameters, which in turn enables more requirements such as degree of flatness to be incorporated into the design process. Finally, we examine some applications of the new FS-KYP to robustness analysis of continuous control systems. Specifically, we introduce a new bisection method to compute the H∞ gain of uncertain polytopic systems.

There are two main approaches to robust stability analysis: the input-output stability framework with scaling or multiplier, and the Lyapunov functions. Analysis methods in these two directions are usually developed independently, and the relationship between the two is not clear except for some special cases. This motivates us to study the relationship between the two approaches. The generalized performance problem refers to certain frequency-domain conditions on a transfer matrix. We prove the equivalent relationship between generalized performance and robust stability under certain assumptions. The definition of generalized performance requires the internal stability of a transfer matrix, which is not a necessity for robust stability. In view of this, we derive new frequency-domain conditions for robust stability without this requirement. Our result contains a version of the circle criterion as a special case. To tackle the generalized performance problem, we propose a version of the Kalman-Yakubovich-Popov (KYP) lemma to transform the frequency-domain conditions into linear matrix inequalities (LMIs). The proposed LMI condition is then connected to the quadratic stability of an uncertain linear system. Combining the derived results gives a clear picture of the relationships between robust stability, generalized performance, quadratic stability, and KYP lemma. The connections not only unify some previous results but also extend those results to more general stability regions and types of uncertainty. In addition to robust stability analysis, we also tackle the corresponding synthesis problem, i.e. robust pole placement. The desired region for robust pole placement can be the intersection or the union of simple regions. (Simple regions are the half plane, the disk, and the outside of a disk.) One contribution of our synthesis result is that the desired region can be non-convex¡Xmost results on robust pole placement focus on convex regions only. Two examples of the longitudinal control of a combat aircraft and the attitude control of a satellite demonstrate the effectiveness of our result.