Showing papers by "Bernard Brogliato published in 2007"

Kalman-Yakubovich-Popov Lemma

Abstract: The Kalman–Yakubovich–Popov Lemma (also called the Yakubovich–Kalman–Popov Lemma) is considered to be one of the cornerstones of Control and Systems Theory due to its applications in absolute stability, hyperstability, dissipativity, passivity, optimal control, adaptive control, stochastic control, and filtering. Despite its broad field of applications, the lemma has been motivated by a very specific problem which is called the absolute stability Lur’e problem [1, 2], and Lur’e’s work in [3] is often quoted as +being the first time the so-called KYP Lemma equations have been introduced. The first results on the Kalman–Yakubovich–Popov Lemma are due to Yakubovich [4, 5] . The proof of Kalman [6] was based on factorization of polynomials, which were very popular among electrical engineers. They later became the starting point for new developments. Using general factorization of matrix polynomials, Popov [7, 8] obtained the lemma in the multivariable case. In the following years, the lemma was further extended to the infinite-dimensional case (Yakubovich [9], Brusin [10], Likhtarnikov and Yakubovich [11]) and discrete-time case (Szego and Kalman [12]).

Passivity-Based Control

Abstract: This chapter is devoted to investigate how the dissipativity properties of the various systems examined in the foregoing chapter can be used to design stable and robust feedback controllers (in continuous and discrete time). We start with a classical result of mechanics, which actually is the basis of Lyapunov stability and Lyapunov functions theory. The interest of this result is that its proof hinges on important stability analysis tools, and allows one to make a clear connection between Lyapunov stability and dissipativity theory. The next section is a brief survey on passivity-based control methods, a topic that has been the object of numerous publications. Then, we go on with the Lagrange–Dirichlet Theorem, state-feedback and position-feedback control for rigid-joint–rigid-link systems, set-valued robust control for rigid-joint–rigid-link fully actuated Lagrangian systems, state and output feedback for flexible-joint–rigid-link manipulators, with and without actuators dynamics, and constrained Lagrangian systems. Regulation and trajectory tracking problems, smooth and nonsmooth dynamical systems, are treated. The chapter ends with a presentation of state observers design for a class of differential inclusions represented by set-valued Lur’e systems.

Comments on "Control of a Planar Underactuated Biped on a Complete Walking Cycle"

Abstract: The authors point out several approximations and flaws in the above article by Chemori and Loria (see ibid., vol. 49, no. 5, p. 838-843, 2004)

Stability of Dissipative Systems

Abstract: In this chapter, various results concerning the stability of dissipative systems are presented. First, the input/output properties of several feedback interconnections of passive, negative imaginary, maximal monotone systems are reviewed. Large-scale systems are briefly treated. Then the conditions under which storage functions are Lyapunov functions are given in detail. Results on stabilization, equivalence to a passive system, input-to-state stability, and passivity of linear delay systems are then provided. The chapter ends with an introduction to \(H_{\infty }\) theory for nonlinear systems that is related to a specific dissipativity property, and with a section on Popov’s hyperstability.