An extension of the LQR/LQG methodology to systems with saturating actuators, referred to as SLQR/SLQG, is obtained. The development i s based on the method of stochastic linearization. Using this method and the Lagrange multiplier technique, solutions to the SLQR and SLQG problems are derived. These solutions are given by Riccati and Lyapunov equations coupled with two transcendental equations. It is shown that, under standard stabilizability and detectability conditions, these equations have a unique solution, which can be found by a simple bisection algorithm. When the level of saturation tends to in nity, these equations reduce to their standard LQR/LQG counterparts.
Stability of a switched system that consists of a set of linear time invariant subsystems and a periodic switching rule is investigated. Based on the Floquet theory, necessary and sufficient conditions are given for exponential stability. It is shown that there exists a slow switching rule that achieves exponential stability if at least one of these subsystems is asymptotically stable. It is also shown that there exists a fast switching rule that achieves exponential stability if the average of these subsystems is asymptotically stable. The results are illustrated by examples.
This is a textbook and reference for readers interested in quasilinear control (QLC). QLC is a set of methods for performance analysis and design of linear plant or nonlinear instrumentation (LPNI) systems. The approach of QLC is based on the method of stochastic linearization, which reduces the nonlinearities of actuators and sensors to quasilinear gains. Unlike the usual - Jacobian linearization - stochastic linearization is global. Using this approximation, QLC extends most of the linear control theory techniques to LPNI systems. A bisection algorithm for solving these equations is provided. In addition, QLC includes new problems, specific for the LPNI scenario. Examples include Instrumented LQR/LQG, in which the controller is designed simultaneously with the actuator and sensor, and partial and complete performance recovery, in which the degradation of linear performance is either contained by selecting the right instrumentation or completely eliminated by the controller boosting.
Absfrucf-A phase locked loop based system that tracks the resonance frequency of a series RLC circuit is investigated. Assuming that the RLC circuit parameters are slowly time-varying, a linear time-invariant model that accurately predicts the performance of the system is developed. The results are illustrated by examples.analysis method is developed. In Section 5, the results are illustrated. In Section 6, conclusions are given. PHASE LOCKED LOOPA phase locked loop [5], [6] is a feedback system that is composed of a phase comparator (PC), a low-pass filter (LPF) and a voltage controlled oscillator (VCO) as shown
A control system that tunes the resonant frequency of a lightly damped resonator to its excitation frequency in the face of detuning disturbances is investigated. The resonance tuning is achieved by adaptively controlling the resonant frequency of the resonator using the error between the excitation frequency and resonant frequency. Assuming that the parameters of the resonator are slowly time-varying, a nonlinear time-varying model that accurately predicts the tuning performance of the system is developed. This developed model is subsequently linearized to obtain a linear time-invariant model that facilitates both analysis and design of the resonance tuning system. Based on the developed linear time-invariant model, guidelines for designing the resonance tuning system are provided. The results are illustrated by examples.
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