L2-stability analysis of feedback systems via positive operator theory Thesis submitted for the Degree of Doctor of Philosophy in the faculty of Engineering

Abstract

This investigation is concerned with developing L2L_2L2?-stability criteria for deterministic continuous?time dynamical systems consisting of a cascade of operators in L2L_2L2?, in a single negative?feedback loop. The problem is treated within the mathematical framework of operator algebras, and the method of analysis generally involves the application of positive?operator theory. Specific stability criteria are derived for various interesting feedback configurations obtained by imposing suitable restrictions (linearity, time?invariance, etc.) on the open?loop operators. Many of these criteria are in the now?familiar “multiplier” form. For the particular case of systems containing a cascade of a linear time?invariant convolution operator GGG and a time?varying gain k(t)k(t)k(t), the obtained criteria take the form of frequency?domain inequalities and involve very general non?causal multipliers. These results are derived by employing bounds on the rate of variation of the gain (specifically, a two?sided bound), which give rise to conditions involving certain “shifting” in the multiplier. As a straightforward extension of these results, criteria imposing less stringent average?variation constraints on k(t)k(t)k(t) are obtained. A by?product of the method of analysis is the derivation of generalised conditions for the factorisation of a linear convolution operator in a Banach algebra into a cascade of two operators-one causal and the other anti?causal; these conditions are obtained by relating factorisability to certain positivity properties of the operators. Combined time–frequency?domain criteria are developed for systems that contain a time?invariant nonlinear memoryless operator FFF in L2L_2L2? within an otherwise linear feedback loop. With the requirement that the nonlinearity belongs to certain well?defined classes (odd and monotonic, power?law functions, or functions with restricted odd asymmetry), it is shown that non?causal multipliers-more general than those permitted by existing criteria-may be used. For the more general case of systems involving an additional time?varying gain k(t)k(t)k(t) in the loop, similar results involving non?causal multipliers are derived by bounding k(t)k(t)k(t). It is shown that these bounds give rise to proportional “shifts” either in the frequency?domain conditions involving the linear part and the multiplier, or in the time?domain conditions involving the kernel of the multiplier alone. For more complicated cases of feedback systems (linear and nonlinear) containing a linear time?varying operator GGG in L2L_2L2?, L2L_2L2?-stability criteria are derived using a state?space description of GGG. A new class of multipliers with non?stationary kernels is introduced, and time?domain stability criteria involving the solution of certain Riccati equations are obtained. By imposing additional requirements such as controllability and differentiability on GGG, simpler criteria involving specific frequency?domain inequalities are derived. As an application, it is shown that the use of periodic multipliers provides improved stability results for the nonlinear damped Mathieu equation with a forcing function. To illustrate the versatility of the techniques developed thus far-even for systems that do not conform to the Lur’e configuration-the problem of developing L2L_2L2?-stability criteria for time?varying feedback systems with multiplicative nonlinearities, which arise in certain flight?control applications, is examined. Interesting results are also obtained for the case of periodic systems. Finally, an attempt is made to provide a step?by?step computational procedure for checking the frequency?domain stability criteria involving multipliers, as derived in the stability?theory literature. This includes determining an allowable region for the phase characteristics of multipliers using only the phase function of the linear part, and constructing suitable multipliers using a few computer?oriented methods. The use of positive?operator theory for stability analysis has required, as an intermediate step, the development of positivity conditions for various compositions of operators in L2L_2L2? (both nonlinear and time?varying). These positivity conditions, as well as the factorisation conditions for linear operators in L2L_2L2? obtained in this thesis, are of independent interest and have applications in several other areas of Mathematical System Theory.