H_∞Control And System Design For Some Special Classes Of Systems

Recently more control systems are designed based on synthesis rather by usingthe analysis method, i.e., the controller is designed directly to meet thespecifications. For linear systems, this type of design is the well-known H_∞control.For nonlinear systems, besides the nonlinear the H_∞control, so-called sum ofsquares (SOS) method is really a synthesis method, it can give the requiredLyapunov function and nonlinear control law directly. There are already somesuccessful applications of these methods for ordinary systems. However, for somespecial classes of systems, they are still less discussed. For example, for the lightlydamped systems, the unstable systems and nonlinear systems, etc.. This includes thedetermination of the performance specifications, the design limititions and thetheoretical works needed to meet these particular design requirements.The contributions of this dissertation is to generalize the above metionedadvanced synthesis methods to some special classes of systems, to present practicaldesign procedures and in the meantime to develop some new theoretical results. Thisdissertation is organized as follows.The H_∞loop shaping method is studied with the control design of the lightlydamped flexible system. The influences of the lightly damped poles on the norm ofthe coprime factor perturbations and on the resulting robustness of the design aregiven. And it is pointed out that the H_∞-norm in the H_∞loop shaping design is justthe stability margin of the closed-loop system and is used to guarantee the robuststability of the system, though the H_∞-norm is traditionally regarded as theperformance index in the H_∞control theory.The H_∞state feedback control and H_∞output feedback control for the unstablesystems are studied and refined on an electromagnetic suspension system. For the H∞state feedback control, it is shown that the design is completed only when therobustness constraints from the Bode integral theorem are considered besidessolving the Riccati equations. For the output feedback design, the requirements forthe mid-frequency range are alway neglected in the ordinary H_∞design, however, itis different for the unstable systems. It is pointed out that the weighting functionsfor the output feedback control design must be determined by the mid-frequencyrange Bode integral constraint with H_∞optimization.For the nonlinear systems, nonlinear H_∞control inherits the design ideas thatfor the linear systems, and the difficulty is to solve the HJI (Hamilton-Jacobi-Issacs)inequalities implicitly. A Taylor series expansion based method is presented in the design of an affine nonlinear electromagnetic suspension system. This nonlinear H_∞design result is verified by using the Hamilton function.The recently emerged SOS method is a novel nonlinear control design method,which belongs to a kind of numerical methods for polynomials. The SOS method ispresented by estimation the region of attraction of the system. A generalizedS-method and the determination of decision variables for set-inclusion problems areproposed. The SOS control design of large attitude maneuvers for satellites is usedas an example. A diagonal dominant procedure for the SOS problem is proposed toreduce the numerical error and speedup its convergence. Similar to the LinearMatrix Inequalities (LMI) method, SOS method solves the difficult nonlinearproblem with numerical methods, and it may have more applications in thenonlinear control field in future.