Research Of H_∞ Control Problems Based On Finite Frequency Ranges

H_∞norm is often used as the performance index in the design of control systems. The conventional H_∞norm is a performance index in entire frequency range,however, in practical engineering,design specifications are usually not given in entire frequency range but rather in finite frequency ranges(e.g.,in the design of servo systems,only the tracking performance and the steady-state accuracy in low frequency range and middle frequency range are taken into consideration),or different specifications are required in different frequency ranges(e.g.,a closed-loop shaping control design typically requires small sensitivity in a low frequency range and small complementary sensitivity in a high frequency range).Thus the conventional H_∞norm is not completely compatible with practical requirements(i.e.,it may bring strong conservatism)and new design ideas are required to cope with this situation.In this dissertation,H_∞control problems based on finite frequency ranges are considered.The main results are summarized as follows.Firstly,the problem to design digital PID controller is considered for discrete-time linear time-invariant(LTI)systems based on the window H_∞norm conception.For a concerned finite frequency band,the discrete-time LTI system can be transformed into an equivalent continuous-time system by inverse bilinear transformation.Therefore,the existing window H_∞norm method in finite frequency band can be applied to design a continuous-time PID controller.Then the digital PID controller is obtained by bilinear transformation.A simulation example is given to show the effectiveness of this approach.Secondly,stabilizing H_∞control in finite frequency ranges and finite frequency tracking problem are investigated.As we know,GKYP lemma has brought a brand new idea to synthesis problems in finite frequency ranges.However,dynamic output feedback control via GKYP lemma with small gain specification does not automatically guarantee the stability of the resulting closed-loop systems.In this dissertation,improvements are made to the existing approach to render the asymptotical stability,and the new approach is applied to research the tracking problem.For the synthesis problem with finite frequency small gain specification via dynamic output feedback control,we add a stability constraint to the design in terms of linear matrix inequality(LMI),without adding any new variables.Furthermore,for the situation that feasible solution can not be found after adding the stability constraint, based on the non-uniqueness of the null space condition,we provide an alternative null space condition to enlarge the feasibility region of this design.Simulation example in tracking problem shows that,although dynamic output feedback control in finite frequency range is conservative,by choosing basis matrix reasonably,the degree of conservatism could be smaller than that of optimal H_∞control in entire frequency range.Finally,switched system with switching strategy based on frequency ranges is discussed.First,based on the assumption that the operating frequency band of system can be detected,using the above methods,several controllers are designed in different frequency ranges respectively.Then,with a switching strategy based on frequency ranges,a switched system satisfying specified control specifications is constructed. This design is not only a substitute to the conventional mixed sensitivity design approach(which relies on the application of weighting functions)but also an enrichment and development to the existing research on switched\hybrid systems.