A Parametric Method For The Design Of Dynamic Compensator Based On Multi-objective Optimization

Quasi-linear systems,considered as a special type of nonlinear systems,are commonly used to be the modeling of robotic systems,spacecrafts rendezvous,attitude control of combined spacecraft and so on,which have great value in engineering applications.Meanwhile,it maintains strong-coupled,highly nonlinear characteristics but can be written in linear format,thus it becomes the bridge and tache between linear systems and general nonlinear systems,and also has importantly theoretical significance.In the past researches,the controller design mainly applied state and static output feedback.Generally,because of complicated and varied work conditions,it is difficult to obtain state variables such that state feedback cannot be implemented,while static output feedback cannot completely achieve the desired control effects resulted in by state feedback.Dynamic compensator,as an important complementary of controller forms,can effectively overcome the drawbacks of state and static output feedback to further improve the control performances.In this paper,necessary condition is presented to judge the controllability of quasi-linear systems for normal and descriptor cases,then,a parametric method,based on the solutions of generalized Sylvester equations,is proposed to design dynamic compensator such that the generally parameterized expressions of dynamic compensator are established.With the proposed parametric method,the closed-loop system yields a linear constant form with an expected eigenstructure.Simultaneously,multi-objective design and optimization is considered based on free parameters provided by parameter method.By utilizing the degrees of design freedom in parameters,robustness and low gain criteria are designed,and then an optimized dynamic compensator is obtained by solving a multi-objective optimization problem.Furthermore,this research also puts forward a parametric method to design minimal order dynamic compensator,which fills the gap of correlative research.The main work of this paper is introduced as follows:1.Necessary condition is presented to judge the controllability of quasi-linear systems in normal and descriptor cases,which is based on the coefficient matrices of quasi-linear systems such that the complicated computation of solving state-transformation-matrix can be avoided.The proposed controllability results naturally expand the PBH criterion,and also are the precondition and foundation in the design of dynamic compensator.2.This paper proposes a parametric method to design dynamic compensator for quasi-linear systems included first-order,second-order,high-order and descriptor cases.The proposed method is based on the solutions of generalized Sylvester equations,and converts the closed-loop system into a linear constant form with an expected eigenstructure which is decided by an arbitrary matrix Λ containing desired eigenvalues.Furthermore,parameteric method establishes a more general parameterized forms of left and right eigenvector matrices,and provides two groups of free parameters which can be used to improve and optimized system performances,then the completely parametric expressions of dynamic compensator,constisting of arbitrary matrix Λ,left and right eigenvector matrices and free parameters,are developed.3.A novel and common approach is considered to solve the minimal order of dynamic compensator.The presented approach directly utilizes the coefficient matrices of quasi-linear systems to determine the minimal order of compensation vector,which reduces the computation load.Then,the parametric expressions of minimal order dynamic compensator are established by using the proposed parameteric method such that the gap of correlative research is filled.4.According to free parameters provided by parametric method,robustness and low gain criteria,which contain overall eigenvalue sensitivity function,disturbance attenuation,robustness degree,H2/H∞ norm,low control gain and low compensation gain,are formulated.Then,an overall objective function which includes each performance objective weighted is established to express the synthetic performances of control systems.Because free parameters are without constraints of physical meanings,optimized interval can be expanded to the global area.By using the degrees of design freedom in parameters,an approximately optimal dynamic compensator can be obtained by solving a multi-objective optimization problem.