现代控制理论chapter5-Linear Feedback Control System.ppt

Modern Control Theory,沈阳航空航天大学自动化学院,Chapter 5,Linear Feedback Control System,outline,5-1 Introduction,5-2 State feedback and output feedback,5-3 Pole-Placement,5-5 State Estimator,5.1.1 Analysis and Design,5-1 Introduction,Given the system,design of the structure and parameters of controller,analysis,Response,controllability,observability,stability,quantitative and qualitative analysis,Analysis,and Design,Control the performance of the system,to meet the expected requirements,goal,foundation,5-2,State feedback and output feedback,5.2.1,state feedback,General expression,LITS expression,or,The introduction of state feedback not only changes the input matrix,but also changes the state matrix.As result,it changes the eigenvalues of the system and the poles of transfer function,and it is clearly important and meaningful.,5.2.2,output feedback,By linear feedback law,State and output equations as,Closed-loop transfer function matrix:,5.2.3,Discussion,on,State,Feedback and Output Feedback,1 The introduction of feedback,does not add a new state variable,namely the closed-loop system and open-loop system have the same dimension.,2 Two forms can maintain the controllability,but not the,observability,.For the state feedback,the system may not be able to maintain the original,observability,;for the output feedback,the closed-loop system will be able to maintain the,observability,.,3 Output feedback is easily realized in project.But state feedback has better features.but with the development of the theory about the observer and the,Kalman,filter,the physical realization of state feedback has been basically solved.So generally speaking,state feedback is a more adaptive feedback,.Of course,we select the form according to the actual situation.,5.2.3,Discussion,on,State,Feedback and Output Feedback,explain,Feedback system,with state estimator,We cannot measure the states,physically,Why?,Answer,We can construct the states based on input and output-state estimator,Feedback control system with state estimator,Note,Reconstruction values of state variables are gradually equal to their value,.,2.,The order of observer-state feedback system is equal to sum of the controlled system order and the estimator order,that is,the system state variables of estimator compose the closed-loop system state variables.,5-3,Pole-placement,5.3.1,question,The dynamic characteristics of a system is determined by the poles,so we can place the poles to meet the response.,principles to select poles,（,1,）,n-dimensional control system,n poles,。,（,2,）,Plural poles must be Conjugate poles,（,3,）,we need to study the impact of quality,as well,as distribution and the relationship between,the zero,when,select the desired pole position.,（,4,）,Must take into account the sensitivity of anti-,jamming,principles to select poles,5.3.2,SISO pole-placement,The system is as following,It is controllable,，,where A,、,B,、,C are,real constant matrices,respectively.A linear state feedback is defined by,The dynamic equations of the feedback systems is as:,Output feedback,prove,The systems characteristic polynomial:,closed-loop system characteristic polynomial by the state feedback:,Find non-singular transformation matrix P,，,-controllable,canonical form,Have been transformed,closed-loop system,characteristic polynomial is,suppose,The state matrix of,Closed-loop system,Put the eigenvalues of the closed-loop system to the desired location based on non-singular transformation.,Specific steps,（,1,）,For SISO(,controllable,canonical form,),the state feedback matrix K is,are characteristic polynomial,coefficients,are hopeful characteristic,polynomial coefficients,（,2,）,non-singular transformation,（,3,）,The non-singular transformation does not change the transfer function,The transfer function based on state feedback is,在按状态反馈组成的闭环系统中，,其闭环零点和开环零点是相同的。,（,4,）,suppose the system is controllable and observable,hence we get,The necessary and sufficient condition of controllable and,observable system is as following:,example,solution,：,（,1,）,transform,to state-space representation,Given the system,please find a real constant vector K so that the,eigenvalues,of the closed-Loop system are-1,review,State feedback,：,The goal,？,How to do it,？,Condition,？,（,2,）,characteristic polynomial,hopeful,characteristic polynomial,（,3,）,transform to,controllable,canonical form,（,4,）,（,5,）,Another method,（,6,）,block diagram,5.3.4,Configurate,states that can not entirely be controlled,linear,nonsingular transformation,If the system states can be controlled,we can arbitrarily use state-feedback to,configurate,poles.Well,if the system states can not completely be controlled,whether we can do this?If not,then the problem can be solved and to what extent?Now to discuss this issue.,idea,The subsystem is completely,controllable,so,we,can put the poles to the hopeful positions based on state,feedback.,The subsystem is not controllable,so state,feedback will not has any effect on,eigenvalues.,prove,characteristic polynomial:,eigenvalues?,the characteristic polynomial,of,closed-loop system:,This equation contains the uncontrollable subsystems characteristic polynomial,so we can achieve the conclusion that the state feedback has no effect on uncontrollable subsystem.,思考：是不是不完全能控的系统就不能极点配置了呢？,Note:,The necessary and sufficient condition for pole placement is that the eigenvalues of uncontrollable subsystem have negative real parts.Such system can be stable(stabilization).,5.3.5,Characteristics of State Feedback,（,1,）,State feedback does not introduce a new,state variable,So it does not increase the,system dimension,。,（,2,）,State feedback does not change the,zeros.,（,3,）,State feedback control can maintain the,systems controllability,but may not,be able to maintain the systems,observability,.,Why?,Due to technical and economic reasons,a number of state variables is often not simple and not easy to measure its physical quantities,and some or even all state variables can not be measured.How to realize state feedback?To solve this problem,we are going to introduce a system that is easy to use measured input and output variables to estimate the states.So we can make the output of the device can approximate the true state variables This is the so-called state remodeling problem,but such a device is often referred to a state estimator or state observer.,5-5 State Estimator,5.5.1,Introduction,1.problem,2.,Equivalence,construct a system,the input of original control system as its input,and its states are the reconstruction of the original state of the system states.It is considered that the structure and the parameters of the simulation system are same to the original system.,As long as the system is stable,that is,the eigenvalues have negative real parts,so,Original system,This is a free response,，,and its solution is,为重构状态,和,真实,状态间的等价指标。,5.5.2,State Estimator,The necessary and sufficient condition for designing estimator is that the system is completely observable.,Theorem,The unobservable subsystem is,asymptotic stable,。,prove,State equation:,construct the system,select proper,，,has negative real,eigenvalues,因此仅当 成立时，对任意 和 ，才有 即,满足等价性指标，而 等价于 的特征值均具有负实部，具 的不能观测部分是渐近稳定的。,5.5.5 Feedback System with State Estimator,Theorem,If and only if the system is completely observable,hence we can set the estimator,eigenvalues,.,prove,Duality,let ,hence set,eigenvalues,to the hopeful positions,。,example,Given the system,Please design a estimator with the,eigenvalues,-3,，,-4,，,-5,。,Two methods,solution,：,（,1,）,characteristic polynomial,（,2,）,The Dual System,It is clear that the dual system is controllable,and can be transformed into controllable canonical form,based on nonsingular transformation matrix,（,3,）,The dual system,feedback gain,The state feedback gain,:,（,4,）,the estimator feedback matrix,so we can get the state equation of the estimator as following,block diagram of feedback system with state estimator,5.5.7 Separation Principle,Using,the state estimates instead of the true state to do state feedback,we get the entire closed-loop system simulation block diagram shown in figure 5-17.The compensator consists of observer and controller,here observer is a full-dimensional observer.,5-17,The mathematical model of system 5-17can be written as:,then we can obtain the state space representation as following,it is a dimensional system,To,take a non-singular matrix,and,Due to non-singular transformation does not change the system characteristic polynomial,we can obtain the characteristic polynomial,of,the closed-loop system shown in the figure as following,we can get the model by,non-singular transformation,Homework,5-1,5-2,5-5,5-9,5-10,