现代控制理论chapter5-Linear Feedback Control System

Modern Control Theory

Chapter 5 Linear Feedback Control System

沈阳航空航天大学 自动化学院

Modern Control Theory

outline 5-1 Introduction 5-2 State feedback and output feedback 5-3 Pole-Placement 5-5 State Estimator

沈阳航空航天大学 自动化学院

Modern Control Theory

5-1 Introduction5.1.1 Analysis and Design analysis

quantitative and qualitative analysisAnalysis and Design

Response ,controllabilit y, observability stability foundation

Given the system, design of the structure and parameters of controller

Control the performance of the systemto meet the expected requirements

goal

沈阳航空航天大学 自动化学院

Modern Control Theory

5-2

State feedback and output feedback

5.2.1 state feedback General expression u (t ) f (r ), x(t ) LITS expressionu (t ) Lr (t ) Kx(t ) or

u(t ) r (t ) Kx

x Ax B( Lr Kx) ( A BK ) x BLr沈阳航空航天大学 自动化学院

Modern Control Theory

x Ax B( Lr Kx) ( A BK ) x BLr

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.

沈阳航空航天大学 自动化学院

Modern Control Theory

5.2.2 output feedbackBy linear feedback law

u r Hy r HCxState and output equations as

x ( A BHC ) x Bu y CxClosed-loop transfer function matrix :

GH C(sI A BHC) 1 B沈阳航空航天大学 自动化学院

Modern Control Theory

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 closedloop system will be able to maintain the observability.沈阳航空航天大学 自动化学院

Modern Control Theory

5.2.3 Discussion on State Feedback and Output Feedback

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.

沈阳航空航天大学 自动化学院

Modern Control Theory

explai n

Feedback system with state estimator Why?

Feedback control system with state estimator

We cannot measure the states physically Answer

We can construct the states based on input and output -----state estimator沈阳航空航天大学 自动化学院

Modern Control Theory

Note 1. Reco

nstruction 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.

沈阳航空航天大学 自动化学院

Modern Control Theory

5-3 Pole-placement5.3.1 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

沈阳航空航天大学 自动化学院

Modern Control Theory

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 antijamming

沈阳航空航天大学 自动化学院

Modern Control Theory

5.3.2 SISO pole-placementThe system is as following x Ax Bu y Cx It is controllable,where A、B、C are n n, n 1,1 nreal constant matrices, respectively. A linear state feedback is defined by

u (t ) r (t ) Kx(t )

The dynamic equations of the feedback systems is as:

x Ax B(V Kx) ( A BK ) x BV y (C DK ) x DV

沈阳航空航天大学 自动化学院

Modern Control Theory

Output feedback

u V Hy

x Ax B(V Hy) [ A BH ( I DH) 1C ]x [B BH ( I DH) 1 D]V y ( I DH) 1Cx ( I DH) 1 DV沈阳航空航天大学 自动化学院

Modern Control Theory

proveThe system’s characteristic polynomial:

f (s) sI A sn a1sn 1 an 1s anclosed-loop system characteristic polynomial by the state feedback :

f (s) sI A BK s a1sn n

n 1

an 1s an

Find non-singular transformation matrix P ,

x Px沈阳航空航天大学 自动化学院

Modern Control Theory

x Ax Bu y Cx

--controllable canonical form a1

0 A PAP 1 0 an

I n 1 an 1

B PB [0 1]T 0 CP 1 [ ] C n n 1 1

u r KP x r Kx 1

KP 1 K沈阳航空航天大学 自动化学院