CHAP.6 DISCRETE TIME CONTROL SYSTEM 离散时间控制系统
离散时间控制系统
6.3 Data-Sampled Control System u(t y(t) Continuous Time System X(0 Hold Device Sample Device u(k) X(k) y(k D/A Digital Computer A/D u(民c) Discrete Model of y() Continuous Time System Figure 6.3 Block Diagram of Discretization Continuous Time System X() D/A Digital Computer A/D Figure 6.4 Discrete Model of Continuous Time System
Figure 6.3 Block Diagram of Discretization Continuous Time System Figure 6.4 Discrete Model of Continuous Time System
6.3.2 Three Basic Assumptions 1.Assumption about the Sampling Operator (K)= y(t),t =kT 0, tk≠kT y(t) kT 0 Figure 6.5 Schematic Diagram of Sampling
1. Assumption about the Sampling Operator = = t k T y t t k T y k k k 0, ( ), ( ) Figure 6.5 Schematic Diagram of Sampling
2.Assumption about the Sampling Period The sampling period should satisfy the conditions that are determined by the Nyquist-Shannon sampling Theorem: 可以从采样信号中完全复现被采样的连续信号的条件是 采样频率0、必须大于或等于连续信号频谱中所含最高次 谐波频率 0。的两倍，即：0、>20。 +(@川 0 c Figure 6.6 The Amplitude Spectrum of Continuous Signal
2. Assumption about the Sampling Period The sampling period should satisfy the conditions that are determined by the Nyquist-Shannon sampling Theorem: 可以从采样信号中完全复现被采样的连续信号的条件是 采样频率 必须大于或等于连续信号频谱中所含最高次 谐波频率 的两倍，即： s 2c s c Figure 6.6 The Amplitude Spectrum of Continuous Signal
3.Assumption about the Holding Operation The zero-order hold is generally used. 零阶保持器将采样信号在每个采样时刻的采样值，一直 保持到下一个采样时刻，从而使采样信号变成阶梯序列信 号。 44,() ↑4，) T Figure 6.7 The Schematic Diagram of Zero-Order Holding Mode
3. Assumption about the Holding Operation The zero-order hold is generally used. 零阶保持器将采样信号在每个采样时刻的采样值，一直 保持到下一个采样时刻，从而使采样信号变成阶梯序列信 号。 Figure 6.7 The Schematic Diagram of Zero-Order Holding Mode
6.3.3 Discretization from the State Solution of Continuous Time System 直接离散化 Theorem 6.2 Consider the continuoustime LTI-system X(t)=LX(t)+B(t),X(t)=X。t≥0 .(6.71) y(t)=CX(t)+Du(t) the discretized model of (6.71)based on the basic assumptionsis xk+D=Gx0)+Hak),X0=X,k=0l.6.72 y(k)=CX(k)+Du(k)
直接离散化
Thevariables have the followingrelations between (6.71)and (6.72). .X()=[X(0-r,4k=[401=r,Jy(k)=[y(]-r(6.73) and the coefficient matrixes have the followingrelations: G=e,H=∫ediB C=[Ctl-r,D=[D1-r.(6.74) Where,T.is the sampling period
the state response of continuous time LTI system(6.71)is X(t)-ex(t)eBu(t)dt 令1=(k+1)T,=kT X(k+)X(Bu(ydt ↓ zero-orderhold )=kT+5)=kDk=012, X(k+1)T)=eX(kT)+-(Bd(kT+Eu(kT) =eX(kT)+e-BdEu(kT) (6.78) i记G=e”H=e-s9Bc 1=T-5 H=eBd(T-1)=-e"Bdi=ediB
the state response of continuous time LTI system (6.71) is = + − − t t t t t t e t e d 0 0 ( ) ( ) ( ) ( ) 0 ( ) X X Bu A A t = (k +1)T t 0 = kT + + − + = + k T kT T k T k T e k T e d ( 1) [( 1) ] X(( 1) ) X( ) Bu( ) A A zero-order hold u( ) = u(k T + ) = u(k T) k = 0,1,2, = + + = + + − + − + T T T T A T k T kT e k T e d k T k T e k T e d k T k T 0 ( ) 0 [( 1) ( )] ( ) ( ) (( 1) ) ( ) ( ) ( ) X B u X X B u A A A T e A G = = − T T e d 0 ( ) H B A t = T − H B B B A A A − − = − = − = T t T T T T t t e d T t e dt e dt 0 0 0 ( ) (6.78) 记 令
then (6.78)can be rewritten as X(k+1)=GX(k)+Hu(k)离散系统的状态方程 When t=kT,the discrete expression of output equation can be y(k)=CX(k)+Du(k) 离散系统的输出方程
When , the discrete expression of output equation can be then (6.78) can be rewritten as X(k +1) = GX(k) + Hu(k) 离散系统的状态方程 y(k) = CX(k) + Du(k) t = kT 离散系统的输出方程
Example 6.8 Consider the continuous time LTI system described by v(t)=[11X(t0 1、 2、 Please discretize the system firstly,and.the sampling period.T=0.02s,then determine the state and output response value at every sampling point,when the initial stateis X(0)=1-1,and the inputis unit step signal. Solution The state transition matrix of the original-continuous time system has been calculated in Chapter 2 as (0)=e 2e'-e4 -e2 -2e+2e2 -e+2e-2
1、 2