Discription of Computer- Chapter 4 Discription of Computer- Controlled System

Contents 4.1 Description of Linear Discrete Systems 4.2 Pulse Response Function 4.3 Pulse Transfer Function 4.4 Open/closed-loop Pulse Transfer Function 4.5 Response of the CCS 4.6 Performance Specifications of the CCS 4.7 State Space Description of the CCS

4.1 Description of Linear Discrete Systems

4.1 Description of Linear Discrete Systems The system considered in the class is the linear time-invariant system, i.e. the relation between the output and input is unchangeable over time. r(kT)→y(kT); r(kT-iT)→y(kT-iT), k=0,1,2,…; i=…,-2,-1,0,1,2,…

4.2 Pulse Response Function Pulse response function is the basis for studying pulse transfer function. G(s) x(t) x*(t) y(t) y*(t) Fig. 4.1 Continuous system with impulse sampling signal input

4.2 Pulse Response Function

4.2 Pulse Response Function

4.2 Pulse Response Function

4.2 Pulse Response Function

4.3 Pulse Transfer Function

4.3 Pulse Transfer Function G(z) X(z) Y(z) Fig. 4.2. Diagram of Pulse Transfer System

4.3 Pulse Transfer Function

4.3 Pulse Transfer Function

4.4 Open/closed-loop Pulse Transfer Function 4.4.1 Laplace transform of the sampled signal G(s) x(t) x*(t) y(t) Y(s) X*(s) X(s)

4.4 Open/closed-loop Pulse Transfer Function

4.4 Open/closed-loop Pulse Transfer Function

4.4 Open/closed-loop Pulse Transfer Function 4.4.2 Properties of X*(s)

4.4 Open/closed-loop Pulse Transfer Function

4.4 Open/closed-loop Pulse Transfer Function 4.4.3 How to get pulse transfer function (1) System with sampler G(s) x(t) x*(t) y(t) y*(t) Fig. 4.3 system with sampler

4.4 Open/closed-loop Pulse Transfer Function (2) System without sampler G(s) x(t) y(t) X(s) Y(s) Fig. 4.4 system without sampler

4.4 Open/closed-loop Pulse Transfer Function (3) The methods to get pulse transfer function G(s) x(t) x*(t) y(t) y*(t)

4.4 Open/closed-loop Pulse Transfer Function 4.4.4 Pulse transfer function and difference equation Pulse transfer function can be converted to the difference equation, and vice versa.

4.4 Open/closed-loop Pulse Transfer Function

4.4 Open/closed-loop Pulse Transfer Function

4.4 Open/closed-loop Pulse Transfer Function

4.4 Open/closed-loop Pulse Transfer Function 4.4.5 Pulse transfer function of the system with ZOH The transfer function of the zero order holder is The transfer function of the system with zero order holder is

4.4 Open/closed-loop Pulse Transfer Function

4.4 Open/closed-loop Pulse Transfer Function 4.4.6 Open loop pulse transfer function of the system (1) Pulse transfer function of cascaded elements without sampler between them G1(s) G2(s) R(s) C1(s) C*(s) C(s) C(z) R(z) R*(s) G(z) T (s) Fig. 4.5 cascade connection without sampler

4.4 Open/closed-loop Pulse Transfer Function

4.4 Open/closed-loop Pulse Transfer Function (2) Pulse transfer function of cascaded elements with sampler between them G1(s) G2(s) R(s) C1(s) C*(s) C(s) C(z) R(z) R*(s) G(z) T C1*(s) C1(z) Fig. 4.6 cascade connection with sampler

4.4 Open/closed-loop Pulse Transfer Function

4.4 Open/closed-loop Pulse Transfer Function

4.4 Open/closed-loop Pulse Transfer Function (3) Pulse transfer function of parallel elements G1(s) U*(s) G2(s) Y(s) Y*(s) U(s)  Y1(s) Y2(s)

4.4 Open/closed-loop Pulse Transfer Function G1(s) U*(s) G2(s) Y(s) Y*(s) U(s)  Y1(s) Y2(s)

4.4 Open/closed-loop Pulse Transfer Function 4.4.7 Closed-loop pulse transfer function of the system (1) Sampler is located after the comparator G (s) E(s) C(s) E(z) E*(s) Ф(z) H (s) R(s) R*(s) R(z) C*(s) C(z) B(s) - Fig. 4.7 sampler is located after the comparator

4.4 Open/closed-loop Pulse Transfer Function

4.4 Open/closed-loop Pulse Transfer Function

4.4 Open/closed-loop Pulse Transfer Function (2) Sampler is located at the feedback channel G (s) E(s) C(s) E*(s) H (s) R(s) R*(s) R(z) C*(s) C(z) B(s) - Fig. 4.8 Sampler is located at the feedback channel

4.4 Open/closed-loop Pulse Transfer Function

4.4 Open/closed-loop Pulse Transfer Function (3) Sampler is located at the forward channel T R*(s) R(z) C*(s) R(s) E*(s) E(s) G (s) E(z) - C(z) B(s) H (s) Fig. 4.9 sampler is located at the forward channel

4.4 Open/closed-loop Pulse Transfer Function

4.4 Open/closed-loop Pulse Transfer Function

4.4 Open/closed-loop Pulse Transfer Function G2(s) E(s) C(s) R(s) - G1(s) E*(s) U(s)

4.4 Open/closed-loop Pulse Transfer Function

Fig. 4.10 Open loop sampling system 4.5 Response of the CCS (1) System response at sampling instant Gh(s) G1(s) Gp(s) H(s) x(t) x*(t) y*(t) Fig. 4.10 Open loop sampling system

4.5 Response of the CCS

4.5 Response of the CCS

4.5 Response of the CCS

The response of the open-loop control system 4.5 Response of the CCS The response of the open-loop control system

4.5 Response of the CCS G2(s) E(s) C(s) R(s) - G1(s) E*(s) U(s) H(s)

4.5 Response of the CCS

4.5 Response of the CCS 0.5 0.5

The response of the open-loop control system 4.5 Response of the CCS The response of the open-loop control system

4.5 Response of the CCS (2) System response between consecutive sampling instants-Laplace transform method G (s) E(s) C(s) E(z) E*(s) Ф(z) H (s) R(s) R*(s) R(z) C*(s) C(z) B(s) -

4.5 Response of the CCS

4.5 Response of the CCS Example 4.9: Assume r(t)=1(t), T=1s, find the analyzing solution of system output c(t). G2(s) E(s) C(s) R(s) - G1(s) E*(s) U(s)

4.5 Response of the CCS

4.5 Response of the CCS

4.5 Response of the CCS

4.5 Response of the CCS

4.5 Response of the CCS

4.5 Response of the CCS

4.5 Response of the CCS Continuous output

4.6 Performance Specifications of CCS Dynamic specifications: Delay time (td) : the time for the response to get to the half of the final value Rising time (tr) : the time for the response to get to 90% of the final value from 10% of the final value for the over-damped system and transmission lag system; the time for the response to get to 100% of the final value from 0 for under-damped systems

4.6 Performance Specifications of CCS Peak time (tp) : the time for the response to get to the first peak Overshoot (σp) : c(tp)-c(∞)/ c(∞)x100% Settling time (ts) : if t≥ts, |c(t)- c(∞)|<Δ(Δ=0.02 c(∞) or 0.05 c(∞)), ts is set as the settling time, which is related to the maximum time constant of the system

4.6 Performance Specifications of CCS Dynamic specifications

4.6 Performance Specifications of CCS Stable state specifications: stable state error Integrated specifications: different performance specifications can be defined in the optimal control.

4.7 State Space Description 4.7.1 Sampling continuous-time signals The sampled version of the signal f is the sequence {f(tk):kZ}, where tk=kh, it is called periodic sampling and h is sampling period. The corresponding frequency fs=1/h(Hz) or s=2/h(rad/s) is called sampling frequency. Nyquist frequency is a very useful frequency fN=1/(2h)(Hz) or N=/h(rad/s).

4.7 State Space Description 4.7.2 Sampling a continuous-time state-space system To find the discrete-time equivalent of a continuous-time system is called sampling a continuous-time. The model obtained is called a stroboscopic model. Consider the system (4.7.1) (4.7.1)

4.7 State Space Description (1) Zero-order-Hold sampling of a system A common situation in computer control is that the D-A converter is a zero-order-hold circuit, i.e. it holds the analog signal constant until a new conversion is commanded.

4.7 State Space Description The state at some future time t is obtained by solving (4.7.1). The state at time t, where tk t  tk+1 ,is

4.7 State Space Description The system equation of the sampled system at the sampling instants is then (4.7.2) The model in (4.7.2) is called a zero-order-hold sampling of the system in (4.7.1), can also be called the zero-order-hold equivalent of (4.7.1).

4.7 State Space Description For periodic sampling with period h, the model of (4.7.2) simplifies to the time-invariant system:

4.7 State Space Description (2) How to compute  and  i. Using the equation directly for 1-order system Example 4.10 First-order system where a  0. we get The sampled system thus becomes

4.7 State Space Description ii. Numerical calculation in MATLAB or MATRIX when a = 1, b = 3, c = 1, d = 1, h = 0.1 in Example 4.10, then a. Using Matlab command line, the result is as follows: >> sys=ss(1,3,1,1); >> sysd=c2d(sys,0.1) The result is  = 1.105,  = 0.3155, c = 1, d = 1.

4.7 State Space Description b. The second method using Matlab, the command is expm. From the following equation, We know that so >> expm([1 3;0 0]*0.1) ans = 1.1052 0.3155 0 1.0000

4.7 State Space Description iii. Series expansion of the matrix exponential Example 4.11 Double integrator The double integrator is described by Hence The discrete-time model of the double integrator is

4.7 State Space Description iv. The Laplace transform of exp(At) is (sI-A)-1 Example 4.12 Motor Simple normalized model of an electrical DC motor is given by The Laplace transform method gives Hence where L-1 is the inverse of the Laplace transform.

4.7 State Space Description (3) The inverse of sampling Get continuous-time system from a discrete-time description. where ln(.) is the matrix logarithmic function. The continuous-time system is thus obtained by taking the matrix logarithm function of a block matrix. *只有当矩阵Ф具有非负实数特征值时，该矩阵才存在对数。

4.7 State Space Description Example 4.13 Inverse sampling

4.7 State Space Description

4.7 State Space Description (4) State-space block diagram We can draw the state-space block diagram for a discrete-time control system, for example:

4.7 State Space Description (5) Solution of the system equation Time-invariant discrete-time systems can be described by the difference equation (4.7.3) For simplicity the sampling time is used as the time unit, h = 1. Assume that the initial condition x(k0) and the input signals u(k0), u(k0+1),…are given. How is the state then evolving? It is possible to solve (4.7.3) by simple iterations.

4.7 State Space Description

4.7 State Space Description Example 4.14 Solution of the difference equation If |λi|<1,j=1,2,then x(k) will converge to the origin. If one of the eigenvalues of Ф has an absolute value larger than 1, then one or both of the states will diverge.

4.7 State Space Description 4.7.3 Changing coordinates in state-space models Assume that T is a nonsingular matrix and define a new state vector z(k) = Tx(k). Then

4.7 State Space Description The invariants under the transformation are of interest. Theorem 3.1 Invariance of the characteristic equation. The characteristic equation is invariant when new states are introduced through the nonsingular transformation matrix T. Coordinates can be chosen to give simple forms of the system equations.

4.7 State Space Description (1) Diagonal Form Assume that  has distinct eigenvalues. Then there exists a T such that where i are the eigenvalues of .

4.7 State Space Description Consider the motor in example 4.12 with h=1. Using the transformation

4.7 State Space Description (2) Jordan Form If  has multiple eigenvalues, then it is generally not possible to diagonalize . Let  be a n  n matrix and introduce the notation where Lk is a k  k matrix. Then there exists a matrix T such that where k1 + k2 + ... + kr = n.

4.7 State Space Description 4.7.4 Transforms between state space model and pulse transfer function

4.7 State Space Description (1) Pulse-transfer function to state-space models If the pulse-transfer function is as following its state-space equation is

4.7 State Space Description

4.7 State Space Description Define state variable: With inverse z-transform Together with

4.7 State Space Description (2) Difference equation to state-space models If the forward formation is or the backward formation is

4.7 State Space Description Then the state-space equation is

4.7 State Space Description

4.7 State Space Description Define state variable: With inverse z-transform Together with

4.7 State Space Description (3) Discrete state-space to Z transfer function

4.7 State Space Description Example: Sampling this continuous-time system to obtain the corresponding discrete system, T = /2. When the input signal u(t) is a unit step, determine the analyzing solution of the y(t). Solution: 1. Compute discrete-time system

4.7 State Space Description When T = /2, The discrete-time system is

4.7 State Space Description 2. Compute the analyzing solution of y(t) The pulse-transfer function can now be introduced

4.7 State Space Description (z-transform of sin kT is ) Step response of the continuous-time system is

Summarization

Summarization ComputeΦ, Γ Inverse sampling Draw block diagram

Summarization (2) Changing coordinates in state-space models (3) Transforms between state space model and pulse transfer function

Homework 1. Find the pulse transfer function of the following systems:

Homework 2. Obtain the state-space equation for the following pulse-transfer function, and then draw the block diagram for the state-space form. 3. Obtain the state-space equation for the following difference equation, and then draw the block diagram for the state-space form.(y(0)=y(1)=u(0)=0)

Homework 4. Compute the z-pulse transfer function. 5. The system’s difference equation is It is assumed that y(k)=0 k<0. when the input is 1(t), determine the analyze solution of y(k).

Homework 6. Try to compute the transform matrix T when the original system and the changed system are described as follows: