1. Automatic Control Theory School of Automation NWPU Teaching Group of Automatic Control Theory

2. Automatic Control Theory Exercises (32) 6 — 5, 6, 7

3. Summary （ 1 ） §6.3.2 The z-transform of typical function ⑴⑵⑶⑷⑸⑹⑺⑻⑴⑵⑶⑷⑸⑹⑺⑻ §6.3.1 Definition of z-T.

4. Summary （ 2 ） 1. linear property §6.3.3 Properties of z-Transform 2. Real shifting theorem Lag 3. Complex shifting theorem Lead 4. Initial-value theorem 5. Final-value theorem 6. Convolution theorem

5. Summary （ 3 ） §6.3.5 Limitations of z-Transform §6.3.4 Inverse z-Transform Long Division Partial-Fraction expansion Residue Expansion of (1) only shows the information of samples; (2) In some cases, the continuous signal may jump on the sampling point.

6. Automatic Control Theory （ Lecture 32 ） Chapter 6 Analysis and Design of Linear Discrete-Time Systems § 6.1 Discrete-Time Control Systems § 6.2 Signal Sampling and Holding § 6.3 z-Transform § 6.4 Mathematical Models of Discrete-Time Systems § 6.5 Stability and Steady-state Errors of Discrete-Time Systems § 6.6 Dynamic Performance Analysis of Discrete-Time Systems § 6.7 Digital Control Design for Discrete-Time Systems

7. Automatic Control Theory （ Lecture 32 ） §6 Analysis and Design of Linear Discrete-Time Systems §6.4 Mathematical Models of Discrete-Time Systems

8. §6.4 Mathematical Models of Discrete-Time Systems (1) §6.4.1 Linear Time-Invariant Difference Equations (1) Definition of difference e(kT) = e(k) Forward difference First-order Second-order nth-order Backward difference First-order Second-order nth-order

9. §6.4 Mathematical Models of Discrete-Time Systems (2) (2) Difference equation The (forward) difference equation of nth-order linear time-invariant discrete system. The equation of the input, output and their higher order differences. (3) To solve difference equations: The (backward) differential equation of n-order linear time-invariant discrete system. Iteration method Z-transform method

10. §6.4 Mathematical Models of Discrete-Time Systems (3) Solution. Example1 The differential equation of a continuous system is ： Obtain the corresponding forward difference equation and its solution.

11. Solution Solution I of the difference equation —— Iteration method §6.4 Mathematical Models of Discrete-Time Systems (4)

12. Solution II of difference equation — Z-transform method §6.4 Mathematical Models of Discrete-Time Systems (5)

13. §6.4.2 Mathematical Models in Complex Domain — Impulse Transfer Function 1.Definition ： The ratio of the z-T. of the output to the z-T. of the input under zero initial condition §6.3 离散系统的数学模型 （ 4 ） §6.4 Mathematical Models of Discrete-Time Systems (6) Convolution formula — The z-transform of unity impulse response sequence

14. 2. The properties of impulse transfer function: (1) G(z) is a complex function of complex variable z ； (2) G(z) depends only on the structure and parameters of the system ； (3) G(z) has a relation with the difference equation of the system ； (4) G(z) is equal to Z[ k * (t) ] ； (5) G(z) ~ zero-pole location in z plane. 3. The limitation of impulse-transfer functions (1) It can not reflect the full information of the system response under non-zero initial conditions; (2) It is only for SISO discrete systems; (3) It is only for linear time-invariant difference equations; §6.3 离散系统的数学模型 （ 4 ） §6.4 Mathematical Models of Discrete-Time Systems (7)

15. Example 2 Consider the discrete system shown in the figure (T=1). Obtain (1)Impulse-transfer function of the system (2)Zero-poles location in z plane; (3)Difference equation of the system. Solution. (1) (3) (2) Zero-poles location in z plane §6.4 Mathematical Models of Discrete-Time Systems (8)

16. §6.4.3 Impulse Transfer Function of Open-Loop Systems (1) Switch between factors (2) No switch between factors §6.4 Mathematical Models of Discrete-Time Systems (9)

17. (3) ZOH in the system ZOH does not change the system order and O.-L. poles but changes the O.-L. zeros. §6.4 Mathematical Models of Discrete-Time Systems (10)

18. §6.4.4 Impulse Transfer Function of Closed-Loop Systems  (z) (The Mason’s formula is generally not applicable.) §6.4 Mathematical Models of Discrete-Time Systems (11) Example 1.

19. §6.4 Mathematical Models of Discrete-Time Systems (12) Example 2.

20. §6.4 Mathematical Models of Discrete-Time Systems (13) Example3. Obtain

21. §6.4 Mathematical Models of Discrete-Time Systems (14) Example 3 Obtain

22. The Mason’s formula can obtain  (z) or C(z) in the following two cases §6.4 Mathematical Models of Discrete-Time Systems (15) I.Single loop (no feedforward）discrete system; at least one actual sampling switch in forward path. II.There are actual or equivalent switches between all factors.

23. Summary §6.4.1 Linear Time-Invariant Difference Equations (1) Definition of difference ① Forward ② Backward §6.4 Mathematical Models of Discrete-Time Systems (2) The difference equation and its solving method ① Iteration ② Z-transformation §6.4.2 Impulse-Transfer Function (1) Definition(2) Properties(3) Limitation §6.4.3 Impulse Transfer Function of Open-Loop Systems (1) Switch between factors (2) No switch between factors (3) With ZOH §6.4.4 Impulse Transfer Function of Closed-Loop Systems (1) General Method (2) Mason’s formula

24. Automatic Control Theory Exercises (32) 6 — 5, 6, 7