若控制误差过大

1、Problem 1. Simply answer the following questions.(a) Numerical approximations of integration (Ch.8 p.8p.15)Let If each integrator can be replaced by its digital equivalent,i.e. , .Hence, 1. Left-side rule: 2. Right-side rule: 3. Trapezoidal rule: 4. Simpsons rule: 5. Step Invariance. (p.15)/ Impulse

2、 Invariance. (p.13)/ Ramp Invariance. (p.43)6. Matched z-transform. (p.26)補充 Numerical approximations of differentiation(1) Backward difference: (2) Forward difference: (b) Anti Windup Schemes of Digital PID control (Ch.8 p.54p.55)若控制誤差過大，經過積分器放大後造成致動器輸出爬升至上限，造成控制輸出無法繼續提升系統輸出，達到積分飽和。此時迴路視同斷路，積分器變成不穩

3、定的裝置，且系統存在迴授路徑因過載而燒掉的危險性。當發生積分飽和時，除非重新給予積分器一個正常的數值以清除累積的積分效應，不然會持續輸出一個極大值。Anti Windup系統架構如右： (或參考上方圖片)(c) Design methods to synthesize digital controllers:1. Frequency Response Method2. Root-Locus Method3. State-Space Method4. Polynomial Approach(d) Controllability & ObservabilityControllability

4、: is controllable if there exists a sequence of input u(0), u(1), u(2), , u(n-1) that can translate the system from any initial state x(0) to any final state x(N) with N finite.Observability: is observable if for any initial state x(0), it can be calculated from the N measurements y(0), y(1), , y(N-

5、1) with input u(0), u(1), , u(N-1), with N finite.(e) Digital Controllers: (Ch.13 p.35p.38)IPC-based controllers:優點：1. 已經考慮了溫度、灰塵、水、震動與EMI等問題2. 較一般PC更小巧3. 耐惡劣環境，壽命大於三年4. 直接記憶存取，記憶體容量大5. 作業系統支援普及(如DOS, Windows, CWIN等)，發展軟體支援多樣化6. 擁有一般PC之功能，如具高速數據處理能力，發展工具多元化，有現成多樣的擴充介面，如數位介面卡、A/D卡與D/A卡，強大的網路功能，高開放性的架

6、構，具親和力與方便操作的人機介面以及價格競爭力高等7. 生產廠商較多，具模組化設計，有較統一的產品規格8. 維修容易缺點：1. 成本高昂，較不適合用於大量低價的商品2. 體積較大，不適合用於小型系統DSP-Based Controllers:優點：1. 採用RISC運算架構，處理速度可快達101600 MIPS以上2. 乘法運算透過專用硬體處理，運算速度快，可縮短複雜算術運算的控制法則與數位訊號處理的執行時間3. 具有定點與浮點運算功能4. 晶片內涵RAM、ROM、Serial I/O、CAM bus、A/D、PWM、Flash memory等周邊元件，產品可靠度高，電路設計簡單5. 體積小，

7、耗電量小，價格便宜，易於商品化與工業產品化6. 具有多處理器平行處理能力缺點：1. 發展與作業系統受侷限，大多受限於C/C+或ASM語言2. 資料儲存量較小，記憶容量有限3. 人機介面較不易處理4. 硬體實現較PC-based控制器困難，無泛用形式，修改變更彈性不高5. 軟體支援較PC-based控制器少，甚至必須自行完成Microcontroller-Based Controllers:優點：1. 價格低廉，適合小型控制器產品，如家電、3C產品等2. 體積小，甚至有SMT/SMD形式，適合消費性產品3. 內涵RAM、ROM、Serial I/O等周邊元件，產品可靠度高，電路設計簡單4. 耗電

8、量少，甚至有省電裝置，可用於電池供電之攜帶型產品系統缺點：1. 運算速度不及IPC-Based與DSP-Based控制器2. 發展工具侷限於C/C+及ASM語言3. 人機介面不易處理4. 硬體實現後，修改彈性不高5. 僅有整數運算或定點小數運算功能(f) Implementation issues of digital controllers: (Ch.13 p.3)1. Selection of sampling rate.2. Interfacing to sensors: prefilter selection, computational delay.3. Choice of A/D

9、and D/A converters.4. Interfacing to nonlinear actuators.5. Interfacing to human operators.6. Numerical errors and digital controller structures.7. Word length selection in digital arithmetic device and memory.8. Selection of digital computation devices.9. Programming: real-time programmingsoftware

10、engineering in digital control systems.(g) 雙自由度補償/控制器之四大結構圖 (Ch. 8 p.13p.14) (Two degree-of-freedom compensation)1. 串聯迴授補償 (Series-feedback compensation)2. 前饋補償器 (Forward / Feed-forward compensation)3. 命令式前饋控制器 (Command-feed-forward compensation)4. 前饋負載干擾 (Forward-load disturbance with series compen

11、sation)(h) Interfacing technique to nonlinear actuator. (Ch.13 p.16)1. All the control algorithms are designed based-on linear models of the plant.2. Nonlinear effects of physical actuators must be taken into account for the control system, especially during startup and shutdown, as well as large ch

12、anges.3. One of nonlinear effects caused by nonlinear actuators is integrator windup.4. Here simple heuristic methods are used to deal with nonlinear actuators.(i) Phase-lag compensator design procedure. (Ch.9 p.20p.21)(Using frequency response method)1. Determine the frequency , at which the phase

13、angle is approximately . The phase margin of the compensated system will occur at approximately the frequency.2. Choose to ensure that little phase lag is introduced at . Actually, the compensator will introduce approximately phase lag, which has been accounted for in step 1.3. At , we want . Since

14、the gain of the compensator at high frequencies is (where a0 is the compensator DC gain), then Solving the last two equations for yields the design is now complete, since the compensator DC gain, pole, and zero are known.(j) Equivalent discrete-time controllerProblem 2. True-False Problem(a)T(b)T(c)F(d)T(e)T(f)F(g)T(h)F(i)T(j)FProblem 3. Cont