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(bǔ)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(guò)大，經(jīng)過(guò)積分器放大後造成致動(dòng)器輸出爬升至上限，造成控制輸出無(wú)法繼續(xù)提升系統(tǒng)輸出，達(dá)到積分飽和。此時(shí)迴路視同斷路，積分器變成不穩(wěn)

3、定的裝置，且系統(tǒng)存在迴授路徑因過(guò)載而燒掉的危險(xiǎn)性。當(dāng)發(fā)生積分飽和時(shí)，除非重新給予積分器一個(gè)正常的數(shù)值以清除累積的積分效應(yīng)，不然會(huì)持續(xù)輸出一個(gè)極大值。Anti Windup系統(tǒng)架構(gòu)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(yōu)點(diǎn)：1. 已經(jīng)考慮了溫度、灰塵、水、震動(dòng)與EMI等問(wèn)題2. 較一般PC更小巧3. 耐惡劣環(huán)境，壽命大於三年4. 直接記憶存取，記憶體容量大5. 作業(yè)系統(tǒng)支援普及(如DOS, Windows, CWIN等)，發(fā)展軟體支援多樣化6. 擁有一般PC之功能，如具高速數(shù)據(jù)處理能力，發(fā)展工具多元化，有現(xiàn)成多樣的擴(kuò)充介面，如數(shù)位介面卡、A/D卡與D/A卡，強(qiáng)大的網(wǎng)路功能，高開(kāi)放性的架

6、構(gòu)，具親和力與方便操作的人機(jī)介面以及價(jià)格競(jìng)爭(zhēng)力高等7. 生產(chǎn)廠商較多，具模組化設(shè)計(jì)，有較統(tǒng)一的產(chǎn)品規(guī)格8. 維修容易缺點(diǎn)：1. 成本高昂，較不適合用於大量低價(jià)的商品2. 體積較大，不適合用於小型系統(tǒng)DSP-Based Controllers:優(yōu)點(diǎn)：1. 採(cǎi)用RISC運(yùn)算架構(gòu)，處理速度可快達(dá)101600 MIPS以上2. 乘法運(yùn)算透過(guò)專(zhuān)用硬體處理，運(yùn)算速度快，可縮短複雜算術(shù)運(yùn)算的控制法則與數(shù)位訊號(hào)處理的執(zhí)行時(shí)間3. 具有定點(diǎn)與浮點(diǎn)運(yùn)算功能4. 晶片內(nèi)涵RAM、ROM、Serial I/O、CAM bus、A/D、PWM、Flash memory等周邊元件，產(chǎn)品可靠度高，電路設(shè)計(jì)簡(jiǎn)單5. 體積小，

7、耗電量小，價(jià)格便宜，易於商品化與工業(yè)產(chǎn)品化6. 具有多處理器平行處理能力缺點(diǎn)：1. 發(fā)展與作業(yè)系統(tǒng)受侷限，大多受限於C/C+或ASM語(yǔ)言2. 資料儲(chǔ)存量較小，記憶容量有限3. 人機(jī)介面較不易處理4. 硬體實(shí)現(xiàn)較PC-based控制器困難，無(wú)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(yōu)點(diǎn)：1. 價(jià)格低廉，適合小型控制器產(chǎn)品，如家電、3C產(chǎn)品等2. 體積小，甚至有SMT/SMD形式，適合消費(fèi)性產(chǎn)品3. 內(nèi)涵RAM、ROM、Serial I/O等周邊元件，產(chǎn)品可靠度高，電路設(shè)計(jì)簡(jiǎn)單4. 耗電

8、量少，甚至有省電裝置，可用於電池供電之?dāng)y帶型產(chǎn)品系統(tǒng)缺點(diǎn)：1. 運(yùn)算速度不及IPC-Based與DSP-Based控制器2. 發(fā)展工具侷限於C/C+及ASM語(yǔ)言3. 人機(jī)介面不易處理4. 硬體實(shí)現(xiàn)後，修改彈性不高5. 僅有整數(shù)運(yùn)算或定點(diǎn)小數(shù)運(yùn)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(bǔ)償/控制器之四大結(jié)構(gòu)圖 (Ch. 8 p.13p.14) (Two degree-of-freedom compensation)1. 串聯(lián)迴授補(bǔ)償 (Series-feedback compensation)2. 前饋補(bǔ)償器 (Forward / Feed-forward compensation)3. 命令式前饋控制器 (Command-feed-forward compensation)4. 前饋負(fù)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