Outline • Advantages/disadvantages. • Design procedures. • Examples. Frequency Response Controller Design M. Sami Fadali Professor of Electrical Engineering UNR 1 2 Design Procedure Frequency Response Design 1. Select Advantage: Familiar design procedure. and obtain the transfer function . Disadvantages: 2. Bilinearly transform 1. Indirect design: controller distortion. into using 2. Requires experience. 3. Familiar criteria (PM, GM) have different values from their analog counterparts for the same performance. MATLAB >> gd=c2d(g,T) >> d2c(gd,'tustin') 3 4 Design Procedure (Cont.) Deign Specifications 3. Draw the Bode plot of , and use analog frequency response methods to design a controller that satisfies the frequency domain specifications • PM, GM , BW • Use frequency response design procedures for analog systems (e.g. lag, lead, lag-lead). 4. Transform the controller back into the zplane using = gain crossover (0 dB magnitude) frequency • BW increases with 5. Verify that the performance obtained is satisfactory. 5 6 Solution Example 6.15 • T = 0.1 s, obtain z-transfer function Z Consider the cruise control system of Example 3.2, where the analog process is • Bilinear transformation Transform the corresponding to the -plane by considering both and . Evaluate the role of the sampling period by analyzing the corresponding Bode plots. . • T = 0.01 s, obtain z-transfer function • Bilinear transformation 7 . 8 Bode Plots , & Discussion For 1 2 • Pole in -plane in the same position as the pole in the s-plane. • Have TF zero while does not (different frequency response from the analog system • Greater influence of the zero on the system dynamics when the sampling period is smaller. • Distortion in the low frequency range is negligible. • Gain as goes to zero is unity as is the DC gain of the analog system. Bode Diagram Magnitude (dB) 0 G1(w) -20 G(s) -40 G 2(w) -60 -80 0 Phase (deg) -45 -90 -135 -180 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 Frequency (rad/sec) 9 10 Example 6.16 Solution • For 10% overshoot, DC motor speed control system: (type 0) analog plant has the transfer function Design a digital controller by using frequency response methods to obtain: (i) zero steady-state error due to a unit step, (ii) an overshoot less than 10%, (iii) a settling time of about 1 we calculate • Choose Z . . 11 . 12 Bilinear Transformation Bode Plot Bode Diagram Gm = 25.7 dB (at 32.2 rad/sec) , Pm = 61.3 deg (at 4.86 rad/sec) 100 , pole-zero cancellation (simple design) Magnitude (dB) 50 C(w)G(w) 0 G(w) -50 -100 -150 0 Phase (deg) -45 -90 -135 -180 -225 -270 -2 10 13 Step Response 0 10 1 10 2 10 3 10 4 10 5 10 Frequency (rad/sec) 14 Example 6.17 DC motor speed control system: (type 0) analog plant has the transfer function Step Response System: Gcl Peak amplitude: 1.07 Overshoot (%): 7.17 At time (sec): 0.56 1.2 -1 10 System: Gcl Settling Time (sec): 0.818 Amplitude 1 0.8 Design a digital controller by using frequency response methods to obtain: (i) zero steady-state error due to a unit step, (ii) an overshoot less than 10%, (iii) a settling time of about 1 s Use 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 Time (sec) 15 16 Solution Design • TF of analog system, ADC and DAC • Cancel two poles with zeros. • Add poles at zero and Z . . . • Poles almost in the same locations as poles of . to make the gain • Consider RHP zero at crossover frequency about 5 rad/s. 17 Bode plots of & 18 Closed-loop Step Response Step Response Bode Diagram Gm = 9.7 dB (at 18.2 rad/sec) , Pm = 59.9 deg (at 3.99 rad/sec) System: Gcl Peak amplitude: 1.08 Overshoot (%): 8.09 At time (sec): 0.4 40 Magnitude (dB) 20 1.2 C(w)G(w) 0 System: Gcl Settling Time (sec): 0.718 1 G(w) -20 Amplitude -40 -60 0 0.8 0.6 Phase (deg) -45 0.4 -90 -135 0.2 -180 -225 0 -1 10 0 10 1 10 2 10 3 10 4 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Time (sec) Frequency (rad/sec) 19 20