Unformatted text preview: EE3530, Spring 2008 Homework # 6 4003 2. Bode de...ned the sensitivity function relating a transfer function G to one of its parameters k as the ratio of percent change in k to percent change in G. We de...ne the reciprocal of Bode' function as s
G Sk = d ln G k dG dG=G = = : dk=k d ln k G dk Thus, when the parameter k changes by a certain percentage, S tells us what percent change to expect in G. In control systems design we are almost always interested in the sensitivity at zero frequency, or when s = 0. The purpose of this exercise is to examine the eect of feedback on sensitivity. In particular, we would like to compare the topologies shown in Fig. 4.31 for connecting three ampli...er stages with a gain of K into a single ampli...er with a gain of 10. (a) For each topology in Fig.4.31, compute 10R.
i so that if K = 10, Y = G (b) For each topology, compute Sk when G = Y =R. [Use the respective i values found in part (a).] Which case is the least sensitive? (c) Compute the sensitivities of the systems in Fig. 4.31(b, c) to 2 and 3 . Using your results, comment on the relative need for precision in sensors and actuators. Figure 4.31: Three ampli...er topologies for problem 4.2 1 EE3530, Spring 2008 Homework # 6 4005 3. Compare the two structures shown in Fig. 4.32 with respect to sensitivity to changes in the overall gain due to changes in the ampli...er gain. Use the relation S= d ln F K dF = : d ln K F dK as the measure. Select H1 and H2 so that the nominal system outputs satisfy F1 = F2 , and assume KH1 > 0. 4008 CHAPTER 4. BASIC PROPERTIES OF FEEDBACK Problems and Solutions for Section 4.2. Control of Steady State Error
5. Consider the second-order plant 1 G(s) = : (s + 1)(5s + 1) 2 K K F1 = ( )2 ; F2 = 1 + KHand error constant with respect to track1 + K 2 H2 (a) Determine the system type 1 ing polynomial reference inputs of the system for P, [D = kp ] PD kI [D = kp +kD s],Fand PID2[D = kp + +kD s] controllers Let kp = 19, 2 F2 s 1 SK = ; SH = kI = 9:5, and kD = 4: + KH1 1 1 + K 2 H2
Solution: Figure 4.32: Block diagrams for Problem 4.3 (b) Determine the system type and error constant of the system with respect to disturbance inputs for each of the three regulators in part (a) with respect to rejecting polynomial disturbances w(t) at the 2 input to the plant. = F2 =) H2 = H1 + 2H1 F1 K (c) Is this system better at tracking references or rejecting disturbances? Explain your response brie y. F SK1 2 F2 for (d) Verify your results = parts (a) and (b) using MATLAB by plotSK = (1 + KH1 )2 + KH1 ting unit step and ramp responses for1both tracking and disturbance rejection. System 2 is less sensitive. Solution: (a) P: Y (s) kp G(s) 19 = = 2 R(s) 1 + k p G(s) 5s + 6s + 20 19 1
2 EE3530, Spring 2008 Homework # 6 4010 CHAPTER 4. BASIC PROPERTIES OF FEEDBACK 6. Consider a system with the plant transfer function G(s) = 1=s(s+1). You wish to add a dynamic controller so that ! n = 2 rad/sec. and 0:5. Several dynamic controllers have been proposed: (1) D(s) = (s + 2)=2, s+2 , s+4 (s + 2) , (3) D(s) = 5 s + 10 (s + 2)(s + 0:1) (4) D(s) = 5 , (s + 10)(s + 0:01) (2) D(s) = 2 (a) Using MATLAB, compare the resulting transient and steady-state responses to reference step inputs for each controller choice. Which controller is best for the smallest rise time and smallest overshoot? (b) Which system would have the smallest steady-state error to a ramp reference input? (c) Compare each system for peak control eort, that is, measure the peak magnitude of the plant input u(t) for a unit reference step input. (d) Based on your results from parts (a) to (c), recommend a dynamic controller for the system from the four candidate designs. Solution: (a) The transfer functions for cases (1)-(4) are, 1) s+2 s+2 Y (s) = 2 = R(s) s + 2s + 2 (s + 1 j1) 2) Y (s) s+2 (s + 2) = 3 = 2 + 5s + 2 R(s) s + 5s (s + :58 j:42)(s + 3:8) 3) Y (s) 5s + 10 = 3 R(s) s + 11s2 + 15s + 10 4) Y (s) (s + 2)(s + :1) = R(s) (s + :11 j:1)(s + 9:9)(s + :87) See attached transient responses.
3 EE3530, Spring 2008 Homework # 6 4016 CHAPTER 4. BASIC PROPERTIES OF FEEDBACK 10. Consider the DC-motor control system with rate (tachometer) feedback shown in Fig. 4.34(a).
0 (a) Find values for K 0 and kt so that the system of Fig. 4.34(b) has the same transfer function as the system of Fig.4.34(a). (b) Determine the system type with respect to tracking r and compute 0 the system Kv in terms of the new parameters K 0 and kt . (c) Does the addition of tachometer feedback with positive kt increase or decrease Kv ? Figure 4.34: Control system for Problem 10 Solution: (a) Using block diagram reduction techniques: 1 to its output. k - Eliminate the second summer by absorbing Kp : This will result in Figure 4.34(b) where - Move the picko point from the input of the K0 = Kp KKm k kkt 0 kt = : Kp kKm m s + kKm kt ) (b) The inner-loop in Fig. 4.34(a) may be reduced to s(1 + which means that the unity feedback system has a pure integrator in the forward loop and hence it is Type 1 with respect to reference kKm input ( r ) and Kv = (1 + kKm kt ) (c) The introduction of kt reduces the velocity constant and therefore makes the error to a ramp larger 4 ...
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