# CHAPTER 7 STEADY-STATE RESPONSE ANALYSES

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1 CHAPTER 7 STEADY-STATE RESPONSE ANALYSES 1. Introduction The steady state error is a measure of system accuracy. These errors arise from the nature of the inputs, system type and from nonlinearities of system components such as static friction, backlash, etc. These are generally aggravated by amplifiers drifts, aging or deterioration. The steady state performance of a stable control system is generally judged by its steady state error to step, ramp and parabolic inputs. Consider a unity feedback system as shown in the Fig. 1. The input is R(s), the output is C(s), the feedback signal H(s) and the difference between input and output is the error signal E(s). Fig.1 Closed-loop unity feedback control system 1

2 The closed loop transfer function (CLTF) is: As we know Therefore, Steady-state error e ss may be found using the Final Value Theorem (FVT) as follows: The above equation shows that the steady state error depends upon the input R(s) and the forward transfer function G(s). The expression for steady-state errors for various types of standard test signals are derived in the following sections: 2. Steady state error and standard test input 2.1 Step input Where K p is the position error constant and equals G(0) or =lim S 0 GS 2.2 Ramp input Where K v is the velocity error constant and equals =lim S 0 S GS 2

3 2.3 Parabolic input Where K a is the acceleration error constant and equals =lim S 0 GS 3. Steady state error and system type The system type refers to the how many poles of G(s) are located at the origin (s = 0). For G(S) given below, the system is of n-type. 3.1 Type zero system (n=0) In that case G(S) = = Thus a system with n = 0, or no integration in G(s) has A constant position error, Infinite velocity error and Infinite acceleration error 3.1 Type one system (n=1) In that case G(S) = = Thus a system with n =1, or with one integration in G(s) has A zero position error, A constant velocity error and Infinite acceleration error 3

4 3.1 Type two system (n=2) In that case G(S) = Thus a system with n = 2, or with one integration in G(s) has A zero position error, A zero velocity error and A constant acceleration error Summary 4

5 Example #1 For the system shown below, find K so that there is 10% error in the steady state Since the system type is 1, the error stated in the problem must apply to a ramp input; only a ramp yields a finite error for in a type 1 system. Thus, 4. Steady state error of non-unity feedback systems Consider the non-unity feedback control system shown in Fig. 2. Fig. 2, non-unity feedback system Add to the block shown in Fig. 2, two feedback blocks H(S) = +1 and H(S) = 1 as shown in Fig. 3. Fig.3 5

6 Example #2 For the system shown below, find The system type Appropriate error constant associated with the system type, and The steady state error for unit step input The equivalent feed-forward transfer function is System Type: 0 Appropriate error constant is K p 6

7 Steady State Error e ss The negative value for E ss implies that the output is larger than the input step. Example #3 7

8 Which represent a type 1 system. = lim S 0 GS= = lim S 0 S GS= 0 = = = lim S 0 S GS= Example #4 Consider the servo system with tachometer feedback shown in Figure. Obtain the error signal E(s) when both the reference input R(s) and disturbance input D(s) are present. Obtain also the steady-state error when the system is subjected to a reference input (unit-ramp input) and disturbance input (step input of magnitude d). 8

9 Using super position: (when D(S) = 0) (when R(S) = 0) The overall output is 9

10 Example 5: For the control system shown below in Fig A, a) Determine the values of gain K and the time constant T so that the system response for unit-step input is as shown in Fig. B. b) With these values of K and T, obtain in part (a), find the rise time and peak time and percentage overshoot. c) Calculate the position error coefficient and the steady-state error. Fig. A From system response (Fig. B), T S = 4 (based on 2% tolerance) and T d = 0.5 since 4 = =4 Therefore, =1 Also, we have the delay time T d = 0.5 Multiply both sides by, then = =0.5 = +0.7 =0.5 But we get that, =1, therefore =0 Solving this equation = = = = / From the system block diagram Fig. B 10

11 The system T.F. = + 3+ The system characteristic equation is + 3+=0 The standard form of 2 nd order system characteristic equation is =0 By comparing 2 =3 = 2 3 Also, = = = The % maximum over shoot M p = 100= % Rise Time: = 1 Β = cos -1 () = = rad = =0.842 Peak Time: = 1 = =1.34 To get the steady-state error and position error coefficient, the system must be unity feedback, so we will add +ve and ve feedback as shown in fig 11

12 Then the unity feedback system will be =lim = = 1 1+ = = = = Example 6: The control system, shown in Fig. below, is subjected to a unit ramp function, a) Determine the value of K and K h such that the system has an overshoot of % and a damped natural frequency of rad/sec. b) Calculate the rise time, peak time and settling time based on ±2% tolerance c) Define the system type d) Calculate the steady-state error of that system e) If an integrator is added in the forward path after the 1 st summing point directly, calculate the new steady-state error. K ( s + 2) 1 s Maximum overshoot = = taking (ln) for both sides = 3.29 = = =. = = 0.25=0.5 Since the damped natural frequency (ω d ) = = 1 = =4 /

13 E The block diagram can be simplified as follows R(S) C(S) The closed loop transfer function = But the general form T.F. of the 2 nd order system is = By comparing the coefficient of both T.F. = =16 ## 2+ =2 =0.125 ## The angle = 0.5 Rise time (T r ) = = Peak time (T p ) = =... = = ## = ## Settling time (T s ) based on ±2% tolerance = =. =2.0. ## The system type = 1 The velocity error coefficient K V = lim =lim = =4 The steady-state error (E ss ) = =0.25 ## +2+ h = When an integrator is added in the forward path after the 1 st summing point directly as shown in Fig. R(S) + 1 Integrator +2+ C(S) The new G(s)H(s) = The new velocity error coefficient K V = lim =lim = The new steady-state error (E ss ) = = = 0 ## h = 13

14 Salman Bin Abdul Aziz University College of Engineering א Department of Electrical Engineering א EE3511 Automatic Control Systems Sheet 5 (Performance of feedback א א א control systems) 1) A thermometer requires 1 min to indicate 98% of the response to a step input. Assuming the thermometer to be a first-order system, find the time constant. 2) Consider the unit-step response of a unity-feedback control system whose openloop transfer function is 1 = +1 Obtain the rise time, peak time, maximum overshoot, and settling time. 3) Consider the closed-loop system given by = +2 + Determine the values of ζ and ω n, so that the system responds to a step input with approximately 5% overshoot and with a settling time of 2 sec. (Use the 2% criterion.) 4) Figure 1 is a block diagram of a space-vehicle attitude-control system. Assuming the time constant T of the controller to be 3 sec and the ratio K/J = 2/9 rad 2 /sec 2 Find the damping ratio of the system, rise time, maximum overshoot and peak time. Fig. 1 5) Consider the system shown in Fig. 2(a). The damping ratio of this system is and the undamped natural frequency is 3.16 rad/sec. To improve the relative stability, we employ tachometer feedback. Fig. 2(b) shows such a tachometerfeedback system. Determine the value of K h, so that the damping ratio of the system is 0.5. Then find the rise time, maximum overshoot and settling time and compare them with those obtained from the original system. 14

15 Fig. 2 6) Referring to the system shown in Fig. 3, determine the values of K and k such that the system has a damping ratio 0.7 and an undamped natural of 4 rad/sec. Fig. 3 7) For the system shown in Fig. 4, determine the values of gain K and velocity feedback constant K h so that the maximum overshoot in the unit-step response is 0.2 and the peak time is 1 sec. With these values of K and K h, obtain the rise time and settling time. Assume that J = 1 kg-m 2 and B = 1 N-m/rad/sec. Fig. 4 8) When the control system shown in Fig. 5 (a) is subjected to a unit-step input, the system output responds as shown in Fig. 5 (b). Determine the values of K and T from the response curve. Fig. 5 15

16 9) For the closed-loop system given by = Calculate the rise time, peak time, maximum overshoot, and settling time when R(s) is considered as unit step input. 10) Figure 6 shows three systems. System I is a positional servo system. System II is a positional servo system with PD control action. System III is a positional servo system with velocity feedback. Compare the unit-step response of the three systems and obtain the best one with respect to the speed of response and maximum overshoot. Fig. 6 11) For a unity feedback control systems given below, find the position (K p ), velocity (K v ) and acceleration (K a ) error coefficients. 12) For the system shown in Fig. 7, find K so that there is 10% E ss 16 Fig. 7

17 13) For the system shown in Fig. 8, find The system type Appropriate error constant associated with the system type, and The steady state error for unit step input Fig. 8 14) Consider the system shown in Fig. 9. Prove that the steady-state error for a unit ramp input is. Also show that the damping ratio is natural frequency is 1 and the undamped Fig. 9 15) Consider the system shown in Fig. 10(a). The steady-state error to a unit-ramp input is e ss =. Show that the steady-state error for following a ramp input may be eliminated if the input is introduced to the system through a proportional-plusderivative filter, as shown in Fig. 10(b), and the value of k is properly set. 17

18 (a) Fig. 10 (b) 16) Consider servo system with tachometer feedback shown in Fig. 11. Obtain the error signal E(s) when both the reference input R(s) and disturbance input D(s) are present. Obtain also the steady-state error when the system is subjected to a reference input (unit-ramp input) and disturbance input (step input of magnitude d). Fig For the control system shown in Fig. 12, Determine the steady-state error for a unit step when K = 0.4 and G p (s) = 1. Select an appropriate value for G p (s) so that the steady-state error is equal to zero for the unit step input. Fig Consider the closed-loop system in Fig. 13. Determine values of the parameters k and a so that the following specifications are satisfied: (a) The steady-state error to a unit step input is zero. (b) The closed-loop system has a percent overshoot of less than 5%. 18

19 Fig For the control system shown in Fig. 15, the system is subjected to an input r(t)=2t. where K 1 and K 2 are constants. (a) Obtain the system type, (b) Calculate the value of K 2 if the steady-state error E SS = and K 1 = 3.0, (c) If K 1 is set to zero and K 2 is reduced to its half value obtained in (b), calculate the steady-state error (E SS ). 2 8 ( s + 10s) Fig. 14 Fig

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