ECE317 : Feedback and Control

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1 ECE317 : Feedback and Control Lecture : Steady-state error Dr. Richard Tymerski Dept. of Electrical and Computer Engineering Portland State University 1

2 Course roadmap Modeling Analysis Design Laplace transform Transfer function Block Diagram Linearization Models for systems electrical mechanical example system Stability Pole locations Routh-Hurwitz Time response Transient Steady state (error) Frequency response Bode plot Design specs Frequency domain Bode plot Compensation Design examples Matlab & PECS simulations & laboratories 2

3 Time response System We would like to analyze (stable) system s property by applying a test input r(t) and observing a time response y(t). Time response is divided as Transient (natural) response Steady-state (forced) response (after yt dies out) 3

4 Ex: Transient & steady-state responses Transient response Steady-state resp Step response Time (sec) 4

5 Typical test inputs Step function (Most popular) Ramp function Sinusoidal input was dealt with earlier freq. response Parabolic function 5

6 Steady-state value for step input G(s) Suppose that G(s) is stable. By the final value theorem: Step response converges to some finite value, called steady-state value 6

7 Steady-state error for input u(t) 7

8 Example revisited For the example on Slide 4: Steady-state error : 1-0.8= Step response Time (sec) 8

9 Performance measures Transient response Peak value Peak time Percent overshoot Delay time Rise time Settling time Steady state response Steady state error (Previous lectures) Next, we will connect these measures with s-domain. (Today s lecture) 9

10 Steady-state error of feedback system Suppose that we want output y(t) to track r(t). Error Steady-state error Assumptions L(s) = Plant(s)*Controller(s) Unity feedback (no block on feedback path) CL system is stable Final value theorem (Suppose CL system is stable!!!) 10

11 Error constants CL system s ability to reduce steady-state error ess Large error constant means large ability. Three error constants Step-error (position-error) constant Ramp-error (velocity-error) constant Parabolic-error (acceleration-error) constant 11

12 Steady-state error for step r(t) Kp 12

13 Steady-state error for ramp r(t) Kv 13

14 Steady-state error for parabolic r(t) Ka 14

15 Zero steady-state error When does steady-state error become zero? (i.e. accurate tracking!) Infinite error constant! For step r(t) L(s) must have at least 1-integrator. (system type 1) For ramp r(t) L(s) must have at least 2-integrators. (system type 2) For parabolic r(t) L(s) must have at least 3-integrators. (system type 3) 15

16 L(s) has 2-integrators. Example 1 Characteristic equation CL system is NOT stable for any K. e(t) will not converge. (Don t use today s results if CL system is not stable!!!) 16

17 L(s) has 1-integrator. Example 2 By Routh-Hurwitz criterion, CL is stable if Step r(t) Ramp r(t) Parabolic r(t) 17

18 L(s) has 2-integrators. Example 3 By Routh-Hurwitz criterion, we can show that CL system is stable. Step r(t) Ramp r(t) Parabolic r(t) 18

19 Integrators in L(s) Integrators in L(s) (i.e. plant and controller) are very powerful to eliminate the steady-state errors. Examples 2 & 3 Lab 5 addition of an integral compensator However, integrators in L(s) tend to destabilize the feedback system. Example 1 19

20 Unity Gain Feedback (H(s)=1) Table 3.1: 20

21 Unity Gain Feedback (H(s)=1) Procedure to determine steady state error: Given G(s) (and H=1) and input type: 21

22 Non-Unity Gain Feedback (H(s) 1) Ideal Gain Correction term 22

23 Non-Unity Gain Feedback (H(s) 1) DC gain of feedback block: Define steady state error: Final value theorem: Closed loop gain: 23

24 Non-Unity Gain Feedback (H(s) 1) Closed loop gain: Closed loop gain, general form: Steady state error: 24

25 Non-Unity Gain Feedback (H(s) 1) Table 3.2: 25

26 Non-Unity Gain Feedback (H(s) 1) Procedure to determine steady state error: Given G(s) and H(s) and input type: 26

27 Example 4 Solution: Steps: 27

28 Example 4, cont d 28

29 Example 4, cont d From Table 3.2: 29

30 Example 4, cont d Ideal output and actual output: Error evolution: 30

31 Example 5 Solution: Steps: 31

32 Example 5, cont d 32

33 Example 5, cont d From Table 3.2: 33

34 Example 6: H 1 Method (H 1 method can also be used for H=1 problems) Solution - Non-Unity Gain Feedback method: Steps: 34

35 Example 6: H 1 method, cont d 35

36 Example 6: H 1 method, cont d From Table 3.2: 36

37 Example 6: H=1 method Solution - Unity Gain Feedback method: From Table 3.1: 37

38 Example 6: H=1 method, cont d Which is the same result obtained by the H 1 method. 38

39 Example 6, cont d, time response 39

40 Summary Steady-state error Stable (!) unity and non-unity gain feedback systems are treated in a consistent manner. The key tool is the final value theorem. Main determinants of steady-state error is: 1) system type 2) input type 40

(b) A unity feedback system is characterized by the transfer function. Design a suitable compensator to meet the following specifications:

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