ECE317 : Feedback and Control


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1 ECE317 : Feedback and Control Lecture : Steadystate 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 RouthHurwitz 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 Steadystate (forced) response (after yt dies out) 3
4 Ex: Transient & steadystate responses Transient response Steadystate 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 Steadystate 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 steadystate value 6
7 Steadystate error for input u(t) 7
8 Example revisited For the example on Slide 4: Steadystate error : 10.8= 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 sdomain. (Today s lecture) 9
10 Steadystate error of feedback system Suppose that we want output y(t) to track r(t). Error Steadystate 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 steadystate error ess Large error constant means large ability. Three error constants Steperror (positionerror) constant Ramperror (velocityerror) constant Parabolicerror (accelerationerror) constant 11
12 Steadystate error for step r(t) Kp 12
13 Steadystate error for ramp r(t) Kv 13
14 Steadystate error for parabolic r(t) Ka 14
15 Zero steadystate error When does steadystate error become zero? (i.e. accurate tracking!) Infinite error constant! For step r(t) L(s) must have at least 1integrator. (system type 1) For ramp r(t) L(s) must have at least 2integrators. (system type 2) For parabolic r(t) L(s) must have at least 3integrators. (system type 3) 15
16 L(s) has 2integrators. 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 1integrator. Example 2 By RouthHurwitz criterion, CL is stable if Step r(t) Ramp r(t) Parabolic r(t) 17
18 L(s) has 2integrators. Example 3 By RouthHurwitz 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 steadystate 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 NonUnity Gain Feedback (H(s) 1) Ideal Gain Correction term 22
23 NonUnity Gain Feedback (H(s) 1) DC gain of feedback block: Define steady state error: Final value theorem: Closed loop gain: 23
24 NonUnity Gain Feedback (H(s) 1) Closed loop gain: Closed loop gain, general form: Steady state error: 24
25 NonUnity Gain Feedback (H(s) 1) Table 3.2: 25
26 NonUnity 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  NonUnity 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 Steadystate error Stable (!) unity and nonunity gain feedback systems are treated in a consistent manner. The key tool is the final value theorem. Main determinants of steadystate 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:
1. (a) The open loop transfer function of a unity feedback control system is given by G(S) = K/S(1+0.1S)(1+S) (i) Determine the value of K so that the resonance peak M r of the system is equal to 1.4.
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