System Stability and Time Responses
Explore the concepts of system stability, time responses, and controller design in the realm of control systems. Learn how to assess, achieve stability, evaluate systems quantitatively, and set design specifications. Experiment with typical test inputs and analyze transient and steady-state responses for system evaluation and controller design.
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Presentation Transcript
Whats to come We will subsequently learn about system stability and how to assess and achieve it. Stability is a necessary requirement, but not sufficient for most control systems. Specifications other than stability How to evaluate a system quantitatively in t-domain? How to give design specifications in t-domain? What are the corresponding conditions in s-domain? (we will design controllers in the s-domain). 2
Temperature controlexample We want to change the room temperature from 10 to 20 degree. Stable system Oscillatory Too long Are we happy with theseresponses? convergencetime 3
Timeresponse 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) 4
Ex: Transient & steady-stateresponses Step response 0.8 Transient response 0.6 Steady-state resp. 0.4 0.2 0 0 2 4 8 6 10 12 Time (sec) 5
Usage of timeresponses Modeling Some parameters in the system may be estimated by time responses. Analysis A system can be evaluated by seeing transient and steady-state responses. (Satisfactory or not?) Design Given design specs in terms of transient and steady- state responses, controllers are designed to satisfy all the design specs. 6
Typical testinputs Step function (Most popular) Ramp function Sinusoidal input will be dealt with later. Parabolic function 7
Steady-state value for stepinput G(s) Suppose that G(s) is stable. By the final value theorem: Step response converges to some finite value, called steady-state value 9
Peak value, peaktime percentovershoot 11
Delay, rise, and settlingtimes Delay time: time to reach 0.5yss Rise time: time to rise from 0.1yss to 0.9yss 2% settling time: time to settle down within 2% of yss (5% settling time is alsopopular.) 12
An examplerevisited For the example in Slide 6: Steady-state error : 1-0.8=0.2 Delay time around 1.5sec Step response 0.8 0.6 Rise time around 5sec Settling time around 8sec Remark: There is no peak in this case, so the following are undefined. peak value peak time percent overshoot 0.4 0.2 0 0 2 12 10 8 6 4 Time (sec) 13
Remarks on timeresponses Speed of response is measured by Rise time, delay time, peak time and settlingtime Relative stability is measured by Percent overshoot Typically . Fast response (short rise time, short peaktime) Large percent overshoot Small stabilitymargin In controller design, we need to take trade-off between response speed and stability. ( No-free-lunch theorem in Control Engineering) 14
Performancemeasures Transient response Peak value time Percentovershoot Delay time Rise time Settling time Steady stateresponse Steady stateerror (Next lecture) Peak Over the next few lectures we will connect these measures with the s-domain. (a later lecture) 15
Summary A stable system time response consists of transient and steady state components. This is also termed natural and forced responses, respectively. Typical test input signals: step, ramp and parabola Typical response specifications: e.g. percent overshoot, settling time Next lecture, transient response (1st and 2nd order systems( 16
