3.Time-Domain Analysis of Control system 2022-2-3
2022-2-3 1 3.Time-Domain Analysis Of Control System
3.1 Introduction Since time is used as an independent variable in most control systems, it is usually of interest to evaluate the time response of the systems In the analysis problem, a reference input signal is applied to a system and the performance of the system is evaluated by studying the response in the time domain The time response of a control system is usually divided into two parts: The transient response and the steady-state response, if c(t) denotes a time response, then, in general, it may be written C(t=c (t+css(t) Where c, (t=transient response 2022-2-3
2022-2-3 2 3.1 Introduction Since time is used as an independent variable in most control systems ,it is usually of interest to evaluate the time response of the systems .In the analysis problem ,a reference input signal is applied to a system ,and the performance of the system is evaluated by studying the response in the time domain.The time response of a control system is usually divided into two parts: The transient response and the steady-state response ,if c(t) denotes a time response ,then ,in general ,it may be written C(t)=ct(t)+css(t) Where ct(t)=transient response
3.2 Typical test signals for time response of control systems e Unlike many electrical circuits and communication systems, The input excitations to many practical control systems are not known ahead of time In many cases the actual inputs of a control system may vary in random fashions with respect to time For the purpose of analysis and design, it is necessary to assume some basic types of input functions so that the performance of a system can be evaluated with respect to this signals By selecting these basic test signals properly not only the mathematical treatment of the problem is systematized, but the responses due to this inputs allow the prediction of the systems performance to other more complex inputs In a design problem performance criteria may be specified with respect to these test signals so that a system may be designed to meet the criteria. To facilitate the time-domain analysis, the following deterministic test signal are often used 2022-2-3 3
2022-2-3 3 3.2 Typical test signals for time response of control systems ¨ Unlike many electrical circuits and communication systems ,The input excitations to many practical control systems are not known ahead of time .In many cases ,the actual inputs of a control system may vary in random fashions with respect to time. ¨ For the purpose of analysis and design ,it is necessary to assume some basic types of input functions so that the performance of a system can be evaluated with respect to this signals .By selecting these basic test signals properly ,not only the mathematical treatment of the problem is systematized ,but the responses due to this inputs allow the prediction of the systems performance to other more complex inputs .In a design problem ,performance criteria may be specified with respect to these test signals so that a system may be designed to meet the criteria. To facilitate the time-domain analysis ,the following deterministic test signal are often used
Test signals r(t) R(S) urpose Impulse r(t)=A(t),t≥0 stability test 0, <0 Step r()=A,t≥0 transient =0,t<0 response test Ramp (t)=A,t≥0 tracking 0,t<0 capability test Parabolic r()=Ar2,t≥0 fast tracking 0.t<0 capability test ∠UL∠-2-5 4
2022-2-3 4 Test signals r(t) R(s) Purpose Impulse stability test Step transient response test Ramp tracking capability test Parabolic fast tracking capability test , 0 0, 0 r t A t t t A , 0 0, 0 r t A t t A s , 0 0, 0 r t At t t 2 A s 2 , 0 0, 0 r t At t t 3 2 A s
3.3 First-Order systems o Unit-impulse response of the first-order system may be found by assuming, 1. e, the intensity of the impulse is equal to one, y(s +I(=~1 Ts+1 (3-3-1) y(t=e/r (3-3-2) 2022-2-3 5
2022-2-3 5 3.3 First –Order Systems ¨ Unit-impulse response of the first-order system may be found by assuming , i.e., the intensity of the impulse is equal to one, 1 1 1 1 Y s R s s s 1 t y t e (3-3-1) (3-3-2)
Unit-impulse response of the first-order syst 0.8 t=e/ 0.6 04 02 2 5 timet 2022-2-3 6
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Unit-step response of the first-order system may be found by assuming R(s)=1ys as Y R TS+1 zS+1八s)szs+1 y()=1-e (3-3-4 Unit-ramp response of the first-order system may be found by assuming R(s)=1/s2 as Y(S=RI Ts+1 τs+1八s2)s2szs+1 (3-3-5) ()=t-(1-e") (3-3-6) 2022-2-3 7
2022-2-3 7 Unit-step response of the first-order system may be found by assuming Rs 1 s as 1 1 1 1 1 1 1 Y s R s s s s s s Unit-ramp response of the first-order system may be found by assuming as 2 R s 1 s 2 2 2 1 1 1 1 1 1 1 Y s R s s s s s s s 1 t y t t e 1 t y t e (3-3-3) (3-3-4) (3-3-5) (3-3-6)
The error signal is then Q|c(0=-(0)y(02=(-c") (3-3-7) As t approaches the infinity, the error signal approaches t le 2022-2-3 8
2022-2-3 8 The error signal is then 1 t e t r t y t e As approaches the infinity, the error signal approaches , i.e., t e (3-3-7)
3.4 Performance of a second-Order System Let's consider a unity feedback system shown below E(S K R(S) G)= s(s+ p) The output y(s can be found as 2022-2-3 9
2022-2-3 9 3.4 Performance of a Second-Order System Let's consider a unity feedback system shown below. The output can be found as Y s
G(S R K R +ps+ k R +250ns+ (3-3-8 where =√Kz=p/2K n OS 2 2022-2-3 10
2022-2-3 10 2 2 2 2 1 2 n n n G s Y s R s G s K R s s p s K R s s s where n K p 2 K (3-3-8)