Main content Test input signals Response of a first-order system Performance of a second-order system Effects of a third pole and a zero on system response Root location and the transient response第一页，共42页。Main content Steady-state error analysis Performance indices The simplification of linear systems Examples and simulation Summary continue第二页，共42页。Introduction Transient response Steady-state response Design specifications How to get compromise?A distinct advantage of feedback control system is the ability to adjust the transient and steady-state responseRefer to P244 Figure 5.1 第三页，共42页。Two performance measures versus Parameter p第四页，共42页。4.1 Test input signals Step input Ramp input Parabolic input Sinusoidal input Unit impulse inputThe standard test input signals commonly used are:第五页，共42页。Representation of test signals Step: Ramp: Parabolic: Sinusoidal:continue Input time domain frequency domain第六页，共42页。StepRampParabolic第七页，共42页。Unit impulse responsecontinueUnit impulse:System impulse response:System response is the convolution integral of g(t) and r(t): 第八页，共42页。Standard test signalcontinueThe standard test signals are of the general form:And its Laplace transform is:第九页，共42页。Performance indices(viewpoint from engineering) Time delay t d Rise time t r Peak time t p Settling time t s Percent overshoot Transient Performance:Steady-state Performance: Steady-state error第十页，共42页。第十一页，共42页。Step response of a control system第十二页，共42页。4.2 Response of a first-order systemThe model of first-order systemorFor example, temperature or speed control system and water level regulating system. 第十三页，共42页。第十四页，共42页。Response of first-order system Unit step response (No steady-state error) Unit impulse response ( transfer function) Unit ramp response (Constant steady-state error) Unit parabolic response ( Infinite steady-state error ) Refer to script 3.1-3.8第十五页，共42页。Unit step response第十六页，共42页。Dynamic Performance:Unit step response第十七页，共42页。Unit impulse response第十八页，共42页。Unit ramp response第十九页，共42页。Unit parabolic response第二十页，共42页。Important conclusion(for nth-order LTI system) From above analysis, we can see that impulse response of a system is the 1st-order derivative of step response or 2nd-order derivative of ramp response of the system. Conclusion: System response for the derivative of a certain input signal is equivalent to the derivative of the response for this input signal. 第二十一页，共42页。4.3 Response and performance of a second-order system Model of 2nd-order system Roots of characteristic equation (Poles)The response depends on and 第二十二页，共42页。第二十三页，共42页。Roots of characteristic equation第二十四页，共42页。Unit step response of 2nd-order system If , 2 positive real-part roots,unstable If , 2 negative real-part roots,underdamped If ,2 equal negative real roots,critically damped If , 2 distinct negative real roots,overdamped If , 2 complex conjugate roots,undamped第二十五页，共42页。Case 1: underdamped Oscillatory response No steady-state error第二十六页，共42页。Case 2: critically damped Mono-incremental response No Oscillation No steady-state error第二十七页，共42页。Case 3: overdamped Mono-incremental response slower than critically damped No Oscillation No steady-state error第二十八页，共42页。第二十九页，共42页。Step response for different damping ratio第三十页，共42页。Performance evaluation( underdamped condition) Performance indices evaluation An example of performance evaluation1 Time delay2 Rise time3 Peak time4 Percent overshoot5 Settling timeRefer to script 3-15 第三十一页，共42页。 Swiftness ClosenessPerformance evaluation第三十二页，共42页。4.4 Effects of a third pole and a zero on 2nd-order system response Effect of a third pole Effect of a third zero Dominant poles第三十三页，共42页。is constantPD control第三十四页，共42页。 PD control can increase damping ratio and reduce the P.O and settling time, and keep the natural frequency constant. PD(比例-微分)控制可以增大系统的阻尼，使阶跃响应的超调量下降，调节时间缩短，且不影响系统稳态误差及系统的自然频率。Performance of PD control第三十五页，共42页。is constantDifferential feedback control第三十六页，共42页。 测速反馈会降低系统的开环增益，从而加大系统在斜坡输入时的稳态误差，但不影响系统的自然频率，并可增大系统的阻尼比, 从而改善系统的动态性能。Performance of Differential feedback control第三十七页，共42页。4.5 Root location and transient response Characteristic roots (modes) Effects of zeros on response Refer to Figure 5.17 (P260)第三十八页，共42页。Impulse response for various root locations in s-plane第三十九页，共42页。When ，(离虚轴越远的点，其响应分量衰减越快。)4.6 High-order system analysis第四十页，共42页。 如果所有的闭环极点中，距虚轴最近的极点周围没有闭环零点，而其它闭环极点又远离虚轴，那么距虚轴最近的闭环极点所对应的响应分量，在系统的时间响应过程中起主导作用，这样的闭环极点就称为闭环主导极点。一对闭环零极点，若它们之间的距离较与其它零极点的距离相比，非常小，则它们的作用可以相互抵消。偶极子主导极点第四十一页，共42页。Assignment E5.2 E5.3 E5.4 E5.8 P5.4第四十二页，共42页。