LINEAR CONTROL SYSTEMS Ali Karimpour Assistant Professor Ferdowsi University of Mashhad

Lecture 13 Ali Karimpour Oct Lecture 13 Time domain analysis of control systems Topics to be covered include: v Transient response of a prototype second order system u Damping ratio and damping factor. u Natural undamped frequency and damped frequency. u Percent overshoot. u Delay time, rise time and settling time. v Different region of S plane. u Constant natural frequency loci. u Constant natural damped frequency loci. u Constant damping factor loci. u Constant damping ratio loci.

Lecture 13 Ali Karimpour Oct Introducing a prototype second order system. معرفی یک سیستم نمونه درجه c e r c r Step response

Lecture 13 Ali Karimpour Oct Step response پاسخ پله

Lecture 13 Ali Karimpour Oct Step response پاسخ پله

Lecture 13 Ali Karimpour Oct Introducing a prototype second order system. معرفی یک سیستم نمونه درجه c e r c r فرکانس طبیعی نسبت میرائی فرکانس طبیعی میرا شده ضریب میرائی

Lecture 13 Ali Karimpour Oct Percent Overshoot درصد فراجهش How can we find P.O. ? چگونه درصد فراجهش را بیابیم؟

Lecture 13 Ali Karimpour Oct Calculation of Percent Overshoot and Peak Time محاسبه درصد فراجهش و لحظه پیک

Lecture 13 Ali Karimpour Oct Peak time لحظه پیک Let n=1 n = 0 n=1 n=3 n=4 n=5 n=2

Lecture 13 Ali Karimpour Oct Calculation of Percent Overshoot محاسبه درصد فراجهش

Lecture 13 Ali Karimpour Oct Percent Overshoot درصد فراجهش 0 100% % % % % % % 1 0%

Lecture 13 Ali Karimpour Oct Rise time زمان صعود How can we find rise time ? چگونه زمان صعود را بیابیم؟

Lecture 13 Ali Karimpour Oct Settling time زمان نشست How can we find settling time ? چگونه زمان نشست را بیابیم؟

Lecture 13 Ali Karimpour Oct v Constant natural frequency loci. نواحی مختلف در صفحه S Different region of S plane. v Constant natural damped frequency loci. v Constant damping factor loci. v Constant damping ratio loci.

Lecture 13 Ali Karimpour Oct مکان فرکانس طبیعی ثابت Constant natural frequency loci. 1 2

Lecture 13 Ali Karimpour Oct مکان فرکانس طبیعی میرا شده ثابت Constant natural damped frequency loci

Lecture 13 Ali Karimpour Oct Constant damping factor loci. مکان ضریب میرائی ثابت

Lecture 13 Ali Karimpour Oct Constant damping ratio loci. مکان نسبت میرائی ثابت

Lecture 13 Ali Karimpour Oct Example 1: Find the loci of a system where damping ratio is greater then and damping factor greater than 2

Lecture 13 Ali Karimpour Oct Example 2: Find the loci that the percent overshoot is less than 16% and settling time is less than 1 sec (according to 5% bound).

Lecture 13 Ali Karimpour Oct Effect of roots loci on step response تاثیر مکان ریشه ها بر پاسخ پله Conjugate root is not shown ریشه مزدوج نشان داده نشده است

Lecture 13 Ali Karimpour Oct Example 3: Find the poles of the following systems and its corresponding step response for k=75, 100, 200 and c e r

Lecture 13 Ali Karimpour Oct c e r k=75 k=100 k=200 k=1000

Lecture 13 Ali Karimpour Oct Exercises + - c e r 1- In the following system set k such that the percent overshoot of system be 4.3% + - c e r 2- In the following system set a and b such that the percent overshoot of system be 4.3% And the steady state error to step input be In the system of problem 1 set k such that a)The error to ramp input be 0.01 b)The percent overshoot of system be 4.3% c)The error to ramp input be 0.01 and the percent overshoot of system be 4.3%

Lecture 13 Ali Karimpour Oct Exercises (Cont.) + - c e r 4- In the following system a)For k=200 derive settling time, rise time and percent overshoot. Confirm your result with step response. b) For k=1000 derive settling time and percent overshoot. Confirm your result with step response. + - c e r 5- In the following system set the k such that the imaginary poles have damping ratio.