LINEAR CONTROL SYSTEMS Ali Karimpour Assistant Professor Ferdowsi University of Mashhad

Ali Karimpour Oct 2009 lecture6 2 Lecture 6 State Space representation Topics to be covered include: v More study on transfer function representation (cont.). v State space representation. u State transition matrix. u Calculation of state transition matrix for time varying system. u Significance of state transition matrix and its property. v State transition equation. v State transition equation determined by state diagram.

Ali Karimpour Oct 2009 lecture6 3 Open loop and closed loop systems Open loop Closed loop Controller سیستمهای حلقه باز و حلقه بسته

Ali Karimpour Oct 2009 lecture6 4 When is the open loop representation applicable? v The model on which the design of the controller has been based is a very good representation of the plant, v The model must be stable, and v Disturbances and initial conditions are negligible. در چه شرایطی کنترل حلقه باز مناسب است؟

Ali Karimpour Oct 2009 lecture6 5 SISO and MIMO سیستم تک ورودی تک خروجی و چند ورودی چند خروجی System u1u1 y1y1 SISO u1u1 u2u2 u3u3 y1y1 y2y2 y3y3 MIMO System

Ali Karimpour Oct 2009 lecture6 6 Sensitivity حساسیت Sensitivity is defined as: M is the system + controller transfer function G is the system transfer function

Ali Karimpour Oct 2009 lecture6 7 Sensitivity حساسیت G(s) K(s) Open loop system r c G(s) K(s) + - r c Closed loop system

Ali Karimpour Oct 2009 lecture6 8 Feedback properties ویژگیهای فیدبک Feedback can effect: a) system gain b)system stability c) system sensitivity d) noise and disturbance

Ali Karimpour Oct 2009 lecture6 9 State Space Models General form of state space model LTI systems مدل فضای حالت If initial condition and input are defined, then x(t), y(t) ? اگر شرائط اولیه و ورودی مشخص باشد مقدار چند است؟ x(t) و y(t) Start with homogeneous equation با معادله همگن شروع کنید.

Ali Karimpour Oct 2009 lecture6 10 State transition matrix ماتریس انتقال حالت If t 0 is not zero

Ali Karimpour Oct 2009 lecture6 11 Determination of state transition matrix تعیین ماتریس انتقال حالت روش اول Method I

Ali Karimpour Oct 2009 lecture6 12 Determination of state transition matrix تعیین ماتریس انتقال حالت روش دوم Method II

Ali Karimpour Oct 2009 lecture6 13 Determination of state transition matrix تعیین ماتریس انتقال حالت روش سوم So i th column of Φ is the response of system to i th position Method III

Ali Karimpour Oct 2009 lecture6 14 Example 1 State transition matrix and x(t) مثال 1 : ماتریس انتقال حالت و را بیابید. x(t)

Ali Karimpour Oct 2009 lecture6 15 State transition matrix for time varying systems ماتریس انتقال حالت برای سیستمهای متغیر با زمان

Ali Karimpour Oct 2009 lecture Φ(0) = I خواص ماتریس انتقال حالت 2- Φ -1 (t)= Φ(-t) 3- Φ(t 2 -t 1 )Φ(t 1 -t 0 ) = Φ(t 2 -t 0 ) 4- Φ k (t) = Φ(kt) Property of state transition matrix

Ali Karimpour Oct 2009 lecture6 17 State transition equation معادله انتقال حالت Convolution integral State transition equation

Ali Karimpour Oct 2009 lecture6 18 Example 2 Find state transition equation for unit step مثال 2 : معادله انتقال حالت را با فرض اعمال پله واحد بیابید.

Ali Karimpour Oct 2009 lecture6 19 Example 2 Find state transition equation for unit step مثال 2 : معادله انتقال حالت را با فرض اعمال پله واحد بیابید.

Ali Karimpour Oct 2009 lecture6 20 Example 3 Find x 1 (s) and x 2 (s) مثال 3 : معادله و را بیابید. x 1 (s) x 2 (s)

Ali Karimpour Oct 2009 lecture6 21 ِ Example 4: Derive x 1 (s) by using state diagram. s s -1 1 u(s) -3

Ali Karimpour Oct 2009 lecture6 22 Exercise 1- Find the state diagram of the following system 2- Find the transfer function of the following system. a) By use of the formula between the two representation. b) By use of state diagram 3- Find y(t) for initial condition [1 3 -1] T and the unit step as input.

Ali Karimpour Oct 2009 lecture6 23 Exercise (cont.) 4- The state transition matrix and state transition equation of following system. Answer is: 5- Suppose x 1 (0)=1, x 2 (0)=3 and x 3 (0)=2 and r(t)=u(t). Find x(t) and c(t) for t≥0