LINEAR CONTROL SYSTEMS Ali Karimpour Associate Professor Ferdowsi University of Mashhad lecture 3 Lecture 3 Different representations of control systems Topics to be covered include: High Order Differential Equation. (HODE model) State Space model. (SS model) Transfer Function. (TF model) State Diagram. (SD model) 2 Dr. Ali Karimpour Feb 2013 lecture 3 High order differential equation (HODE). معادالت ديفرانسيل مرتبه باال dny d n 1 y dy d mu d m1u du an 1 n 1 ...... a1 a0 m bm1 m1 ..... b1 b0 n dt dt dt dt dt dt y is the output u is the input HODE model 3 Dr. Ali Karimpour Feb 2013 lecture 3 Example 1: A high order differential equation (HODE). يک معادله ديفرانسيل مرتبه باال:1 مثال خروجی ورودی HODE model 4 Dr. Ali Karimpour Feb 2013 lecture 3 Example 2: Another high order differential equation. مثالی ديگر از معادله ديفرانسيل مرتبه باال:2مثال خروجی ورودی eb HODE model 5 Dr. Ali Karimpour Feb 2013 lecture 3 State Space Models (SS) معادالت فضای حالت For continuous time systems SS model For linear time invariant continuous time systems SS model 6 Dr. Ali Karimpour Feb 2013 lecture 3 State Space Models معادالت فضای حالت General form of LTI systems in state space form LTI فرم کلی فضای حالتی سيستمهای x1 a11 x a 2 21 . . . . x n an1 a1n x1 b11 b12 . . b1 p u1 b b . . b . . a2 n x2 21 22 2 p u 2 . . . . . . . . . . . . . . . . . . . . . . ann xn bn1 bn 2 . . bnp u p a12 . . a22 . . an 2 SS model 7 Dr. Ali Karimpour Feb 2013 lecture 3 Example 3: A linear time invariant continuous time systems )LTI( يک سيستم خطی غير متغير با زمان:3مثال output SS model 8 Dr. Ali Karimpour Feb 2013 lecture 3 Example 3: Continue ادامه:3مثال The equations can be rearranged as follows: ساده سازی di(t ) 1 v(t ) dt L 1 dv(t ) 1 1 1 v(t ) i(t ) v f (t ) dt C R1C R1C R2C c(t ) v(t ) We have a linear state space model with مدل خطی فضای حالتی 9 Dr. Ali Karimpour Feb 2013 lecture 3 A demonstration robot containing several servo motors در رباتdc مثالی از موتور 10 Dr. Ali Karimpour Feb 2013 lecture 3 Example 4: DC motor DC موتور:4 مثال J - be the inertia of the shaft e(t) - the electrical torque ia(t) k1 ; k2 R - the armature current - constants - the armature resistance Let x1 (t ) (t ) And 0 d x1 (t ) dt x2 (t ) 0 HODE model x2 (t ) (t ) 1 0 x ( t ) 1 k1 va (t ) k1 k 2 x2 (t ) JR JR SS model 11 Dr. Ali Karimpour Feb 2013 lecture 3 Different representations نمايشهای مختلف HODE model SS model TF model SD model 12 Dr. Ali Karimpour Feb 2013 lecture 3 Linear high order differential equation معادالت ديفرانسيل مرتبه باالی خطی The study of differential equations of the type described above is a rich and interesting subject. Of all the methods available for studying linear differential equations, one particularly useful tool is provided by Laplace Transforms. مطالعه معادالت ديفرانسيل مرتبه باالی خطی فوق موضوع بسيار جالبی .بوده و يک روش خاص حل آن استفاده از تبديل الپالس است 13 Dr. Ali Karimpour Feb 2013 lecture 3 Table : Laplace transform table جدول تبديل الپالس u(t) - u(t-τ) 14 Dr. Ali Karimpour Feb 2013 lecture 3 Table : Laplace transform properties. خواص تبديل الپالس y(t-τ) u(t-τ) 15 Dr. Ali Karimpour Feb 2013 lecture 3 Transfer Function model مدل تابع انتقال dny d n 1 y dy d mu d m 1u du a ...... a a y b ..... b b0u n 1 1 0 m 1 1 n n 1 m m 1 dt dt dt dt dt dt Taking laplace transform zero initial condition s n y ( s ) an 1s n 1 y ( s ) ...... a1sy ( s ) a0 y ( s ) s mu ( s ) bm 1s m 1u ( s ) ..... b1su ( s ) b0u ( s ) A( s) y ( s) B( s)u ( s) G( s) y ( s ) B( s ) u ( s) A( s) TF model Or Input-output model HODE model TF model 16 Dr. Ali Karimpour Feb 2013 lecture 3 Different representations نمايشهای مختلف HODE model SS model TF model SD model 17 Dr. Ali Karimpour Feb 2013 lecture 3 But how can we change SS model to TF model? اما چگونه معادالت فضای حالت را به تابع انتقال تبديل کنيم Use Laplace transform Then Let initial condition zero SS model TF model 18 Dr. Ali Karimpour Feb 2013 lecture 3 Different representations نمايشهای مختلف HODE model SS model TF model SD model 19 Dr. Ali Karimpour Feb 2013 lecture 3 TF model properties خواص مدل تابع انتقال 1- It is available just for linear systems. 2- It is derived by zero initial condition. 3- It just shows the relation between input and output so it may lose some information. 4- It can be used to show delay systems but SS can not. 20 Dr. Ali Karimpour Feb 2013 Example 5: A thermal system يک سيستم حرارتی:5مثال lecture 3 Input signal 1 k Output signal Input signal is an step function so: t (Time) t 0 1 1 u (s) u (t ) s t0 k y(s) 0 ? g ( s) t u ( s) s 1 k k t 0 y( s) y(t ) k ke s s 1/ 21 0 t 0 Dr. Ali Karimpour Feb 2013 Example 6: A system with pure time delay سيستمی با تاخير ثابت:6 مثال lecture 3 h(s) e sTd g (s) k s 1 ke sTd h( s ) g ( s ) s 1 22 Dr. Ali Karimpour Feb 2013 lecture 3 State diagram دياگرام حالت x1 dx 2 dt x1 ( s ) sx2 ( s ) x2 (0) x2 (s) s 1 x1 (s) x2 (0) x2(0) s -1 x1(s) s -1 SD model x2(s) 23 Dr. Ali Karimpour Feb 2013 lecture 3 State diagram to state space دياگرام حالت به معادالت حالت x1 x2 x 2 4 x1 5 x2 r c x1 SD model SS model 24 Dr. Ali Karimpour Feb 2013 lecture 3 Different representations نمايشهای مختلف HODE model SS model TF model SD model 25 Dr. Ali Karimpour Feb 2013 lecture 3 Different representations نمايشهای مختلف HODE SS x1 x2 x 2 4 x1 5 x2 r c 5c 4c r c x1 SD TF G(s) 1 s 2 5s 4 Mason’s rule Discussed in this lecture Will be discussed in next lecture 26 Dr. Ali Karimpour Feb 2013 lecture 3 Exercises 2-1 Find the SS model and TF function model for example 1. 2-2 Find the SS model and TF function model for example 2. 2-3 Find the TF function model for example 3. 2-4 Find the TF function model for example 4. a) Suppose angular position as output b) Suppose angular velocity as output 2-5 In example 5 find the output a) the input is e-2t b) the input is unit step 27 Dr. Ali Karimpour Feb 2013 lecture 3 Exercises (Continue) 2-6 In the different representation slide show the validity of red directions 2-7 Change the equations derived in example 2 to one order 3 HODE. 28 Dr. Ali Karimpour Feb 2013