EE 529 Circuits and Systems Analysis Mustafa Kemal Uyguroğlu Lecture 9

Eastern Mediterranean University State vector a listing of state variables in vector form

Eastern Mediterranean University State equations System dynamics Measurement Read-out map Output vector Input vectorState vector

Eastern Mediterranean University x:n-vector (state vector) u:p-vector (input vector) y:m-vector (output vector) A :nxn B :nxp C :mxn D :mxp n n m m n p n p System matrix Input (distribution) matrix Output matrix Direct-transmission matrix

Eastern Mediterranean University Solution of state eq’ns Consists of: Free response Forced sol’n & (Homogenous sol’n)(particular sol’n)

Eastern Mediterranean University Homogenous solution Homogenous equation has the solution State transition matrix X(0)

Eastern Mediterranean University State transition matrix An nxn matrix  (t), satisfying

Eastern Mediterranean University Determination of  (t): transform method Laplace transform of the differential equation:

Eastern Mediterranean University Determination of  (t): transform method

Eastern Mediterranean University Determination of  (t): time-domain solution Scalar case  where

Eastern Mediterranean University Determination of  (t): time-domain solution For vector case, by analogy  where Can be verified by substitution.

Eastern Mediterranean University Φ(t 2 -t 0 ) Properties of TM  (0)=I  -1 (t)=  (-t) Ф(t 2 -t 1 )Φ(t 1 -t 0 )= Φ(t 2 -t 0 ) [Φ(t)] k = Φ(kt) Φ(t)Φ(-t) Φ(t 1 -t 0 )Φ(t 2 -t 1 ) t0t0 t1t1 t2t2 Φ(t) Φ(kt)

Eastern Mediterranean University General solution Scalar case

Eastern Mediterranean University General solution Vector case

Eastern Mediterranean University General solution: transform method L { }  

Eastern Mediterranean University Inverse Laplace transform yields:

Eastern Mediterranean University For initial time at t=t 0

Eastern Mediterranean University The output y(t)=Cx(t)+Du(t) Zero-input response Zero-state response

Eastern Mediterranean University Example  Obtain the state transition matrix  (t) of the following system. Obtain also the inverse of the state transition matrix  -1 (t). For this system the state transition matrix  (t) is given by since

Eastern Mediterranean University Example The inverse ( sI-A ) is given by Hence Noting that  -1 (t)=  (-t), we obtain the inverse of transition matrix as:

Eastern Mediterranean University Exercise 1 Find x 1 (t), x 2 (t) The initial condition

Eastern Mediterranean University Exercise 1 (Solution)

Eastern Mediterranean University Example 2

Eastern Mediterranean University Exercise 2 Find x 1 (t), x 2 (t) The initial condition Input is Unit Step

Eastern Mediterranean University Exercise 2 (Solution)

Eastern Mediterranean University Matrix Exponential e At

Eastern Mediterranean University Matrix Exponential e At

Eastern Mediterranean University The transformation where 1, 2,…, n are distinct eigenvalues of A. This transformation will transform P -1 AP into the diagonal matrix

Eastern Mediterranean University Example 3

Eastern Mediterranean University  Method 2:

Eastern Mediterranean University Matrix Exponential e At

Eastern Mediterranean University Matrix Exponential e At

Eastern Mediterranean University Example 4

Eastern Mediterranean University Laplace Transform

Eastern Mediterranean University