EE 529 Circuits and Systems Analysis Lecture 9

EE 529 Circuits and Systems Analysis Lecture 9 Mustafa Kemal Uyguroğlu

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

State equations St Ou ve cto r Inp ut ate ve cto r System dynamics Measurement Read-out map tp ut ve c to r Eastern Mediterranean University

x: n-vector (state vector) u: p-vector (input vector) y: m-vector (output vector) n n A: nxn p n B: nxp n m C: mxn p m D: mxp 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 Eastern Mediterranean University X(0)

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

Properties of TM (0)=I ) t ( ΦΦ -1(t)= (-t) t 0 Ф(t 2 -t 1)Φ(t 1 -t 0)= Φ(t 2 -t 0) [Φ(t)]k= Φ(kt) t 1 t 2 Φ(t 1 -t 0) Φ(t 2 -t 0Φ(t ) 2 -t 1) Φ(t) Φ(kt)Φ(t) 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 (s. I-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

Eastern Mediterranean University

Eastern Mediterranean University