State Space Analysis Hany Ferdinando Dept of Electrical

State Space Analysis Hany Ferdinando Dept. of Electrical Engineering Petra Christian University State Space 2 - Hany Ferdinando

Overview State Transition Matrix Time Response Discrete-time evaluation State Space 2 - Hany Ferdinando 2

State Transition Matrix The solution of is If the initial condition x(0), input u(t) and the state transition matrix F(t) are known the time response of x(t) can be evaluated State Space 2 - Hany Ferdinando 3

State Transition Matrix The F(t) is inverse Laplace Transform of F(s) and When the input u(t) is zero, then State Space 2 - Hany Ferdinando 4

State Transition Matrix From the equation above we can expand the matrix into (for example, two elements) State Space 2 - Hany Ferdinando 5

State Transition Matrix The f 11(s) can be evaluated from the relation between X 1(s) and x 1(0), the f 12(s), f 21(s) and f 22(s) can be evaluated with the same procedure State Space 2 - Hany Ferdinando 6

Time Response It is the time response of X(t). First, find F(t) from F(s). It is simply the inverse Laplace Transform of F(s). Do the inverse Laplace Transform for each element of F(s). State Space 2 - Hany Ferdinando 7

Example i(t) State Space 2 - Hany Ferdinando 8

Example If x 1 = v. C and x 2 = i. L then State Space 2 - Hany Ferdinando 9

Example For R = 3, L = 1 and C = 0. 5, State Space 2 - Hany Ferdinando 10

Example x 1(0)/s I(s) 1/C 1/L s-1 X 1(s) x 2(0)/s s-1 R -R/L X 2(s) V(s) -1/C State Space 2 - Hany Ferdinando 11

Example When U(s) = 0 x 1(0)/s 1/L s-1 X 1(s) x 2(0)/s s-1 -R/L X 2(s) -1/C State Space 2 - Hany Ferdinando 12

Example f 11(s) is transfer function of X 1(s)/x 1(0). Here, use the Mason Gain Formula to get f 11(s) State Space 2 - Hany Ferdinando 13

Example D 1(s) is path cofactor of D, D is 1 + 3 s-1 + 2 s-2 D 1(s) = 1 + 3 s-1 State Space 2 - Hany Ferdinando 14

Example With the same procedures, find the f 12(s), f 21(s) and f 22(s)! State Space 2 - Hany Ferdinando 15

Example State Space 2 - Hany Ferdinando 16

Example Then the X(t) can be calculated with State Space 2 - Hany Ferdinando 17

Discrete-time Evaluation For discrete-time, use the approximation State Space 2 - Hany Ferdinando 18

Discrete-time Evaluation State Space 2 - Hany Ferdinando 19

Example With the same example above and T = 0. 2 s, State Space 2 - Hany Ferdinando 20

Matlab Use function expm to calculate the y(t) A = [0 -2; 1 -3]; T = 0. 2 psy = expm(A*T) State Space 2 - Hany Ferdinando 21