solution of linear state- space equations

Solution of Linear State-Space Equations

2

Outline

• Laplace solution of linear state-space equations.• Leverrier algorithm.• Systematic manipulation of matrices to obtain the solution.

3

Linear State-Space Equations

1. Laplace transform to obtain their solution x(t).2. Substitute in the output equation to obtain

the output y(t).

4

Laplace Transformation• Multiplication by a scalar (each matrix entry).• Integration (each matrix entry).

5

State Equation

6

Matrix Exponential

7

Zero-input Response

8

Zero-state Response

9

Solution of State Equation

10

State-transition Matrix

• LTI case φ (t − t0) = matrix exponential• Zero-input response: multiply by statetransition matrix to change the systemstate from x(0) to x(t).

• State-transition matrix for time-varyingsystems φ (t, t0)– Not a matrix exponential (in general).– Depends on initial & final time (notdifference between them).

11

Output

12

Example 7.7

x1 = angular position, x2 = angular velocityx3 = armature current. Find:a)The state transition matrix.b)The response due to an initial current of 10 mA.c)The response due to a unit step input.d)The response due to the initial condition of (b) together with the input of (c)

13

a) The State-transition Matrix

14

State-transition Matrix

15

Matrix Exponential

16

b) Response: initial current =10 mA.

17

c) Response due to unit step input.

18

Zero-state Response

19

d) Complete Solution

20

The Leverrier Algorithm

21

Algorithm

22

Remarks

• Operations available in hand-held calculators(matrix addition & multiplication, matrix scalarmultiplication).• Trace operation ( not available) can be easily22

p ) yprogrammed using a single repetition loop.• Initialization and backward iteration starts with:Pn-2 = A + an-1 In an-2 = − ½ tr{Pn-2 A}

23

Partial Fraction Expansion

24

Resolvent Matrix

25

Example 7.8

Calculate the matrix exponential for thestate matrix of Example 7.7 using theLeverrier algorithm.

26

Solution

27

(ii) k = 0

28

Check and Results

29

Partial Fraction Expansion

30

Constituent Matrices

31

Matrix Exponential

32

Properties of Constituent Matrices