Space Equations Outline Laplace solution of linear statespace equations Leverrier algorithm Systematic manipulation of matrices to obtain the solution 2 Linear StateSpace Equations ID: 713776
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Slide1
Solution of Linear State-Space EquationsSlide2
Outline
• Laplace solution of linear
state-space equations
.• Leverrier algorithm.• Systematic manipulation of matrices to obtain the solution.
2Slide3
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).
3Slide4
Laplace Transformation
• Multiplication by a scalar (each matrix entry).
•
Integration (each matrix entry).
4Slide5
State Equation
5Slide6
Matrix Exponential
6Slide7
Zero-input Response
7Slide8
Zero-state Response
8Slide9
Solution of State Equation
9Slide10
State-transition Matrix
• LTI case
φ
(t − t0) = matrix exponential• Zero-input response: multiply by state
transition matrix to change the system
state 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 (not
difference between them).
10Slide11
Output
11Slide12
Example 7.7
12
x
1
= angular position,
x
2
= 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)Slide13
a) The State-transition Matrix
13Slide14
State-transition Matrix
14Slide15
Matrix Exponential
15Slide16
b) Response
:
initial
current =10 mA.16Slide17
c) Response due to unit step input.
17Slide18
Zero-state Response
18Slide19
d) Complete Solution
19Slide20
The Leverrier Algorithm
20Slide21
Algorithm
21Slide22
Remarks
• Operations available in hand-held calculators
(matrix addition & multiplication, matrix
scalarmultiplication).• Trace operation ( not available) can be easily22p ) y
programmed using a single repetition loop.
• Initialization and backward iteration starts with:
Pn-2
= A + an-1 In an-2 = − ½ tr{
P
n
-2
A
}
22Slide23
Partial Fraction Expansion
23Slide24
Resolvent Matrix
24Slide25
Example 7.8
Calculate the matrix exponential for the
state matrix of Example 7.7 using the
Leverrier algorithm.25Slide26
Solution
26Slide27
(ii) k = 0
27Slide28
Check and Results
28Slide29
Partial Fraction Expansion
29Slide30
Constituent Matrices
30Slide31
Matrix Exponential
31Slide32
Properties of Constituent Matrices
32