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# Control Systems, State Space Form and Convolution Integrals

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A control system is represented in the state space form by the following equations in the attached file.

a) Solve for the state vector using the convolution integral
b) Find the output, y(t)

https://brainmass.com/math/linear-algebra/control-systems-state-space-convolution-integrals-154235

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Problem:

A control system is represented in the state space form by the following equations:

a) Solve for the state-transition matrix
b) Solve for the state vector using the convolution integral
c) Find the output y(t)

Solution:

a) For a simple linear differential equation with constant coefficients of form
, a = constant (1)
and with initial condition
(2)
the solution is given by
(3)
This formula can be written in another form, using the convolution integral:
(4)
where
(5)
is the convolution formula.
Equation (3) is also known as "Duhamel's formula".
For a system of linear differential equations with constant coefficients of form:
(6)
the solution is given by a similar formula, using the exponential of matrix:
(7)
where
(8)
We consider now the two-dimensional case, as in our problem.
If (1) and (2) are the eigenvalues of matrix (A), the we can build the matrix of the eigenvectors:
(9)
so that, by a linear transformation, the matrix (A) can be brought in a diagonal form (when possible):
(10)
The converse formula is
(11)
and it is easy to prove that
(12)
where
(13)
Introducing (12) in (8), we will get
(14)

(15)
and this will be introduced in (7) in order to find out the solution of the given problem.
We are going now to compute the eigenvalues of matrix (A), the eigenvectors and the exponential.
The eigenvalues are the roots of the characteristic polynomial:

(16)
The eigenvectors will be determined by solving the systems:
k = 1,2 (17)
For 1 = -4 :
(18)
This is an homogeneous system, so that the solution depends on one parameter. Taking one of the equations (both are equivalent, since the determinant is null), we will have:

 (19)
For 2 = -1 :

The matrix of eigenvectors will be
(20)
Its inverse is
(21)
The exponential of the transition matrix will be, according to (15):

(22)
The homogeneous system of ODE will have the solution:

(23)
(always check if X1(0) = X0)

b) We are going to compute now the convolution integral (Duhamel):

where the integral is to be applied to all the elements of the product of matrices.

(24)
The final solution of our system is:

(25)
(it can be easily checked that X (0) = X0)

c) The output of the system will be:

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