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Defining the State for an n-th Order Differential Equation

Consider the n-th order linear differential equation:

q s y t p s u t
where s t and where
q s s n n - 1 s n 1 1 s 0
p s n - 1 s n 1 1 s 0

One way to define state variables is by introducing the auxiliary variable w which satisfies the differentialequation:

q s w t u t

The state variables can then be chosen as derivatives of w . Furthermore the output is related to this auxiliary variable as follows:

y t p s w t

The proof in the next three equations shows that the introduction of this variable w does not change the system in any way. The first equation uses a simple substition based on the differential equation . Then the order of p s and q s are interchanged. Lastly, y is substituted in place of p s w t (using output equation ). The result is the original equation describing our system.

p s q s w t p s u t
q s p s w t p s u t
q s y t p s u t

Using this auxillary variable, we can directly write the A , B and C matrices. A is the companion-form matrix; its last row (except for a 0 in the first position) contains the alpha coefficients from the q s :

A 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 n - 1

The B vector has zeros except for the n -th row which is a 1 .

B 0 0 1

C can be expressed as

C 0 1 2 n - 1

When all of these conditions are met, the state is

x w s w s 2 w s n 1 w

In conclusion, if the degree of p is less than that of q , we can obtain a state-space representation by insertingthe coefficcients of p and q in the matrices A , B and C as shown above.

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Source:  OpenStax, State space systems. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10143/1.3
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