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Defining the State for an n-th Order Differential Equation

Module by: Thanos Antoulas, JP Slavinsky. E-mail the authors

Summary: 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 q s y t p s u t
(1)
where s=ⅆⅆ⁢t s ⅆ ⅆ t and where
q⁢s=sn+ α n - 1 ⁢sn−1+…+α1⁢s+α0 q s s n α n - 1 s n1 … α1 s α0
(2)
p⁢s= β n - 1 ⁢sn−1+…+β1⁢s+β0 p s β n - 1 s n1 … β1 s β0
(3)

One way to define state variables is by introducing the auxiliary variable ww which satisfies the differential equation:

q⁢s⁢w⁢t=u⁢t q s w t u t
(4)

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

y⁢t=p⁢s⁢w⁢t y t p s w t
(5)

The proof in the next three equations shows that the introduction of this variable ww 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⁢sps and q⁢sqs are interchanged. Lastly, yy is substituted in place of p⁢s⁢w⁢t 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 p s q s w t p s u t
(6)
q⁢s⁢p⁢s⁢w⁢t=p⁢s⁢u⁢t q s p s w t p s u t
(7)
q⁢s⁢y⁢t=p⁢s⁢u⁢t q s y t p s u t
(8)

Using this auxillary variable, we can directly write the AA, BB and CC matrices. AA is the companion-form matrix; its last row (except for a 00 in the first position) contains the alpha coefficients from the q⁢sqs:

A=( 0100…0 0010…0 000⋱…0 ⋮⋮⋮⋮⋱⋮ 0000…1 − α 0 − α 1 − α 2 − α 3 …− α n - 1 ) A 0100…0 0010…0 000⋱…0 ⋮⋮⋮⋮⋱ ⋮ 0000…1 α 0 α 1 α 2 α 3 … α n - 1
(9)

The BB vector has zeros except for the nn-th row which is a 11.

B=00⋮1 B 0 0 ⋮ 1
(10)

CC can be expressed as

C=β0β1β2⋮ β n - 1 C β0 β1 β2 ⋮ β n - 1
(11)

When all of these conditions are met, the state is

x=ws⁢ws2⁢w⋮sn−1⁢w x w s w s 2 w ⋮ s n 1 w
(12)

In conclusion, if the degree of pp is less than that of qq, we can obtain a state-space representation by inserting the coefficcients of pp and qq in the matrices AA, BB and CC as shown above.

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