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Defining the State for an n-th Order Differential Equation

Module by: Thanos Antoulas, JP Slavinsky

Summary: Defining the State for an n-th Order Differential Equation

Consider the n-th order linear differential equation:

qsyt=psut q s y t p s u t (1)
where s=ⅆⅆt s ⅆ ⅆ t and where
qs=sn+ α n - 1 sn-1+…+α1s+α0 q s s n α n - 1 s n1 … α1 s α0 (2)
ps= β n - 1 sn-1+…+β1s+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:

qswt=ut 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:

yt=pswt 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 psps and qsqs are interchanged. Lastly, yy is substituted in place of pswt p s w t (using output equation). The result is the original equation describing our system.

psqswt=psut p s q s w t p s u t (6)
qspswt=psut q s p s w t p s u t (7)
qsyt=psut 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 qsqs:

A=0100…00010…0000⋱…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=wsws2w⋮sn-1w 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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