Defining the State for an n-th Order Differential Equation 2.11 2001/01/17 2003/10/28 22:40:45.942 US/Central Thanos Antoulas aca@rice.edu John Paul Slavinsky jps@alumni.rice.edu Elizabeth Chan lizychan@rice.edu Thanos Antoulas aca@rice.edu John Paul Slavinsky jps@alumni.rice.edu order state n-th order differential equation diff eq 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 ⅆ ⅆ t and where q s s n α n - 1 s n1 … α1 s α0 p s β n - 1 s n1 … β1 s β0 One way to define state variables is by introducing the auxiliary variable w which satisfies the differential equation: 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 ps and qs 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 qs: A 0100…0 0010…0 000⋱…0 ⋮⋮⋮⋮⋱ ⋮ 0000…1 α 0 α 1 α 2 α 3 … α 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 β0 β1 β2 ⋮ β 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 inserting the coefficcients of p and q in the matrices A, B and C as shown above.