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Generating state space in controllable form from differential equations

Nasser M. Abbasi

July 2, 2015 page compiled on July 2, 2015 at 5:09pm

This note shows examples of how to generate states space A,B, C, D  from differential equations. The state space will be in the controllable form.

Every transfer function which is proper is realizable. Which means the transfer function         N(s)
G (s) = D(s)   has its numerator polynomial N (s)  of at most the same order as the numerator D (s)  . Therefore G (s) = -2s2--
        s +s+1   is proper but         --s3---
G(s) =  s2+s+1   is not. To use this method, we start by writing

            ˜
G (s) = k + G (s)

Where  ˜
G (s)  is strict proper transfer function. A strict proper transfer function is one which has N (s)  polynomial of order at most one less than D (s)  . If G (s)  was already a strict proper transfer function, then k  above will be zero.

Converting a proper G (s)  to strict proper is done using long division. Then the result of the division is moved directly to A, B, C,D  in some specific manner. If G (s)  was already strict proper then of course the long division is not needed.

The following two examples illustrate this method. The first one uses the differential equation

y′′(t) + 3y′(t) + 2y(t) = u(t)

And the second example uses

y′′′(t) + 6y ′′(t) − 2y′(t) − 7y(t) = 4u′′′(t) + 3u ′′(t) + 2u′(t) + 4u(t)

References

1.
Lecture notes, ECE 717 Linear systems, Fall 2014, University of Wisconsin, Madison by Professor B. Ross Barmish
2.
Linear system theory and design, Chi-Tsong Chen.


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Figure 1: Example one



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Figure 2: Example two