Page 185 - Numerical Analysis Using MATLAB and Excel

Chapter 5 Differential Equations, State Variables, and State Equations

5.7 Expressing Differential Equations in State Equation Form

A first order differential equation with constant coefficients has the form

dy
a ------ + a yt() = xt() (5.93)
1
dt 0
In a second order differential equation the highest order is a second derivative.

An nth−order differential equation can be resolved to first−order simultaneous differential equa-
n
tions with a set of auxiliary variables called state variables. The resulting first−order differential
equations are called state space equations, or simply state equations. The state variable method offers
the advantage that it can also be used with non−linear and time−varying systems. However, our
discussion will be limited to linear, time−invariant systems.
State equations can also be solved with numerical methods such as Taylor series and Runge−
Kutta methods; these will be discussed in Chapter 9. The state variable method is best illustrated
through several examples presented in this chapter.

Example 5.14
A system is described by the integro−differential equation

di 1 t jωt
Ri + L----- + --- - ∫ it = e (5.94)
d
dt C – ∞
Differentiating both sides and dividing by we obtain
L
2
d t R di 1 1 jωt
------- + ---- ----- + -------- i = ---jωe (5.95)
dt 2 L dt LC L
or
2
d t R di 1 1 jωt
-
------- = – --- ----- – -------- i + ---jωe (5.96)
dt 2 L dt LC L
Next, we define two state variables x 1 and x 2 such that

x = i (5.97)
1
and

----- =
x = di dx 1 x · (5.98)
-------- =
2
dt dt 1
Then,
2
·
⁄
x = d idt 2 (5.99)
2
5−24 Numerical Analysis Using MATLAB® and Excel®, Third Edition

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