ECE 3620Lecture 4 - Systems of differential equations and state-space representations To see how systems of differential equations may be handled. To see the state-space form of differential equations.

pp. 784-790

Simultaneous differential equations. Simultaneous equations arise when there are several interacting phenomena. Simultaneous differential equations arise when there are interacting dynamics in a system. We have already seen one example of multiple equations.

One way to solve a set of linear equations such as $\Abf \xbf = \ybf$ for each $x_i$ is using Cramer's Rule. ($\Abf_i$ = $\Abf$ with $\ybf$ substituted into its $i$-th column)

$$\boxed{ x_i = \frac{det(\Abf_i)}{det(\Abf)} = \frac{\vert\Abf_i\vert}{\vert\Abf\vert}}$$

State-space descriptions

State-space description are very important representations of differential equations. The basic idea is this: to write a high-order differential equation, introduce a bunch of auxiliary (state) variables, and write instead a system of first-order differential equations. For example:

$$\dddot{y} + a_2\ddot{y} + a_1 \dot{y} + a_0 y = f$$

Write

$$\begin{eqnarray*} x_1 &=& y \\ x_2 &=& \dot{y} \\ x_3 = &=& \ddot{y} \end{eqnarray*}$$

Then we can write

$$\begin{eqnarray*} \xdot_1 &=& x_2 \\ \xdot_2 &=& x_3 \\ \xdot_3 &=& -a_0 x_1 - a_1 x_2 -a_2 x_3 + f \end{eqnarray*}$$

The first two of these follow almost trivially; the last is the one that contains the information from the differential equation.

The key feature of these equations is that the RHS must depend only upon the state variables and the input.

Write in matrix form. Allows the powerful results of linear algebra to be applied. Show a nonlinear example. From the state variables, all of the information about the system is given; given the state the evolution of the system into the future can be determined.

Show circuit example:

$$\begin{eqnarray*} \xdot_1 &=& i_1 - i_2 - x_2 \\ &=& (f-x_1) - 0.5 x_1 - x_2 \\ &=& -1.5 x_1 - x_2 + f \end{eqnarray*}$$

$$\begin{eqnarray*} 2\xdot_2 &=& x_1 - v_3 \\ &=& x_1 -5x_2 \end{eqnarray*}$$

(Simplify). Comment on ``canonical'' choices for state variables for circuits and mechanical systems.