0.18 Matrix representation of systems

A system is described by the following differential equation:

The state of this system is

The state $x(t)$ (a vector) is composed of two state variables $x_1(t)$ and $x_2(t)$ . We would like to be able to talk about the time-varying state in terms of these state variables. That is, we'd like an expression where $\frac{d^{1}x(t)}{dt^1}$ can be written in terms of $x_1(t)$ and $x_2(t)$ . From state equation above, we see that $\frac{d^1{x}_1(t)}{dt^1}$ simply equals $\frac{d^{1}y(t)}{dt^1}$ . In the same equation we also notice that $\frac{d^{1}y(t)}{dt^1}$ equals $\frac{d^1{x}_2(t)}{dt^1}$ . Therefore, the derivative of the first state variable exactly equals the second state variable.

We can follow the same process for $x_2(t)$ . Again from state equation , we see that the first derivative of $x_2(t)$ equals the second derivative of $y(t)$ . At this stage, we can bring in information from the system's differential equation. That equation (the first one in this example) also contains the second derivative of $y(t)$ . If we solve for it we get

We already know that $\frac{d^{1}y(t)}{dt^1}$ equals $x_2(t)$ and that $y(t)$ equals $x_1(t)$ . Putting all of this together, we can get an expression for $\frac{d^1{x}_2(t)}{dt^1}$ in terms of the state variables and the input variable.

The important thing to notice here is that by looking at the time-varying behavior of the state, we have been able to reduce the complexity of the problem. Instead of one second-order differential equation we now have two first-order differential equations.

Think about a case where we might have 5, 10, or even 20 state variables. In such an instance, it would be difficult to work with so many equations. For this reason (and in order to have a more compact notation), we represent these state variable equations in terms of matrices. The set of equations above can be written as:

By letting $x(t)=\begin{pmatrix}{x}_1(t)\\ x_2(t)\\ \end{pmatrix}$ , $A=\begin{pmatrix}0 & 1\\ -2 & -3\\ \end{pmatrix}$ , $B=\begin{pmatrix}0\\ 1\\ \end{pmatrix}$ , we can rewrite this equation as:

This is called a state equation .

Let's develop state and output equations for the following circuit diagram:

There are two energy-storage elements in this diagram: the inductor and the capacitor. As we know that energy-storage elements give systems memory, it makes sense that the state variables should be the current $i_L$ flowing through the inductor and the voltage $v_C$ across the capacitor. By using Kirchoff's laws around the left and center loops, respectively, we can find the following two equations:

These equations can easily be rearranged to have the derivatives on the left-hand side equaling linearcombinations of state variables and inputs on the right. These are the state equations. The figure also quicklytells us that the output $y$ is equal to the voltage across the capacitor, $v_C$ .

We can now rewrite the state and output equations in matrix form:

In this example we'll find the state and output equations for the following circuit, as well as represent the system using the compact notation described above.

Here, $u$ and $y$ are the input and output currents, respectively. $x_1$ and $x_2$ are the state variables. Using Kirchoff's laws and the $i$ - $v$ relation of a capacitor, we can find the following three equations:

Through simple rearranging and substitution of the terms, we find the state and output equations:

State equations:

Output equation:

This equations can be more compactly written as:

The simple oscillator is defined by the following differential equation:

The states are $x_1=y$ (which is also the output equation) and $x_2=\frac{d^{1}y}{dt^1}$ . These can be rewritten in state equation form as:

The compact matrix notation is: