Difference equations and system representations

## Difference equations

Whereas continuous systems are described by differential equations, discrete systems are described by difference equations. From the digital control schematic, we can see that the difference equations show the relationship between the input signal e(k) and the output signal u(k). Suppose we are interested in the kth output signal u(k). Then, to get this output signal, we have the computer that computes some function that considers past input signal e(0) to e(k) and output signals u(0) to u(k-1), which can be expressed as a function of the form

We assume that the function f is linear and depends only on a finite number of signals e's and u's. Then the basic structure of the difference equation can be written as

For a while, we will consider the following difference equation (1).

(1)

## Finding transfer function using z-transform

Recall that a transfer function for the continuous system we have been considering so far was derived by first taking the Laplace transform of differential equations and then solved for Output/Input in terms of s. To derive the transfer function in discrete form, the mathematical tool very similar to the Laplace transform called z-transform will be used.

The z-transform is defined by

where f(k) is the amplitude of a sample, and the value k = 0, 1, 2, 3, ... refer to the discrete sample time.

Analogously, this can lead to the relation

By using these relations, we can easily find the discrete transfer function of a given difference equation.

Suppose we are going to find the transfer function of the system defined by the above difference equation (1), first, apply the above relations to each of u(k), e(K), u(k-1), and e(k-1) and you should arrive at

After few steps of algebra, you will have the following transfer function in discrete form (2).

(2)

Note: When finding a transfer function, zero initial conditions must be assumed.

## Derivation of state-space from difference equations

As you see in the continuous Modeling Tutorial page, another way to represent the system is to use the state-space form. The basic structure of the discrete state-space is

Matrices F, G, H and J are the discrete version of A, B, C, and D for a continuous case.

If you rewrite the above difference equation (1) to the state-variable form by adding

we can express the system in state-space form

By letting the output be u(k-1), the output equation can be written as

## Matlab representation

Now we will show you how to enter equations derived above into Matlab.

### 1. Transfer Function

Recall for continuous transfer function, the numerator and the denominator matrices are entered in descending powers of s. The same thing applies to discrete transfer functions. The numerator and the denominator matrices will entered in descending powers of z. For example, we enter the above transfer function (2) as follows:
```
numDz=[1 -0.95];
denDZ=[1 -0.75];
```

### 2. State-Space

For discrete state-space models, we will do exactly the same as what we did for continuous models. For example, the above discrete state-space model will be entered as
```
F=[1 0;
1 0.75];

G=[0;
-0.95];

H=[0 1];

J=[0];
```

Note: It is possible to convert from the state-space to transfer function, or vice versa using Matlab. To learn about conversion, see Conversion.