This page explains the meaning and image of the proper transfer function, which is often used in control engineering.
Note: If you have not learned the basics of transfer functions, please see this page first.
- Proper transfer function satisfies “Degree of numerator ≤ Degree of denominator.”
- Proper transfer functions are strictly realizable in the real world.
- Improper transfer functions are not strictly realizable in the real world.
- However, improper systems can often be approximated by “almost equal systems.”
Definition of Proper Transfer Function
Suppose a linear system is represented by the following transfer function.
$$G(s)=\frac{N(s)}{D(s)}$$
In this case, the transfer function is
- Proper if the degree of $N(s) \leq$ the degree of $D(s)$
- Improper otherwise (i.e. if the degree of $D(s) \lt$ the degree of $N(s)$)
Furthermore, proper transfer functions can be further classified into two types:
- Strictly proper if the degree of $N(s) \lt$ the degree of $D(s)$
- Biproper if the degree of $N(s) =$ the degree of $D(s)$
Basically, classical control deals only with proper transfer functions.
Example of Proper Transfer Function
As a simple example, consider a system whose differential equation is expressed as
$$\ddot{y} + \dot{y} + y = \dot{u} + u.$$
The transfer function $G(s)$ for this would be
$$Y(s) = \ubg{\frac{s+1}{s^2 + s + 1}}{\large G(s)}U(s).$$
The degree of the denominator $D(s)$ is 2, and the degree of the numerator $N(s)$ is 1. Thus, this transfer function is proper.
Eventually, we see that the degrees are determined as follows.
- The degree of $D(s)$: Number of times to differentiate output in differential equation
- The degree of $N(s)$: Number of times to differentiate input in differential equation
Therefore, we can say that a system is proper if the differential equation satisfies
$$\text{Number of times to differentiate }\textbf{input} \leq \text{Number of times to differentiate }\textbf{output}.$$
Intuitive Meaning of “Proper”
Uh-huh. I understand the formula, but what does “Proper” ultimately mean?
In a nutshell, proper transfer functions are strictly realizable in the real world, while improper transfer functions are not. Let’s take a closer look at this.
First, let’s start with why improper transfer functions are not strictly realizable (because it is easier to understand.) As a simple example, consider a system whose differential equation is
$$\dot{y} + y = \ddot{u} + \dot{u} + u.$$
The transfer function $G(s)$ for this is
$$Y(s) = \ubg{\frac{s^2 + s+1}{s + 1}}{\large G(s)} U(s).$$
Since the degree of the denominator $D(s)$ is 1, and the degree of the numerator $N(s)$ is 2, this transfer function is improper. This can be further organized as follows:
$$\begin{align}Y(s) &= \frac{s^2 + s+1}{s + 1}U(s) \\\\ &= \frac{s(s+ 1)+1}{s + 1} U(s) \\\\ &= \biggl( \ubgd{ \vphantom{ \frac{1}{s+1} } s}{\text{Imroper}}{\text{Term}} \ + \ \ubgd{\frac{1}{s+1}}{\text{Proper}}{\text{Term}} \biggl) U(s)\end{align}$$
We have now divided the improper transfer function into an improper term and a proper term. Further expansion yields
$$Y(s) = \ubgd{sU(s)}{\text{Derivative}}{\text{of Input}} + \frac{1}{s + 1}U(s).$$
Like this, output from an improper transfer function contains time derivatives (or nth-order time derivatives) of input. These time derivatives are what prevent its realization in the real world.
Isn’t it easy to just differentiate the input?
You may think so, but strict time differentiation is not feasible in the real world. Let us review the definition of differentiation. The time derivative of a function $f(t)$ is defined as
$$\dot{f}(t) = \lim _{\Delta t \rightarrow 0} \frac{f(t+\Delta t) – f(t)}{\Delta t}.$$
The key point is that we need information about the future, \(f(t+\Delta t)\), although the limit is applied. Of course, there is no way to obtain future information, so “strict” time differentiation is not feasible.
For these reasons, improper transfer functions cannot be strictly realized in the real world. On the other hand, proper transfer functions can be strictly realized because they do not have the problem.
When should we care whether a transfer function is proper or not?
Naturally, for practical use, the transfer function must be proper. However, there are limited situations when one should care whether the transfer function is proper or not.
First of all, if the plant is a real-world system (e.g., a physical phenomenon), there is no need to worry about it. This is because the transfer function of a real-world system is always proper. If it were improper, it would not exist, right?
On the other hand, we should care whether a human-designed system is proper or not. For example, a controller is a system designed by humans as they wish, so it may be improper.
For example, a PD controller is improper and not strictly feasible because it involves differential manipulation of the error (i.e., the input to the controller).
Oh, but isn’t PD control used in practice?
That’s right. In fact, in most cases, an improper controller can be realized by replacing the differentiation with “not strictly but almost differentiation.”
Typical examples would be a differential circuit, which differentiates an electrical signal, or a numerical differentiation, where the differentiation is approximated in a program. These are not strict differentiations because they always include errors, but in most cases, they are sufficient for practical use.
Thus, when designing a controller, we should be careful whether it is proper or not. And if it is not proper, we need to consider whether it can be substituted by non-strict means.
This page explained the meaning and image of the proper transfer function. To conclude, a summary is shown here.
- Proper transfer function satisfies “Degree of numerator ≤ Degree of denominator.”
- Proper transfer functions are strictly realizable in the real world.
- Improper transfer functions are not strictly realizable in the real world.
- However, improper systems can often be approximated by “almost equal systems.”
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