# Mathematical Models of Control Systems

In this blog, we will discuss about Mathematical Modelling of Control Systems and what are the Types of Mathematical Models of Control Systems with the help of an appropriate example. So let’s start…

In general, control systems are so complexed in nature that it is very difficult to study them directly. Therefore, we need a simplified version of the complex control system to study and analyse the dynamic response of the system.

“The process of getting a simplified mathematical model of a complex control system by using certain physical laws is called as Mathematical Modelling of Control Systems.

That’s why modelling is one of the important aspects of studying and analyzing the control system.

## Mathematical Models of Control Systems

Mathematical Models of Control Systems are defined as a set of mathematical equations which represents the dynamics of the system accurately or at least close to the actual dynamic response of the system.

These mathematical models are obtained after a lot of approximations and assumptions, which are:

• All physical systems are assumed to be linear.
• Distributed parameters are considered as lumped parameters which gives us ordinary differential equation rather than partial differential equation.
• Physical parameters of the system do not change with time.
• Uncertainties, disturbances or noises are neglected.
• Environmental conditions do not affect the system.

For example: resistance of each component of the system is assumed to be constant with respect to increasing or decreasing temperature.

Hence, physical models are based on a lot of assumptions and approximations. That’s why these physical models are not the exact replica of the actual control system but these models are very closed to the real control system and can be used to study the system accurately.

Now as we know that control system consists of several components and each of these components has their respective characteristics. So, to obtain the mathematical model of control system, characteristics of each component needs to be expressed mathematically.

This can be done by using certain physical laws. For example, ordinary electrical network can be analysed through the application of Ohm’s law and Kirchhoff’s law.

Similarly, mechanical systems for translational and rotational motion can be analysed by using Newton’s law of motion and d’Alembert’s law.

Hence by using certain physical laws, the governing equation between input and output of the system can be obtained.

These mathematical models are based on certain physical laws. For one physical law, there is one corresponding mathematical model which represents input-output relationship. So, for one physical model, there may be more than one mathematical model depending on a particular physical law.

Now, let’s see what are the types of mathematical model of control system.

## Types of Mathematical Models of Control Systems

Basically, there are three types of mathematical models which are:

1. Differential Equation Model
2. Transfer Function Model
3. State Space Model

In this blog, we will only study about differential equation model and transfer function model. There is a whole separate chapter about state space model. So, we will study about state space model in another blog.

### Differential Equation Model

Differential equation model is a time domain mathematical model of control system which describe the relationship between input and output of a control system in terms of the rates of change of system variables.

These models can be used to analyse the stability of the system and its transient response.

Example:

Now let’s understand the Differential Equation Model and Transfer Function Model by the example of RLC network as shown in the figure below.

In the given RLC network, a resistor, an inductor and a capacitor are connected in series. Voltage Vi(t) is given at the input terminals and output voltage V0(t) is obtained at the capacitor terminals. The current flowing through the circuit is i(t).

Now, we must find the relationship between the input and output voltage.

Applying KVL

$$V_{i}\left(t\right)=Ri\left(t\right)+L\frac{di\left(t\right)}{dt}+V_{0}\left(t\right)\;\;\;\;\;\_\_\_\left(1\right)$$

Current through the capacitor is given by

$$i\left(t\right)=C\frac{dV_0\left(t\right)}{dt}\;\;\;\;\;\_\_\_\left(2\right)$$

As same current i(t) is flowing through R and L also. Therefore, putting value of i(t) from eqn(2) into eqn(1), we get

$$V_i\left(t\right)=RC\frac{dV_0\left(t\right)}{dt}+LC\frac{d^2V_0\left(t\right)}{dt^2}+V_0\left(t\right)$$

Taking LC common, we get

$$V_i\left(t\right)=\left[\frac RL\frac{dV_0\left(t\right)}{dt}+\frac{d^2V_0\left(t\right)}{dt^2}+\frac1{LC}V_0\left(t\right)\right]\cdot\left(LC\right)$$

The above equation can be rewritten as

$$\left(\frac1{LC}\right)V_i\left(t\right)=\frac{d^2V_0\left(t\right)}{dt^2}+\left(\frac RL\right)\frac{dV_0\left(t\right)}{dt}+\left(\frac1{LC}\right)V_0\left(t\right)\;\;\;\;\;\_\_\_\left(3\right)$$

As the above eqn(3) is in the form of a differential equation therefore eqn(3) is called as the Differential Equation Model of the given electrical system.

The above equation is a second order differential equation.

### Transfer Function Model

Transfer function model of a control system is an s-domain mathematical model which describe the relationship between input and output of a control system in terms of complex frequency domain (s-domain) functions.

These models can be used to analyse the frequency response of the system and design controllers.

Example:

We will understand the transfer function model by using the same RLC network that we discussed above.

Now, taking Laplace Transform of both sides of eqn(3), we get

$$\left(\frac1{LC}\right)V_i\left(s\right)=s^2V_0\left(s\right)+\left(\frac RL\right)sV_0\left(s\right)+\left(\frac1{LC}\right)V_0\left(s\right)$$

$$\Rightarrow\left(\frac1{LC}\right)V_i\left(s\right)=\left(s^2+\frac RLs+\frac1{LC}\right)V_0\left(s\right)$$

$$\Rightarrow\frac{V_0\left(s\right)}{V_i\left(s\right)}=\frac{\left(\frac1{LC}\right)}{\left(s^2+\frac RLs+\frac1{LC}\right)}=\frac{Output}{Input}=Transfer\;Function\;\;\;\;\;\_\_\_\left(4\right)$$

As the above eqn(4) is in the form of a transfer function therefore eqn(4) is called as the Transfer Function Model of the given electrical system.

Transfer function model of the given RLC network (second order electrical system) is shown in figure below:

Both eqn(3) and eqn(4) describes the characteristics of the given RLC network. It means that by using any one of the above models, we can study and analyse the given RLC network.

### State Space Model

State space model of a control system is a mathematical representation of a physical system as a set of input, output and state variables related by first-order differential equations.

Hence, we can say that state space models describe the behavior of a control system in terms of a set of first-order differential equations.

These models are used to analyse the system’s stability, controllability and observability.

We will study about state-space models in detail in a separate blog.

In our next blog, we will talk about Mathematical Modelling of Mechanical Systems and how to write mathematical models for mechanical systems by using d’Alembert’s Law.

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