What is mathematical modeling

Mathematical models are an essential part for simulation and design of control systems. The purpose of the mathematical model is to be a simplified representation of reality, to mimic the relevant features of the system being analyzed.

Through mathematical modeling phenomena from real world are translated into a conceptual world. This process is initiated by observing the phenomena, applying a mathematical model to it and predict its behavior through simulation.

There are two main categories of mathematical modeling: theoretical and experimental modeling.

In theoretical modeling, the system is described using equations derived from physics. In order to be able to model the system in such a way, several simplifications have to be applied. For example, when modeling the suspension of a vehicle, we assume that the stiffness of the spring is constant, even if in reality is not.

Systems which are modeled entirely based on physical principles (equations) are called white-box models. This means that the user has all the details concerning how the system works.

Examples of physics laws and principles used for white-box modeling:

• mechanical systems
  • Newton’s law
  • Hooke’s law
• electrical systems
  • Ohm’s law
  • Kirchhoff’s law
• hydrodynamics
  • Bernoulli’s law
  • conservation of mass law
• thermodynamics
  • ideal gas law
  • entropy

An example of theoretical modeling is the equation describing the behavior of a RL electric circuit.

Legend:
E [V] – battery voltage
S – switch
R [Ω] – resistance
L [H] – inductance
i [A] – electric current
u [V] – voltage drop

The system’s behavior can be modeled with the differential equation:

\[L \cdot \frac{di}{dt} + R \cdot i = E\]

In many cases, the mathematical model of the system, ends up to being too complicated and requires simplification. The drawback of the simplification process is that it could ignore some relevant physical phenomena which could be critical for the control system design.

Experimental modeling, also called system identification, is based on measurements. The mathematical model of the system is derived from several sets of measurements, each recording the system’s response (output) for different stimulus and perturbations (inputs).

System which are modeled entirely based on experimental data (input-output measurements) are called black-box models. This means that the user can observe the response (output) of the model for a certain stimulus (input) but has no information about the internal mechanism (principles).

Black-box models can be constructed as artificial neural networks, trained based on the input-output measurements of the system.

In order to have a representative black-box model, based on artificial neural networks, we need a lot of data-set for training, which needs to cover all the possible working scenarios of the system. If the measurement data-set doesn’t have a considerable coverage of the real behavior, the output of the model could be unrealistic for certain inputs.

Most of the time, it’s better to benefit from the advantages of both methods. In this case we’ll end up with gray-box models. For this type of models we know what structure of system we are analyzing but we don’t have the right parameters for it.

A gray-box model is for example a transfer function or a state-space model. We can have a system that it’s behaving as a first-order system, with the transfer function H(s), but we don’t know the gain (K) and the time constant (T). By using experimental data we can estimate the parameter K and T.

\[H(s) = \frac{K}{T \cdot s + 1}\]

There are different “flavors” of grey-box models, mainly depending on the model structure. In the table below we can summarize the characteristics of each type of mathematical models.

Model type	Characteristics	Consists of
White-box	governing physical laws know parameters know	set of linear / non-linear differential equations
Light-gray-box	some of the physical governing laws known model structure known parameters unknown	set of linear / non-linear differential equations with parameters estimation transfer function with parameters estimation state-space model with parameters estimation
Dark-gray-box	some of the physical governing laws known model structure unknown parameters unknown	neuro-fuzzy models with parameters estimation
Black-box	model structure unknown parameters unknown	artificial neural networks

Even if theoretical modeling, if done properly, delivers more information about the system being analyzed, experimental modeling could be the right method for modeling due to the following reasons:

• if the system is complex, deriving the mathematical equations can be very hard
• most of the parameters used in the mathematical equations are not know so the overall behavior of the modeled system is uncertain
• not all the physical phenomena are captured or well known

In our Systems Modeling category we are going to focus on theoretical modeling. This method makes the link between mathematics and physics and gives the user a clear understanding how mathematics is used and applied in real engineering topics.

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