- Mathematical Modelling Definition: Mathematical modelling of control systems is the process of creating block diagrams to determine system performance and transfer functions.
- Mechanical Systems: These can be linear (involving force, velocity, and displacement) or rotational (involving torque, angular velocity, and angular displacement).
- Electrical Systems: These involve voltage, current, and charge, with parameters like resistance, capacitance, and inductance.
- Force-Voltage Analogy: In this analogy, mass (M) is similar to inductance (L), force is similar to voltage (V), and displacement (x) is similar to charge (Q).
- Force-Current Analogy: Here, mass (M) is similar to capacitance (C), force is similar to current (I), and displacement (x) is similar to flux (ψ).
Mathematical Modelling of Control System
Control engineering models many physical domains, including:
- Mechanical systems
- Electrical systems
- Electronic systems
- Thermal systems
- Hydraulic systems
- Chemical systems
Mathematical modelling of a control system starts with physical laws, parameters, assumptions, inputs, outputs and initial conditions. The resulting equations can be represented as differential equations, state-space models, transfer functions or block diagrams for analysis and controller design.
The sections below model translational and rotational mechanical systems, then compare their equations with electrical RLC networks.
Mathematical Modelling of Mechanical Systems
This page considers translational motion and rotational motion. Translational does not mean linear; a translational model can contain nonlinear forces.
In a translational mechanical system, three common variables are:
- Applied force, represented by F
- Linear velocity, represented by v
- Linear displacement, represented by x
The corresponding element parameters are:
- Mass, represented by M
- Viscous damping coefficient, represented by B
- Spring stiffness, represented by K
In a rotational mechanical system the corresponding variables are:
- Torque, represented by T
- Angular velocity, represented by ω
- Angular displacement, represented by θ
The rotational example groups its parameters into two entries:
- Moment of inertia, represented by J
- Viscous rotational damping B and torsional stiffness K
Consider the translational mass-spring-damper system in the figure.
Displacement x is measured from the chosen reference position. Newton’s second law states that the applied force equals the mass times acceleration.
The spring and viscous damper contribute restoring and damping forces, giving the force balance shown below.
Substitute F1, F2 and F3 into the force balance. Taking the Laplace transform with zero initial displacement and velocity gives the displayed force-to-displacement transfer function.
This transfer function is one form of the mathematical modeling of a mechanical control system.
Mathematical Modelling of Electrical System
In an electrical system the variables used here are:
- Voltage, represented by V.
- Current, represented by I.
- Charge, represented by Q.
The linked overview covers active and passive components. The three RLC parameters used in this model are passive:
- Resistance, represented by R.
- Capacitance, represented by C.
- Inductance, represented by L.
Mechanical-electrical analogies match systems that have differential equations of the same form. Two consistent mappings are used below.
Force Voltage Analogy : Consider the series combination of a resistor, inductor and capacitor.
The source voltage V drives the same current through all three elements. Applying KVL expresses the source voltage in terms of charge, resistance, capacitor value and inductance.
Comparing this equation with the mechanical force balance gives the following mappings.
- Mass M is analogous to inductance L.
- Force F is analogous to voltage V.
- Displacement x is analogous to charge Q.
- Viscous damping coefficient B is analogous to resistance R.
- Spring stiffness K is analogous to inverse capacitance 1/C.
This mapping is the force-voltage analogy.
Force Current Analogy : Now consider a parallel combination of resistor, inductor and capacitor.
The same voltage E appears across each branch. Applying KCL expresses the source current in terms of voltage integral or flux, resistance, capacitance and inductor value.
Comparing this equation with the mechanical force balance gives the mobility mappings below.
- Mass M is analogous to capacitance C.
- Force F is analogous to current I.
- Displacement x is analogous to flux linkage ψ, the time integral of voltage.
- Viscous damping B is analogous to conductance 1/resistance R.
- Spring stiffness K is analogous to inverse inductance 1/L.
This mapping is the force-current analogy.
For the rotational mechanical system, θ is angular displacement. Newton’s rotational law relates applied torque to angular acceleration through the moment of inertia.
The rotational spring and damper contribute the remaining torques shown in the balance.
Substitute T1, T2 and T3 into that balance. Taking the Laplace transform with zero initial angle and angular velocity gives the displayed torque-to-angle transfer function.
The linked electrical control system page gives wider control context, but this equation models a rotational mechanical system.
