Laplace Transforms and Systems Modeling

Sources

1. Control Tutorials for MATLAB & Simulink, “Introduction: System Modeling” - Accessed on 2024-10-05
2. Byju’s, “Laplace Transform” - Accessed on 2024-10-05
3. Electrical4U, “Transfer Function of Control System” - 2024-04-17
4. Tutorialspoint, “Modelling of Mechanical Systems” - Accessed on 2024-10-05
5. Dr. Nhut Ho, University Northridge “Modeling Electrical Systems” - Accessed on 2024-10-05

1. System Modeling

• The first step to designing control system processes is to produce the right mathematical model to represent the system to be controlled.
  • This can either be derived from physical laws or from experimental data.

Basic Parts of Dynamic Systems

Dynamic systems are systems that evolve over time according to a fixed rule, typically expressed as first-order differential equations.

Example 1

$$\dot{x}=\frac{dx}{dt}=f(x(t),u(t),t)$$

• $f$ is the function producing the rate of change $\frac{dx}{dt}$
  • It can be time-invariant if it does not depend on time, thus, $\dot{x}=f(x,u)$
• $x(t)$ (or the state variables) is the configuration of the system at time $t$
  • may depend on time even if the function $f$ itself is time-invariant
• $u(t)$ (or the control inputs) is the vector containing the external inputs
  • may depend on time even if the function $f$ itself is time-invariant

Given the initial state $x(t_0)$ and the time history of inputs $u(t)$ between $t_0$ and $t_1$, we can integrate the system equation (at Example 1) to determine any of the system’s future state.

$n$ is referred to as the system order, which is used for determining the dimensionality of the state-space. It also indicates the minimum number of state variables needed to solve state equations or predict the system’s future behavior.

Linearity of Dynamic Systems

Although, in reality, most physical systems are nonlinear, many of them are also approximately linear at a certain range of values.

Before the introduction of more advanced computers, scientists and engineers only analyzed linear time-invariant (LTI) systems. As a result, control theory was predominantly founded on these assumptions. Moreover, they solved many engineering challenges using LTI techniques.

State-Space Representation

Standard State-Space Representation for LTI Systems

$$\begin{array}{rl}\dot{x}& =Ax+Bu\\ y& =Cx+Du\end{array}$$

• $x$ is the state variables vector

• $u$ is the input/control vector

• $\dot{x}$ is the time derivative of the state vector

• $A$ is the system matrix

• $B$ is the input matrix

• $y$ is the output vector

• $C$ is the output matrix

• $D$ is the feedforward matrix

• The output vector $y$ and its equation is useful due to the existence of state variables which are often not directly observed.
• $C$ defines which states variable the controller can use.

2. Laplace Transforms

• Laplace transform is useful for reducing differential equations into algebraic ones. For this reason, they play a large role in control system engineering.
• It is an integral transform that converts a derivative function with $t$ to a complex function with $s$. It is defined by the formula

$$F(s)={\int }_0^{+\infty }f(t)\cdot e^{-s\cdot t}\cdot dt$$

• The purpose of Laplace transform is to transform equations into easier problems.
• Laplace transform is used in engineering fields for applications such as electrical circuit analysis, system modeling, and digital signal processing because these usually involved differential equations.

3. Transfer Functions

• Transfer function pertains to the ratio of the Laplace transform, assuming that the initial conditions are non-existent, of an output to input of a specific system.
  • The impulse input is related to the output because of the resulting transfer function that can be extracted.
  • $G(s)=\frac{C(s)}{R(s)}$
• Poles and zeros are useful to know where the transfer function approaches infinity or zero. In this regard, they can also play a major role in influencing a system’s behavior.
• In control system analysis, the Laplace transform allows for a consistent representation of various signal types.
• To find the transfer function, we would need to get equations for the system, then their Laplace transform (with initial conditions set to 0). Afterward, we determine a system output and input, then produce the ratio needed in a transfer function (output divided by input).
• Although systems have varying kinds of signals, they can be uniformly represented using their Laplace form.

4. Modelling of Mechanical Systems

Translational Mechanical Systems

• Unlike rotational mechanical systems, they only move along a straight path.
• 3 basic elements of translational mechanical systems
  1. mass
     1. $F=M\frac{d^{2}x}{d{t}^2}$
  2. spring
     1. $F=Kx$
  3. damper
     1. $F=B\frac{dx}{dt}$
• The net force is 0 due to applied and opposing forces being in directions that counter act one another.

Rotational Mechanical Systems

• In contrast to translational mechanical systems, they move about a fixed axis.
• Three basic elements of rotational mechanical systems
  1. moment of inertia
     • $T=J\frac{d^2\theta }{d{t}^2}$
  2. torsional spring
     1. $T_k=K\theta$
  3. dashpot
     • $T=B\frac{d^2\theta }{d{t}^2}$