# 1: Mathematical Models of Physical Systems

Learning Objectives

• 1. Obtain mathematical model of a physical system from its component descriptions.
• 2. Obtain system transfer function from its differential equation model.
• 3. Obtain a physical system model in the state variable form.
• 4. Linearize a nonlinear dynamic system model about an operating point.

This chapter describes the process of obtaining the mathematical description of a dynamic system, i.e., a system whose output changes over time. The models of continuous-time systems are primarily described in terms of linear and nonlinear differential equations.

Physical systems of interest to engineers include electrical, mechanical, electromechanical, thermal, and fluid systems, among others. Using lumped parameter assumption, their behavior is mathematically described by ordinary differential equations (ODEs) models. These equations are, in general, nonlinear, but are often linearized about an operating point for analysis and design purposes.

To model physical systems with interconnected components, individual component models can be assembled to obtain the system model. In the case of electrical systems, these elements include resistors, capacitors, and inductors. For mechanical systems, these include inertias (masses), springs, and dampers (or friction elements). For thermal systems, these include thermal capacitance and thermal resistance. For hydraulic and fluid systems, these include reservoir capacity and flow resistance.

To model physical systems, where properties (or entities) flow in and out of a system boundary, e.g., a hydraulic reservoir, conservation laws and/or balance equations may be used to describe system dynamics in terms of rate of change of an accumulated property. Let Q represent the accumulated property, and let $$q_{in}$$ and $$q_{out}$$ represent the inflow and outflow rates, then the relevant dynamic equation is described as: $\frac{dQ}{dt}=q_{in}-q_{out}+g-c$ where $$g$$ and $$c$$ denote the internal generation and consumption of that property.

The Laplace transform is commonly used to convert a set of linear differential equations into algebraic equations that can then be manipulated to obtain an input-output description in the form of a transfer function. The transfer function forms the basis for analysis and design of control systems by conventional methods in the frequency domain. In contrast, the modern control theory is established on time-domain analysis involving the state equations that describe system behavior as time derivatives of a set of state variables.

Linearization of nonlinear models is accomplished using Taylor series expansion about a critical point. The resulting linear model is effective in the neighborhood of the critical point. The linear systems theory is well-estabished and is relied upon for controller design, aimed to modify the system behavior.