Page 76 - Modern Control Systems

50 Chapter 2 Mathematical Models of Systems

2.1 INTRODUCTION

To understand and control complex systems, one must obtain quantitative
mathematical models of these systems. It is necessary therefore to analyze the rela-
tionships between the system variables and to obtain a mathematical model.
Because the systems under consideration are dynamic in nature, the descriptive
equations are usually differential equations. Furthermore, if these equations can be
linearized, then the Laplace transform can be used to simplify the method of solu-
tion. In practice, the complexity of systems and our ignorance of all the relevant
factors necessitate the introduction of assumptions concerning the system opera-
tion. Therefore we will often find it useful to consider the physical system, express
any necessary assumptions, and linearize the system. Then, by using the physical
laws describing the linear equivalent system, we can obtain a set of linear differen-
tial equations. Finally, using mathematical tools, such as the Laplace transform, we
obtain a solution describing the operation of the system. In summary, the approach
to dynamic system modeling can be listed as follows:

1. Define the system and its components.
2. Formulate the mathematical model and fundamental necessary assumptions based on
basic principles.
3. Obtain the differential equations representing the mathematical model.
4. Solve the equations for the desired output variables.
5. Examine the solutions and the assumptions.
6. If necessary, reanalyze or redesign the system.

2.2 DIFFERENTIAL EQUATIONS OF PHYSICAL SYSTEMS

The differential equations describing the dynamic performance of a physical system
are obtained by utilizing the physical laws of the process [1-3].This approach applies
equally well to mechanical [1], electrical [3], fluid, and thermodynamic systems [4].
Consider the torsional spring-mass system in Figure 2.1 with applied torque T a{t).
Assume the torsional spring element is massless. Suppose we want to measure the
torque T s(t) transmitted to the mass m. Since the spring is massless, the sum of the
torques acting on the spring itself must be zero, or

Ut) - T s{t) = 0,

which implies that T s{t) = T a(t). We see immediately that the external torque T a(t)
applied at the end of the spring is transmitted through the torsional spring. Because
of this, we refer to the torque as a through-variable. In a similar manner, the angular
rate difference associated with the torsional spring element is

<a{t) = to s(t) - a> a{t).