Introduction to Control Systems (Iqbal)

This chapter describes the process of obtaining the mathematical description of a dynamic system, i.e., a system whose behavior changes over time. The system is assumed to be assembled from components. The system model is based on the physical laws that govern the behavior of various system components. Physical systems of interest to engineers include, for example, electrical, mechanical, electromechanical, thermal, and fluid systems. By using lumped parameter assumption, their behavior is mathematically described in terms of ordinary differential equation (ODE) models. These equations are nonlinear, in general, but can be linearized about an operating point for analysis and design purposes. Models of interconnected components are assembled from individual component descriptions. The components in electrical systems include resistors, capacitors, and inductors. The components used in mechanical systems include inertial 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.

State Variable Models

State variable models are time-domain models that express system behavior as time derivatives of state variables, i.e., the variables to express the process state. The state variables are typically selected as the natural variables associated with the energy storage elements in the process, but alternate variables can also be used. The state equations of the system describe the time derivatives of the state variables. When the state equations are linear, they are expressed in a vector-matrix form.

In the case of electrical networks, capacitor voltages and inductor currents serve as natural state variables. In the case of mechanical systems, positions and velocities of inertial masses serve as natural state variables. In thermal systems, heat flow is a natural state variable. In hydraulic systems, the head (height of the liquid in the reservoir) is a natural state variable.

The number of state variables determines the system order, however, the choice of state variables for a system model is not unique. For example, in a mechanical system model, position and momentum can serve as state variables in place of position and velocity.