The first step in control design is the development of a mathematical model for the process or system under consideration. In the modeling of systems, we assume a cause and effect relationship described by the simple input/output diagram below. An input is applied to a system, and the system processes it to produce an output. In general, a system has the three basic components listed below.

1. Inputs: These represent the variables under the designer's disposal. The designer produces these signals directly and applies them to the system under consideration. For example, the voltage source to a motor and the external torque input to a robotic manipulator both represent inputs. Systems may have single or multiple inputs.
2. Outputs: These represent the variables which the designer ultimately wants to control and that can be measured by the designer. For example, in a flight control application an output may be the altitude of the aircraft, and in automobile cruise control the output is the speed of the vehicle.
3. System or Plant: This represents the dynamics of a physical process which relate the input and output signals. For example, in automobile cruise control, the output is the vehicle speed, the input is the supply of gasoline, and the system itself is the automobile. Similarly, an air conditioning system regulates the temperature in a room. The output from the system is the air temperature in the room. The input is cool air added to the room. The system itself is the room full of air with its air flow characteristics. Note that the system could be mechanical, thermal, fluid, electrical, electro-mechanical, etc.



Figure 1: Block diagram representation of a system.

In this lecture we will show by examples how mathematical models for simple engineering systems can be developed. Notice that in each example four steps are taken. First, a diagram of all system components and externally applied inputs is drawn. From this diagram, the inputs and outputs are identified. Then a diagram is made of the system components in which all internal signals are shown. Finally, the differential equations governing the system dynamics are obtained. These equations form the mathematical model of the system.

Example 1 Thermal system: Oil, tank and heater described in lecture #2.

The oil, tank and heater system was described in the previous lecture. Now, we develop a mathematical model (differential equation) for this system which describes its dynamics. The different signals and components needed to derive a mathematical model for the thermal system are shown in Figure 1.

We first identify the input and output.
Input: voltage to heater, v, output: oil temperature, T.



Figure 2: Diagram of the thermal system signals and components.

We apply the heat balance equation to obtain the differential equation:

Energy supplied by the heater = Energy stored by oil + Energy lost to the surrounding environment by conduction



where, k is a contant provided by the heater manufacturer, v is the voltage to the heater, c is thermal capacitance of the oil (see Table 2.2, page 35 in the text book), T is the oil temperature, Ta is the ambient tempearture (environment), R is the thermal resistance of the tank wall (heat conduction) and t is the time. Equation (1) is the differential equation that describes the dynamics of our thermal system. Note that the input and output appear in this equation. If we know the input, v then we can solve equation (1) for the output, T.

Example 2 Mechanical system: Spring-mass-damper system.

In this example we model the spring-mass-system shown in Figure 3(a). The mass, m is subjected to an external force f. Let's suppose that we are interested in controlling the position of m. The way to control the position of the mass is by chossing f.

We first identify the input and output.
Input: external force, f, output: mass position, x.



Figure 3: (a) Diagram of the mechanical system components. (b) Free body diagram of the mechanical system.
Click here to see an animation for the analysis.


We apply Newton's second law to obtain the differential equation of this mechanical system. Using the free-body-diagram shown in Figure 3(b), we have



where, b is the damping coefficient and k is the spring stiffness (see Table 2.2, page 35 in the text book). Equation (2) is the differential equation that describes the dynamics of the spring-mass-damper system. Note that the input and output appear in this equation. If we know the input, f then we can solve equation (2) for the output, x.

The thermal and mechanical systems described in this lecture can be represented by the following block diagram:



Figure 4: Block diagram representation of the thermal or mechanical system.