CHAPTER
5 RESPONSEOF FIRST-ORDER SYSTEMS
Before discussing a complete control system, it is necessary to become familiar with the responses of some of the simple, basic systems that often are the building blocks of a control system. This chapter chapter and the three that follow describe in detail the behavior of several basic systems and show that a great variety of physical systems can be represented by a combination of these basic systems. Some of the terms and conventions that have become well established in field of automatic control will also be introduced. By the end of this part of the book, systems for which a transient must be calculated will be of high-order and require calculations that are time-consuming if done by hand. The reader should start now using Chap. 34 to see how the digital computer can be used to simulate the dynamics of control systems. TRANSFER
FUNCTION
We shall develop the transfer function for a or der der sy st em by by considering the unsteady-state behavior of an ordinary in-glass thermometer. A cross-sectional view of the bulb is shown in Fig. 5 . Consider the thermometer to be located in a flowing stream of fluid for which the temperature x varies with time. Our problem is to calculate the response or the time variation of the thermometer reading y for a particular change in MERCURY
THERMOMETER.
*In order that the result of the analysis of the thermometer be general and therefore applicable to other first-order systems, the symbols and y have been selected to represent surrounding temperature and thermometer reading, respectively.
49
5 0
O P E N - L O O P SY S T E M S
FIGURE
5-l
Cross-sectional
view
of
thermometer.
The following assumptions* will be used in this analysis: 1. All the resistance to heat transfer resides in the film surrounding the bulb (i.e., the resistance offered by the glass and mercury is neglected). 2. All the thermal capacity is in the mercury. Furthermore, at any instant the mercury assumes a uniform temperature throughout. 3. The glass wall containing the mercury does not expand or contract during the transient response. (In an actual thermometer, the expansion of the wall has an additional effect on the response of the thermometer reading. (See Iinoya and Altpeter (1962) .) It is assumed that the thermometer is initially at steady state. This means that, before time zero, there is no change in temperature with time. At time zero the thermometer will be subjected to some change in the surrounding temperature By applying the unsteady-state energy balance Input rate
output rate = rate of accumulation
we get the result hA(x -y)-0 = where A = surface area of bulb for heat transfer, C = heat capacity of mercury, m = mass of mercury in bulb, = time, hr h = film coefficient of heat transfer, For illustrative purposes, typical engineering units have been used.
p a r a m e t e r s because all *Making the first two assumptions is often referred to a s t h e l u m p i n g the resistance is “lumped” into one location and all the capacitance into another. As shown in the analysis, these assumptions make it possible to represent the dynamics of the system by an ordinary differential equation. If such assumptions were not ma&, the analysis would lead to a partial differential equation, and the representation would be to as a distributed-parumeter system. In Chap. 21, distributed-parameter systems will be considered in detail.
RESPONSE OF FIRST-ORDER SYSTEMS
Equation (5.1) states that the rate of flow of heat through the film resistance surrounding the bulb causes the internal energy of the mercury to increase at the same rate. The increase in internal energy is manifested by a change in temperature and a corresponding expansion of mercury, which causes the mercury column, or “reading” of the thermometer, to rise. The coefficient will depend on the flow rate and properties of the surrounding fluid and the dimensions of the bulb. We shall assume that is constant for a particular installation of the thermometer. Our analysis has resulted in Eq. (5. which is a first-order differential equation. Before solving this equation by means of the transform, deviation variables will be introduced into Eq. (5.1). The reason for these new variables will soon become apparent. Prior to the change in the thermometer is at steady state and the derivative is zero. For the steady-state condition, Eq. (5.1) may be written = 0
(5.2)
The subscript is used to indicate that the variable is the steady-state value. Equation (5.2) simply states that = or the thermometer reads the true, bath temperature. Subtracting Eq. (5.2) from Eq. (5.1) gives =
(5.3)
= dyldt because Notice that d(y is a constant. If we define the deviation variables to be the differences between the variables and their steady-state values
(5.3) becomes hA(X
If we let
=
Y) =
(5.4)
Eq. (5.4) becomes dt
Taking the
(5.5)
transform of Eq. (5.5) gives Y(s) =
(5.6)
Eq. (5.6) as a ratio of Y( S) to X(S) gives 1
+1 The parameter time.
is called
(5.7)
ti me constant of the system and has the units of
52
LINEAR OPEN-LOOP
The expression on the right side of Eq. (5.7) is called the of the system. It is the ratio of the transform of the deviation in thermometer reading to the transform of the deviation in the surrounding temperature. In examining other physical systems, we shall usually attempt to obtain a transfer function. Any physical system for which the relation between transforms of input and output deviation variables is of the form given by Eq. (5.7) is called a s y s t e m . Synonyms for first-order system are first-order lag and single exponential stage. The naming of all these terms is motivated by the fact that Eq. (5.7) results from a first-order, linear differential equation, Eq. (5.5). In Chap. 6 is a discussion of a number of other physical systems which are first-order. By reviewing the steps leading to Eq. one can discover that the introduction of deviation variables prior to taking the transform of the differential equation results in a transfer function that is free of initial conditions because the initial values of X and Y are zero. In control system engineering, we are primarily concerned with the deviations of system variables from their steady-state values. The use of deviation variables is, therefore, natural as well as convenient. PROPERTIES OF TRANSFER FUNCTIONS. In general, a transfer function relates two variables in a physical process; one of these is the cause (forcing function or input variable) and the other is the effect (response or output variable). In terms of the example of the mercury thermometer, the surrounding temperature is the cause or input, whereas the thermometer reading is the effect or output. We may write Transfer function = G(s) = where G(s) = symbol for transfer function X(s) = transform of forcing function or input, in deviation form = transform of response or output, in deviation form The transfer function completely describes the dynamic characteristics of the system. If we select a particular input variation X(t) for which the transform is X(s), the response of the system is simply Y(s) = By taking the inverse of Y(s), we get Y(t), the response of the system. The transfer function results from a linear differential equation; therefore, the principle of superposition is applicable. This means that the transformed response of a system with transfer function G(s) to a forcing function X(s) = where
and
+
are particular forcing functions and a and Y(s) = = =
+ +
are constants, is
