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(28.10)
b,,z
Writing this in the more compact matrix form, we have i = Ax + Bu (28.11) In this expression, there are m different inputs where m 5 n. The nature of the linear physical system expressed by Eq. (28.11) is completely stated by the matrices A and B. For the time-invariant system, the elements of A and B are constants. The outputs of interest to the control engineer may differ from the state variables (xi). The most general statement for relating the output to the state variables is y = cx (28.12) where y is the vector of outputs (yt, ~2, . . . , yP) chosen by the control engineer for some practical reason. The matrix C is a p x it matrix containing constant elements. The way in which the matrix C is selected will be clarified in the example to follow. In summary, the state-space description for a linear timeinvariant system is given by Eqs. (28.11) and (28.12).
STATE-SPACE REPRESENTATION OF PHYSICAL. SYSTEMS
Example 28.1. For the two-tank noninteracting liquid level system shown in Fig. 28.1, obtain the state-space description as expressed by Eqs. (28.11) and (28.12). The output y of interest is the level in tank 2. Notice that streams enter both tanks. For this example, let the state variables be the physical variables h r and h 2, which are the levels in tanks 1 and 2. These state variables am called physical variables because they can be easily measured or observed. (In another example, we shall consider a different set of state variables.) For the liquid-level system shown in Fig. 28.1 we may write (28.13) (28.14) or
--hl + -$q dt = RlAl dh2 1 1 -hl - -hz + $4, dr = RlA2 A&
(28.15) (28.16)
These equations can be written as follows 1; = Ah+Bu where h= A= -1 RlAl 1 RIAZ 0 -1 RzA2. B= (28.17)
r 1 A1
0 1 A2
FIGURE28-1 Liquid-level system for Examples 28.1 and 28.2:
Al = l,A2 = 0.5, R, = 0.5, R2 = 2l3
STATE-SPACE
METHODS
If the output is to be the level in tank 2 (h2), we have y = Ch where y = yt = h2 C=[O 11
In this case y is a scalar (i.e., a 1 X 1 matrix).
The choice of output can be stated in many ways. Regardless of the choice, the output is related to the state variables by Eq. (28.12). To see how the matrix C depends on the choice of output, consider the following examples: If y is to be a scalar that is the arithmetic average of the levels in the two tanks one can show that, C = [0.5 0.51. If the output is to be ht and h2, one can show tha c =
1 0 0 1
Selection of State Variables
To the beginner, the selection of state variables may seem mysterious. The state variables of a system are the smallest set of variables that contain sufficient information to permit all future states to be determined. Although the number of state variables is fixed, the actual selection of these state variables is not unique. If possible, it is convenient to choose state variables that are directly related to physical variables which can be measured or observed (e.g., temperature, level, composition, position, velocity, etc.) For mechanical systems, transducers are available for measuring velocity; for this reason, velocity is considered a physical variable. On the other hand, since the measurement of rate of change of composition is not easily made, this variable is not usually considered a physical variable. If one solves a dynamic problem by means of an analog computer or by means of a simulation language such as TUTSIM or ACSL which involves simulated integrator blocks, one legitimate set of state variables is the output from each integrator. In the control literature, the types of state variables have been classified as follows.
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