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STATE VARIABLES
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A linear physical system can be described mathematically by:
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an nth order differential equation a transfer function n first-order differential equations a matrix differential equation
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So far, we have used the first two mathematical representations for describing physical systems. The third and fourth representations are referred to as state variable descriptions. To illustrate these four methods of description, consider the familiar secondorder process relating an output y to an input u. The four expressions for this process are listed below. 1. nth order differential equation (n = 2)
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zd*y dy rdt2+25rdt+y=u
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2. Transfer function. The transfer function corresponding to Eq. (28.1) is
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(28.1)
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(28.2) U(s) 3. 12 first-order differential equations (n = 2). Equation (28.1) can be expressed by the following differential equations: (28.3~) ii:1 = x* 1 . 25 1 (28.3b) i* = -T;zx1- ,7x* + -Ju wherext = y andx2 = j In Eqs. (28.3~) and (28.3b), xt and x2 are the state variables. To see that Eqs. (28.3) are the equivalent to Eq. (28. l), differentiate both sides of Eq. (28.3~); the result is Xl = i* (28.4) In Eq. (28.3b), we may now replace i2 by X 1 and x2 by i 1; the result is Xl = Since xt = y, we may write il=j a n d Using these expressions in Eq. (28.5) gives
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--$x1-
Y(s) -
1 7232 + 2578 + 1
Y. 1 TX + -pu
fl = jj + u 72
(28.6)
STATE-SPACE REPRESENTATION OF PHYSICAL SYSTEMS
Equation (28.6) is, of course, the same as Eq. (28.1). We shall see later that other choices for x 1 and ~2 are possible; at this point, the reader is asked to accept Eqs. (28.3~) and (28.3b) as a valid description of the second-order system under consideration. 4. Matrix differential equation. Equations (28.3~) and (28.3b) can be written as one matrix differential equation as follows: . where i = Ax+bu (28.7)
- 0 A= -1 72
-25 7
0 1 72
u is a scalar
The representation given by Eqs. (28.3) and the representation given by Eq. (28.7) are exactly the same; Equation (28.7) is in a more compact form. The state variables xt and x2 are represented by the column vector x. The coefficients of the state variables on the right sides of Eqs. (28.3~) and (28.3b) are the elements of the matrix A. In this example, there is only one input or forcing term, U, which is a scalar. Each term on the right side of Eq. (28.7) must be a vector containing two elements (i.e., a 2 X 1 matrix). In order for the expression given by Eq. (28.7) to agree with Eqs. (28.3~) and (28.3b), the coefficient of u must be a vector with the upper element zero. With some practice, the reader will be able to look at a matrix expression such as Eq. (28.7) and quickly see the equivalent set of differential equations. The output y in representations 1 or 2 often represents a physical variable of interest, such as the temperature of a process or the position of a mechanical system. The alternate state variable representation given by Eqs. (28.3) or Eq. (28.7) contains two state variables, one of which is y and the other the derivative of y (i.e., j). In this case only y may be of interest to the control engineer; j is available, but may not be of interest since it cannot always be measured easily. (For example, there is no easy way to measure the rate of change of temperature if y represents temperature.)
State-Space Description
In general, a physical system can be described by state variables as follows
il = fl(XlJ29 f 2 = f2(XlJZt . . .,X,,Ul,U2, . . * ,X,,Ul,U2, . . .,u,) . . . ,u,)
(28.8)
xn = fn(XlJ2,
f. .,X,,Ul,U2,
f. .,u,>
STATE-SPACE
METHODS
where xl, ~2,. . . ,x, are n state variables and u r , ~2, . . . ,um are m inputs or forcing terms. The above set of equations may be written as a matrix expression as follows i = f(x, u) If the system parameters vary with time, the vectorfwill contain explicit functions of time. An example for an element off might be the expression on the right side of the following equation: ii1 = 2txt + X2 -f u1+ U2 In this chapter, we shall be concerned with time-invariant systems for which ii is a linear combination of state variables and the coefficients are constant. For the time-invariant case, we may write the general term iiin Eq. (28.8) as follows: ii = UilXl + lZi2X2 + + UinXn + bilZ.41 + . + bimk!m for i = 1,2,3, . . . . n The equivalent matrix expression for Eq. (28.9) is
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