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Combining Eqs. (28.38) and (28.36) leads to
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= -(KJq)x1 - Kcx2 + (K&I)r
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i3 =
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(28.39) (28.40)
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-ax1 - Kcx2 + cxr
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where (Y = Kc/r1
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Summarizing the state variable equations given by Eqs. (28.32), (28.33), and (28.40) and using the definition of xg in Eq. (28.37) give * il = x2 i2 where A = K&q a = l/T1
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b = 11~ a = Kc/r,
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= -abxl - ( a +
b)x2 +Axg +AK,r
i3 = -CYX~
- Kcx2 + cxr
The A and b terms in x = Ax + br are
If m is required as a function of t, it can always be found by solving Eq. (28.37) for m; thus
m = x3 +
SUMMARY
:. State-space representation is an alternative to the kansfer function representation of a physical system that we have used up to this point. A transfer function that relates an output variable to an input variable represents an nth-order differential equation. In the state-space representation, the &-order differential equation is written as n first-order differential equations in terms of n state variables. These n differential equations can also be written in a more compact form as a matrix differential equation:
i = Ax + Bu
For an nth-order dynamic system, the number of state variables is fixed at n, but the selection of the variables is not unique. Of the many sets of state variables that one can choose, we discussed three sets that are useful in control theory; namely, physical variables, phase variables, and canonical variables. The state-space representation gives all of the dynamic detail of a system (e.g., the dependent variable and its successive derivatives for the case of phase variables). Whether or not this detail is needed depends on the problem being solved. We shall see the value of state-space representation in multivariable control and in nonlinear control in later chapters.
STATE-SPACE REPRESENTATION OF PHYSICAL SYSTEMS
APPENDIX 28A ELEMENTARY MATRIX ALGEBRA
The purpose of this section is to provide in a convenient location a review of some of the elementary operations of matrix algebra for use in state-space methods. It is expected that the reader has had a course in linear algebra discussing the concepts of a vector and a matrix and the operations performed on them. . VECTORS. An n-dimensional column vector is an ordered series of elements (numbers): x 1, ~2, . . . , x,, and is written as
Xl x2 x =
Multiplication of a vector by a scalar (Ax) results in a vector for which each element is multiplied by A.
MATRICES. A matrix is a rectangular array of elements (numbers) that takes the
form:
alI1
a22 an2
... ...
in which tie elements are written a ij . The subscript i refers to the ith row and j to the jth column. A is called an n X m matrix where IZ is the number of rows and m is the number of columns. If n = m, the matrix is called a square matrix. If m = 1, the matrix is a column vector (n X 1). If n = 1, the matrix is a row vector (1 X m). The transpose of a matrix, AT, is a matrix for which the rows and columns of the matrix A are interchanged. If the diagonal elements (a ij) of a square matrix am unity and all off-diagonal elements are zero, then the matrix is called a unit matrix and is given the symbol I. If A = AT for a square matrix, the matrix A is said to be symmetrical. When two matrices are added (or subtracted), the corresponding elements are added (or subtracted), thus
a11 + hl
a12 + b12
a22 +.!m
. . .
aim + blm a2m + km
A+B=
+ b21
an1 + bnl
a,,2 + b,,2
. . . anm + brim
STATE-SPACE
METHODS
The product of two matrices C = AB is a matrix whose elements are obtained by the expression m Cij = for i = 1 . . . n Ix aikbkj
a n d j = l... p whereAisannXmmatrixandBisanmXpmatrix.ThematrixCisannXp matrix. INVERSE OF A MATRIX. The inverse of a matrix is related tb the concept of division for numbers. The inverse of a number x is written l/x or x-l. The product of a number x and its inverse is equal to unity. The inverse of a matrix A is written A- and the product of a matrix and its inverse is equal to the unit matrix; thus A-IA = I The expression used for matrix inversion for the examples used in this chapter takes the form:
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