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barcode reader integration with asp.net i to STATESPACE METHODS in Software
i to STATESPACE METHODS Code 128C Recognizer In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Code 128C Generator In None Using Barcode generator for Software Control to generate, create Code 128 Code Set B image in Software applications. Combining Eqs. (28.38) and (28.36) leads to
Recognize Code 128B In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. Printing Code 128 Code Set B In Visual C#.NET Using Barcode generation for VS .NET Control to generate, create Code 128A image in .NET applications. = (KJq)x1  Kcx2 + (K&I)r
Printing Code128 In VS .NET Using Barcode encoder for ASP.NET Control to generate, create Code 128B image in ASP.NET applications. ANSI/AIM Code 128 Encoder In .NET Using Barcode generator for .NET Control to generate, create Code 128 Code Set B image in VS .NET applications. i3 = ANSI/AIM Code 128 Drawer In VB.NET Using Barcode drawer for .NET Control to generate, create USS Code 128 image in .NET applications. Generate Code128 In None Using Barcode generation for Software Control to generate, create Code 128 Code Set B image in Software applications. (28.39) (28.40) Printing DataMatrix In None Using Barcode creator for Software Control to generate, create DataMatrix image in Software applications. Creating UCC128 In None Using Barcode encoder for Software Control to generate, create GS1128 image in Software applications. * 7 Barcode Drawer In None Using Barcode generator for Software Control to generate, create barcode image in Software applications. Bar Code Drawer In None Using Barcode creation for Software Control to generate, create bar code image in Software applications. ax1  Kcx2 + cxr
Make DUN  14 In None Using Barcode creator for Software Control to generate, create ITF14 image in Software applications. Print EAN / UCC  13 In None Using Barcode printer for Office Word Control to generate, create GTIN  128 image in Word applications. where (Y = Kc/r1
Barcode Scanner In Visual C#.NET Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in .NET framework applications. Painting UPCA Supplement 5 In None Using Barcode maker for Online Control to generate, create UPCA Supplement 5 image in Online applications. 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 Bar Code Maker In Java Using Barcode drawer for Eclipse BIRT Control to generate, create barcode image in BIRT applications. Bar Code Creation In None Using Barcode generation for Font Control to generate, create bar code image in Font applications. b = 11~ a = Kc/r, Draw EAN / UCC  13 In ObjectiveC Using Barcode creation for iPad Control to generate, create EAN13 image in iPad applications. Code39 Printer In ObjectiveC Using Barcode drawer for iPad Control to generate, create Code 39 Extended image in iPad applications. = 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
:. Statespace 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 nthorder differential equation. In the statespace representation, the &order differential equation is written as n firstorder 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 nthorder 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 statespace 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 statespace representation in multivariable control and in nonlinear control in later chapters. STATESPACE 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 statespace 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 ndimensional 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 offdiagonal 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
STATESPACE
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 xl. 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 AIA = I The expression used for matrix inversion for the examples used in this chapter takes the form:

