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barcode reader integration with asp.net = e* x(O) + in Software
= e* x(O) + Decoding Code 128B In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Code 128 Creator In None Using Barcode drawer for Software Control to generate, create Code128 image in Software applications. e*(  )Bu(T)d
Scanning Code 128A In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. Draw Code 128 Code Set A In C# Using Barcode maker for VS .NET Control to generate, create ANSI/AIM Code 128 image in .NET framework applications. (29.4) Code 128C Creation In VS .NET Using Barcode creator for ASP.NET Control to generate, create Code128 image in ASP.NET applications. Make Code 128 Code Set A In .NET Framework Using Barcode generation for .NET framework Control to generate, create ANSI/AIM Code 128 image in .NET applications. TRANSFER FUNCTION MATRIX
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Barcode Generation In None Using Barcode generation for Software Control to generate, create bar code image in Software applications. Bar Code Generation In None Using Barcode drawer for Software Control to generate, create barcode image in Software applications. MATRIX
Encoding EAN8 In None Using Barcode drawer for Software Control to generate, create EAN / UCC  8 image in Software applications. UPC  13 Creation In None Using Barcode encoder for Online Control to generate, create EAN13 image in Online applications. i = Ax
UPC A Generator In Java Using Barcode encoder for Eclipse BIRT Control to generate, create UPCA image in Eclipse BIRT applications. Code 128 Code Set B Generator In .NET Using Barcode generator for Reporting Service Control to generate, create Code 128A image in Reporting Service applications. (29.5) Make GS1  12 In None Using Barcode generation for Office Word Control to generate, create GTIN  12 image in Microsoft Word applications. UCC  12 Creation In None Using Barcode generator for Online Control to generate, create UPC Code image in Online applications. Let us now turn our attention to the solution of the matrix differential equation
Bar Code Generator In Java Using Barcode generator for Java Control to generate, create bar code image in Java applications. ANSI/AIM Code 39 Generation In None Using Barcode drawer for Online Control to generate, create Code 39 Extended image in Online applications. This is Eq. (29.1) for the case of no inputs (i.e., u = 0). The initial conditions for Eq. (29.5) may be expressed as x(O). One can show that the solution to Eq. (29.5) with initial conditions x(O) is given by x(t) = I + At + g2 + . . .
+ $tk} x(O) The infinite series of matrix terms within the braces is given the symbol ek. This symbol is chosen to recall that the infinite series of the scalar term ear is 1 + at + gt2 + . . . + Etk Using the symbol ek, we may write Eq. (29.6) as x(t) = eA x(0) (29.7) ak
The symbol eAr is an IZ X n matrix in which each element contains a power series of t. The solution to Eq. (29.1) can be shown to be . x(t) = e* x(O) + e*(  )Bu(T)dT i0 (29.8) Notice that Eq. (29.8) resembles Eq. (29.4), which is the solution for the scalar differential equation. Since e Ar is awkward and perhaps misleading as to its nature, e is sometimes replaced by +(t); thus t)(t) = eAr
(transition
matrix) (29.9) Either of the terms +(t) and eAr can be used for the transition matrix. In this book, we shall use eAr. Example 29.1. Solution of a matrix differential equation.
Solve the following
matrix differential equation i=  1 0 where u(t) is a unitstep function and
2 x+
x(O) = [ :, I
STATESPACE
METHODS
One can show that
,*r =
emr 0
,t _ ,2t
e2r
In the next section, the method used to obtain the elements of this matrix will be developed. Applying Eq. (29.8) gives x(t) = [ o,] +[ x(t) = e(t7)  (33e2( 7) 0*5e2( 7) ]I: 1 f, 2e + 0.5ee2  0.5ee2
Determining e * t One method for determining the elements of the transition matrix e Ar is to use Laplace transforms. Consider the matrix differential equation of JZq. (29.1) i = Ax + Bu If we take the Laplace transform of each side, we obtain sX(s)  x(O) = AX(s) + BU(s) or 1 sX(s)  AX(s) = x(O) + BU(s) Solving for X(s) gives (~1  A)X(s) = x(O) + BU(s) (29.10) To obtain an expression for X(s), premultiply both sides of Eq. (29.10) by (81  A) ; thus (d A) @1  A)X(s) = @I A) x(O) + (d A) BU(s) This equation becomes X(s) = (sI  A) x(O) + (s1  A) BU(s) To obtain x(t) from Eq. (29.1 l), we may take the inverse transform; thus x(t) = Ll{ (s1  A) x(O)} + Ll{ (s1  A) BU(s)} By comparing Eqs. (29.8) and (29.12), we see that eAt = Ll{ (s1  A) } (29.13) (29.12) (29.11) TRANSFER
FUNCTION MATRIX
e*( T)~U(T)dT = L { ($1  A) BU(s)} TRANSFER FUNCTION MATRIX
(29.14) When x(O) = 0, a case frequently used in control applications, we obtain from Eq. (29.11) . (29.15) X(s) = (sI A) BU(s) This may be written X(s) = G(s)U(s) where G(s) = @I A) B (transfer function matrix) (29.17) (29.16) The term G(s) is called the transfer &nction matrix and serves the same purpose as the transfer function for the scalar case; namely, it relates a set of state variables X(s) to a set of inputs U(s). If we prefer to relate the output to the input as expressed by Eq. (29.2), we may proceed as follows. Taking the Laplace transform of both sides of Eq. (29.2) gives Y(s) = CX(s) Combining Eqs. (29.15) and (29.18) gives * Y(S) = C(sI A) BU(s) We may now write Y(s) = Gl(s)W) where Gl(s) = C(sIA) B (29.20) The term Gt(s) in Eq. (29.20) is also a transfer function matrix that relates the output vector Y to the input vector U. Example 29.2. Determine the transfer function matrix for the 2tank liquidlevel system shown in Fig. 29.1. As developed in Example 28.1 [Eq. (28.17)] of the previous chapter, this system is described by li = Ah+Bu where A= (29.18) (29.19)

