 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
barcode reader integration with asp.net STATESPACE in Software
STATESPACE Decoding Code128 In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Code128 Generator In None Using Barcode maker for Software Control to generate, create Code 128 Code Set A image in Software applications. METHODS
ANSI/AIM Code 128 Scanner In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. Paint Code 128 In C# Using Barcode creation for .NET framework Control to generate, create Code 128 Code Set B image in VS .NET applications. STATE VARIABLES
ANSI/AIM Code 128 Drawer In .NET Using Barcode encoder for ASP.NET Control to generate, create Code 128 Code Set A image in ASP.NET applications. Code 128B Generator In VS .NET Using Barcode creation for Visual Studio .NET Control to generate, create Code 128 Code Set A image in VS .NET applications. A linear physical system can be described mathematically by: Drawing Code 128 Code Set B In Visual Basic .NET Using Barcode generation for VS .NET Control to generate, create Code 128B image in .NET applications. Painting EAN13 In None Using Barcode generator for Software Control to generate, create GTIN  13 image in Software applications. an nth order differential equation a transfer function n firstorder differential equations a matrix differential equation Bar Code Maker In None Using Barcode generation for Software Control to generate, create bar code image in Software applications. Universal Product Code Version A Creation In None Using Barcode maker for Software Control to generate, create GS1  12 image in Software applications. 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) Code39 Creation In None Using Barcode generation for Software Control to generate, create Code 39 Extended image in Software applications. Code 128C Drawer In None Using Barcode maker for Software Control to generate, create Code 128B image in Software applications. zd*y dy rdt2+25rdt+y=u
Create NW7 In None Using Barcode encoder for Software Control to generate, create Codabar image in Software applications. Draw Data Matrix In .NET Using Barcode encoder for .NET Control to generate, create DataMatrix image in VS .NET applications. 2. Transfer function. The transfer function corresponding to Eq. (28.1) is
Scan Bar Code In Visual C# Using Barcode scanner for .NET framework Control to read, scan read, scan image in .NET applications. Code128 Reader In C# Using Barcode recognizer for .NET framework Control to read, scan read, scan image in VS .NET applications. (28.1) Decoding Barcode In Java Using Barcode Control SDK for Eclipse BIRT Control to generate, create, read, scan barcode image in BIRT reports applications. Code 128C Drawer In None Using Barcode creation for Microsoft Word Control to generate, create ANSI/AIM Code 128 image in Word applications. (28.2) U(s) 3. 12 firstorder 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 UCC128 Drawer In ObjectiveC Using Barcode maker for iPad Control to generate, create GTIN  128 image in iPad applications. Universal Product Code Version A Scanner In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. $x1 Y(s)  1 7232 + 2578 + 1
Y. 1 TX + pu
fl = jj + u 72
(28.6) STATESPACE 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 secondorder 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.) StateSpace 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,>
STATESPACE
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 timeinvariant systems for which ii is a linear combination of state variables and the coefficients are constant. For the timeinvariant 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

