 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
all k in Visual Studio .NET
all k Recognize QR Code 2d Barcode In .NET Framework Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in .NET applications. QR Code ISO/IEC18004 Printer In .NET Framework Using Barcode creation for Visual Studio .NET Control to generate, create Quick Response Code image in VS .NET applications. (7.45) Denso QR Bar Code Decoder In .NET Using Barcode recognizer for .NET Control to read, scan read, scan image in VS .NET applications. Paint Barcode In Visual Studio .NET Using Barcode creator for Visual Studio .NET Control to generate, create barcode image in Visual Studio .NET applications. then the system is said to be asymptotically stable; that is, if, undriven, its state tends to zero from any finite initial state q,. It can be shown that if all eigenvalues of A are distinct and satisfy the condition (7.45), then the system is also BIB0 stable. Scanning Barcode In Visual Studio .NET Using Barcode recognizer for .NET framework Control to read, scan read, scan image in Visual Studio .NET applications. Encode QRCode In C#.NET Using Barcode drawer for VS .NET Control to generate, create QR Code JIS X 0510 image in Visual Studio .NET applications. STATE SPACE ANALYSIS
Making Denso QR Bar Code In .NET Framework Using Barcode maker for ASP.NET Control to generate, create Quick Response Code image in ASP.NET applications. Printing QR Code In VB.NET Using Barcode encoder for .NET framework Control to generate, create QR Code JIS X 0510 image in VS .NET applications. [CHAP. 7
Painting Matrix 2D Barcode In .NET Framework Using Barcode creator for Visual Studio .NET Control to generate, create 2D Barcode image in VS .NET applications. Generating DataMatrix In .NET Using Barcode encoder for .NET framework Control to generate, create ECC200 image in Visual Studio .NET applications. 76 SOLUTIONS OF STATE EQUATIONS FOR CONTINUOUSTIME LTI SYSTEMS .
Bar Code Generation In .NET Using Barcode creation for .NET Control to generate, create bar code image in .NET framework applications. ANSI/AIM Code 93 Printer In .NET Framework Using Barcode creation for VS .NET Control to generate, create USS 93 image in VS .NET applications. Laplace Transform Method: Consider an Ndimensional state space representation
Encode Data Matrix ECC200 In Java Using Barcode creation for Android Control to generate, create Data Matrix 2d barcode image in Android applications. Make Bar Code In Java Using Barcode maker for Android Control to generate, create barcode image in Android applications. where A, b, c, and d are N x N, N X 1, 1 X N, and 1 X 1 matrices, respectively. In the following we solve Eqs. ( 7 . 4 6 ~ )and (7.46b) with some initial state q(0) by using the unilateral Laplace transform. Taking the unilateral Laplace transform of Eqs. ( 7 . 4 6 ~ and ) (7.466) and using Eq. (3.441, we get Drawing Code 128C In ObjectiveC Using Barcode printer for iPhone Control to generate, create USS Code 128 image in iPhone applications. UPCA Supplement 2 Creation In Java Using Barcode printer for BIRT Control to generate, create UPC Code image in BIRT applications. Rearranging Eq. (7.47~1, have we ( s I  A)Q(s) 1D Barcode Drawer In Java Using Barcode drawer for Java Control to generate, create 1D image in Java applications. USS Code 39 Decoder In Java Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications. = q(0) GS1  12 Encoder In C# Using Barcode maker for Visual Studio .NET Control to generate, create GS1  12 image in .NET framework applications. UCC  12 Decoder In C#.NET Using Barcode reader for .NET Control to read, scan read, scan image in .NET applications. + bX(s) (7.49) Premultiplying both sides of Eq. (7.48) by (sI  A)' yields
Q(S) = (SI A)'~(o+ ( S I  A)'~x(s) ) Substituting Eq. (7.49) into Eq. (7.47b1, we get
Taking the inverse Laplace transform of Eq. (7.501, we obtain the output y(t). Note that c(sI  A)'q(0) corresponds to the zeroinput response and that the second term corresponds to the zerostate response. System Function H(s): As in the discretetime case, the system function H(s) a continuoustime LTI system of is defined by H(s) = Y(s)/X(s) with zero initial conditions. Thus, setting q(0) = 0 in Eq. (7.501, we have Thus, CHAP. 71
STATE SPACE ANALYSIS
C. Solution in the Time Domain: Following
we define
where k!= k(k
 1) . . 2.1.If
t = 0,then
Eq. (7.53)reduces to
e0 = I
where 0 is an N x N zero matrix whose entries are all zeros. As in e  a r at e , we can show that
eA(tr) ea('') (7.54) = ea'ea' = (7.55) (7.56) =eA~eAr =eAreA~
Setting Thus, in Eq. (7.55),we have
eAteAr
= eAteAt = e O = I
which indicates that eA' is the inverse of eA'. The differentiation of Eq. (7.53)with respect to
yields
which implies
eAt
= AeA' = e A t ~
Now using the relationship [App. A, Eq. (A.70)] and Eq.(7.581,we have
STATE SPACE ANALYSIS
[CHAP. 7
Now premultiplying both sides of Eq. (7.46a) by eA', obtain we
ePA'q(t = L  A ' ~ q ( +)e A ' b x ( t ) ) t
e P A ' q ( t  e P A ' A q ( t= e  A ' b x ( t ) ) ) From Eq. (7.59)Eq. (7.60) can be rewritten as
[ e  A ' q ( t ) = C A ' b x ( t ) ] Integrating both sides of Eq. (7.61)from 0 to I , we get
Hence
e  * ' q ( t ) = q ( 0 ) + /'e*'bx(r) d i
(7.62) Premultiplying both sides of Eq. (7.62) by eA' and using Eqs. (7.55)and (7.561, we obtain
If the initial state is q(t,,)and we have x( t ) for t
2 I,, then
which is obtained easily by integrating both sides of Eq. (7.61) from t , to t . The matrix function eA' is known as the statetransition matrix of the continuoustime system. Substituting Eq. (7.63) into Eq. (7.466),we obtain D. Evaluation of eA': Method 1: As in the evaluation of An, by the CayleyHamilton theorem we have
When the eigenvalues A , of A are all distinct, the coefficients b,, b , , . . ., b N  , can be found from the conditions For the case of repeated eigenvalues see Prob. 7.45.
CHAP. 71
STATE SPACE ANALYSIS
Method 2: Again, as in the evaluation of An we can also evaluate eA' based on the diagonalization of A. If all eigenvalues A, of A are distinct, we have eA' = P
where P is given by Eq. (7.30). Method 3: We could also evaluate eA' using the spectral decomposition of A, that is, find constituent matrices E, (k = 1,2,. . ., N ) for which A=A,El
+ A2E2+ . . . +ANEN
( 7.69) where A, ( k = 1,2,. . ., N ) are the distinct eigenvalues of A. Then, when eigenvalues A, of A are all distinct, we have e A t= e A ~ ' E l e A ~ + ~ ., . +eAN'E, + ' . (7.70) Method 4: Using the Laplace transform, we can calculate eA'. Comparing Eqs. (7.63)and (7.49),we see that E. Stability: From Eqs. (7.63) and (7.68) or (7.70), we see that if all eigenvalues A, of the system matrix A have negative real parts, that is, Re{A,) < 0 all k (7.72) then the system is said to be asymptotically stable. As in the discretetime case, if all eigenvalues of A are distinct and satisfy the condition (7.721, then the system is also B I B 0 stable. Solved Problems
STATE SPACE REPRESENTATION 7.1. Consider the discretetime LTI system shown in Fig. 71. Find the state space representation of the system by choosing the outputs of unitdelay elements 1 and 2 as state variables q,[n] and q2[n], respectively. From Fig. 71 we have 4 1 b + 11 = 4 2 M 42[n + 11 = 2 q , [ n l + 3 q 2 b I + x b I y b I = 2q,[nl+ 3 q h I + x b I

