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has rank N. in Visual Studio .NET
has rank N. Recognizing QR In VS .NET Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in VS .NET applications. Encode QR Code In Visual Studio .NET Using Barcode drawer for VS .NET Control to generate, create QR Code image in .NET framework applications. We prove this by contradiction. Suppose that the rank of M, is less than N. Then there exists an.initia1 state q[O] = q, f 0 such that Decoding QR Code ISO/IEC18004 In Visual Studio .NET Using Barcode reader for .NET framework Control to read, scan read, scan image in VS .NET applications. Bar Code Encoder In .NET Using Barcode maker for .NET framework Control to generate, create barcode image in .NET framework applications. Moq, = 0 Scanning Barcode In .NET Using Barcode recognizer for VS .NET Control to read, scan read, scan image in VS .NET applications. Print QR Code In C#.NET Using Barcode maker for VS .NET Control to generate, create QR Code image in .NET framework applications. CHAP. 71
QR Code Maker In .NET Framework Using Barcode maker for ASP.NET Control to generate, create Denso QR Bar Code image in ASP.NET applications. QR Code ISO/IEC18004 Printer In Visual Basic .NET Using Barcode printer for Visual Studio .NET Control to generate, create QR Code 2d barcode image in VS .NET applications. STATE SPACE ANALYSIS
EAN13 Maker In VS .NET Using Barcode printer for .NET Control to generate, create EAN / UCC  13 image in VS .NET applications. Bar Code Maker In Visual Studio .NET Using Barcode maker for .NET Control to generate, create bar code image in .NET applications. Now from Eq. (7.65), for x(t ) = 0 and to = 0, GS1 128 Encoder In .NET Framework Using Barcode maker for .NET framework Control to generate, create EAN 128 image in .NET applications. Make Uniform Symbology Specification ITF In VS .NET Using Barcode maker for .NET Control to generate, create 2/5 Interleaved image in VS .NET applications. However, by the CayleyHamilton theorem eA'can be expressed as
Create Code 128B In None Using Barcode printer for Software Control to generate, create Code128 image in Software applications. Generating GTIN  13 In .NET Framework Using Barcode generation for ASP.NET Control to generate, create EAN13 image in ASP.NET applications. Substituting Eq. (7.136) into Eq.(7.135),we get
Generate 2D Barcode In Visual C#.NET Using Barcode generator for Visual Studio .NET Control to generate, create Matrix Barcode image in .NET applications. Painting USS128 In .NET Using Barcode generation for ASP.NET Control to generate, create UCC128 image in ASP.NET applications. in view of Eq. (7.134).Thus, qo is indistinguishable from the zero state and hence the system is not observable. Therefore, if the system is to be observable, then Mo must have rank N. Data Matrix Creation In C#.NET Using Barcode generator for Visual Studio .NET Control to generate, create DataMatrix image in VS .NET applications. GS1  12 Maker In None Using Barcode maker for Software Control to generate, create UPCA image in Software applications. 7.53. Consider the system in Prob. 7.50.
Read UPC Symbol In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. Code39 Encoder In Java Using Barcode generator for Eclipse BIRT Control to generate, create Code39 image in BIRT reports applications. ( a ) Is the system controllable
( 6 ) Is the system observable
From the result from Prob. 7.50 we have
Now and by Eq. (7.128) the controllability matrix is
M, = [b Ab] = [1 11 and IM,I = 0. Thus, it has a rank less than 2 and hence the system is not controllable. ( b ) Similarly, and by Eq. (7.133) the observability matrix is
and J M J =  2 # 0. Thus, its rank is 2 and hence the system is observable. Note from the result from Prob. 7.50(b) that the system function H(s) has polezero cancellation. As in the discretetime case, if H(s) has polezero cancellation, then the system cannot be both controllable and observable. 7.54. Consider the system shown in Fig. 722. Is the system controllable ( b ) Is the system observable ( c ) Find the system function H(s). STATE SPACE ANALYSIS
[CHAP. 7
Fig. 722 In matrix form
where Now and by Eq. (7.128) the controllability matrix is
M,= [b Ab] = [: :] and lM,I = 0. Thus, its rank is less than 2 and hence the system is not controllable. ( b ) Similarly, and by Eq. (7.133) the observability matrix is
and lMoI= 0. Thus, its rank is less than 2 and hence the system is not observable. By Eq. (7.52) the system function H(s) is given by H ( S )= c ( s ~  A )  ' ~
Note that the system is both uncontrollable and unobservable.
CHAP. 71
STATE SPACE ANALYSIS
Supplementary Problems
Consider the discretetime LTI system shown in Fig. 723. Find the state space representation of the system with the state variables q , [ n ] and q , [ n ] as shown. Fig. 723 Consider the discretetime LTI system shown in Fig. 724. Find the state space representation of the system with the state variables q , [ n ] and q , [ n ] as shown. Fig. 724 STATE SPACE ANALYSIS
[CHAP. 7
7.57. Consider the discretetime LTI system shown in Fig. 725. ( a ) Find the state space representation of the system with the state variables q , [ n ] and q 2 [ n ] as shown. ( b ) Find the system function H ( z ) . Find the difference equation relating x [ n ] and y [ n ] . Fig. 725 CHAP. 71
STATE SPACE ANALYSIS
7.58. A discretetime LTI system is specified by the difference equation
y [ n ] + y [ n  11  6 y [ n  21 = 2x[n
 1 1 + x [ n  21 Write the two canonical forms of state representation for the system.
7.59. Find A for " ( a ) Using the CayleyHamilton theorem method. ( b ) Using the diagonalization method.
Am. A"  2(+)" + 3 0 ) " 6 ( f )"  6 ( f)" 3($)"  2(;)" (i)" (4)" +
7.60. Find A for " ( a ) Using the spectral decomposition method.
( b ) Using the ztransform method.
Am. An=
Given a matrix
[(3)" 0 0 f(2)"+ ;(3)" ;(2)"  ; (  3 ) " f(2)"  + (  3 ) " j(2)"  j (  3 ) " ( a ) Find the minimal polynomial m ( A ) of A. ( b ) Using the result from part ( a ) , find An.
Ans.
( a ) m(A) = ( A  3XA
+ 3) = A' STATE SPACE ANALYSIS
[CHAP. 7
Consider the discretetime LTI system with the following state space representation: (a) Find the system function H(z). (b) Is the system controllable (c) IS the system observable
Ans.
( 2  1)* (b) The system is controllable. (c) The system is not observable.
(a) H ( z ) = Consider the discretetime LTI system in Prob. 7.55. (a) (b) (c) (dl Is the Is the IS the Is the system system system system The The The The asymptotically stable BIBO stable controllable observable Am. (a) (b) (c) (dl
system is asymptotically stable. system is BIBO stable. system is controllable. system is not observable. The controllability and observability of an LTI system may be investigated by diagonalizing the system matrix A. A system with a state space representation v [ n + 1 1 = Av[n] + *bx[n] y [ n ]= b [ n ] (where A is a diagonal matrix) is controllable if the vector Ib has no zero elements, and it is observable if the vector e has no zero elements. Consider the discretetime LTI system in Prob. 7.55. Let d n ] = Tq[n].Find the matrix T such that the new state space representation will have a diagonal system matrix. Write the new state space representation of the system. Using the result from part ( b ) , investigate the controllability and observability of the system.

