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N = O O in VS .NET
N = O O QRCode Scanner In Visual Studio .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET applications. Make QR Code JIS X 0510 In .NET Framework Using Barcode printer for .NET Control to generate, create QR Code JIS X 0510 image in Visual Studio .NET applications. ( a ) Show that the matrix N is nilpotent of index 3.
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1 0 0 Thus, N is nilpotent of index 3. ( b ) Since the diagonal matrix A can be expressed as 21, we have
that is, A and N commute. Since A and N commute, then, by the result from Prob. 7.46
Now [see App. A, Eq. (A.4911
and using similar justification as in Prob. 7.44(b),we have
Thus, CHAP. 71
STATE SPACE ANALYSIS
7.48. Using the state variables method, solve the secondorder linear differential equation y"(t) + 5y'(t)+ 6 y ( t )= x ( t ) (7.127) with the initial conditions y(0) = 2, yl(0) = 1 , and x ( t ) = e  ' u ( t ) (Prob. 3.38). Let the state variables q J t ) and q,(t) be qdt)=Y(I) ~ 2f ) ( =~'(t) Then the state space representation of Eq. (7.127) is given by [Eq. (7.19)l
q ( t ) = Aq(t) + b x ( t ) with
A=[: ~ t t =)c q ( t ) b=[y] c = [ l O] q'ol
[ q d o 0 ) ]= 2( ) Thus, by Eq.(7.65) with d = 0. Now, from the result from Prob. 7.39
c ~ q ( 0= [ I ) ~](e~'[: +e''[: ]:[)I: Thus, 7.49. Consider the network shown in Fig. 720. The initial voltages across the capacitors C, and C, are f V and 1 V, respectively. Using the state variable method, find the voltages across these capacitors for t > 0 . Assume that R , = R , = R, = 1 0 and C ,= C 2 =1 F. Let the state variables q , ( t ) and q2(t ) be
STATE SPACE ANALYSIS
[CHAP. 7
Fig. 720 Applying Kirchhoffs current law at nodes 1 and 2, we get
Substituting the values of R , , R 2 , R,, C , , and C 2 and rearranging, we obtain
41(t) =  2 9 1 ( t ) + q 2 ( t ) 42(t) = 9 l ( t )292(t) In matrix form
il(t) =Aq(t) with Then, by Eq. (7.63) with x ( t ) = 0 and using the result from Prob. 7.43, we get
7.50. Consider the continuoustime LTI system shown in Fig. 721. (a) Is the system asymptotically stable ( b ) Find the system function H(s). ( c ) IS the system B I B 0 stable From Fig. 721 and choosing the state variables q , ( t ) and q 2 ( t )as shown, we obtain
41(t) = 9 2 ( t ) + x ( t ) 42(t) = 29l(t) +92(t) x ( t ) ~ ( t =) 9 L t )  % ( t ) CHAP. 71
STATE SPACE ANALYSIS
In matrix form
where Now Thus, the eigenvalues of A are A , =  1 and A t = 2. Since Re{A,) > 0, the system is not asymptotically stable. ( b ) By Eq. (7.52) the system function H(s) is given by Note that there is polezero cancellation in H(s) at s = 2. Thus, the onIy pole of H(s) is  1 which is located in the lefthand side of the splane. Hence, the system is B I B 0 stable. Again it is noted that the system is essentially unstable if the system is not initially relaxed. 7.51. Consider an Nthorder continuoustime LTI system with state equation
The system is said to be controllable if it is possible to find an input x ( t ) which will drive the system from q(t,) = q , to q(t ,) = q , in a specified finite time and q , and q , are any finite state vectors. Show that the system is controllable if the controllability matrix defined by has rank N.
STATE SPACE ANALYSIS
[CHAP. 7
We assume that r, and q[Ol = 0. Then, by Eq. (7.63) we have
Now, by the CayleyHamilton theorem we can express eA' as
Substituting Eq. (7.130) into Eq. (7.129) and rearranging, we get
Let Then Eq. (7.131) can be rewritten as
For any given state q , we can determine from Eq. (7.132) unique Pk's (k = 0,1,. . . , N  I), and hence x(t), if the coefficients matrix of Eq. (7.132) is nonsingular, that is, the matrix has rank N.
7.52. Consider an Nthorder continuoustime LTI system with state space representation
q ( t ) = Aq(t) + b x ( t ) ~ ( t= c 4 t ) ) T h e system is said to be observable if any initial state q(t,) can be determined by examining the system output y ( t ) over some finite period of time from to to t , . Show that the system is observable if the observability matrix defined by

