SYSTEMS ANALYSIS TECHNIQUES
SYSTEMS ANALYSIS TECHNIQUES ECSE 6400
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This 13 page Class Notes was uploaded by Immanuel Brakus PhD on Monday October 19, 2015. The Class Notes belongs to ECSE 6400 at Rensselaer Polytechnic Institute taught by Staff in Fall. Since its upload, it has received 12 views. For similar materials see /class/224785/ecse-6400-rensselaer-polytechnic-institute in ELECTRICAL AND COMPUTER ENGINEERING at Rensselaer Polytechnic Institute.
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Date Created: 10/19/15
5 Solutions to 51 AX t Bu 2 Topics Be able to Ref Section 25 derive the matrix solution to 120 AX 2 Bu t use a series expansion to nd the solution to X0 Ax t Bu 2 nd the solution using modal decomposition show the connection between states representatzons and operators F all 2004 Direct Solution Given X t AX t verify that xt eA XO is a solution How is eAt defined What if A was a scalar a eat eAt Now verify by direct substitution direct solution cont Derive the solution to 320 Axt Bu r Derivation Ans xtequotx0je quot39BuIdI Premulti ply both sides of by the integrating factor e note Ae A e A A Next integrate both s1des Example had a state variable representation 0 1 o t 2 0 2 X0 2 quot0 yt1 1Xt Find eA z 1 Solution A2t2 m Aktk k eA IAt by recognizing each series t eAtze 0 O e2t For the diagonal case when X10 0 0 0 0 X2 0 0 O A 0 0 0 0 39 kn thenM e1 0 0 0 0 0 e t 0 0 0 At Consider 2 S 323S2 51S2 This has a state variable representation t 2 1 0 t t x0 0 1xlt luo y1 2 0Xt ut 0 Auxwrulfw Find XU ifx0 Solution 22 kk eArIAtAtNAt I 1quot by recognizing each series 72 22 e e te 0 equot2 Given the initial conditions find Xt eAJ Xt What are the eigenvalues of A J1 01 1 1 Express xt wrt the basis L 2 1 1 T the basis Modal Decomposition Given Xt Axt Bu t let X0 M lt10 What is the differential equation for qt What is thenew operator under this transformation How can you uncouple the qt s 1Ibglow do you find M Denote the columns of M y Xi AnsA tilxi0 Example Recall HS 2 4 had a state variable US s2 33 2 representation 150 O 2 ut 0 l 2 3 Xt yt1 0Xt Us m8dal decomposition to solve for Xt When u t Solution Find the eigenvalues of A AM 712 Find the corresponding eigenvectorsn 1 Express xt in terms of its modes Find the initial condition vector q0 Ammwxxm Find xt in terms of its modes What is xt if x0 lies along one of the elgenvectors Example Repeat the second Example using Modal Decomposition 22m 1 x0 0 W xlt0 1 0 1 1 3 Find Xt if ut 0 Solution Find the eigenvalues of A Amfwfq Find the eigenvectors of A to find M Find qt What are the initial conditions q0w An no 39 3rquot What is the representation of Xt wrt the new ba31s the columns of M note 39 The solutlon IS in the directlon of the eigenvectors 39 No initial condition along an eigenvector gt the mode ass001ated with that eigenvector IS not present in the solution Xt Give a geometric interpretation of control and Signal analys1s