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PPT On State Transition Matrix

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Published in: Physics
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In control theory, the state-transition matrix is a matrix whose product with the state vector at an initial time gives at a later time . The state-transition matrix can be used to obtain the general solution of linear dynamical systems.

Aswathi K / Abu Dhabi

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  1. Lesson 6 State transition matrix linear system by Meiling CHEN Linear system 1. Analysis 1
  2. X(t) = AX(t) -k Bbl (t) dt y (t) = CX(t) -k Du (t) The behavior of x(t) et y(t) : 1. Homogeneous solution of x(t) 2. Non-homogeneous solution of x(t) linear system by Meiling CHEN
  3. Homogeneous solution = Ax(t) sx(s) - x(0) = Ax(s) — eAtx(0) State transition matrix A x(to) — ent x(0) x(0) —e At x(to) x(t)=e e x(to ) linear system by Meiling CHEN — to ) 3
  4. Properties 1, 2. 3. 4, 5. x(0) = — to ) linear system by Meiling CHEN
  5. Non-homogeneous solution X(t) = 14x(t) + Bu (t) dt y(t) = Cx(t) + Du (t) x(t) + J — T)Bu (T)dT Convolution Homogeneous linear system by Meiling CHEN
  6. Zero-input response Zero-state response linear system by Meiling CHEN 6
  7. Example 1 let u(t) x(0) Ans.' — 2e—t + 2e linear system by Meiling CHEN A)
  8. u(t) let 0] Using Maison 's gain.formula A = 3s—1 + 2s X2(s) linear system by Meiling CHEN
  9. How to find State transition matrix A Methode 1: Methode 2: Methode 3: Cayley-Hamilton Theorem linear system by Meiling CHEN 9
  10. Methode 1: 1 0 0 o 0 3 —2 1 01 adj(sl — A) linear system by Meiling CHEN 10
  11. Methode 2: —6 diagonal matrix linear system by Meiling CHEN 11
  12. 8.4 System Characteristic Equation and Eigenvalues Diagonization The characteristic equation and eigenvalues (characteristic values) are introduced in Chapter 7. Here, we give a general definition of 'the system eigenvalues are show that they are identical to the eigenvalues of the system matrix A The eigenvalues are defined in linear algebra as scalars, A, satisfying Av = /v, where the vectors v are called the eigenvectors. This system of n linear algebraic equations ((81 — A)v = 0) has a solution v 0 if and only ifthe corresponding determinant is equal to zero (Appendix A), that is e s 'des contain the oopyliqhted material from Linear Dynamic Systems and Signals, Prentice Hall 2003, 8-65 Prepared by Professor Zoran Galic
  13. Modal Transformation Diagonization One of the most interesting similarity transformations is the one that puts matrix A into diagonal form. Assume that P = V = [VI, , where Vi are 'the eigenvectors. We then have V-IAV = A A = It is easy to show that the elements .Xi, i = 1, n, on the matrix diagonal of A are the roots of the characteristic equation 1>41 - Al = — Al = O, that is, they are the eigenvalues. This can be shown in a straightfonvard way Thi s state transformation is known as the modal transformatiom 8-70 e s 'des contain the oopyliqhted material from Linear Dynamic Systems end Signals, Prentice Hall 2003, Prepared by Professor Zoran Gajic
  14. Case 1: (81 (121 e distinct 3 1+4 depend linear system by Meiling CHEN 14
  15. In the case of A matrix is phase-variable form and 13 • • • In P-IAP = A = = peAtp I linear system by Meiling CHEN Vandermonde matrix for phase-variable form 15
  16. Case 1: 1 o 0 (81 1 o distinct O 2 O O O o 1-1 o 1-2 o O O linear system by Meiling CHEN depend — 123 16
  17. (131 I o O o I O 1 P -IAP = I o O O I o linear system by Meiling CHEN O O 2 17
  18. Case 3: distinct Jordan form VI P -IAP = Jordan Generalized eigenvectors form (Ill (Ill (Ill ID-IAP = Alt Alt Alt Alt Alt Alt linear system by Meiling CHEN 18
  19. Example: 1-1 (81 (81 3 -1 1 1 1-3 1 — 1/71 12 21 22 12 21 22 o = peAtp I 2 te linear system by Meiling CHEN 19
  20. Method 3: Cayley-Hamilton Theorem One of the most powerful theorems of linear algebra and matrix analysis is the Cayley—Hamilton theorem. It simply says the following. Theorem 8.1 Ever s uare matrix satisfies its characteristic e uation. The proof of this theorem is also simple. It follows from the definition of the matrix inversion, and is given in Appendix A. Hence, if the characteristic equation of A is given by = 1 • • • al X CIO then, in addition to the above single algebraic equation, the algebraic equations are satisfied = + an—I AR — following n X n e s des contain the oopyiqhted material from Linear Dynamic Systems and Signa,€ Prentice Hall 2003, Prepared by Professor Zoran Gajic 8-71
  21. -k any f (A) — al A — aol — al A — ao linear system by Meiling CHEN 21
  22. Exampıe: let f (A) AI 00 : -O, 4-1,4 -2 Âli 00 PO -F I 100 f(Â2) f (A) AI 00 100 (2-2100) 2100 + (2100 linear system by Meiling CHEN 100 1 2101 22
  23. Example: 2 1 -po+P112 -Po -P12 -1,12 -2 -2 f(-l) f(-2) - e 2 = 2e 0 — 2e—2t + 2e linear system by Meiling CHEN 23
  24. We can extract more information from this equation by taking the derivative with respect to which leads to — xf = ao[k] +01[klXI) Exarnple 8.12: Consider a discrete-time system state matrix that has multiple eigenvalues, that is k—l —045 Ad — o 2 —0.5 with = —0.5. In this case, we have only one algebraic equation e s 'des contain the oopyliqhted material from Linear Dynamic Systems and Signals, Prentice Hall 2003, Prepared by Professor Zoran Galic 8-77
  25. Having obtained Ol[k], we get from the original equation ao[k] = - = (-0.5)k - In this case the state transition matrix corresponding to Ad is given by —0+501[k] O ao[k] —O.5a1[k o (—0.5)k ao[k] — 0.5a1 [k] 2QI[k] (—0.5)k ao[k] - 0.5011k] Example 8.13: For a 2 X 2 continuous-time system matrix, the state transition matrix is obtained as follows e + al (t)A Let and be distinct eigenvalues of A, then ao(t) and (t) satisfy 8-78 e s 'des contain the copyiqhted material from Linear Dynamic Systems and Signals, Prentice Hall 2003, Prepared by Professor Zoran Gajic
  26. e e ao(t) + al ao(t) + 01 Consider then 1 —3. Hence ao(t) — QI(t) I (3e-t e 31), o —3 1 —t e e = ao(t) — 301(t) This system of algebraic equations has the solution ao(t) — which implies 01 (t) — e s 'des contain the oopyliqhted material from Linear Oynamic Systems and Signa,€ Prentice Hall 2003, Prepared by Professor Zoran Gajic 8-79
  27. —t o.5(3e = —1.5(e Example 8.14: Consider a multiple eigenvalues that is The eigenvalues of this matrix e Prepared by Professo ao(t) 01 (t) = —301(t) ao(t) — 401 (t) —t o.5(e 2) continuous-time system whose system matrix has o o 1 O 1 —3 are given by = X2 The state 'transition matrix eAt IS given by e e s des contain the oopynqhted material from Linear Dynamic Systems and Signals, Prentice Hall 2003, Zoran Gaoc 8-80
  28. In this case, we have only one equation for three unknowns, that is e — + 01 (t) A + X By taking the derivatives of this equation we can generate two new equations (ext = ao(t) + + te 'M = 01 (t) + and For for az(t) in the previous two equations we obtain al (t) = e —t (1 + t -F 0.5t2). Hence, we have ao(t) e s des contain the copyiohted material from Linear Dynamic Systems and Signals, Prentice Hall 2003, 2 = 2az(t) —o ao(t) -h al t e O.5t2e- = —1, the last equation implies 02(t) Using 'the value -F t) and 8-81 Prepared by Professor Zoran Gajic
  29. e O O o 0 0 — 01 al o —3Q1 01 .01 o 01 — 302 o — 302 8Q2 — 302 6Q2 ao — 302 It is interesting to observe that the procedures for calculating the coefficients ai[k] and ate(t) always produce real values for these coefficients eve n in cases of complex conjugate eigenvalues. However, in such cases, we have to deal with complex numbers, but the final results for ai[k] and Problems 8.8—8.9. e s des contain the oopyriqhted material from Linear Dynamic Systems and Signa,€ Prentice Hall 2003. must be real, see Prepared by Professor Zoran Gajic 8-82

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