solutions2HW6

# solutions2HW6 - 6.2 Draw phase portraits for the systems...

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Unformatted text preview: 6.2 Draw phase portraits for the systems with the following A-matrices: -3 45 a) 0 -2 Solution: Eigenvalues: —2, — 8 ~1 1 Modal matrix: M =[ :I 1 0 Phase Portrait ~8 —6 b ) lo 2] Solution: Eigenvalues: 2, — 8 Modal matrix: M: *3 1 5 0 Phase Poﬁrait )04 c —40 Solution: Eigenvalues: 4}, — 4f Modal matrix: M=[ 0'5 0'5 ] 0.5} —0.5j Phase Portlail d) [3 ‘3] Solution: Eigenvalues: 4 — 4}, 4 + 4j . 0. Modal matrix: M = 05 5 0.5 j \$5} Phase Portrait _‘_. 4 ,, 2 0 ‘2 —5 "—3 5 10 e) [on] Solution: Eigenvalues: 0,1 1 Modal matrix: M:[? 0] Phase Portrait l O l 0 Solution: Eigenvalues: 0,1 _ 0 1 Modal matrlx: M: 1 Phase Portrait 00 g) 00 Solution: Eigenvalues: 0, 0 1 0 Modal matrix: M=[0 1] Phase Parliait 6.7 6.8 —l l 0 h) 0 —l 0 0 O —1 Solution: The matrix has two regular eigenvectors [l 0 0]T,[0 D If, and one generalized eigenvector. Eigenvalues are M = it] = A] = w]. Suppose \$1403) is the state-transition matrix for AU). Deﬁne a (nonsingular) change of variables by x(t)= Man) such that (in) = M—‘AM§(:)\$ Ergo). Determine an expression for (13203:) in terms of CIJA(I,1:). Does the result depend on whether M is time-varying or not? Solution: (Mm) = Arum—Hr) = M2(i)Z“(r)M" So, M'lmA (mm = 3024(1) = (1)305) This holds for time-varying as well. For every system £0): Ax(t) there is deﬁned a system p(t)=—ATp(t) called the adjoint system. Show that if 613140,?) is the state transition matrix for the original system, then the state transition matrix for the adjoint system is \$30,!) . Solution: Let the state transition matrix for the adjoint system be (DPUJ). Since (DP (at) is the transition matrix, it satisﬁes @PUJ) 2 ___AT‘DP(I T) dr CDPCEJ): I We must show that (bl—Ant) satisﬁes the same differential equation and initial conditions, since these equations deﬁne a unique solution. Since ¢A(r,t) = 43:03) , d _ _i —l =_ —l i —] d! CDACCJ) — dr (13A (LT) (DA (£,T)[dt ¢A(I,TJ]¢A (IJ). Since [%¢A(r,r):l = ACE/10,1), gammy) = —¢;‘(z,r)A = 43,10,014“ By transposing, -§;[¢A(t,r)f = -ATcI>§(z,r), and that) = I. This shows that (DPUJ) 2 (1)} (1,3) . ...
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solutions2HW6 - 6.2 Draw phase portraits for the systems...

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