To determine The diagonalized state space representation of the given system. Explanation Given Information: System state space representation is given by, x ˙ = [ − 5 − 5 4 2 0 − 2 0 − 2 − 1 ] x + [ − 1 2 − 2 ] r ; y = [ − 1 1 2 ] x . Concept Used: A system x ˙ = A x + B u ; y = C x + D u , can be transformed to a similar system z ˙ = P − 1 A P z + P − 1 B u ; y = C P z + D u . Where z = P − 1 x . Calculation: System matrix is given as, A = [ − 5 − 5 4 2 0 − 2 0 − 2 − 1 ] . To find out the eigen values, λ I − A = [ λ + 5 5 − 4 − 2 λ 2 0 2 λ + 1 ] det ( λ I − A ) = ( λ + 5 ) [ λ ( λ + 1 ) − 4 ] − 5 [ − 2 ( λ + 1 ) ] − 4 [ − 4 ] = 0 ⇒ ( λ + 5 ) ( λ 2 + λ − 4 ) + 10 ( λ + 1 ) + 16 = 0 ⇒ λ 1 = − 1 , λ 2 = − 2 , λ 3 = − 3 For λ = − 1 , A x = λ x ⇒ [ − 5 − 5 4 2 0 − 2 0 − 2 − 1 ] [ x 1 x 2 x 3 ] = − [ x 1 x 2 x 3 ] − 5 x 1 − 5 x 2 + 4 x 3 = − x 1 ⇒ − 4 x 1 − 5 x 2 + 4 x 3 = 0 ( 1 ) 2 x 1 − 2 x 3 = − x 2 ⇒ 2 x 1 + x 2 − 2 x 3 = 0 ( 2 ) − 2 x 2 − x 3 = − x 3 ⇒ x 2 = 0 ( 3 ) Substitute this value in first equation, − 4 x 1 + 4 x 3 = 0 ⇒ x 1 = x 2 ( 4 ) So, one choice of eigen vector is, x = [ 1 0 1 ] . Similarly, for λ = − 2 , A x = λ x ⇒ [ − 5 − 5 4 2 0 − 2 0 − 2 − 1 ] [ x 1 x 2 x 3 ] = − 2 [ x 1 x 2 x 3 ] − 5 x 1 − 5 x 2 + 4 x 3 = − 2 x 1 ⇒ − 3 x 1 − 5 x 2 + 4 x 3 = 0 ( 5 ) 2 x 1 − 2 x 3 = − 2 x 2 ⇒ 2 x 1 + 2 x 2 − 2 x 3 = 0 ( 6 ) − 2 x 2 − x 3 = − 2 x 3 ⇒ x 3 = 2 x 2 ( 7 ) Substitute this value in fifth equation, − 3 x 1 − 5 x 2 + 8 x 2 = 0 ⇒ x 1 = x 2 ( 8 ) So, one choice of eigen vector is, x = [ 1 1 2 ]

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Chapter 5, Problem 45P

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Chapter 5, Problem 44P

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Ch. 5 - Prob. 1RQ Ch. 5 - Name three basic forms for interconnecting...Ch. 5 - For each of the forms in Question 2, state...Ch. 5 - Besides knowing the basic forms as discussed in...Ch. 5 - For a simple, second-order feedback control system...Ch. 5 - Prob. 6RQ Ch. 5 - Prob. 7RQ Ch. 5 - How are summing junctions shown on a signal-flow...Ch. 5 - If a forward path touched all closed loops, what...Ch. 5 - Name five representations of systems in state...

Ch. 5 - Prob. 11RQ Ch. 5 - Which form of the state-space representation leads...Ch. 5 - When the system matrix is diagonal, what...Ch. 5 - What terms lie along the diagonal for a system...Ch. 5 - Prob. 15RQ Ch. 5 - Prob. 16RQ Ch. 5 - For what kind of system would you use the observer...Ch. 5 - Describe state-vector transformations from the...Ch. 5 - Prob. 19RQ Ch. 5 - Prob. 20RQ Ch. 5 - Prob. 21RQ Ch. 5 - Find the closed-loop transfer function, T(s) =...Ch. 5 - Find the equivalent transfer function, T(s) =...Ch. 5 - Reduce the system shown in Figure P5.4 to a single...Ch. 5 - Reduce the block diagram shown in Figure P5.6 to a...Ch. 5 - Find the unity feedback system that is equivalent...Ch. 5 - 8. Given the block diagram of a system shown in...Ch. 5 - 9. Reduce the block diagram shown in Figure P5.9...Ch. 5 - Reduce the block diagram shown in Figure P5.10 to...Ch. 5 - 11. For the system shown in Figure P5.11, find the...Ch. 5 - 12. For the system shown in Figure P5.12, find the...Ch. 5 - Prob. 13P Ch. 5 - For the system of Figure P5.14, find the value of...Ch. 5 - 15. For the system shown in Figure P5.15, find K...Ch. 5 - For the system of Figure P5.16, find the values of...Ch. 5 - Find the following for the system shown in Figure...Ch. 5 - 18. For the system shown in Figure P5.18, find ,...Ch. 5 - Prob. 19P Ch. 5 - Prob. 20P Ch. 5 - Find the transfer function G(s) = Eo(s)/T(s) for...Ch. 5 - Prob. 22P Ch. 5 - Prob. 23P Ch. 5 - State Space SS 24. Given the system below, draw a...Ch. 5 - Prob. 25P Ch. 5 - Using Mason’s rule, find the transfer function,...Ch. 5 - Using Mason’s rule, find the transfer function,...Ch. 5 - Prob. 28P Ch. 5 - Use block diagram reduction to find the transfer...Ch. 5 - State Space SS 30. Represent the following systems...Ch. 5 - Prob. 31P Ch. 5 - State Space SS 32. Repeat Problem 31 and represent...Ch. 5 - Prob. 33P Ch. 5 - Prob. 34P Ch. 5 - Repeat Problem 34 for the system shown in Figure...Ch. 5 - Prob. 37P Ch. 5 - State Space SS 38. Consider the rotational...Ch. 5 - Prob. 40P Ch. 5 - Prob. 41P Ch. 5 - State Space SS 42. Consider the subsystems shown...Ch. 5 - Prob. 43P Ch. 5 - Prob. 44P Ch. 5 - State Space SS 45. Diagonalize the following...Ch. 5 - Prob. 46P Ch. 5 - Prob. 48P Ch. 5 - Prob. 51P Ch. 5 - Figure P5.33 shows a noninverting operational...Ch. 5 - Figure P5.34 shows the diagram of au inverting...Ch. 5 - Prob. 54P Ch. 5 - A car active suspension system adds an active...Ch. 5 - Prob. 58P Ch. 5 - Prob. 60P Ch. 5 - Some medical procedures require the insertion of a...Ch. 5 - Prob. 62P Ch. 5 - Prob. 64P Ch. 5 - Prob. 65P Ch. 5 - The purpose of an Automatic Voltage Regulator is...Ch. 5 - 68. Integrated circuits are manufactured through a...Ch. 5 - Prob. 69P Ch. 5 - Prob. 72P Ch. 5 - Prob. 73P Ch. 5 - Assume ideal operational amplifiers in the circuit...Ch. 5 - Parabolic trough collector. Effective controller...

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