Explanation Given Information: System state space representation is given by, x ˙ = − 10 − 3 7 18.25 6.25 − 11.75 − 7.25 − 2.25 5.75 x + 1 3 2 r ; y = 1 − 2 4 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 = − 10 − 3 7 18.25 6.25 − 11.75 − 7.25 − 2.25 5.75 . To find out the eigen values, λ I − A = λ + 10 3 − 7 − 18.25 λ − 6.25 11.75 7.25 2.25 λ − 5.75 det λ I − A = 0 ⇒ λ 3 − 2 λ 2 − 5 λ + 6 = 0 ⇒ λ 1 = 1 , λ 2 = − 2 , λ 3 = 3 For λ = 1 , A x = λ x ⇒ − 10 − 3 7 18.25 6.25 − 11.75 − 7.25 − 2.25 5.75 x 1 x 2 x 3 = x 1 x 2 x 3 − 10 x 1 − 3 x 2 + 7 x 3 = x 1 ⇒ − 11 x 1 − 3 x 2 + 7 x 3 = 0 ( 1 ) 18.25 x 1 + 6.25 x 2 − 11.75 x 3 = x 2 ⇒ 18.25 x 1 + 5.25 x 2 − 11.75 x 3 = 0 ( 2 ) − 7.25 x 1 − 2.25 x 2 + 5.75 x 3 = x 3 ⇒ − 7.25 x 1 − 2.25 x 2 + 4.75 x 3 = 0 ( 3 ) By solving these equations, x 2 = x 1 x 3 = 2 x 1 So, one choice of eigen vector is, x = 1 1 2 . Similarly, for λ = − 2 , A x = λ x ⇒ − 10 − 3 7 18.25 6.25 − 11.75 − 7.25 − 2.25 5.75 x 1 x 2 x 3 = − 2 x 1 x 2 x 3 − 10 x 1 − 3 x 2 + 7 x 3 = − 2 x 1 ⇒ − 8 x 1 − 3 x 2 + 7 x 3 = 0 ( 4 ) 18.25 x 1 + 6.25 x 2 − 11.75 x 3 = − 2 x 2 ⇒ 18.25 x 1 + 8.25 x 2 − 11.75 x 3 = 0 ( 5 ) − 7

7th Edition

ISBN: 9781118170519

Author: Norman S. Nise

Publisher: WILEY

See similar textbooks

Concept explainers

Entropy

In physics, entropy means the amount of disorder or the degree of randomness associated with a thermodynamic system. The system is not self-organized when the entropy of the system is high. For instance, when a system is at an elevated temperature, its m…

Videos

Question

Chapter 5, Problem 46P

To determine

Diagonalized state space representation of given system.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 5, Problem 45P

Chapter 5, Problem 48P

Chapter 5 Solutions

Control Systems Engineering

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...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

Recommended textbooks for you

Elements Of Electromagnetics
Mechanical Engineering
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Oxford University Press
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:9780134319650
Author:Russell C. Hibbeler
Publisher:PEARSON
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:9781259822674
Author:Yunus A. Cengel Dr., Michael A. Boles
Publisher:McGraw-Hill Education
Control Systems Engineering
Mechanical Engineering
ISBN:9781118170519
Author:Norman S. Nise
Publisher:WILEY
Mechanics of Materials (MindTap Course List)
Mechanical Engineering
ISBN:9781337093347
Author:Barry J. Goodno, James M. Gere
Publisher:Cengage Learning
Engineering Mechanics: Statics
Mechanical Engineering
ISBN:9781118807330
Author:James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:WILEY