A linear time-invariant system (A, b, c) is modelled by the state-space equations x(t) = Ax(t) + bu(t) y(t) = c¹x(t) where x(t) is the n-dimensional state vector, and u(t) and y(t) are the system input and output respectively. Given that the system matrix A has n distinct non-zero eigenvalues, show that the system equations may be reduced to the canonical form (t) = A5 (t) + b₁u(t) y(t) = c(t) where A is a diagonal matrix. What properties of this canonical form determine the controllability and observability of (A, b, c)? Reduce to canonical form the system (A, b, c) having 1 1 -2] A = -1 2 1 01 -1 b = 1 0 and comment on its stability, controllability and observability by considering the ranks of the appropriate Kalman matrices [b Ab A²b] and [e ATC (A¹)²c].
A linear time-invariant system (A, b, c) is modelled by the state-space equations x(t) = Ax(t) + bu(t) y(t) = c¹x(t) where x(t) is the n-dimensional state vector, and u(t) and y(t) are the system input and output respectively. Given that the system matrix A has n distinct non-zero eigenvalues, show that the system equations may be reduced to the canonical form (t) = A5 (t) + b₁u(t) y(t) = c(t) where A is a diagonal matrix. What properties of this canonical form determine the controllability and observability of (A, b, c)? Reduce to canonical form the system (A, b, c) having 1 1 -2] A = -1 2 1 01 -1 b = 1 0 and comment on its stability, controllability and observability by considering the ranks of the appropriate Kalman matrices [b Ab A²b] and [e ATC (A¹)²c].
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please only answe if you can show all the steps
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,