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
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![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) = A(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
2 1
01 -1
b =
1
C =
1
0
and comment on its stability, controllability and
observability by considering the ranks of the
appropriate Kalman matrices [b Ab A²b]
and [c ATC (A¹)³c].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F28762982-18d2-44c2-a100-f0debae2592e%2Fc077aca9-657c-454b-9a39-be2c232aa2f4%2Fcer6uzq_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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) = A(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
2 1
01 -1
b =
1
C =
1
0
and comment on its stability, controllability and
observability by considering the ranks of the
appropriate Kalman matrices [b Ab A²b]
and [c ATC (A¹)³c].
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