Test: State Variable Analysis

QUESTION: 1

Which one of the following information is necessary to formulate the problem of control systems optimization?

A.
System state equation, output equation, control vector and the performance index only.
B.
System state equation, output equation, control vector, constraints of the problem, the performance index and system parameters.
C.
System state equation, output equation, control vector and constraints of the problem only.
D.
System state equation, output equation and the control vector only.

Solution:

QUESTION: 2

For arbitrary pole placement, the following combination is necessary.

A.
State feedback and stability
B.
Output feedback and observability
C.
Output feedback and controllability
D.
State feedback and controllability

Solution:

QUESTION: 3

For a linear time invariant system, an optimal controller can be designed if

A.
the system is unstable but observable
B.
the system is controllable and observable
C.
the system is stable and unobservable
D.
the system is uncontrollable but stable

Solution:

An optimal controller for an LTI system can be designed provided the system is both controllable and observable

QUESTION: 4

The eigen values of linear system are the location of

A.
zeros of the system
B.
poles of the system
C.
both (a) and (b)
D.
finite poles and zeros

Solution:

Eigen value are given by | sI - A | = 0, which is the location of poles.

QUESTION: 5

If the system matrix of a linear time invariant continuous system is given by

Its characteristic equation will be given by

A.
s² + 5s + 2 = 0
B.
S² - 2s - 5 = 0
C.
s² + s + 3 = 0
D.
s² + 2s + 5 = 0

Solution:

Characteristic equation is

or, s² + 5s + 2 = 0

QUESTION: 6

The state-variable description of a linear autonomous system is is a two-dimensional state vector and A is a matrix given by

The poles of the system are located at

A.
- 5 and +5
B.
- 5j and +5j
C.
-5 and -5
D.
+5 and +5

Solution:

Poles of given system are given by

QUESTION: 7

Assertion (A): The eigen values of a linear continuous-data time invariant system controls the stability of the system.
Reason (R): The roots of the characteristic equation are the same as the eigen values of system matrix A of the state equations.

A.
Both A and R are true and R is a correct explanation of A.
B.
Both A and R are true but R is not a correct explanation of A.
C.
A is true but R is false.
D.
A is false but R is true.

Solution:

QUESTION: 8

The vector matrix differential equation of a system is given by

The state transition matrix of the system is

A.
B.
C.
D.

Solution:

We have:

QUESTION: 9

The system matrix of a continous time system is given by:

The characteristic equation is

A.
s³ + 12s² - 70s + 120 = 0
B.
s³ - 17s² + 10s + 50 = 0
C.
s³ + s² + 2s + 20 = 0
D.
s³ + 15s² + 75s + 125 = 0

Solution:

The characteristic equation | sI - A | = 0.

Now, | sI - A | = 0
or, (s + 5) [(s + 5)² + 0] + 1 (0 - 0) + 0 = 0
or, (s + 5)³ = 0 or s³ + 15s² + 75s + 125 = 0