Which one of the following information is necessary to formulate the problem of control systems optimization?
For arbitrary pole placement, the following combination is necessary.
For a linear time invariant system, an optimal controller can be designed if
An optimal controller for an LTI system can be designed provided the system is both controllable and observable
The eigen values of linear system are the location of
Eigen value are given by  sI  A  = 0, which is the location of poles.
If the system matrix of a linear time invariant continuous system is given by
Its characteristic equation will be given by
Characteristic equation is
or, s^{2} + 5s + 2 = 0
The statevariable description of a linear autonomous system is is a twodimensional state vector and A is a matrix given by
The poles of the system are located at
Poles of given system are given by
Assertion (A): The eigen values of a linear continuousdata 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.
The vector matrix differential equation of a system is given by
The state transition matrix of the system is
We have:
The system matrix of a continous time system is given by:
The characteristic equation is
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 s^{3} + 15s^{2} + 75s + 125 = 0
A system is described by the following equations:
The transfer function of the system is
Transfer function is given by
Given,
and
Now,
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