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Control Systems - State Variable Analysis - Important Short Questions, Answers, Tutorial Problems: State Variable Analysis

__STATE VARIABLE ANALYSIS__

**1.
****Define
state variable.**

The state
of a dynamical system is a minimal set of variables(known as state variables)
such that the knowledge of these variables at t-t_{0} together with the
knowledge of the inputs for t > t_{0} , completely determines the
behavior of the system for t > t_{0}

**2.
****Write the
general form of state variable matrix.**

The most
general state-space representation of a linear system with m inputs, p outputs
and n state variables is written in the following form:

= AX + BU

Y = CX +
DU

Where =
state vector of order n X 1.

U = input
vector of order n X1.

A=System
matrix of order n X n.

B=Input
matrix of order n X m

C =output
matrix of order p X n

D =
transmission matrix of order p X m

**3.
****Write the
relationship between z-domain and s-domain.**

All the
poles lying in the left half of the S-plane, the system is stable in S-domain.
Corresponding in Z-domain all poles lie within the unit circle. Type equation
here.

**4. What are the methods available for the stability
analysis of sampled data control system?**

The following
three methods are available for the stability analysis of sampled data control
system

1. Juri‘s
stability test. 2. Bilinear transformation. 3. Root locus technique.

**5.
****What is
the necessary condition to be satisfied for design using state feedback?**

The state
feedback design requires arbitrary pole placements to achieve the desire
performance. The necessary and sufficient condition to be satisfied for
arbitrary pole placement is that the system is completely state controllable.

**6.
****What is
controllability?**

A system
is said to be completely state controllable if it is possible to transfer the
system state from any initial state X(t_{0}) at any other desired state
X(t), in specified finite time by a control vector U(t).

**7.
****What is
observability?**

A system
is said to be completely observable if every state X(t) can be completely
identified by measurements of the output Y(t) over a finite time interval.

**8.
****Write the
properties of state transition matrix.**

The
following are the properties of state transition matrix

1. Φ (0) = e^{Ax0}
= I (unit matrix).

2. Φ (t) = e^{At}
= (e^{-At})^{-1} = [Φ(-t)]^{-1}.

3. Φ (t_{1}+t_{2})
= e^{A(t1+t2)} = Φ(t_{1}) Φ(t_{2}) = Φ(t_{2})
Φ(t_{1}).

**9.
****Define
sampling theorem.**

Sampling
theorem states that a band limited continuous time signal with highest
frequency f_{m}, hertz can be uniquely recovered from its samples
provided that the sampling rate F_{s} is greater than or equal to 2f_{m}
samples per second.

**10. What is sampled data control system?**

When the
signal or information at any or some points in a system is in the form of
discrete pulses, then the system is called discrete data system or sampled data
system.

**11. What is Nyquist rate?**

The
Sampling frequency equal to twice the highest frequency of the signal is called
as Nyquist rate. f_{s}=2f_{m}

**12. What is similarity transformation?**

The
process of transforming a square matrix **A**
to another similar matrix **B** by a
transformation **P ^{-1}AP = B**
is called similarity transformation. The matrix P is called transformation
matrix.

**13. What is meant by diagonalization?**

The
process of converting the system matrix **A**
into a diagonal matrix by a similarity transformation using the modal matrix **M** is called diagonalization.

**14. What is modal matrix?**

The modal
matrix is a matrix used to diagonalize the system matrix. It is also called
diagonalization matrix.

If A =
system matrix.

M = Modal
matrix

And M^{-1}=inverse
of modal matrix.

Then M^{-1}AM
will be a diagonalized system matrix.

**15. How the modal matrix is determined?**

The modal
matrix M can be formed from eigenvectors. Let m_{1}, m_{2}, m_{3}
…. m_{n} be the eigenvectors of the n^{th} order system. Now
the modal matrix M is obtained by arranging all the eigenvectors column wise as
shown below.

Modal
matrix , M = [m_{1}, m_{2}, m_{3} …. m_{n}].

**16. What is the need for controllability test?**

The
controllability test is necessary to find the usefulness of a state variable.
If the state variables are controllable then by controlling (i.e. varying) the
state variables the desired outputs of the system are achieved.

**17. What is the need for observability test?**

The
observability test is necessary to find whether the state variables are
measurable or not. If the state variables are measurable then the state of the
system can be determined by practical measurements of the state variables.

**18. State the condition for controllability by
Gilbert’s method.**

**Case (i) when the eigen values are distinct**

Consider
the canonical form of state model shown below which is obtained by using the
transformation X=MZ.

= ΛZ + U

Y=Z + DU

Where, Λ
= M^{-1}AM; = CM , = M^{-1}B and M = Modal matrix.

In this
case the necessary and sufficient condition for complete controllability is
that, the matrix must have no row with all zeros. If any row of the matrix is
zero then the corresponding state variable is uncontrollable.

**Case(ii) when eigen values have multiplicity**

In this
case the state modal can be converted to Jordan canonical form shown below

= JZ + U

Y=Z + DU Where, J = M^{-1}AM

In this
case the system is completely controllable, if the elements of any row of that
correspond to the last row of each Jordan block are not all zero.

**19. State the condition for observability by
Gilbert’s method.**

Consider
the transformed canonical or Jordan canonical form of the state model shown
below which is obtained by using the transformation, X =MZ

= ΛZ + U

Y=Z + DU (Or)

= JZ + U

Y=Z + DU where =CM
and M=modal matrix.

The
necessary and sufficient condition for complete observability is that none of
the columns of the matrix be zero. If any of the column is of has all zeros
then the corresponding state variable is not observable.

**20. State the duality between controllability and
observability.**

The
concept of controllability and observability are dual concepts and it is
proposed by kalman as principle of duality.The principle of duality states that
a system is completely state controllable if and only if its dual system is
completely state controllable if and only if its dual system is completely
observable or viceversa.

**21. What is the need for state observer?**

In
certain systems the state variables may not be available for measurement and
feedback. In such situations we need to estimate the unmeasurable state
variables from the knowledge of input and output. Hence a state observer is
employed which estimates the state variables from the input and output of the
system. The estimated state variable can be used for feedback to design the
system by pole placement.

**22. How will you find the transformation matrix, P _{o}
to transform the state model to observable phase variable form?**

Compute
the composite matrix for observability,Q_{0}

Determine
the characteristic equation of the system |λI -A |=0.

Using the
coefficients a_{1},a_{2},….a_{n-1} of characteristic
equation form a matrix, W. Now the transformation matrix, P_{0} is
given by P_{0}=W Q_{0}^{T}.

**23. Write the observable phase
variable form of state model.**

The
observable phase variable form of state model is given by the following
equations

= A_{0}Z
+ B_{0}u.

Y =C_{0}Z + Du

Where, A_{0}
= , B_{0} = and C_{0} = [ 0 0 ….. 0 1 ]

**24. What is the pole placement by state feedback?**

The pole
placement by state feedback is a control system design technique, in which the
state variables are used for feedback to achieve the desired closed loop poles.

**25. How control system design is carried in state
space?**

In state
space design of control system, any inner parameter or variable of a system are
used for feedback to achieve the desired performance of the system. The
performance of the system is related to the location of closed loop poles.
Hence in state space design the closed loop poles are placed at the desired
location by means of state feedback through an appropriate state feedback gain
matrix, K.

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