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-t₀ together with the knowledge of the inputs for t > t₀ , completely determines the behavior of the system for t > t₀

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(t₀) 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 (t₁+t₂) = e^A(t1+t2) = Φ(t₁) Φ(t₂) = Φ(t₂) Φ(t₁).

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^-1AP = 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^-1AM will be a diagonalized system matrix.

15. How the modal matrix is determined?

The modal matrix M can be formed from eigenvectors. Let m₁, m₂, m₃ …. 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 = [m₁, m₂, m₃ …. 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^-1AM; = CM , = M^-1B 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^-1AM

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,Q₀

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

Using the coefficients a₁,a₂,….a_n-1 of characteristic equation form a matrix, W. Now the transformation matrix, P₀ is given by P₀=W Q₀^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

= A₀Z + B₀u.

Y   =C₀Z + Du

Where, A₀ = , B₀ = and C₀ = [ 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.