UNIT 04 STATE VARIABLE ANALYSIS | Basic Signal and Systems -AKTU

In this section, we will examine UNIT 04 STATE VARIABLE ANALYSIS | Basic Signal and Systems -AKTU, subject “Electrical and Electronics Engineering”. I hope this article is useful to you as you prepare for your AKTU exams.

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Important Questions For Basic Signals and Systems : 
*Unit-01     *Unit-02    
*Unit-03    *Unit-04 
*Unit-05    *Short-Q/Ans
*Question-Paper with solution 21-22 

Q1. What is state variable analysis ? Also define state and state variable. 

Ans.
A. State variable analysis : It is best to construct a set of n first-order differential equation using a set of auxiliary variables called state variables, and this method is known as State Variable Analysis. An n^th order differential equation is typically not suitable for computer solution. 

B. Definition of the following terms: 

i. State: The state of a dynamic system is the smallest set of variables such that the knowledge of these variables att = t₀ with the of knowledge the input for t ≥ t, completely determines the behaviour of the system for any time t ≥ t₀.

ii. State variables: State variables are the factors that contribute to the dynamical system’s determination of its state. 

iii. State vector: These n state variables may be regarded as n components of a vector x if n variables are required to fully characterize the behaviour of a particular system. State vector refers to such a vector.

iv. State space: The n-dimensional space whose coordinate axes consists of the x₁ axis, x₂ axis…..,x_n axis is called state space. Any state can be represented by a point in the state space. 

v. State equation: A state space representation, also known as a state equation, is a mathematical representation of a physical system that consists of output, input, and state variables connected by a differential equation. 

Q2. Obtain the state space representation of the system y + 65 + 11y + 6y = 6u  where, y → output , u → input

Ans. 

Q3. What is transfer function ? Also derive the expression for transfer function of a state model.

Ans. A. Transfer function: Under the premise that all initial conditions are zero, it is defined as the ratio of the system’s Laplace transform’s output to input.

B. Derivation: 

Let us consider a vector matrix differential equation

2. Now, taking Laplace transform with zero initial conditions

3. For a single-input-single-output system, Y and U are scalars. 

4. Now transfer matrix can be given as  

Q4. Explain state transition matrix, its physical significance and properties. 

Ans. A. State transition matrix: The matrix ф(t) = exp (At) is an n xn matrix and it helps in transition from initial state X(0) to any other staterit) for t > 0, hence ф() is called state transition matrix. 

Derivation: 

1. Consider homogeneous equation, with u(t) = 0

Q5. Find the state transition matrix for a system matrix 

Ans. 

Q6. Define controllability and observability in state variable analysis 

Ans. A. Controllability:

1. If the desired state of the system can be changed over a specific time period by applying input, the system is said to be controllable.

2. Consider n^th order multiple input linear time invariant system represented by its state equation as,  

3. The necessary and sufficient condition for the system to be completely state controllable is that the rank of the composite matrix Q_c is n.

4. The composite matrix Q_c is given by, 

B. Observability:

1. If all of a system’s state can be ascertained using information about the output of the system at a particular time, the system is said to be observable. 

2. State equation

3. The system is completely observable if and only if the rank of the composite matrix Q_o is n. 

4. Composite matrix Q_o is given by, 

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