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Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE) PDF Download

Q1: Consider the state-space description of an LTI system with matricesPrevious Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)For the input, sin(ωt), ω > 0, the value of ω for which the steady-state output of the system will be zero, is ___ (Round off to the nearest integer).  (2023)
(a) 0
(b) 1
(c) 2
(d) 3
Ans:
(c)
Sol: We have, transfer function
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)From eqn. (1), we get
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Now, condition for output is zero,
−ω+ 4 = 0
⇒ ω = 2rad/sec.

Q2: The state space representation of a first-order system is given as
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)where, x is the state variable, u is the control input and y is the controlled output. Let u = −Kx be the control law, where K is the controller gain. To place a closed-loop pole at -2, the value of K is _________.   (2021)
(a) 1
(b) 2
(c) 4
(d) 6
Ans: 
(a)
Sol: Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Characteristic equation,
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
Q3: Consider a state-variable model of a system
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)where y is the output, and r is the input. The damping ratio ξ and the undamped natural frequency ωn (rad/sec) of the system are given by  (2019)
(a) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)

(b) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
(c) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
(d) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
Ans: (a)
Sol: Characteristic equation is,
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
Q4: Consider the system described by the following state space representation
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)If u(t) is a unit step input and Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE) the value of output y(t) at t = 1 sec (rounded off to three decimal places) is_________    (SET-2  (2017))
(a) 1.284
(b) 1.862
(c) 2.366
(d) 0.655
Ans:
(a)
Sol: Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
Q5: The transfer function of the system Y(s)/U(s) whose state-space equations are given below is:  (SET-1(2017))
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)(a) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)

(b) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
(c) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
(d) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
Ans: (d)
Sol: Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Transfer function = C[sI − A]−1 B + D
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
Q6: Consider a linear time invariant system Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)with initial condition x(0) at t = 0. Suppose α and β are eigenvectors of (2 x 2) matrix A corresponding to distinct eigenvalues  λ1 and λ2 respectively. Then the response x(t) of the system due to initial condition x(0) = α is  (SET-2 (2016))
(a) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)

(b) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
(c) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
(d) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
Ans: (a)
Sol: Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
Eigen values are λ1 and  λ2
we can write,
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Response due to initial conditions,
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
Q7: In the signal flow diagram given in the figure,  u1 and u2 are possible inputs whereas y1 and y2 are possible outputs. When would the SISO system derived from this diagram be controllable and observable?  (SET-1(2015))
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)(a) When uis the only input and yis the only output
(b) When u2 is the only input and y1 is the only output
(c) When u1 is the only input and yis the only output
(d) When uis the only input and yis the only output
Ans: 
(b)
Sol: Equations from the flow diagram, Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Considering the SISO cases:[/latex]
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
Q8: Consider the system described by following state space equations
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE) If u is unit step input, then the steady state error of the system is  (SET-3(2014))
(a) 0
(b) 1/2
(c) 2/3
(d) 1
Ans: 
(a)
Sol: Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Transfer function of the given system is given by
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Given, input = unit step
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE) ∴ Final Value
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE) ∴ Error = Final value - Initial value
ess = 0

Q9: The second order dynamic system
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE) has the matrices P, Q and R as follows :
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)The system has the following controllability and observability properties:  (SET-2 (2014))
(a) Controllable and observable
(b) Not controllable but observable
(c) Controllable but not observable
(d) Not controllable and not observable
Ans: 
(c)
Sol: Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)For controllability,
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Also, for observability,
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)∴ System is controllable but not observable.

Q10: The state transition matrix for the system 
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE) is  (SET-2(2014))
(a) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)(b) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)(c) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)(d) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Ans:
(c)
Sol: Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)State transition matric is given by,
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)∴ State transition matrix Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)

=[et0tetet]Q11: The state variable formulation of a system is given as      
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)

The system is  (2013)
(a) controllable but not observable
(b) not controllable but observable
(c) both controllable and observable
(d) both not controllable and not observable
Ans: 
(a)
Sol: Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)The system is controllable.
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)The system is not observable. 

Q12: The state variable formulation of a system is given as      
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)

The response y(t) to the unit step input is  (2013)
(a) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)

(b) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
(c) e2tete−2t−e−t
(d) 
1−e−t
Ans: (a)
Sol: Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
Q13: The state variable description of an LTI system is given by
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)where y is the output and u is the input. The system is controllable for  (2012)
(a) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)

(b) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
(c) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
(d) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
Ans: (d)
Sol: Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)For stable system to be controllable, the metric Qc must be non singular.
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)
Q14: The system Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE) is  (2010)
(a) Stable and controllable
(b) Stable but uncontrollable
(c) Unstable but controllable
(d) Unstable and uncontrollable
Ans:
(c)
Sol: Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Transfer function = C[sI − A]−1 B
So, denominator of equation (i) gives pole of the system.
(s+1)(s−2) = 0
s = −1 and 2
One pole lies in RHS of s-plane. Hence, system is unstable.
For controllability, Qc is defined as
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Hence the system is controllable.

Q15: A system is described by the following state and output equations
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE) when u(t) is the input and y(t) is the output
The state-transition matrix of the above system is  (2009)
(a) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)(b) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)(c) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)(d) Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)Ans:
(b)
Sol: Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)State transition matrix
Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE)

The document Previous Year Questions- State Variable Analysis - 1 | Control Systems - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Control Systems.
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FAQs on Previous Year Questions- State Variable Analysis - 1 - Control Systems - Electrical Engineering (EE)

1. What is state variable analysis in control systems?
Ans.State variable analysis refers to a mathematical approach used in control systems to model and analyze dynamic systems. It involves the use of state variables, which represent the system's state at any given time, allowing for a comprehensive understanding of the system's behavior and the design of control strategies.
2. How do state variables differ from traditional variables in system analysis?
Ans.State variables are specific variables that describe the state of a system at a particular time, capturing essential information about the system's dynamics. In contrast, traditional variables may not provide a complete representation of the system's behavior over time. State variables are particularly useful in systems that exhibit multi-dimensional behavior.
3. What are the benefits of using state variable analysis in engineering?
Ans.The benefits of using state variable analysis include improved understanding of system dynamics, the ability to handle multi-input and multi-output systems, and enhanced control design capabilities. It also facilitates the use of modern control techniques, such as state feedback and observers, leading to more robust system performance.
4. Can state variable analysis be applied to nonlinear systems?
Ans.Yes, state variable analysis can be applied to nonlinear systems, although it may involve more complex mathematical techniques. Nonlinear state-space representations can capture the dynamics of such systems, allowing engineers to analyze and design controllers that accommodate nonlinear behaviors.
5. What are the common applications of state variable analysis in real-world systems?
Ans.Common applications of state variable analysis include control system design in aerospace, robotics, automotive systems, and electrical circuits. It is widely used in industries for designing systems that require precise control and stability, such as in automatic flight control systems and industrial automation.
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