BEE063 – DIGITAL CONTROL SYSTEM
Section-Wise Solved Answers (2024–25)
SECTION A
(Attempt all questions) – 2 × 7 = 14 marks
(a) What is PID controller?
A PID controller is a control device that combines Proportional (P), Integral (I), and Derivative (D) actions to control a system. The proportional part reduces present error, the integral part eliminates steady-state error by considering past errors, and the derivative part predicts future error to improve system stability and transient response.
(b) What is pulse transfer function?
The pulse transfer function is the discrete-time equivalent of the continuous-time transfer function. It represents the relationship between the Z-transform of the sampled output and sampled input of a digital control system when initial conditions are zero.
(c) What is bi-linear transformation?
Bi-linear transformation is a mathematical technique used to convert a continuous-time system (s-plane) into a discrete-time system (z-plane). It maps the left half of the s-plane into the inside of the unit circle in the z-plane, preserving system stability.
(d) What is the mathematical relationship between Z-transform and Laplace transform?
The relationship between Z-transform and Laplace transform is given by
z=esTz = e^{sT}z=esT
where T is the sampling period. This relation connects continuous-time analysis with discrete-time analysis.
(e) Define state transition matrix.
The state transition matrix describes how the state of a system evolves with time in the absence of inputs. It is used in state-space analysis to compute the future state of a discrete-time system from its initial state.
(f) What type of stability is measured using root locus method?
The root locus method measures relative stability of a control system. It shows how the closed-loop poles move in the z-plane as system parameters vary.
(g) What is characteristic equation of a discrete control system?
The characteristic equation of a discrete control system is obtained by setting the denominator of the closed-loop pulse transfer function equal to zero. It determines system stability in the z-plane.
SECTION B
(Attempt any three) – 7 × 3 = 21 marks
(a) Explain lifted state-space models in detail
Lifted state-space models represent a discrete-time system over multiple sampling intervals. Instead of describing the system at a single sampling instant, the lifted model captures system behavior over a block of samples. This approach is widely used in predictive control and digital system analysis as it simplifies controller design and improves computational efficiency.
(b) Derive the relationship between discrete state-space model and pulse transfer function
A discrete state-space model is represented as
x(k+1)=Ax(k)+Bu(k)x(k+1) = Ax(k) + Bu(k)x(k+1)=Ax(k)+Bu(k) y(k)=Cx(k)y(k) = Cx(k)y(k)=Cx(k)
Taking Z-transform and eliminating the state variable gives the pulse transfer function:
G(z)=C(zI−A)−1BG(z) = C(zI - A)^{-1}BG(z)=C(zI−A)−1B
This equation establishes the relationship between state-space representation and pulse transfer function.
(c) State and prove Nyquist sampling theorem
Nyquist sampling theorem states that a continuous-time signal can be perfectly reconstructed from its samples if the sampling frequency is at least twice the maximum frequency present in the signal.
Mathematically:
fs≥2fmaxf_s \ge 2f_{max}fs≥2fmax
If this condition is violated, aliasing occurs, leading to distortion.
(d) Explain system identification techniques for discrete-time models
System identification involves developing a mathematical model of a system using measured input-output data. Common techniques include parametric methods like least squares estimation and non-parametric methods like frequency response analysis. These techniques are essential when system parameters are unknown.
(e) Discuss stability analysis using bi-linear transformation
Bi-linear transformation converts stability conditions from the s-plane to the z-plane. Since the left half of the s-plane maps inside the unit circle, a discrete system is stable if all poles lie inside the unit circle. This method allows use of continuous-time stability tools for digital systems.
SECTION C
Q3. Attempt any one
(a) Discuss sampling process and data reconstruction
Sampling is the process of converting a continuous signal into a discrete signal by taking values at regular intervals. Data reconstruction is the reverse process where the original signal is recovered using filters such as zero-order hold or sinc interpolation. Proper sampling prevents loss of information.
(b) Explain Jury stability test and canonical forms
Jury stability test is used to determine the stability of discrete-time systems by checking whether all roots of the characteristic equation lie inside the unit circle.
Canonical forms include controllable canonical form, observable canonical form, and diagonal form, which simplify system analysis and controller design.
Q4. Attempt any one
(a) Explain Lyapunov stability theorem
Lyapunov stability theorem states that a system is stable if a positive definite Lyapunov function exists whose difference decreases with time. This method does not require solving system equations and is widely used for nonlinear and discrete systems.
(b) Design of lead compensator using Bode plot
A lead compensator improves system stability and transient response. It adds positive phase margin by shifting the Bode magnitude and phase curves. The design involves selecting appropriate zero and pole locations based on desired phase margin.
Q5. Attempt any one
(a) Controllability and observability of discrete state-space models
A system is controllable if its states can be driven to any desired value using suitable inputs. It is observable if internal states can be determined from outputs. Mathematical tests involve controllability and observability matrices. These properties are essential for stability analysis.
(b) Root locus-based controller for sampled data system
Root locus method is used to design controllers by analyzing pole movement in the z-plane. By adjusting controller parameters, closed-loop poles can be placed inside the unit circle to achieve desired stability and performance.
Q6. Attempt any one
(a) Design of lag compensator using Bode plot
Lag compensator improves steady-state accuracy by increasing low-frequency gain. It is designed using Bode plot by placing the pole and zero close to each other at low frequencies.
(b) Frequency response analysis for discrete-time systems
Frequency response analysis studies system behavior using magnitude and phase plots in the z-plane. It helps determine stability margins and resonance characteristics of digital control systems.
Q7. Attempt any one
(a) Time response parameters of second-order system
Key parameters include rise time, peak time, settling time, and overshoot. These depend on damping ratio and natural frequency and are used to evaluate system performance.
(b) Robust control techniques for discrete-time systems
Robust control techniques ensure system stability despite parameter variations and uncertainties. Methods include H-infinity control and adaptive control strategies.
