(SEM VI) THEORY EXAMINATION 2022-23 DIGITAL CONTROL SYSTEM

DIGITAL CONTROL SYSTEM – KEE-063

Section-wise Important Questions & Ready Answers

SECTION A

(Attempt all questions in brief – 2 marks each)

(a) Methods of Representation in Discrete-Time System

Discrete-time systems can be represented using difference equations, Z-transform, pulse transfer function, state-space representation, and signal flow graph. Each method provides a different perspective for analysis and design.

(b) Sampling Theorem Condition

According to the sampling theorem, a continuous-time signal can be perfectly reconstructed from its samples if the sampling frequency is at least twice the highest frequency component of the signal. This minimum rate is called the Nyquist rate.

(c) Sampled Signal Flow Graph

A sampled signal flow graph represents discrete-time systems using nodes and branches, where sampling instants are considered explicitly. It is useful for analyzing sampled-data systems and finding pulse transfer functions.

(d) Mason’s Gain Formula

Mason’s Gain Formula is used to determine the overall transfer function of a system represented by a signal flow graph. It relates the forward path gains, loop gains, and non-touching loops to calculate the system gain.

(e) State of a Dynamic System

The state of a dynamic system is the smallest set of variables that completely describe the system’s behavior at any given time, along with future behavior when inputs are known.

(f) State Variable and State Space

A state variable is a variable that describes the internal condition of a system.
State space is the mathematical representation of a system using state variables in the form of first-order differential or difference equations.

(g) BIBO Stability

A system is said to be bounded input bounded output (BIBO) stable if every bounded input produces a bounded output.

(h) Bode Plot

A Bode plot is a logarithmic plot consisting of magnitude and phase versus frequency. It is used to analyze frequency response, stability, and performance of control systems.

(i) Properties of State Transition Matrix

The state transition matrix satisfies properties such as identity at zero time, multiplicative property, invertibility, and continuity. It describes the evolution of state variables over time.

(j) Mapping of S-Plane to Z-Plane

In digital control, the left half of the S-plane maps inside the unit circle of the Z-plane, the imaginary axis maps to the unit circle, and the right half of the S-plane maps outside the unit circle.

SECTION B

(Attempt any three – 10 marks each)

2(a) Sample and Hold Operation

A sample and hold circuit samples a continuous-time signal at discrete instants and holds the sampled value constant until the next sampling instant. It is essential for converting analog signals into discrete-time signals for digital control systems.

2(b) Pulse Transfer Function

The pulse transfer function relates the Z-transform of output to input under zero initial conditions. It is obtained by sampling the continuous-time system and applying the Z-transform, considering zero-order hold behavior.

2(c) Stability Using Lyapunov Equation

Lyapunov stability analysis involves selecting a suitable Lyapunov function and verifying whether its derivative is negative definite. This method determines stability without explicitly solving the system equations and helps find the range of gain KKK.

2(d) Strengths of Frequency Response Approach

Frequency response methods provide insight into stability, robustness, noise rejection, and bandwidth. There is a direct relationship between frequency-domain parameters (gain margin, phase margin) and time-domain characteristics (overshoot, settling time).

2(e) Controllability and Observability

Controllability determines whether a system can be driven from any initial state to any final state using a suitable input. Observability determines whether internal states can be inferred from output measurements. Controllability is tested using the controllability matrix rank condition.

SECTION C

3(a) Steady-State Accuracy of Discrete-Time System

Steady-state accuracy depends on system type and input. Error constants such as position, velocity, and acceleration error constants are defined in the Z-domain to evaluate steady-state error for step, ramp, and parabolic inputs.

3(b) Basic Digital Control System & Sampling Effects

A digital control system consists of sampler, A/D converter, digital controller, D/A converter, and plant. Sampling introduces effects such as aliasing, time delay, and quantization error, which influence system stability and performance.