About this previous year paper
Here is your Advance Control System (EEE052) – Section A, B and C in brief, clear paragraph format (exam-ready, not too long) based on your uploaded paper
SECTION A (Brief Answers)
Properties of STM
State Transition Matrix (STM) describes the solution of state equations. It satisfies properties such as Φ(0) = I (identity matrix), Φ(t1+t2) = Φ(t1)Φ(t2), and its inverse exists as Φ⁻¹(t) = Φ(–t). It is used to determine system response.
Jury Stability Criterion
Jury stability criterion is used to determine stability of discrete-time systems. It checks whether all roots of the characteristic equation lie inside the unit circle in the z-plane.
Properties of Z-Transform
Z-transform has properties like linearity, time shifting, scaling, convolution, and initial/final value theorems. It is mainly used in discrete-time system analysis.
Controllability and Observability
A system is controllable if state variables can be driven to any desired value using suitable input. It is checked using controllability matrix. Observability means internal states can be determined from output measurements, checked using observability matrix.
Bilinear Transformation
Bilinear transformation is a mapping from s-plane to z-plane given by
z = (1 + sT/2) / (1 – sT/2).
It preserves stability and is used in digital control design.
Lyapunov Stability Conditions
A system is stable if a Lyapunov function V(x) is positive definite and its derivative is negative definite. This ensures asymptotic stability.
Canonical Variables
Canonical variables simplify state-space representation. They provide systematic form but may lack physical meaning.
STM and Formula
STM is given by Φ(t) = e^(At) for a linear time-invariant system.
Stability of Given System
For F(z) = 8Z⁴ + 4Z³ + 2Z² + 4Z, Jury test must be applied. Since constant term is zero, one root is at origin, indicating marginal or unstable behavior depending on other roots.
Popov’s Criterion
Popov’s criterion is a frequency-domain method used to determine stability of nonlinear feedback systems using a Popov plot.
SECTION B (Medium Length Answers)
Fuzzy Logic and Evolution
Fuzzy logic deals with reasoning under uncertainty using degrees of membership instead of binary logic. It evolved from classical set theory and was introduced by Zadeh. It is widely used in control systems, washing machines, and automotive systems.
Linear Quadratic Equation and Hamilton-Jacobi Equation K)
Linear Quadratic Regulator (LQR) minimizes a quadratic cost function of states and control inputs. The Hamilton-Jacobi equation provides optimal control law by minimizing performance index. It leads to Riccati equation solution.
Unit Step Response
Unit step response is obtained by applying R(s) = 1/s to transfer function and taking inverse Laplace transform. It shows transient and steady-state behavior.
Uniqueness of State Equation Solution
The solution of state equation is unique if matrix A and input functions are continuous. It follows from existence and uniqueness theorem of differential equations.
SISO vs MISO Systems
SISO has single input and single output, simpler analysis. MISO has multiple inputs and single output, used in complex control systems.
Non-Linearity Types
Non-linearity occurs when output is not proportional to input. Types include saturation, dead zone, hysteresis, relay, and backlash nonlinearity.
Describing Function of On-Off Nonlinearity
Describing function approximates nonlinear element using first harmonic component. For relay with hysteresis, it depends on amplitude and hysteresis width.
SECTION C (15-Mark Style Answers)
Pontryagin’s Minimum Principle
Pontryagin’s principle provides conditions for optimal control. It defines Hamiltonian function and states that optimal control minimizes Hamiltonian at every instant. It is used to solve optimal trajectory problems.
Neural Networks and Comparison with Fuzzy Networks
Neural networks are computational models inspired by human brain, used for learning and pattern recognition. They consist of neurons, weights, and activation functions. Fuzzy networks use fuzzy rules and membership functions. Neural networks learn from data, while fuzzy systems use rule-based reasoning.
Riccati Equation
Riccati equation arises in optimal control problems like LQR. It is given by
AᵀP + PA – PBR⁻¹BᵀP + Q = 0.
Solution of Riccati equation gives optimal state feedback gain matrix.