Course Details
Subject {L-T-P / C} : EE3302 : Advanced Control Systems { 3-1-0 / 4}
Subject Nature : Theory
Coordinator : Dr. Shubhobrata Rudra
Syllabus
Introduction to Advanced Control Systems [1L]: Purpose of feedback control, Definitions: Systems, dynamical Vs Static systems, linear Vs nonlinear systems, time varying Vs time invariant systems, continuous Vs discrete systems, etc. Modelling approaches of dynamical systems: Transfer function modelling vs state space modelling.
Introduction to state space modelling [2L]: Motivating examples to understand the advantages of state space modelling compare to transfer function modelling, the concept of state, state variables and state model, Motivating examples: Mechanical systems and electrical networks, Definitions: state, state vector and state space.
State space representation [3L]: General form of state space representation of LTI systems, Examples: Mechanical systems, Electrical networks, electromechanical systems, pneumatic and hydraulic systems, state space representation of nonlinear systems.
Realization of the transfer function from a given State model [2L]: TF from SS, characteristics polynomial, concepts of eigenvalues and pole, basic concepts of zeros.
Realization of the state space model from transfer function [4L]
Concepts of non-uniqueness of state space realization of state space model.
Canonical Realizations of state space representation: controller canonical form, observer canonical form, observability canonical form, controllability canonical form, diagonal canonical form, Jordan canonical form, realization of state model from transfer function matrix.
• Review of Linear Algebra [5L]: Concepts of Field, vector space, vector subspace, linear combination, spanning set, linear independence and dependence, basis, normed linear space, eigenvalues, eigenvector, rank space and null space, the concept of the equilibrium point, C-H Theorem, foundation and representation, similarity transformation, diagonalisation, transformation to controller canonical and Observer canonical forms, sign definite matrix and it's properties.
• Solutions of LTI sate equations [3L]: solution of homogeneous state equations, Computation methods of STM, properties of STM, solution of the non-homogenous equation.
MID-SEMESTER
• Linearization and stability [4L]: Linearization, concepts of equilibrium points, notions of stability for dynamical systems: Internal and external stability, asymptotic stability, exponential stability, Lyapunov stability: definitions, Lyapunov’s first method, Lyapunov’s direct method, concepts of quadratic form and definiteness, Lyapunov stability/instability theorems, Lyapunov equation.
• Controllability and Observability [2L]: controllability and observability: Definitions, different approach of testing controllability and observability, Controllability and observability duality, controllability and observability: Pole-zero cancellation.
• State feedback controller design [3L]: philosophy of state feedback controller design, process of pole assignment: Direct comparison, transformation techniques, Ackermann’s formula, tracking of reference inputs, LQR Design.
• Asymptotic Observer Design [2L]: philosophy of observer design, building a full order observer, Observer design methods, concepts of separation principle, reduced order observers.
• Introduction to Digital Control systems [3L]: advantages of digital control system, components of the digital control system, modelling the sampler, modelling the ZoH, Z-transform, inverse Z-transform, the notion of stability in discrete domain, the impact of sampling on the controllability and observability.
• Analysis of Nonlinear Systems: [4L] Properties of Nonlinear systems, phase plane analysis, Describing function analysis, the existence of a unique solution, Lyapunov stability analysis.
• Nonlinear Controller Design [2L]: preliminary ideas, problems with gain scheduling, state feedback linearization
Course Objectives
- Providing a detailed analysis of the linear systems and highlights the critical tools of linear algebra that are essential for analysing the system behaviour in the state space.
- Discussing the salient features of the nonlinear systems in detail, along with providing a clear view of the stability analysis tools for the different nonlinear systems.
- Conducting a detailed discussion on the different approaches of the state feedback control law design for both linear and nonlinear systems.
Course Outcomes
Construct a state model that possesses a special canonical form either from the first principle equation or transfer function representation of dynamical systems. <br />Analyse the solution of the input-to-state equation <br />Simplify a control analysis problem by application of linear algebraic tools <br />Carry out the stability Analysis of the autonomous dynamical systems <br />Evaluate the fitness of a given system for an intended control application from the findings of its controllability (observability) test data <br />Devise a suitable feedback control law for autonomous dynamical systems.
Essential Reading
- K.Ogata, Modern Control Engineering, Pearson
- G. F. Franklin, J.G. Powell and M.L. Workman, , Feedback Control of Dynamic Systems, Pearson Higher Education
Supplementary Reading
- M. Gopal, Digital Control and State Variable Methods, Tata McGraw-Hill
- S.H. Zak, Systems and Control, Oxford Univ. Press