Mathematics of Engineering Systems
Complete course reference covering 12 chapters from signal spaces to control theory. Worked examples, homework solutions, and past exams.
12 chapters building from functional analysis foundations through transform methods to control system design and analysis.
Mathematical foundations: vector spaces, functional analysis, and distribution theory
Course overview, systems as maps, applications in control, signal processing, and communications. Linearization of nonlinear systems.
Normed linear spaces, metric spaces, Banach spaces, inner product spaces, Hilbert spaces. Orthogonality, separability, and signal expansions using Fourier, Haar, and polynomial bases.
Dual spaces of normed linear spaces, weak and weak* convergence. Distribution theory, test functions, the Schwartz space, and the Dirac delta as a distribution.
System theory and the transform methods that power frequency-domain analysis
System properties: linearity, time-invariance, causality. LTI (convolution) systems, BIBO stability, transfer functions, frequency response, Bode plots, feedback systems, and state-space descriptions.
All four Fourier transforms: DDFT, CDFT, DCFT, and CCFT. Fourier series, DFT properties, FFT algorithm, Plancherel/Parseval theorems, transforms of distributions, and the sampling/bandwidth tradeoff.
Input-output relations for LTI systems via Fourier analysis. Computing transfer functions for convolution systems using Fourier transforms.
Two-sided and one-sided Laplace and Z-transforms. Properties: linearity, convolution, shift, differentiation, scaling. Initial and final value theorems. Inverse transforms, systems analysis, causality, stability, and minimum-phase systems.
Design and realization of control systems, state-space methods, and sampling
Transfer function shaping via feedback control: PID controllers, Bode plot analysis, root locus method, Nyquist stability criterion, system gain and passivity, predictive and feedforward control.
Realizations of transfer functions: controllable, observable, and modal canonical forms. Zero-state equivalence, algebraic equivalence, and discretization of continuous-time systems.
Sampling of continuous-time and discrete-time signals. The Nyquist-Shannon sampling theorem and reconstruction from samples.
Stability analysis and structural properties of linear systems
Stability criteria for linear systems. Lyapunov's direct method for general nonlinear systems. Linearization for stability analysis. Discrete-time stability.
Controllability and observability of linear systems. Feedback and pole placement, observer design, canonical forms, and Riccati equations for stabilizing controllers.
Reference material on integration theory and complex analysis
Measurable spaces, Borel sigma-fields, Lebesgue integration, Fatou's lemma, monotone and dominated convergence theorems, differentiation under the integral, and Fubini's theorem.
Cauchy's integral formula and the residue theorem. Essential tools for contour integration used in the Nyquist criterion and inverse Laplace/Z-transforms.
Complete worked solutions for all assignments
Practice with previous year examinations