This course will teach fundamentals of control design and analysis using state-space methods. This includes both the practical and theoretical aspects of the topic. By the end of the course, you should be able to design controllers using state-space methods and evaluate whether these controllers are robust to some types of modeling errors and nonlinearities. You will learn to: Design controllers using state-space methods and analyze using classical tools. Understand impact of implementation issues (nonlinearity, delay). Indicate the robustness of your control design. Linearize a nonlinear system, and analyze stability.

Covers the same material as 18.03 with more emphasis on theory. First order equations, separation, initial value problems. Systems, linear equations, independence of solutions, undetermined coefficients. Singular points and periodic orbits for planar systems. Topics include first order equations, separation, initial value problems, systems, linear equations, independence of solutions, undetermined coefficients, and singular points and periodic orbits for planar systems.

Input-output and state-space models of linear systems driven by deterministic and random signals; time- and transform-domain representations. Sampling, discrete-time processing of continuous-time signals. State feedback and observers. Probabilistic models; stochastic processes, correlation functions, power spectra, and whitening filters. Detection; matched filters. Least-mean square error estimation; Wiener filtering.

Mathematical introduction to neural coding and dynamics. Convolution, correlation, linear systems, Fourier analysis, signal detection theory, probability theory, and information theory. Applications to neural coding, focusing on the visual system. Hodgkin-Huxley and related models of neural excitability, stochastic models of ion channels, cable theory, and models of synaptic transmission.

This course gives a mathematical introduction to neural coding and dynamics. Topics include convolution, correlation, linear systems, game theory, signal detection theory, probability theory, information theory, and reinforcement learning. Applications to neural coding, focusing on the visual system are covered, as well as, Hodgkin-Huxley and other related models of neural excitability, stochastic models of ion channels, cable theory, and models of synaptic transmission. Visit the Seung Lab Web site.

This course offers an advanced introduction to numerical linear algebra. Topics include direct and iterative methods for linear systems, eigenvalue decompositions and QR/SVD factorizations, stability and accuracy of numerical algorithms, the IEEE floating point standard, sparse and structured matrices, preconditioning, linear algebra software. Problem sets require some knowledge of MATLAB®.

This course introduces the student to the study of linear algebra. Practically every modern technology relies on linear algebra to simplify the computations required for internet searches, 3-D animation, coordination of safety systems, financial trading, air traffic control, and everything in between. Upon completion of this course, the student will be able to: Define and identify linear equations; Write a system of equations in matrix-vector form; Explain the geometric interpretation of a system of linear equations; Solve linear equations using a variety of methods; Define general, particular, and homogeneous solutions; Identify how many solutions a linear system has; Correctly manipulate vectors algebraically and perform matrix-vector and matrix-matrix multiplication; Define linear combination and span; Define and distinguish between singular and nonsingular matrices and calculate a matrix inverse; Define and compute LU decompositions; Relate invertibility of matrices to solvability of linear systems; Define and characterize Euclidean space; Define and compute dot and cross-products; Define and identify vector spaces and subspaces; Define spanning set and determine the span of a set of vectors; Define and verify linear independence; Define basis and dimension; Show that a set of vectors is a basis; Define and compute column space, row space, nullspace, and rank; Define and identify isomorphisms and homomorphisms; Use row and column space to solve linear systems; State the rank-nullity theorem; Define inner product, inner product space, and orthogonality; Interpret inner products geometrically; Define determinants using the permutation expansion; State the properties of determinants, such as that the determinant of the product is the product of the determinants; Compute the determinant using cofactor expansions, row reduction, and Cramer's Rule; Define and compute the characteristic polynomial of a matrix; Define and compute eigenvalues and eigenvectors; Explain the geometric significance of eigenvalues and eigenvectors; Define similarity and diagonalizability; Identify similar matrices; Identify some necessary conditions for diagonalizability. (Mathematics 211; See also: Computer Science 105)

Provides ways to conceptualize and analyze manufacturing systems and supply chains in terms of material flow, information flow, capacities, and flow times. Fundamental building blocks: Inventory and Queuing Models, Forecasting and Uncertainty, Optimization, Process Analysis, Linear Systems and System Dynamics. Factory Planning: Flow Planning, Bottleneck Characterization, Buffer and Batch-Size Tactics, Seasonal Planning, Dynamics and Learning for Various Process Flow Topologies and for Various Market Contexts.

Numerical methods for solving problems arising in heat and mass transfer, fluid mechanics, chemical reaction engineering, and molecular simulation. Topics: numerical linear algebra, solution of nonlinear algebraic equations and ordinary differential equations, solution of partial differential equations (e.g. Navier-Stokes), numerical methods in molecular simulation (dynamics, geometry optimization). All methods are presented within the context of chemical engineering problems. Familiarity with structured programming is assumed. This course focuses on the use of modern computational and mathematical techniques in chemical engineering. Starting from a discussion of linear systems as the basic computational unit in scientific computing, methods for solving sets of nonlinear algebraic equations, ordinary differential equations, and differential-algebraic (DAE) systems are presented. Probability theory and its use in physical modeling is covered, as is the statistical analysis of data and parameter estimation. The finite difference and finite element techniques are presented for converting the partial differential equations obtained from transport phenomena to DAE systems. The use of these techniques will be demonstrated throughout the course in the MATLAB® computing environment.

Numerical methods for solving problems arising in heat and mass transfer, fluid mechanics, chemical reaction engineering, and molecular simulation. Topics: numerical linear algebra, solution of nonlinear algebraic equations and ordinary differential equations, solution of partial differential equations (e.g. Navier-Stokes), numerical methods in molecular simulation (dynamics, geometry optimization). All methods are presented within the context of chemical engineering problems. Familiarity with structured programming is assumed.

Advanced introduction to applications and theory of numerical methods for solution of differential equations, especially of physically-arising partial differential equations, with emphasis on the fundamental ideas underlying various methods. Topics include finite differences, spectral methods, finite elements, well-posedness and stability, particle methods and lattice gases, boundary and nonlinear instabilities.

Systems manipulate signals. There are a few simple systems which will perform simple functions upon signals. Examples include amplification (or attenuation), time-reversal, delay, and differentiation/integration.

We introduce the time-frequency spectrogram and implement it in MATLAB. We use it to view whale songs. We then model a sensing system containing multipath distortion as a LTI system and describe its impulse response. We use convolution to add multipath distortion to the whale sounds.