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CDS 90 abc
Senior Thesis in Control and Dynamical Systems
9 units (009)

first, second, third terms
Prerequisites: CDS 110 ab or CDS 140 ab (may be taken concurrently).
Research in control and dynamical systems, supervised by a Caltech faculty member. The topic selection is determined by the adviser and the student and is subject to approval by the CDS faculty. First and second terms: midterm progress report and oral presentation during finals week. Third term: completion of thesis and final presentation. Not offered on a pass/fail basis.
Instructor:
Murray
CDS 101
Design and Analysis of Feedback Systems
6 units (204)

first term
Prerequisites: Ma 1 and Ma 2 or equivalents.
An introduction to feedback and control in physical, biological, engineering, and information sciences. Basic principles of feedback and its use as a tool for altering the dynamics of systems and managing uncertainty. Key themes throughout the course will include input/output response, modeling and model reduction, linear vs. nonlinear models, and local vs. global behavior. This course is taught concurrently with CDS 110 a, but is intended for students who are interested primarily in the concepts and tools of control theory and not the analytical techniques for design and synthesis of control systems.
Instructors:
Murray, MacMynowski
CDS 110 ab
Introductory Control Theory
12 units (309) first, 9 units (306) second terms
Prerequisites: Ma 1 and Ma 2 or equivalents; ACM 95/100 may be taken concurrently.
An introduction to analysis and design of feedback control systems, including classical control theory in the time and frequency domain. Modeling of physical, biological, and information systems using linear and nonlinear differential equations. Stability and performance of interconnected systems, including use of block diagrams, Bode plots, the Nyquist criterion, and Lyapunov functions. Robustness and uncertainty management in feedback systems through stochastic and deterministic methods. Introductory random processes, Kalman filtering, and norms of signals and systems. The first term of this course is taught concurrently with CDS 101, but includes additional lectures, reading, and homework that is focused on analytical techniques for design and synthesis of control systems.
Instructors:
Murray, MacMynowski, Burdick
CDS 140 ab
Introduction to Dynamics
9 units (306)

second, third terms
Prerequisites: ACM 95/100 ab or equivalent.
Basics topics in dynamics in Euclidean space, including equilibria, stability, Lyapunov functions, periodic solutions, PoincareBendixon theory, Poincare maps. Attractors and structural stability. Introduction to simple bifurcations and eigenvalue crossing conditions. Discussion of bifurcations in applications, invariant manifolds, the method of averaging and singular perturbation theory. Additional topics may include Hamiltonian and Lagrangian systems.
Instructors:
Murray, MacMynowski, Staff
CDS 150
Stochastic System Analysis and Bayesian Updating
9 units (306); third term

Recommended prerequisite: ACM/EE 116
This course focuses on a probabilistic treatment of uncertainty in modeling a dynamical system's inputoutput behavior, including propagating uncertainty in the input through to the output. It covers the foundations of probability as a multivalued logic for plausible reasoning with incomplete information that extends Boolean logic, giving a rigorous meaning for the probability of a model for a system. Approximate analytical methods and efficient stochastic simulation methods for robust system analysis and Bayesian system identification are covered. Topics include: Bayesian updating of system models based on system timehistory data, including Markov Chain Monte Carlo techniques; Bayesian model class selection with a recent informationtheoretic interpretation that shows why it automatically gives a quantitative Ockham's razor; stochastic simulation methods for the output of stochastic dynamical systems subject to stochastic inputs, including Subset Simulation for calculating small "failure" probabilities; and Bayes filters for sequential estimation of system states and model parameters, that generalize the Kalman filter to nonlinear dynamical systems.
Instructor:
Beck
CDS 190
Independent Work in Control and Dynamical Systems
Units to be arranged

first, second, third terms; maximum two terms
Prerequisites: CDS 110 ab or CDS 140 ab.
Research project in control and dynamical systems, supervised by a CDS faculty member.
CDS 201
Linear Algebra and Applied Operator Theory
9 units (306)

first term
Linear spaces, subspaces, spans of sets, linear independence, bases, dimensions; linear transformations and operators, examples, nullspace/kernel, rangespace/image, onetoone and onto, isomorphism and invertibility, ranknullity theorem; products of linear transformations, left and right inverses, generalized inverses. Adjoints of linear transformations, singularvalue decomposition and MoorePenrose inverse; matrix representation of linear transformations between finitedimensional linear spaces, determinants, multilinear forms; metric spaces: examples, limits and convergence of sequences, completeness, continuity, fixedpoint (contraction) theorem, open and closed sets, closure; normed and Banach spaces, inner product and Hilbert spaces: examples, CauchySchwarz inequality, orthogonal sets, GramSchmidt orthogonalization, projections onto subspaces, best approximations in subspaces by projection; bounded linear transformations, principle of superposition for infinite series, wellposed linear problems, norms of operators and matrices, convergence of sequences and series of operators; eigenvalues and eigenvectors of linear operators, including their properties for selfadjoint operators, spectral theorem for selfadjoint and normal operators; canonical representations of linear operators (finitedimensional case), including diagonal and Jordan form, direct sums of (generalized) eigenspaces. Schur form; functions of linear operators, including exponential, using diagonal and Jordan forms, CayleyHamilton theorem. Taught concurrently with ACM 104.
Instructor:
Beck
ACM/CDS 202
Geometry of Nonlinear Systems
9 units (306)

third term
Prerequisites: CDS 201 or AM 125 a.
Basic differential geometry, oriented toward applications in control and dynamical systems. Topics include smooth manifolds and mappings, tangent and normal bundles. Vector fields and flows. Distributions and Frobenius's theorem. Matrix Lie groups and Lie algebras. Exterior differential forms, Stokes' theorem.
Instructor:
Murray
CDS 205
Geometric Mechanics
9 units (306)

third term
Prerequisites: CDS 202, CDS 140.
The geometry and dynamics of Lagrangian and Hamiltonian systems, including symplectic and Poisson manifolds, variational principles, Lie groups, momentum maps, rigidbody dynamics, EulerPoincare equations, stability, and an introduction to reduction theory. More advanced topics (taught in a course the following year) will include reduction theory, fluid dynamics, the energy momentum method, geometric phases, bifurcation theory for mechanical systems, and nonholonomic systems. Not offered 201213.
CDS 212
Introduction to Modern Control
9 units (306)

first term
Prerequisites: ACM 95/100 abc or equivalent; CDS 110 ab or equivalent.
Introduction to modern control systems with emphasis on the role of control in overall system analysis and design. Examples drawn from throughout engineering and science. Open versus closed loop control. Statespace methods, time and frequency domain, stability and stabilization, realization theory. Timevarying and nonlinear models. Uncertainty and robustness.
Instructor:
Doyle
CDS 213
Robust Control
9 units (306)

third term
Prerequisites: CDS 212, CDS 201.
Linear systems, realization theory, time and frequency response, norms and performance, stochastic noise models, robust stability and performance, linear fractional transformations, structured uncertainty, optimal control, model reduction, m analysis and synthesis, real parametric uncertainty, Kharitonov's theorem, uncertainty modeling.
Instructor:
Doyle
Ae/CDS/ME 251 ab
Closed Loop Flow Control
9 units

(306 a, 135 b).
Prerequisites: ACM 100 abc, Ae/APh/CE/ME 101 abc or equivalent..
This course seeks to introduce students to recent developments in theoretical and practical aspects of applying control to flow phenomena and fluid systems. Lecture topics in the second term drawn from: the objectives of flow control; a review of relevant concepts from classical and modern control theory; highfidelity and reducedorder modeling; principles and design of actuators and sensors. Third term: laboratory work in open and closedloop control of boundary layers, turbulence, aerodynamic forces, bluff body drag, combustion oscillations and flowacoustic oscillations.
Instructors:
Colonius, McKeon
CDS 270
Advanced Topics in Systems and Control
Hours and units by arrangement
Topics dependent on class interests and instructor. May be repeated for credit.
CDS 280
Advanced Topics in Geometric Mechanics or Dynamical Systems Theory
Hours and units by arrangement
Prerequisites: instructor's permission.
Topics will vary according to student and instructor interest. Examples include chaotic transport theory, invariant manifold techniques, multidimensional geometric perturbation theory, the dynamics of coupled oscillators, rigidbody dynamics, numerical methods in dynamical systems theory. May be repeated for credit. Not offered 201213.
CDS 300 abc
Research in Control and Dynamical Systems
Hours and units by arrangement
Research in the field of control and dynamical systems. By arrangement with members of the staff, properly qualified graduate students are directed in research.
Instructor:
Staff
Published Date:
July 28, 2022