Difference between revisions of "CDS course discussion, Apr 2014"
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* CDS 112  Control system design  * CDS 112  Control system design  
−  * CDS 140   +  * CDS 140  Introduction to dynamics 
* CDS 212  Feedback control theory  * CDS 212  Feedback control theory  
    
−  * CDS  +  * CDS 240  Nonlinear dynamical systems 
* CDS 213  Robust control  * CDS 213  Robust control  
* CDS 270  Advanced topics (nonlinear control, adaptive control, system ID)  * CDS 270  Advanced topics (nonlinear control, adaptive control, system ID)  
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=== Proposed CDS course requirements ===  === Proposed CDS course requirements ===  
CDS graduate major:  CDS graduate major:  
−  * Core: {{deleteCDS 201}} {{insertACM 104}}, {{deleteCDS 202}} {{insertACM 113}}, ACM/EE 116, CDS  +  * Core: {{deleteCDS 201}} {{insertACM 104}}, {{deleteCDS 202}} {{insertACM 113}}, ACM/EE 116, CDS 140{{deletea}}, {{insertCDS 212}} 
* Depth: 45 units in CDS or other advanced courses in systems theory, dynamical systems, and/or applied mathematics.  * Depth: 45 units in CDS or other advanced courses in systems theory, dynamical systems, and/or applied mathematics.  
* Breadth: 27 units in a particular area outside of CDS. Courses taken to satisfy the focus must represent a coherent program of advanced study in the chosen area.  * Breadth: 27 units in a particular area outside of CDS. Courses taken to satisfy the focus must represent a coherent program of advanced study in the chosen area.  
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* {{insertACM 113. Mathematical optimization}}  * {{insertACM 113. Mathematical optimization}}  
    
−  * CDS  +  * CDS 140{{deletea}}  Dynamical systems 
* CDS 212  Feedback control theory  * CDS 212  Feedback control theory  
    
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* Ae/CDS/ME 251a  Closed loop flow control (alt years)  * Ae/CDS/ME 251a  Closed loop flow control (alt years)  
    
−  * CDS 140b   +  * CDS {{delete140b}} {{insert240}}  Nonlinear systems (alt years) 
* {{deleteCDS 150  Stochastic system analysis}}  * {{deleteCDS 150  Stochastic system analysis}}  
* {{deleteCDS 205  Geometric mechanics (alt years)}}  * {{deleteCDS 205  Geometric mechanics (alt years)}}  
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* Convert CDS 110b into CDS 112. Coordinate with CDS 212, but offer as separate courses  * Convert CDS 110b into CDS 112. Coordinate with CDS 212, but offer as separate courses  
* Attempt to cooffer CDS 113/213 as an integrated class  * Attempt to cooffer CDS 113/213 as an integrated class  
−  * Remove CDS 150  +  * Remove CDS 150, 201, 202, 205 from catalog 
−  
* Update CDS PhD requirements as described above  * Update CDS PhD requirements as described above  
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{{delete'''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: MacMartin, Burdick, Murray.}}  {{delete'''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: MacMartin, Burdick, Murray.}}  
−  '''CDS 110 {{deleteab}}. Introductory Control Theory.'''  +  '''CDS 110 {{deleteab}}. Introductory Control Theory.''' 12 units (309) first{{delete, 9 units (306) second}} term{{deletes}}. Prerequisites: Ma 1abc and Ma 2/102 or equivalents; {{deleteACM 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. {{deleteRobustness 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.}} {{insertDesign of feedback controls in state space and frequency domain based on stability, performance and robustness specifications.}} Instructors: MacMartin, Doyle, Burdick, Murray. 
−  {{insert'''CDS 112. Control System Design.''' 9 units (306); second term. Prerequisite: CDS 110. Optimizationbased design of control systems, including optimal control and receding horizon control. Robustness and uncertainty management in feedback systems through stochastic and deterministic methods. Introductory random processes, Kalman filtering, and norms of signals and systems. }}  +  {{insert'''CDS 112. Control System Design.''' 9 units (306); second term. Prerequisite: CDS 110. Optimizationbased design of control systems, including optimal control and receding horizon control. Robustness and uncertainty management in feedback systems through stochastic and deterministic methods. Introductory random processes, Kalman filtering, and norms of signals and systems. Instructors: Burdick, MacMartin, Doyle, Murray}} 
−  '''CDS 140 ab. Introduction to Dynamics.''' 9 units (306); second, third  +  '''CDS 140 {{deleteab}}. Introduction to Dynamics.''' 9 units (306); second{{delete, third}} term{{deletes}}. Prerequisites: Ma 2/102 or equivalent, {{insertACM 104}}. Basics topics in dynamics {{insertfor continuous state systems in continuous and discrete time}} {{deletein Euclidean space}}, including equilibria/invariant sets, stability, Lyapunov functions/invariants, attractors and periodic solutions. Introduction to structural stability, bifurcations and eigenvalue crossing conditions. Instructors: Murray, MacMartin. 
+  
+  {{insert'''CDS 240. Nonlinear Dynamical Systems.''' 9 units (306); third term. Prerequisites: CDS 140. Analysis of nonlinear dynamical systems modeled using differential equations, including invariant and center manifolds, bifurcations, limit cycles, regular and singular perturbations, the method of averaging, input/output stability. Additional advanced topics may be included based on student and instructor interests. Instructors: Murray, MacMartin, Colonius, McKeon.}}  
{{delete'''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.}}  {{delete'''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. Prerequisite: CDS 110 ab or CDS 140 ab. Research project in control and dynamical systems, supervised by a CDS faculty member.  +  '''CDS 190. Independent Work in Control and Dynamical Systems.''' Units to be arranged; first, second, third terms; maximum two terms. Prerequisite: CDS 110 {{deleteab}} or CDS 140 {{deleteab}}. Research project in control and dynamical systems, supervised by a CDS faculty member. 
{{delete'''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.}}  {{delete'''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.}} 
Latest revision as of 00:15, 29 April 2014
This page keeps track of proposed changes to CDS courses that would be implemented in 201415 and future academic years. In the pages below, insertions are marked in blue and deletions are in red and stroked out.
Background: With the establishment of the new CMS PhD program, we are trying to better align the CDS program with the structure of the CMS program and also update our course structure to better serve other options that make use of CDS courses.
CMS course requirements
 Core requirements: 7 common courses taken by all CMS students (first year)
 Depth requirement: 3 courses in a given area (e.g., Feedback & control, Inference & statistics, Information systems, Networked systems, Optimization, Uncertainty quantification)
 Breadth requirement: 3 courses from mathematics, science, engineering, or economics
Track  Fall  Winter  Spring 
Core 



Feedback and control 



Proposed CDS course requirements
CDS graduate major:
 Core:
CDS 201ACM 104,CDS 202ACM 113, ACM/EE 116, CDS 140a, CDS 212  Depth: 45 units in CDS or other advanced courses in systems theory, dynamical systems, and/or applied mathematics.
 Breadth: 27 units in a particular area outside of CDS. Courses taken to satisfy the focus must represent a coherent program of advanced study in the chosen area.
 Qualifying exam: taken at the start of
secondthird quarter of the first year  Candidacy exam: taken no later than the end of the second year of studies
CDS graduate minor: 54 units of advanced courses with a CDS listing.
CDS undergraduate minor: CDS 110a, CDS 140a and 9 additional units chosen from CDS 110b, CDS 140b, CDS 212 or CDS 270 + complete a senior thesis.
Track  Fall  Winter  Spring 
Core 



Advanced (depth) 

 
Service (UG/nonmajor) 



Implementation plan
The implementation of the changes above will be spread across 23 years for continuity.
201415
 Convert CDS 110a into CDS 110. Coteach with CDS 101 (to satisfy CNS requirements)
 Convert CDS 110b into CDS 112. Coordinate with CDS 212, but offer as separate courses
 Attempt to cooffer CDS 113/213 as an integrated class
 Remove CDS 150, 201, 202, 205 from catalog
 Update CDS PhD requirements as described above
201516
 Remove CDS 101 as a course offering
 CDS 112 not offered; attempt to cooffer CDS 112/212 as an integrated class
201617
 RMM on sabbatical
 Offer CDS 112/212 either as two classes or one integrated class, depending on 201516 experience
Catalog entries
CDS 90 abc. Senior Thesis in Control and Dynamical Systems. 9 units (009); first, second, third terms. Prerequisite: 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: MacMartin, Burdick, Murray.
CDS 110 ab. Introductory Control Theory. 12 units (309) first, 9 units (306) second terms. Prerequisites: Ma 1abc and Ma 2/102 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. Design of feedback controls in state space and frequency domain based on stability, performance and robustness specifications. Instructors: MacMartin, Doyle, Burdick, Murray.
CDS 112. Control System Design. 9 units (306); second term. Prerequisite: CDS 110. Optimizationbased design of control systems, including optimal control and receding horizon control. Robustness and uncertainty management in feedback systems through stochastic and deterministic methods. Introductory random processes, Kalman filtering, and norms of signals and systems. Instructors: Burdick, MacMartin, Doyle, Murray
CDS 140 ab. Introduction to Dynamics. 9 units (306); second, third terms. Prerequisites: Ma 2/102 or equivalent, ACM 104. Basics topics in dynamics for continuous state systems in continuous and discrete time in Euclidean space, including equilibria/invariant sets, stability, Lyapunov functions/invariants, attractors and periodic solutions. Introduction to structural stability, bifurcations and eigenvalue crossing conditions. Instructors: Murray, MacMartin.
CDS 240. Nonlinear Dynamical Systems. 9 units (306); third term. Prerequisites: CDS 140. Analysis of nonlinear dynamical systems modeled using differential equations, including invariant and center manifolds, bifurcations, limit cycles, regular and singular perturbations, the method of averaging, input/output stability. Additional advanced topics may be included based on student and instructor interests. Instructors: Murray, MacMartin, Colonius, McKeon.
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. Prerequisite: 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, EulerPoincaré 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.
CDS 212. Introduction to Modern Control. 9 units (306); third term. Prerequisites: ACM 95/100 abc or equivalent; CDS 110 ab or equivalent, ACM 104. 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, Murray, Burdick.
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.
ACM 216. Markov Chains, Discrete Stochastic Processes and Applications. 9 units (306); second term. Prerequisite: ACM/EE 116 or equivalent. Stable laws; Markov chains; classification of states; ergodicity; Von Neumann ergodic theorem; mixing rate; stationary/equilibrium distributions and convergence of Markov chains; Markov chain Monte Carlo and their applications to scientific computing; Metropolis Hastings algorithm; coupling from the past; martingale theory and discrete time martingales; rare events; law of large deviations; Chernoff bounds.
ACM 217/EE 164. Advanced Topics in Stochastic Analysis. 9 units (306); third term. Prerequisite: ACM 216 or equivalent. The topic of this course changes from year to year and is expected to cover areas such as stochastic differential equations, stochastic control, statistical estimation and adaptive filtering, empirical processes and large deviation techniques, concentration inequalities and their applications. Example of selected topics for stochastic differential equations include continuous time Brownian motion, Ito's calculus, Girsanov theorem, stopping times, and applications of these ideas to mathematical finance and stochastic control.
Ae/CDS/ME 251 ab. Closed Loop Flow Control. 9 units; (306 a, 135 b). Prerequisites: ACM 100abc, Ae/APh/CE/ME 101abc 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. Instructor: 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 2701: System identification. 6 units (204); third term.
 CDS 2702: Nonlinear and adaptive control. 6 units (204); third term.
 CDS 2703: Flight control. 6 units (204); third term.
CDS 280. Advanced Topics in Geometric Mechanics or Dynamical Systems Theory. Hours and units by arrangement. Prerequisite: 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 2013–14.
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.