Courses 2024-25

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.

An introduction to analysis and design of feedback control systems in the time and frequency domain, with an emphasis on state space methods, robustness, and design tradeoffs. Linear input/output systems, including input/output response via convolution, reachability, and observability. State feedback methods, including eigenvalue placement, linear quadratic regulators, and model predictive control. Output feedback including estimators and two-degree of freedom design. Input/output modeling via transfer functions and frequency domain analysis of performance and robustness, including the use of Bode and Nyquist plots. Robustness, tradeoffs and fundamental limits, including the effects of external disturbances and unmodeled dynamics, sensitivity functions, and the Bode integral formula.

Basic system concepts; state-space and I/O representation. Properties of linear systems, including stability, performance, robustness. Reachability, observability, minimality, state and output-feedback. Brief introduction to optimal control and control of networked and nonlinear systems. Motivating case studies from tech, biology, neuroscience, and medical systems.

Research project in control and dynamical systems, supervised by a CDS faculty member.

Advanced topics in optimization-based design of control, optimal control, and estimation/filtering. Optimal control theory using calculus of variations, Hamilton-Jacobi-Bellman equation, Pontryagin's maximum principle, and optimal control applications including reinforcement learning and model predictive control. Kalman filtering, Bayesian filtering, and nonlinear filtering methods for autonomous systems.

Scalable analysis and synthesis of robust control systems. Motivation throughout from case studies in tech, neuro, bio, med, and socioeconomic networks. Co-design of sparse and limited (delayed, localized, quantized, saturating, noisy) sensing, communications, computing, and actuation using System Level Synthesis (SLS). Layering, localization, and distributed control. Computational scalability exploiting sparsity and structure. Uncertainty, including noise, disturbances, parametric uncertainty, unmodeled dynamics, and structured uncertainty (LTI/LTV). Tradeoffs, robustness versus efficiency, conservation laws and hard limits in time and frequency domain. Advanced topics, depending on class interest, can include interplay between automation, optimization, control, modeling and system identification, and machine learning, and nonlinear dynamics and sum of squares, global stability, regions of attraction.

This course studies nonlinear dynamical systems beginning from first principles. Topics include: existence and uniqueness properties of solutions to nonlinear ODEs, stability of nonlinear systems from the perspective of Lyapunov, and behavior unique to nonlinear systems; for example: stability of periodic orbits, Poincaré maps and stability/invariance of sets. The dynamics of robotic systems will be used as a motivating example.

This course studies nonlinear control systems from Lyapunov perspective. Beginning with feedback linearization and the stabilization of feedback linearizable system, these concepts are related to control Lyapunov functions, and corresponding stabilization results in the context of optimization based controllers. Advanced topics that build upon these core results will be discussed including: stability of periodic orbits, controller synthesis through virtual constraints, safety-critical controllers, and the role of physical constraints and actuator limits. The control of robotic systems will be used as a motivating example.

Advanced topics in robotic motion planning and navigation, including inertial navigation, simultaneous localization and mapping, Markov Decision Processes, Stochastic Receding Horizon Control, Risk-Aware planning, robotic coverage planning, and multi-robot coordination. Course work will consist of homework, programming projects, and labs. Given in alternate years.

Mathematical treatment of data-driven machine learning methods for controlling robotic and dynamical systems with various uncertainties. Gradient and least-squares estimators and variants for dynamical systems for system identification and residual learning. Adaptive control methods for online adaptation and combination with deep learning. Learning-based control certificates such as neural Lyapunov functions and neural contraction metrics.

Research in the field of control and dynamical systems. By arrangement with members of the staff, properly qualified graduate students are directed in research.