ECE 7850 Hybrid Systems: Theory and Applications (Spring 2017)
Course Info
Instructor:  Wei Zhang, 404 Dreese Labs 
Time:  TuTh 11:10AM  12:30PM 
Location:  Caldwell Labs 0119 
Webpage:  ECE 7850 webpage 

Course Description
Hybrid dynamical systems are characterized by coupled continuous and discrete dynamics. It switches between many operating modes where each mode is governed by its own characteristic dynamical law. Mode transitions are triggered by variables crossing specific thresholds (state events), by the elapse of certain time periods (time events), or by external inputs (input events). This course will introduce the students to the recent theoretical and application advances in modeling, control, and estimation of hybrid systems. The main topics include stability analysis and stabilization, optimal control, reachability analysis, safety verification, and estimation of hybrid systems. Emphasis will be placed on connecting the stateoftheart theories and methods to real world applications in electrical, mechanical, and aerospace engineering.
Special Notes
 Prerequisite: ECE 5750  Linear System Theory or consent of instructor
 While some prior exposure to control theory is certainly helpful, the course is sufficiently selfcontained. Motivated students specializing in other technical areas with a solid math background should be able to follow course.
 This is not a seminar type of course. Students are required to understand the mathematical derivations and proofs for most of the results covered in the class.
Grading Policy
The students will be evaluated based on several homework assignments (30%), a Midterm Exam (30%) and a final project (40%). The homework will be designed to solidify the students' understanding in the existing theories and tools in the literature, while the final project is primarily for the students to extend the theory to solve real world problems of their interest.
Tentative Topics
 Introduction to Hybrid Systems
 Examples and modeling frameworks of hybrid systems
 Solution and execution:
 Background: Basic nonsmooth analysis, differential inclusion
 Hybrid state trajectory, Filippov solution, zeno phenomena
 Stability Analysis and Stabilization
 Background: Lyapunov theory, Linear Matrix Inequality, Sprocedure
 Stability analysis: stability under arbitrary switching, stability under constrained switching, MultipleLyapunov function
 Stabilization: LMI based synthesis using multipleLyapunov function; controlLyapunov function approach
 Discrete Time Optimal Control
 Background: dynamic programming, Model Predictive Control (MPC), multiparametric programming
 Switched LQR problem
 MPC of switched Piecewise Affine Systems
 Infinitehorizon optimal control and its connection to stability/stabilization
 Reachability Analysis and Computation
 Background: Differential games and HamiltonJacobiIsaacs (HJI) equations
 HJI based reachability analysis
 Discretetime reachability through polyhedral operations
 Applications in safety verification in robotics and intelligent transportation
 Continuous Time Optimal Control
 Background: Hilbert space, weak topology, chattering lemma, basic calculus of variation
 Theory of numerical optimization in infinitedimensional space
 Optimality functions and master algorithms
 Epiconvergence and consistent approximation
 Applications to optimal control of switched nonlinear systems
 Lecture 1: Course Info and Hybrid System Examples. [Blank Version] [Annotated Version]
 Lecture 2: Modeling Frameworks for Hybrid Systems. [Blank Version] [Annotated Version]
Finite state automaton, differential equation/inclusion, Caratheodory solutions, hybrid automaton, other hybrid system models
 Lecture 3: Solution Notions of Hybrid Systems. [Blank Version] [Annotated Version]
Execution of hybrid automaton, Zeno phenomenon, Filippov solution, existence/uniques/computation of sliding motion
 Lecture 4: Basic Lyapunov Stability. [Blank Version] [Annotated Version]
Basic stability concepts, Lyapunov theorems, Lyapunov equation for linear system, converse Lyapunov theorems
 Lecture 5: Semidefinite Programming for Stability Analysis [Blank Version] [Annotated Version]
Properties of Symmetric matrices, Shur complement lemma, Linear Matrix Inequalities, semidefinite programming, Sprocedure
 Lecture 6: Stability Analysis of Switched and Hybrid Systems [Blank Version] [Annotated Version]
Stability under arbitrary switching, stability under slow switching, stability under statedependent switching, Multiple Lyapunov Functions, Computation of piecewise quadratic Lyapunov functions for piecewise linear systems
 Lecture 7: Switching Stabilization via ControlLyapunov Function [Blank Version]
Classical ControlLyapunov Function Approach, Switching Stabilization Problem, Switching Stabilization via Control Lyapunov Function, Special Case: Quadratic Switching Stabilization, Special Case: Piecewise Quadratic Switching Stabilization
 Lecture 8: Discrete Time Optimal Control and Dynamic Programming [Blank Version]
Discretetime optimal control problems, Dynamic Programming, Bellman Equation, Convergence of Value Iterations, Connection between Optimal Control and Stabilization Problems
 Lecture 9: Model Predictive Control of Linear and Hybrid Systems: Basic Formulation and Algorithms [Blank Version] [Annotated Version]
Formulation of General MPC Problems, Linear MPC Problems, Linear MPC Example: Cessna Citation Aircraft, DiscreteTime Hybrid System Models, MPC of Hybrid Systems
 Lecture 10: Explicit Model Predictive Control [Blank Version]
Online MPC vs. explicit MPC, Introduction Multiparametric Programming, Linear Explicit MPC, Hybrid Explicit MPC
 Lecture 11: Model Predictive Control: Theoretical Aspects [Blank Version] [Annotated Version]
Theoretical issues of model predictive controller, Persistent Feasibility of MPC, Stability of MPC, Stability Without Terminal Constraint/Cost, Connection to Unconstrained Problem
 Lecture 12: ContinuousTime Optimal Control (Background) [Blank Version] [Annotated Version]
Basics on finite dimentionsl optimization, directional derivative, calculus of variation, EulerLagrange equation
 Lecture 13: ContinuousTime Switched Optimal Control [Blank Version] [Annotated Version]
UCSB (Prof. Hespanha)
UC Berkeley (Prof. Sastry and Prof. Tomlin)
Cornell (Prof. KressGazit)
RPI (Prof. Julius)
TU Delft (Prof. Schutter)
University of Porto

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