EE291E: Hybrid Systems, Spring 2001

The hybrid phenomena captured by such mathematical models are manifested in a great diversity of complex engineering applications such as real-time systems, embedded software, robotics, mechatronics, aeronautics, and process control. The high-profile and safety-critical nature of such applications has fostered a large and growing body of work on formal methods for hybrid systems: mathematical logic, computational models and methods and automated reasoning tools supporting the formal specification and verification of performance requirements for hybrid systems, and the design and synthesis of control programs for hybrid systems that are provably correct with respect to formal specifications.

This course investigates modeling, analysis and synthesis of various classes of hybrid systems.  An introduction to computational and simulation tools for hybrid systems will be given. The course consists of lectures, a handful of homeworks, and a final project.

• Instructors:
• T. John Koo, koo@eecs.berkeley.edu, 333 Cory Hall, (510) 643-5798

Office hour: 3:30-4:30, 467 Cory Hall
• S. Shankar Sastry, sastry@eecs.berkeley.edu, 269M Cory Hall, (510) 642-1857

 
• CCN: 25551

 
• Lectures: TuTh 2:00-3:30, 531 Cory Hall (Hogan Room)

Date	Lecturer	Contents
January 16	T. John Koo	Introduction
January 18	T. John Koo	Background: Discrete systems
January 23	T. John Koo	Background: Continuous systems
January 25	T. John Koo	Modeling: Hybrid systems
January 30	T. John Koo	Analysis: Hybrid systems
February 1	Jie Liu	Simulation: Ptolemy II
February 6	Tunc Simsek	Simulation: Lambda-SHIFT
February 8	T. John Koo	Modeling: Composition of hybrid systems
February 13	T. John Koo	Analysis: Stability
February 15	T. John Koo	Analysis: Stability of hybrid systems
February 20	T. John Koo	Analysis: Stability of hybrid systems
February 22	T. John Koo	Analysis: Reachability
February 27	T. John Koo	Analysis: Bisimulation
March 1	T. John Koo	Analysis: Time automata
March 6	T. John Koo	Analysis: Time automata
March 8	T. John Koo	Analysis: Rectangular automata
March 13	Howard Wong-Toi	Analysis: Linear hybrid automata/Computation: HYTECH
March 15	T. John Koo	Analysis: Linear hybrid systems
March 20	T. John Koo	Project discussion
March 22	T. John Koo	Project discussion
March 27		No Class
March 29		No Class
April 3	T. John Koo	Synthesis: Controller synthesis
April 5	T. John Koo	Synthesis: Control mode switching
April 10	S. Shankar Sastry	Synthesis: Optimal control
April 12	S. Shankar Sastry	Synthesis: Dynamical games
April 17	T. John Koo	Synthesis: Controller synthesis
April 19	T. John Koo	Computation: Reachable set
April 24	Jianghai Hu	Advanced Topics: Stochastic hybrid systems
April 26	Slobodan Simic	Advanced Topics: Geometric theory of hybrid systems
May 1	Johan Eker	Advanced Topics: Embedded control systems

• Homeworks: The homeworks capture both theory and applications.

 
• Projects: Potential project topics will be announced during the course. A project could be both theoretical and experimental in nature. You are encouraged to do a project related to an existing or potential research topic, but your own project ideas are also very welcome.

 
• Hybrid Systems Seminars: Together with EE222 Nonlinear Systems: Analysis, Stability, and Control there is a series of lectures on various topics in hybrid systems and nonlinear control.

 
• Miscellaneous:
• Prerequisites Some familiarity with basic concepts in dynamical systems and logic is needed. Knowledge about control theory and automata theory will be useful.
• Textbook Lecture notes will be made available on this homepage. Lecture notes of Spring 1999 (UC Berkeley report No. UCB/ERL M99/34) are available.
• Grading Grades will be awarded on the basis of homework assignments and the written and oral project presentation.
• Bibliography References and further reading are discussed in the context of each lecture, see the homepage of each lecture above.
• EE291E, Spring 1999 and 2000 There are several major differences between this year's Hybrid Systems course and its predecessors. Most of the material that is presented is either new or will be presented in a different context. The homepages of the previous courses are available as http://robotics.eecs.berkeley.edu/~lygeros/Teaching/ee291E.html and http://robotics.eecs.berkeley.edu/~johans/ee291e.html.
• Hybrid Systems in Stanford University and University of Pennsylvania Recently, courses on Hybrid Systems were taught in Stanford Univeristy by Prof. Claire T. Tomin in  Aeronautics and Astronautics Department and University of Pennsylvania by Prof. Rajeev Alur and Prof. George Pappas in  Computer and Information Sciences Department and Electrical Engineering Department.

Lecture Notes

Lecture 1: Introduction

Topics: Motivations, Examples: Bouncing Ball, Thermostat, Automated Highway Systems,  Unmanned Aerial Vechicles,  Modeling of Hybrid Automaton.
Lecture notes: Presentation

Reference:
• Chapter 2 and references: T. John Koo, ``Hybrid System Design and Embedded Controller Synthesis for Multi-Modal Control,'' Ph.D. Thesis, Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, CA, 2000.
• Lecture 1: John Lygeros and S. Shankar, EE291E, Spring 1999.
• Lecture 1: Karl Johansson, Tom Henzinger and Luca de Alfaro,  EE291E, Spring 2000.

Lecture 2: Background: Discrete systems

Topics: Models of Computation, Relations and Languages, Finite Automaton
Reference:
• Lecture 2: John Lygeros and S. Shankar, EE291E, Spring 1999.
• Edward A. Lee and Alberto Sangiovanni-Vincentelli , ``A Framework for Comparing Models of Computation ,'' IEEE Transactions on CAD, Vol. 17, No. 12, December 1998 .
• John E. Hopcroft and Jeffrey D. Ullman, ``Introduction to Automata Theory, Languages and Computation,'' Addison-Wesley, 1979.
• Harry R. Lewis and Christos H. Papadimitriou, ``Elements of the Theory of Computing,'', 2nd edition, Prentice Hall, 1998.

Lecture 3: Background: Continuous systems

Topics: Ordinary Differential Equations, Solutions in the sense of Caratheodory, Existence and Uniqueness Theorems, Continuous Dependence on Initial Conditions.
Reference:
• Lecture 2: John Lygeros and S. Shankar, EE291E, Spring 1999.
• Chapter 3: Shankar Sastry, ``Nonlinear Systems: Analysis, Stability, and Control,'' Springer, 1999.

Lecture 4: Modeling: Hybrid systems

Topics: Autonomous Hybrid Automaton, Hybrid Time Trajectory, Types of Execution, Definitions of  Properties - Deterministic, Blocking, and Zeno.
Reference:
• Lecture 3, Lecture 4, Lecture 7: John Lygeros and S. Shankar, EE291E, Spring 1999.

Lecture 5: Analysis: Hybrid systems

Topics: Characterization of Properties - Deterministic, Non-Blocking, and Zeno.
Reference:
• Lecture 3, Lecture 4: Karl Johansson, Tom Henzinger and Luca de Alfaro,  EE291E, Spring 2000.

Lecture 6: Simulation: Ptolemy II

Topics: Hierarchical Hybrid Systems - Modeling, Composition, Simulation
Lecture notes: Presentation by Jie Liu
Reference:
• Ptolemy II: ptolemy.eecs.berkeley.edu
• Modelica: www.modelica.org
• SHIFT: www.path.berkeley.edu/shift
• Dymola: www.dynasim.se
• OmSim: www.control.lth.se/~cace/omsim.html
• ABACUSS: yoric.mit.edu/abacuss/abacuss.html
• Stateflow: www.mathworks.com/products/stateflow

Lecture 7: Simulation: Lambda-SHIFT

Topics: Dynamical Network of Hybrid Automata
Lecture notes:  Presentation by Tunc Simsek

Lecture 8: Modeling: Composition of hybrid systems

Topics: Open Hybrid Automaton, Composition
Reference:
• Lecture 5: Karl Johansson, Tom Henzinger and Luca de Alfaro,  EE291E, Spring 2000.
• Lecture 8: John Lygeros and S. Shankar, EE291E, Spring 1999.

Lecture 9: Analysis: Stability

Topics: Equilibrium Point, Lyapunov Stability, Stability Properties - Uniform, Asymptotic, and Exponential, Lyaunov Stability Thorems, Exponential Stability Theorem and Its Converse, Lyapunov Equation
Reference:
• Lecture 6: Karl Johansson, Tom Henzinger and Luca de Alfaro,  EE291E, Spring 2000.
• Lecture 10: John Lygeros and S. Shankar, EE291E, Spring 1999.
• Chapter 5: Shankar Sastry, ``Nonlinear Systems: Analysis, Stability, and Control, '' Srpinger, 1999.

Lecture 10: Analysis: Stability of hybrid systems

Topics: Lyapunov Inequality, Computation of Linear Matrix Inequalities,  Equilibrium Point of Hybrid Automata, Stability of Hybrid Automata, Lyapunov's Stability Theorem for Hybrid Automata, Quadratic Lyapunov Stability
Reference:
• Lecture 6, Lecture 7:Karl Johansson, Tom Henzinger and Luca de Alfaro,  EE291E, Spring 2000.
• Lecture 10,  Lecture 11 : John Lygeros and S. Shankar, EE291E, Spring 1999.
• Chapter 2: Stephen Boyd, Laurent El Ghaoui, Eric Feron, Venkataramanan Balakrishnan, ``Linear Matrix Inequalities in System and Control Theory,'' SIAM, 1994.
• Michael S. Branicky, ``Multiple Lyapunov Functions and Other Analysis Tools for Switched and Hybrid Systems,'' IEEE Transactions on Automatical Control, Vol. 43, No. 4, pp.475-482.

Lecture 11: Analysis: Stability of hybrid systems

Topics: Lyapunov's Stability Theorem for Hybrid Automata, Stability of Switched Linear Systems, Common Quadratic Lyapunov Function
Reference:
• Lecture 6, Lecture 7:Karl Johansson, Tom Henzinger and Luca de Alfaro,  EE291E, Spring 2000.
• Lecture 10,  Lecture 11 : John Lygeros and S. Shankar, EE291E, Spring 1999.
• Mikael Johansson and Anders Rantzer, ``Computation of Piecewise Quadratic Lyapunov Functions for Hybrid Systems,'' IEEE Transactions on Automatical Control, Vol. 43, No. 4, pp.555-559.
• Daniel Liberzon, João Hespanha and A. S. Morse,``Stability of Switched Linear Systems: a Lie-Algebraic Condition,'' Syst. & Contr. Lett., 37(3):117--122, June 1999.

Lecture 12: Analysis: Reachability

Topics: Transistion  Systems, Predecessor, Reachability Problem, Forward and Backward Reachability Algorithms.
Reference:
• Lecture 12: John Lygeros and S. Shankar, EE291E, Spring 1999.
• Thomas A Henzinger, ``The Theory of Hybrid Automata,'' Proceedings of the 11th Annual IEEE Symposium on Logic in Computer Science (LICS 1996), pp. 278-292.
• Gerardo Lafferriere, George J. Pappas, and Sergio Yovine, ``A New Class of Decidable Hybrid Systems,'' Hybrid Systems : Computation and Control, Lecture Notes in Computer Science,volume 1569, Springer, 1999.

Lecture 13: Analysis: Bisimulation

Topics:  Equivalence Relation, Quotient Space, Quotient Transition Systems, Bisimulation, Bisimulation Algorithm.
Reference:
• Lecture 13: John Lygeros and S. Shankar, EE291E, Spring 1999.
 

Lecture 14: Analysis: Time Automata

Topics:  Definition of Time Automaton,  Time Automata and Transition Systems, Region Equivalence.
Reference:
• Lecture 14: John Lygeros and S. Shankar, EE291E, Spring 1999.
• R. Alur, ``Timed Automata, '' 11th International Conference on Computer-Aided Verification, LNCS 1633, pp. 8-22, Springer-Verlag, 1999.

Lecture 15: Analysis: Time Automata

Topics:  Region Equivalence, Quotient Transition Systems, Complexity Analysis.
Reference:
• Lecture 15: John Lygeros and S. Shankar, EE291E, Spring 1999.
• R. Alur, ``Timed Automata,'' 11th International Conference on Computer-Aided Verification, LNCS 1633, pp. 8-22, Springer-Verlag, 1999.

Lecture 16: Analysis: Rectangular Automata

Topics:  Definition of Rectangular Automata, Initialization, Differential Inclusion, Decidability.
Reference:
• Lecture 16: John Lygeros and S. Shankar, EE291E, Spring 1999.
• P. Kopke, T. Henzinger, A. Puri and P. Varaiya, ``What's Decidable About Hybrid Automata ,'' 7th Annual ACM Symposioum on Theory of Computing (STOCS): 372-382, 1995.
• A. Puri and P. Varaiya, ``Decidable Hybrid Systems,'' Computer and Mathematical Modeling,23(11/12):191-202, 1996.

Lecture 17: Computation: HYTECH

Topics:  Linear  Hybrid Automata and HYTECH
Lecture notes: Presentation by Howard Wong-Toi
Reference:
• HYTECH Homepage
• T.A. Henzinger, P.-H. Ho, and H. Wong-Toi. ``HyTech: A Model Checker for Hybrid Systems,'' Software Tools for Technology Transfer 1:110-122, 1997.

Lecture 18: Synthesis: Controller Synthesis

Topics:  Hybrid Automaton with Inputs,  Safety and Liveness Properties, Controlled Invariant Set, Controller Sysnthesis Problem
Reference:
• Claire Tomlin, John Lygeros, and Shankar Sastry, ``A Game Theoretic Approach to Controller Design for Hybrid Systems ,'' Proceedings of the IEEE, Volume 88, Number 7, July 2000.

Lecture 19: Synthesis: Control Mode Switching

Topics:  Control Mode, Mode Switching Problem, Consistent Mode Switching Condition,  Control Mode Graph
Lecture notes: Presentation by T. John Koo
Reference:
• T. J. Koo, G. J. Pappas, and S. Sastry, ``Mode Switching Synthesis for Reachability Specifications,'' Hybrid Systems: Computation and Control, Lecture Notes in Computer Science, Springer, 2001.

Lecture 20: Synthesis: Optimal Control

Topics:  Optimal Control, Calculus of Variations, Hamilton-Jacobi-Bellmen Equation
Lecture notes: Lectures in Optimal Control and Dynamical Games by S. Shankar Sastry
Reference:
• Claire Tomlin, John Lygeros, and Shankar Sastry, ``A Game Theoretic Approach to Controller Design for Hybrid Systems ,'' Proceedings of the IEEE, Volume 88, Number 7, July 2000.

Lecture 21: Synthesis: Dynamical Games

Topics:  Dynamical Games, Least Restrictive Control, Envelope Protection, Conflict Resolution
Lecture notes: Lectures in Optimal Control and Dynamical Games by S. Shankar Sastry
Reference:
• Claire Tomlin, John Lygeros, and Shankar Sastry, ``A Game Theoretic Approach to Controller Design for Hybrid Systems ,'' Proceedings of the IEEE, Volume 88, Number 7, July 2000.

Lecture 22: Synthesis: Controller Synthesis

Topics:  Controller  Synthesis for Least Restrictive Control, Envelope Protection and Conflict Resolution
Lecture notes: Lectures in Optimal Control and Dynamical Games by S. Shankar Sastry
Reference:
• Claire Tomlin, John Lygeros, and Shankar Sastry, ``A Game Theoretic Approach to Controller Design for Hybrid Systems ,'' Proceedings of the IEEE, Volume 88, Number 7, July 2000.

Lecture 23: Computation: Reachable Set

Topics:  Reach Set, Reachable Set, Optimal Control,  Maximum Principle
Reference:
• P. Varaiya, ``Reach Set Computation Using Optimal Control, '' December 1998.

Lecture 24: Advanced Topics: Stochastic Hybrid Systems

Topics: Markov Stochastic Processes, Stochastic Differential Equation, Embedded Markov Chain, Invariant  Distribution, Stochastic Stability
Lecture notes: Presentation by Jianghai Hu
Reference:
• Jianghai Hu, John Lygeros, and Shankar Sastry, ``Toward a Theory of Stochastic Hybrid Systems.''
• P. G. Doyle and J. L. Snell, ``Random Walks and Electric Networks.''

Lecture 25: Advanced Topics: Geometric Theory of  Hybrid Systems

Topics: Regular Hybrid Systems, Hybrifold, Zeno, Classification of Zeno in Dimension Two,  Stability of  Equilibria
Lecture notes: Presentation and Figures by Slobodan Simic
Reference:
• Slobodan Simic, Karl Henrik Johansson, Shankar Sastry and John Lygeros, ``Towards a Geometric Theory of Hybrid System,'' in Hybrid Systems 2000, Pittsburgh, PA, March 2000.
• J. Lygeros, K. H. Johansson, S. N. Simic, J. Zhang, S. Sastry, ``Dynamical Properties of Hybrid Automata,'' Submitted to IEEE Transactions on Automatic Control.

Lecture 26: Advanced Topics: Embedded Control System

Topics: Embedded Computers, Real-Time Control Systems, Wireless Distributed Systems, Hierarchical Hybrid System Models,
Hybrid Control Systems
Lecture notes: Presentation and by Johan Eker

Course Projects

Project 1: Control and Stability of a Lower Limb Anthropomorphic Exoskeleton

Abstract: This paper introduces a novel control method for a human performance augmentation anthropomorphic lower limb exoskeleton.  The exoskeleton will enable a human to walk with a heavy load for a prolonged period of time.  A switching controller creates a virtual impedance to ensure the machine does not lose its balance.  The use of the operational space formulation allows for a common control model and analysis algorithm to accommodate the different dynamic models adopted by the system throughout walking. Asymptotic stability of the hybrid system is verified using the theory of Lyapunov stability for hybrid systems. An region of unsafe initial conditions is characterized in the system state-space.

Project 2: Transition Control via Sliding Mode Control: Hybrid Model of a Force-Controlled Pneumatic System

Abstract: The purpose of this project is to define a transition controller for a pneumatic system consisting of a linear actuator, electro-proportional pneumatic valve and proportional valve signal amplifier.  Due to the difficulties characterizing the non-linearities present in such a described system, separate sliding controllers were implemented.  One problem with such a scheme is with the transitioning.

The reason I chose to use a sliding mode control on this system is multifold.  First, with sliding control, the model becomes a simplified one-dimensional control problem.  Second, the non-linearities present, which are unknown, can be accounted for using this method.  One problem I encountered with using sliding control is in the complexity of the control law due to a discontinuous flow model.  To deal with this, I will introduce a type of gain scheduling.

This paper will first show results without a safety controller and show how bad transitioning can deter the objective of sliding control.  Then, this paper will present the progress of developing a safety controller and its benefits.

Project 3: Robust Optimal Mode Switching for a Class of Hybrid Systems

                   Xiaotian Sun

Abstract: Today's complex control systems often operate in different modes with  different agents and different objectives.  A control mode is defined  as the operation of a continuous system under a controller that guarantees to track a certain class of output trajectories.  In the robust optimal mode switching problem, we try to determine a finite mode sequence that steers the system from an initial control mode to a desired final control mode, and at the same time, minimizes the worst-case cost function.  By imposing a consistent mode switching condition, we can establish a control mode graph and avoid nested reachability computation.  After establishing the control mode graph, we will be able to associate a worst-case cost to each link in the graph by solving a max-min problem.  Then the worst-case optimal mode sequence may be obtained by searching a shortest path from the initial mode to the final mode on the graph.  This shortest path problem can be solved online efficiently by dynamic programming or linear programming.

Project 4: Optimal Control for Shifting in a Bicycle Race

                   Lance Doherty

Abstract: We compute the optimal control strategy for shifting gears in a bicycle race. Depending on the cadence of the pedals, the cyclist outputs power to accelerate or maintain the bicycle speed. As a function of oxygen consumption, power output has a maximum at a particular cadence - it is in the interest of the cyclist to maintain a cadence close to the optimal to maximize power output by shifting gears. The problem is solved using the methodology of Hedlund and Rantzer [1] by defining a lower bound on the objective function over a state-space grid and maximizing it with linear programming.

Project 5: State Estimation and Fault Detection in Stochastic Hybrid Systems

                    Mario Micheli

Abstract: In this paper we study the problem of state estimation for a particular class of Stochastic Hybrid Systems (SHS). Precisely, given a SHS whose continuous-time evolution is given by a set of stochastic, Gaussian linear systems, and whose discrete-time evolution is described by a known probability distribution (not necessarily Markov; in fact, it may depend on the continuous-time evolution) we formulate an algorithm that yields and estimate for the continuous state and a posterior probability distribution on the discrete state, given a sequence of linear, noisy measurements of the continuous state variable only. We also illustrate an  application to fault detection in Stochastic Hybrid Systems. The algorithm we illustrate is an adaptation to Stochastic Hybrid Systems of a sampling-based estimation method for the so called Conditional Dynamical Linear Models (also known as Switching Kalman Filters).

Project 6: Hybrid Modeling of MEMS

                    Jason Clark

Abstract:  Hybrid system techniques will be used to simulate a microelectromechanical system. The test case is based on a micro motor designed at UC Berkeley [1]. The device is particular class of MEMS called an impact actuator. Currently, research and commercial software packages have been successful at modeling MEMS mainly to the extent of simple structural dynamics such as vibration analysis and nonlinear deflections using such methods as finite element analysis and nodal analysis [2]. What makes the  modeling and simulation of MEMS complex is coupling of multi-scales and multi-energy domains such as electromagnetic, mechanic, thermal, digital/analog electrical, etc.  Recently, impact actuation has been shown to be a promising method for obtaining large deflections out of these micro devices. However, modeling impact and friction in MEMS has remained a challenge. While using the measured coefficient of restitution for polysilicon [3] a first attempt to model impact collisions as a continuous time system in a nodal analysis package simulation for MEMS, Sugar [4], ran into problems with the Zeno phenomenon.  The problem was due to the system of ODE's having discontinuous right-hand sides and  ever decreasing adaptive time steps during impact chattering. These small time steps  alone made the simulation duration impractical. This problem can be addressed in  Ptolemy [5] by modeling the impact actuator system as a heterogeneous system. The  system consists of three types of models of computation: (1) continuous time (the electro-mechanical transfer function), (2) discrete event  (the digital waveform input), and (3)  finite state machine (the impact state versus non-impact state). The transfer function for  the electrostatics and mechanics is obtained from Sugar. This transfer function takes  voltage as input and generates electrostatic forces based on structural geometry. The main reason that a finite state machine approach is used here is that it provides an efficient way  to handle Zeno behavior.

[1]  http://www-bsac.EECS.Berkeley.EDU/~yeh/video.html
[2]  J V Clark, D Bindel, N Zhou, S Bhave, Z Bai, J Demmel, K S J Pister, Sugar:
Advancements in a 3D Multi-Domain Simulation Package for MEMS, Proceedings
of the Microscale Systems: Mechanics and Measurements Symposium, June 4, 2001,
Portland OR, USA.
[3]  A P Lee, Impact Actuation of Polysilicon Micromechanical Structures, UC Berkeley
Ph.D. dissertation 1992.
[4]  www-bsac.eecs.Berkeley.edu/~cfm
[5]  http://ptolemy.eecs.berkeley.edu/

Project 7: Reachable Set Computation for Piecewise Affine Vector Fields

                    Michael Reiser

Abstract:  The goal of this project is to generate an over-approximation of the reachable set of a system defined by piecewise affine vector fields. Such an algorithm is intended to serve as a component of a general method for determining the reachable set of Hybrid Systems with non-linear vector fields. The approximation algorithm is first discussed and then a MATLAB software tool is explained. The utility of the tool is shown by several experimental results. Necessary extensions to the tool and comparisons to other approaches are then presented.

Project 8: Formal Modeling of an Autonomous Model Helicopter

                    Judy Liebman and Cedric Ma

Abstract: An autonomous model helicopter requires a complicated hybrid controller. This controller must accomplish changes in the flight modes as well as maintaining the stability of the vehicle.  In this project we use Berkeley's BEAR helicopter as an example platform.  We model this platform in detail using UML in order to clarify the software structure and timing requirements of the system.  This model is needed in order to analyze the jitter tolerated by the current control system and in order to reevaluate the event triggered design for embedded control systems.  Next, we present possible synchronous reformulations of the helicopter control software and switching technique using Giotto.  The suggested software model presents a new way of integrating a complicated embedded system in which strict timing requirements are guaranteed.

Project 9: Giotto: a Time-Triggered Language for Embedded Programming

Abstract: Giotto is a programming language that offers a principled, tool-supported design methodology for implementing embedded control systems on platforms of possibly distributed sensors, actuators, CPUs, and networks.  Giotto is based on the organizing principle that time-triggered task invocations together with time-triggered mode switches can form the abstract essence of programming real-time control systems.  In this talk, I will discuss two recent additions to Giotto:  (1) nonharmonic schedules, in which higher frequencies need not evenly divide lower frequencies; and (2) schedule constraint graphs, which provide a means of checking the correctness of a Giotto implementation.