AAE 668 & ECE 695Hybrid Systems: Theory and Applications

Lecture 1: Introduction

Inseok Hwang and Jianghai Hu

Time: TTh 10:30-11:45amLocation: ARMS 3115

Office hour: Tu 9:30-10:30am (Hwang), Tu 3-4pm (Hu)Email: {ihwang,jianghai}@purdue.edu

August 22, 2016

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References

Books:

• Hybrid Systems: Foundations, Advanced Topics and Applications,J. Lygeros, S. Sastry and C. Tomlin, 2012

• Switching in Systems and Control, D. Liberzon, Springer, 2003

• Hybrid Dynamical Systems: Modeling, Stability, and Robustness,R. Goebel, R. G. Sanfelice and A. R. Teel, 2012

• Predictive Control for Linear and Hybrid Systems, F. Borrelli, A.Bemporad and M. Morari, 2013.

Survey paper:

• Hybrid Dynamical Systems: An Introduction to Control and Verification,H. Lin and P. J. Antsaklis, 2014

Other topic-specific papers to be given later on

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Hybrid Systems

Hybrid systems: dynamical systems with interacting continuous-time anddiscrete-time dynamics

• State consists of a continuous state x and a discrete state (mode) q

• Dynamics of continuous state: x = f (q, x , u)• Dependent on discrete state and external continuous input u

• Dynamics of discrete state: q+ = g(x , q, σ)• Discrete transitions triggered by

• Continuous state x reaching a subset (called guard)• External discrete input σ

• After transition, continuous state x may jump via a reset map

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Example 1: Bouncing Ball

Hybrid system with one mode (free fall) andone mode-transition event (bounce)1

1Related example: billiards4 / 13

Example 2: (Simple) Room Temperature ControlTurn on/off heater to keep temperature near 20◦C

Control stategy:

• turn heater on if T ≤ 18

• turn heater off if T ≥ 22

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Example 3: Water Tank

xi : volume of water in tank iri : minimum allowed volumevi : outflow rate of tank iw : inflow rate by pump

Goal: keep xi ≥ riPump modes: q = {1, 2}

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Example 4: Aircraft Collision Avoidance

Flying modes:

• Cruising (straight)

• Left turn

• Right turn

• Climbing

• Descending

Goal: find (mode-switching) protocol to avoid A/C collision

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Example 4: Supervisory Switching Control

• Discrete state q: id of the controller chosen by the supervisor

• Continuous state x : internal state of the plant

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Example 5: Variable Structure Control

Given a nonlinear system x = f (x , u), design (discontinuous) control u sothat it is piecewise smooth on different parts of the state space

Example: sliding mode control

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Example 6: Networked Control Systems

Plant dynamics x = Ax + Bu

Plant state x transmitted over the network

• State transmitted to controller over the network at times t0, t1, . . .

• Controller stores received state value between transmissions:

x(t) = x(tk), t ∈ [tk , tk+1)

• Linear stored state feedback controller: u(t) = K x(t)

Stabilization: can we make x(t)→ 0?

Event-triggered control:

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Example 7: B. Subtilis Cell

A stochastic (probabilistic) hybrid system

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Topics Covered in This Course

Topics:

• Continuous and discrete-time system models

• Hybrid systems models and solution properties

• Reachability analysis

• Stability analysis

• Optimal control

• Estimation and identification

• Advanced topics

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Big Picture

This course will be mostly from the control theoretical point of view

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