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Robust Hybrid and Embedded Systems Design. Jerry Ding, Jeremy Gillula, Haomiao Huang, Michael Vitus, and Claire Tomlin. MURI Review Meeting Frameworks and Tools for High-Confidence Design of Adaptive, Distributed Embedded Control Systems Berkeley, CA December 2, 2009. Hybrid System Model.

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robust hybrid and embedded systems design

Robust Hybrid and Embedded Systems Design

Jerry Ding, Jeremy Gillula, Haomiao Huang, Michael Vitus, and Claire Tomlin

MURI Review Meeting

Frameworks and Tools for High-Confidence Design of Adaptive, Distributed Embedded Control Systems

Berkeley, CA

December 2, 2009

backwards reachable set
Backwards Reachable Set

All states for which, for all possible control actions, there is a disturbance action which can drive the system state into a region G(0) in time t

Backwards Reachable Set

Reachability as game: disturbance attempts to force system into unsafe region, control attempts to stay safe

reachable set propagation
Reachable Set Propagation

Theorem [Computing ]:

where is the unique Crandall-Evans-Lions viscosity solution to:

[Mitchell, Bayen, Tomlin 2005]

backwards reachable set safety
Backwards Reachable Set: Safety

unsafe

Backwards Reachable Set

Safety Property can be encoded as a condition on the system’s reachable set of states

In blue, system will stay safe

In red, system may become unsafe

On boundary, apply control to stay out of red

computation
Computation

Ian Mitchell’s level set computational toolbox for Matlab

available at:

  • http://www.cs.ubc.ca/~mitchell/ToolboxLS/

v

y

d

u

body frame

v

wind frame

  • Used for a variety of applications
  • Handles 3 dimensions easily, up to 5 tractably
  • Library of level set functions

inertial frame

5

backwards reachable set capture
Backwards Reachable Set: Capture

Backwards Reachable Set

desired

Capture property can also be encoded as a condition on the system’s reachable set of states

maneuver sequencing using reachable sets
Maneuver Sequencing Using Reachable Sets

Maneuver sequencing is accomplished by stringing together capture sets, starting from the target set and working backwards

Target Set

Unsafe Set

Avoid sets can be combined with capture sets to guarantee safety

experimental platform starmac
Experimental Platform: STARMAC

The Stanford Testbed of Autonomous Rotorcraft for Multi-Agent Control

example collision avoidance
Example: Collision Avoidance
  • Pilots instructed to attempt to collide vehicles

[Gabe Hoffmann]

example quadrotor back flip

Impulse

Example: Quadrotor Back-Flip

Recovery

Drift

  • Divide flip into three modes
  • Difficult problem:
    • Hitting some target sets while avoiding some unsafe sets
  • Solution:
    • Analyze rotational dynamics and vertical dynamics separately
back flip method 1
Back-flip: Method (1)

Recovery

Drift

Impulse

  • Identify target region in rotational state space for each mode
  • Use reachable sets to calculate capture basinfor each target
    • Dynamic game formulation accounts for worst-case disturbances
  • Verify that target of each mode is contained by capture basin of next mode
back flip method 2
Back-flip: Method (2)
  • Identify unsafe region in vertical state space for final mode
  • Use reachable sets to propagate unsafe set for each mode
    • Dynamic game formulation accounts for worst-case disturbances
  • Verify that control keeps state out of unsafe set
assumptions and dynamics
Assumptions and Dynamics
  • Assumptions:
    • 2D flip
    • Linear drag
  • System Dynamics:
back flip recovery mode
Back-Flip: Recovery Mode
  • Controller:
  • Target set:
  • Calculate reachable sets using closed-loop dynamics and worst-case disturbances
back flip drift mode
Back-Flip: Drift Mode
  • No control input
  • Target set:
  • Calculate reachable sets using closed-loop dynamics and worst-case disturbances
  • But what if motors don’t turn off instantly?
back flip motor turn off 1
Back-Flip: Motor Turn Off (1)
  • Model motor turn off as linear decay in angular acceleration
  • Linear regression to get parameters:
back flip motor turn off 2
Back-Flip: Motor Turn Off (2)
  • Calculate forward reachable set for the motors turning off

Convex Hull

2D Projection

back flip drift mode motor turn off
Back-Flip: Drift Mode & Motor Turn Off
  • Target set:
  • Calculate motor turn off set
  • Ensure motor turn off set is contained in drift set
back flip impulse mode
Back-Flip: Impulse Mode
  • Controller:
  • Target set:
  • Calculate reachable sets using closed-loop dynamics and worst-case disturbances
back flip vertical conditions
Back-Flip: Vertical Conditions
  • Drift Mode:
    • Dynamics:
    • Decouples as 3 independent 2D systems
    • Use reachable sets to calculate unsafe starting conditions
  • Impulse Mode:
    • Assume no loss of altitude during impulse
back flip results2
Back-Flip: Results
  • Assumptions Validated
  • Safety Guaranteed
  • Reachability Demonstrated
reachability with sampling and quantization
Reachability with sampling and quantization

In many embedded control applications, use digital controller to control continuous dynamics

Safety and capture results available in discrete and continuous domain

Problem becomes more difficult at interface:

Continuous behavior:

Continuous state evolution

Discrete behavior:

Mode switching

Sampling, quantization

25

continuous time verification methods
Continuous Time Verification Methods

Problems:

How to implement the safe continuous time control law in a digital controller?

Does the discretized control law still ensure safety?

Issues:

Sampling

Quantization

Switched mode control

26

infinite horizon unsafe set comparisons
Infinite Horizon Unsafe Set: Comparisons

Unsafe Initial Condition

∞ Horizon Unsafe Set without quantization and sampling

∞ Horizon Unsafe Set with quantization and sampling

27

reachavoid set for two mode system
Reachavoid Set for Two Mode System

Time horizon N = 12 (2 minutes)

Reachavoid Set Over 2 min

Infinite Horizon Unsafe Set

Desired Target Set

28

next steps
Next steps
  • Transitions with state dependent guards at sampling instants
  • Considerations for partial state information
  • Overapproximations methods for continuous time reachable sets
  • Parametrization of reachable sets by quantized control values
  • Methods for robust optimal control
back flip vertical conditions 1
Back-Flip: Vertical Conditions (1)
  • Initial unsafe set:
  • Recovery Mode:
    • Dynamics:
    • Assume nominal trajectory
    • Calculate the constrained reachable set within the nominal trajectory