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Reachability Based Controller Synthesis for Switched Systems

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### Reachability Based Controller Synthesis for Switched Systems

ICRA 2010 workshop

Formal methods for robotics and automation

May 3, 2010

Jerry Ding Eugene Li Haomiao Huang

Prof. Claire Tomlin

Outline

- Motivation
- Switched System Model
- Reachable Set Computation
- Control Law Synthesis

Motivation

- Modern robotics applications commonly use digital controllers to control continuous dynamics

- Overall system dynamics features
- High level logic, e.g. maneuver library
- Low level controls, e.g. motor control

Linear Temporal Logic

[Kress-Gazit, Fainekos, Pappas, IEEE Trans. Robot. 2009]

Discrete Polygonal Planning

Receding Horizon Control

[Belta, Isler, Pappas, IEEE Trans. Robot. 2005]

[Wongpiromsarn, Topcu, Murray, HSCC 2010]

Problem Formulation

- Given:
- Switched system dynamics
- Target states we want to reach
- Unsafe states we want to avoid

Target set

Unsafe set

Mode

Mode

Problem Formulation

- Compute set of states (q, x) that can be controlled to target set while avoiding unsafe set over finite horizon
- Call this reach-avoid set

Target set

Unsafe set

Reach-avoid set

Mode

Mode

Problem Formulation

- For any (q,x) in the reach-avoid set, automatically synthesize a feedback policy that achieves the specifications

Target set

Unsafe set

Reach-avoid set

Mode

Mode

Reachability based methods

- This type of problem is naturally handled by reachability based analysis tools
- Recovering implementable control law from reachable sets in general non-trivial
- We describe an automatic controller synthesis procedure in the case where
- 1) System is switched
- 2) Disturbance does not affect discrete transitions

Outline

- Motivation
- Switched System Model
- Reachable set computation
- Control Law Synthesis

Switched System Model – Dynamics

Discrete State Space

Continuous State Space

Continuous Dynamics

Reset Relations

Switched System Model – Inputs

- Sampled-data system for practical implementation
- Quantized input to ease computation and analysis

Switching Signal

Piece-wise constant

Continuous Input

Disturbance

Time-Varying

0

T

2T

3T

4T

5T

Switched System Model – Control and Disturbance Policies

- On sampling interval [kT, (k+1)T], define

One step control policy

One step disturbance strategy

(k+1)T

(k+1)T

kT

kT

Reach-avoid Problem Specification

- Reach-avoid problem:

Given safe initial condition

Choose control policy so that

regardless of disturbance strategy

1)

2)

- Denote the set of feasible initial conditions by
- Reachability problem: compute
- Synthesis problem: synthesize

Outline

- Motivation
- Switched Mode System Model
- Reachable set computation
- Control Law Synthesis

Reach-avoid Set Computation – Building Blocks

- Fix input level ui in mode qi, compute one step unsafe reachable set
- Take into account all possible realizations of di

This set can be computed numerically using Level Set Methods

[Mitchell, et al, TAC, 2005]

Unsafe Region

Safe Trajectory

Unsafe

Trajectory

One step unsafe reachable set for fixed input

Level Set Representation:

Mode

Reach-avoid Set Computation – Building Blocks

- Fix input level ui in mode qi, compute one step target reachable set
- Take into account all possible realizations of di

Target Region

This set can be computed similarly as unsafe set

Trajectory reaches target in one step

Trajectory does not reach target

In one step

One Step Target Reachable Set

Mode

Reach-avoid Set Computation – Step 1

- Compute the one step reach-avoid set using set difference

Target Region

Trajectory reaches target in one step while avoiding

unsafe region

Unsafe Region

For

and

One step reach-avoid set for fixed input

Let

then

Mode

Reach-avoid Set Computation – Step 2

- Take union over possible inputs ui in mode qi

Target Region

One step reach-avoid set for input 1

One step reach-avoid set for input 2

One step reach-avoid set for mode qi over all input levels

Unsafe Region

Mode

Reach-avoid Set Computation – Step 3

- Take union over possible mode switches in mode qi

Target set

Unsafe set

Reach-avoid set

Mode

Mode

Reach-avoid set for mode 1 over all input levels

Reach-avoid set for mode 2 over all input levels

Reach-avoid Set Computation – Step 3

- Take union over possible mode switches in mode qi

Target set

Unsafe set

Reach-avoid set

Mode

Mode

One step reach-avoid sets under switching

Reach-avoid Set Computation – Step 4

- Iterate to compute the reach-avoid set over [0,NT]
- By induction, can show that

- Let

Target set

j time step reach-avoid set

Unsafe set

Initialization:

for

to

end

One step reach-avoid set computation

Return:

Outline

- Motivation
- Switched Mode System Model
- Reachable set computation
- Control Law Synthesis

Reach-avoid control law synthesis

- Compute and store the reach-avoid sets

and those corresponding to particular inputs

- These sets define an explicit state feedback policy for the reach-avoid problem

- Number of reachable sets required is given by

Number of quantization levels in mode qi

Length of time horizon

Number of discrete modes

Reach-avoid control law synthesis

- At time k < N

Step 1: Obtain state measurement

Target set

Unsafe set

State Space

j time step

reach-avoid set

Reach-avoid control law synthesis

- At time k < N

Step 2: Find minimum time to reach

Target set

Unsafe set

State Space

j time step

reach-avoid set

Reach-avoid control law synthesis

- At time k < N

Step 3: Find set of possible control inputs

is the set of states that can safely reach

within one step using an admissible input

Target set

One step reach-avoid set for input

Unsafe set

One step reach-avoid set for input

State Space

j time step

reach-avoid set

Reach-avoid control law synthesis

- Over entire time horizon

Step 1: Obtain state measurement

Step 2: Find minimum time to reach

Step 3: Find set of possible control inputs

Step 4: Choose and apply control input

Step 5: Iterate steps 1 through 4

Outline

- STARMAC Quadrotor Platform
- Problem Set-Up
- Reach-avoid Set
- Experimental Results
- Conclusion and Future Work

Gumstix PXA270, or ADL PC104

Carbon Fiber Tubing

Low Level Control

Atmega128

Fiberglass Honeycomb

GPS

Novatel Superstar II

Sensorless Brushless DC Motors

Axi 2208/26

Electronic Speed Controllers

Castle Creations Phoenix-25

Inertial Meas. Unit

Microstrain3DM-GX1

UltrasonicRanger

Senscomp Mini-AE

Battery

Lithium Polymer

STARMAC BackgroundProblem Set-Up

- Target Set: +/- 0.2 m for position, +/- 0.2 m/s for velocity
- Unsafe Set: +/- 1 m/s for velocity
- Time Step: 0.1 seconds, 25 time steps
- Input Range: 9 possible inputs, [-10, -7.5, -5, -2.5, 0, 2.5, 5, 7.5, 10]

Reach-avoid Set - Plots

Reach-avoid at Time Step 1 for All Inputs

Conclusion

- Proposed automatic controller synthesis method for switched systems
- Nonlinear continuous dynamics, up to 3-4 state dimensions
- Differential game setting – possibly large disturbances
- Directly handles state and input constraints
- Provides explicit feedback policy that can be implemented in sampled-data system
- Possible applications:
- Target/obstacle problems
- Stabilization problems
- Safety/invariance problems

Future Work

- Mode transitions with state dependent guards
- Multiple inputs within each mode
- Approximation methods for continuous time reachable sets
- Stochastic reachability problems

Thank You

- Acknowledgements
- Patrick Bouffard
- Jeremy Gillula
- Haomiao Huang
- Tony Mercer
- Michael Vitus

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