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Reachability-based Controller Design for Switched Nonlinear Systems

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### Reachability-based ControllerDesign for Switched NonlinearSystems

Outline

EE 291E / ME 290Q

Jerry Ding

4/18/2012

Hierarchical Control Designs

- To manage complexity, design of modern control systems commonly done in hierarchical fashion
- e.g. aircraft, automobiles, industrial machinery
- Low level control tend to use continuous abstractions and design methods
- ODE model
- Stability, trajectory tracking
- Linear/Nonlinear control methods
- High level control tend to use discrete abstractions and design methods
- Finite state automata, discrete event systems
- Logic specifications of qualitative behaviors: e.g. LTL
- Model checking, supervisory control

Challenges of Interfacing Layers of Control

- Problem becomes more difficult at interface:
- Closed loop behavior results from composition of discrete and continuous designs
- Discrete behaviors may not be implemented exactly by continuous controllers
- Continuous designs may be unaware of high level specifications
- In safety-critical control applications, specifications often involves stringent requirements on closed-loop behavior
- Current design approaches involve a mixture of heuristics and extensive verification and validation

Hybrid Systems Approach

- Capture closed-loop system behavior through hybrid system abstraction

Hybrid Systems Approach

- Formulate design methods within the framework of hybrid system theory

- Challenges:
- Nonlinear dynamics, possibly with disturbances
- Controlled switching: switching times, switching sequence, switching policy
- Autonomous switching: discontinuous vector fields, state resets

Reachability-Based Design for Switched Systems

- Consider subclass of hybrid systems with:
- Controlled switches, no state resets
- Fixed mode sequence
- Variable mode sequence
- Nonlinear continuous dynamics, subject to bounded disturbances
- Design controllers to satisfy reachability specifications
- Reach-avoid problem: Given target set R, avoid set A, design a controller to reach R while avoiding A
- Methods based upon game theoretic framework for general hybrid controller design
- [Lygeros, et al., Automatica, 1999]
- [Tomlin, et al., Proceedings of the IEEE, 2000]

Outline

- Switched Systems with Fixed Mode Sequences:
- Design of Safe Maneuver Sequence for Automated Aerial Refueling (AAR)
- Switched Systems with Variable Mode Sequences:
- Sampled-data switched systems
- Controller synthesis algorithm for reach-avoid problem
- Application example: STARMAC quadrotor experiments

Discrete Transitions

Detach

1

Rejoin

High Level Objective:

Visit waypoint sets Ri,

i = 1,…,6, in sequence

Precontact

Postcontact

Detach

2

Start

Contact

End

Continuous Dynamics

Relative States:

x1, x2 = planar coordinates of tanker in UAV reference frame

x3 = heading of tanker relative to UAV

Controlled inputs:

u1 = translational speed of UAV

u2 = turn rate of UAV

Disturbance inputs:

d1 = translational speed of Tanker

d2 = turn rate of Tanker

Low Level Objective:

Avoid protected zone A around tanker aircraft

Maneuver Sequence Design Problem

- Given waypoint sets Ri, protected zone A, design continuous control laws Ki(x) and switching policies Fi(x) such that
- 1) The hybrid state trajectory (q, x) passes through the waypoint sets qi× Riin sequence
- 2) The hybrid state trajectory (q, x) avoids the protected zones qi× A at all times
- Design approach:
- Select switching policy as follows: in maneuver qi, switch to next maneuver if waypoint Ri is reached
- Use reachable sets as design tool for ensuring
- safety and target attainability objectives for each maneuver
- compatibility conditions for switching between maneuvers

Computation of Reachable Sets

- Use terminal condition to encode avoid set

- Unsafe set computation (Mitchell, et al. 2005):
- Let be the viscosity solution of

Then

- Capture set computation similar
- Numerical toolbox for MATLAB is available to approximate solution
- [Ian Mitchell, http://www.cs.ubc.ca/~mitchell/ToolboxLS/, 2007]

Maneuver Design Using Reachability Analysis

- For mode qN
- 1) Design a control law to drive RN -1 to RN
- 2) Compute capture set to first time instant N such that

Maneuver Design Using Reachability Analysis

- For mode qN
- 3) Compute unsafe set, and verify safety condition
- Modify control law design as necessary

Maneuver Design Using Reachability Analysis

- For modes qk, k < N
- 3) Iterate procedures 1-3 recursively
- For q1 , R0 = X0 , where X0 is the initial condition set

Properties of Control Law

- Continuous control laws designed in this manner satisfy a reach-avoid specification for each maneuver:
- Reach waypoint set Ri at some time, while avoiding protected zone A at all times
- Furthermore, they satisfy a compatibility condition between maneuvers
- This ensures that whenever a discrete switch take place, the specifications of next maneuver is feasible
- Execution time of refueling sequence is upper bounded by

Specifications for Aerial Refueling Procedure

- Target Sets of the form

- Avoid sets of the form

- Control laws of the form

Simulation of Refueling Sequence

Input bounds

Unsafe Set

For Detach 1

Collision Zone

Target Set Radius

Target Set

Collision Set Radius

Capture Set

For Detach 1

Accounting for Disturbances

- Capture sets and unsafe sets can be modified to account for fluctuations in tanker velocity using disturbance set

Unsafe set for contact maneuver without disturbances

Collision Zone

In UAV Coordinates

Reachable set slice at relative angle 0

Unsafe set for contact maneuver with 10% velocity deviation

Rescaled coordinates: distance units in tens of meters

Outline

- Switched Systems with Fixed Mode Sequences:
- Design of Safe Maneuver Sequence for Automated Aerial Refueling (AAR)
- Switched Systems with Variable Mode Sequences:
- Sampled-data switched systems
- Controller synthesis algorithm for reach-avoid problem
- Application example: STARMAC quadrotor experiments

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 for finite representation of control policy

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

Outline

- Switched Systems with Fixed Mode Sequences:
- Design of Safe Maneuver Sequence for Automated Aerial Refueling (AAR)
- Switched Systems with Variable Mode Sequences:
- Sampled-data switched systems
- Controller synthesis algorithm for reach-avoid problem
- Application example: STARMAC quadrotor experiments

Problem Formulation

- Given:
- Switched system dynamics; for simplicity, assume that
- Target set R
- Avoid set A

Target set

Avoid set

Mode

Mode

Problem Formulation

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

Target set

Avoid 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

Avoid set

Reach-avoid set

Mode

Mode

One Step Capture and Unsafe sets

- For each , compute one step capture and unsafe sets assuming
- over one sampling interval

- One step capture set

- One step unsafe set

where is solution of on

Reach-avoid Set Computation – Step 1

- For each , compute one step reach-avoid set using set difference

Mode

Mode

For sets represented by level set functions

The set difference is represented by

Reach-avoid Set Computation – Step 2

- Compute feasible set for one step reach-avoid problem, by taking union over

Mode

Mode

For sets represented by level set functions

The set union is represented by

Reach-avoid Set Computation – Iteration

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

Initialization:

for

to

end

Return:

Reach-avoid control law synthesis

- At time k < N

Step 1: Obtain state measurement

Step 2: Find minimum time to reach

Reach-avoid control law synthesis

- At time k < N

Step 3: Find a control input such that

Step 4: Apply input and iterate steps 1-3

Explicit Form of Control Laws

- Explicit control laws given by

for

where

- Number of reachable sets required is given by

Number of quantization levels in mode qi

Length of time horizon

Number of discrete modes

- Switched Systems with Fixed Mode Sequences:
- Design of Safe Maneuver Sequence for Automated Aerial Refueling (AAR)
- Switched Systems with Variable Mode Sequences:
- Sampled-data switched systems
- Controller synthesis algorithm for reach-avoid problem
- Application example: STARMAC quadrotor experiments

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 Quadrotor PlatformExperiment Setup

- Objectives:
- Drive a quadrotor to a neighborhood of 2D location in finite time, while satisfying velocity bounds

- Disturbances: model uncertainty, actuator noise

- System model

Reach-avoid Problem Set-Up

- Target Set: +/- 0.2 m for position, +/- 0.2 m/s for velocity
- Avoid Set: +/- 1 m/s for velocity
- Time Step: 0.1 seconds, 25 time steps
- Pitch and roll commands:
- Disturbance bounds:

Reach-avoid Set - Plots

Reach-avoid at Time Step 1 for All Inputs

Experimental Results

- Moving car experiment

References

- John Lygeros, Claire Tomlin, and S. Shankar Sastry. Controllers for reachability specifications for hybrid systems. Automatica, 35(3):349 – 370, 1999.
- Claire J. Tomlin, John Lygeros, and S. Shankar Sastry. A game theoretic approach to controller design for hybrid systems. Proceedings of the IEEE, 88(7):949–970, July 2000.
- Jerry Ding, Jonathan Sprinkle, S. Shankar Sastry, and Claire J. Tomlin. Reachability calculations for automated aerial refueling. In 47th IEEE Conference on Decision and Control, pages 3706–3712, Dec. 2008.
- Jerry Ding, Jonathan Sprinkle, Claire Tomlin, S. Shankar Sastry, and James L. Paunicka. Reachability calculations for vehicle safety during manned/unmanned vehicle interaction. AIAA Journal of Guidance, Control, and Dynamics, 35(1):138–152, 2012.

References

- Jerry Ding and Claire J. Tomlin. Robust reach-avoid controller synthesis for switched nonlinear systems. In 49th IEEE Conference on Decision and Control (CDC), pages 6481–6486, Dec. 2010.
- Jerry Ding, Eugene Li, Haomiao Huang, and Claire J. Tomlin. Reachability-based synthesis of feedback policies for motion planning under bounded disturbances. In IEEE International Conference on Robotics and Automation (ICRA), pages 2160 –2165, May 2011.

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