Reachability-based Controller Design for Switched Nonlinear Systems EE

Reachability-based Controller Design for Switched Nonlinear Systems EE 291 E / ME 290 Q 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

Automated Aerial Refueling Procedures

Discrete Transitions Postcontact Detach 2 Contact High Level Objective: Visit waypoint sets Ri, i = 1, …, 6, in sequence Precontact Detach 1 Rejoin End Start

Continuous Dynamics • Relative States: • x 1, x 2 = planar coordinates of tanker in UAV reference frame • x 3 = heading of tanker relative to UAV • Controlled inputs: • u 1 = translational speed of UAV • u 2 = turn rate of UAV • Disturbance inputs: • d 1 = translational speed of Tanker • d 2 = 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× Ri in 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

Capture sets and Unsafe sets

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/Toolbox. LS/, 2007]

Maneuver Design Using Reachability Analysis • For mode q. N • 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 q. N • 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 q 1 , R 0 = X 0 , where X 0 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

Capture Set and Unsafe Set Computation Result Precontact (Mode q 2) Time Horizon

Simulation of Refueling Sequence Input bounds Collision Zone Unsafe Set For Detach 1 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 Rescaled coordinates: distance units in tens of meters Reachable set slice at relative angle 0 Unsafe set for contact maneuver with 10% velocity deviation

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 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 2 T 3 T 4 T 5 T

Switched System Model – Control and Disturbance Policies • On sampling interval [k. T, (k+1)T], define One step control policy k. T One step disturbance strategy (k+1)T k. T (k+1)T

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 Mode Avoid set 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 Mode Avoid set Reach-avoid set Mode

Problem Formulation • For any (q, x) in the reach-avoid set, automatically synthesize a feedback policy that achieves the specifications Target set Mode Avoid set Reach-avoid set 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 For sets Mode 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 For sets The set union Mode represented by level set functions is represented by

Reach-avoid Set Computation – Iteration • Iterate to compute the reach-avoid set over [0, NT] Initialization: for to end Return: • By induction, can show that

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 Length of time horizon Number of discrete modes Number of quantization levels in mode qi

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

STARMAC Quadrotor Platform High Level Control Carbon Fiber Tubing Gumstix PXA 270, or ADL PC 104 Low Level Control Fiberglass Honeycomb Atmega 128 GPS Novatel Superstar II Sensorless Brushless DC Motors Axi 2208/26 Inertial Meas. Unit Microstrain 3 DM-GX 1 Electronic Speed Controllers Ultrasonic Ranger Senscomp Mini-AE Battery Lithium Polymer Castle Creations Phoenix-25

Experiment Setup • Objectives: • Drive a quadrotor to a neighborhood of 2 D 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 Set - Plots Reach-avoid at Time Step 1 for All Inputs

Reach-avoid Set - Plots

Experimental Results

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 47 th 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 49 th 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.