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www.mpc.berkeley.edu Borrelli (UC Berkeley) Iterative Learning MPC - - PowerPoint PPT Presentation

Oct 06, 2022 135 likes •1.19k views

Iterative Learning Model Predictive Control Francesco Borrelli Email: fborrelli@berkeley.edu University of California Berkeley, USA www.mpc.berkeley.edu Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC Slide 1 Acknowledgements

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 1

Iterative Learning Model Predictive Control

Francesco Borrelli Email: fborrelli@berkeley.edu University of California Berkeley, USA

www.mpc.berkeley.edu

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 2

Acknowledgements

Learning MPC: Ugo Rosolia

Adaptive MPC: Monimoy Bujarbaruah, George Xiaojing Zhang Autonomous Drift: Edo Jelavick, George Xiaojing Zhang Analog Optimization: Sergey Vicky Autonomous Drift: Edo Jelavick, Yuri Glauthier Analog Optimization: Sergey Vicky Connected Cars: Jacopo Guanetti, Jongsang Suh, Roya Firoozi, Yeojun Kim , Eric Choi BARC: Jon Gonzales, Tony Zeng, Charlott Vallon

Research Sponsors: Hyundai Corporation Ford Research Labs, Siemens, Mobis, Komatzu National Science Foundation Office of Naval Research

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 3

Iterative Learning Model Predictive Control

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 4

Iterative Learning Model Predictive Control

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 5

Now Available on Amazon

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 6

Constrained Infinite-Time Optimal Control “Solved” as..

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 7

Repeated Solution of Constrained Finite Time Optimal Control

Predictive Controller:

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 8

Repeated Solution of Constrained Finite Time Optimal Control

Approximates the `tail' of the cost Approximates the `tail' of the constraints N constrained by computation and forecast uncertainty Robust and stochastic versions subject of current research

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 9

Repeated Solution of Constrained Finite Time Optimal Control

Predictive Controller: Predictive Control: Theory & Computation

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 10

Repeated Solution of Constrained Finite Time Optimal Control

Predictive Controller: Predictive Control Classical Theory

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 11

Terminal cost: Control Lyapunov function Terminal constraint set: Control Invariant set

Predictive Control Theory: Sufficient conditions to guarantee

Convergence to the desired equilibrium point/region Constraint satisfaction at all times

Control Invariant Set Control Lyapunov Function

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 12

Repeated Solution of Constrained Finite Time Optimal Control

Predictive Controller: Predictive Control Computation

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 13

Offline π(∙) and Online π(x) Computation

Option 1 (Offline Based): “Complex” Offline, “Simple” Online π0(∙) often piecewise constant or affine disturbance feedback Dynamic Programing is one choice Sampling model reduction/aggregation required for n>5 Option 2 (Online Based): “Simple” Offline, “Complex” Online Compute on-line π 0(x(t)) with a “sophisticated” algorithm Interior point method solver is one choice Convexification required for real-time embedded control

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 14

Online Based Excellent, (non-) convex open-source solvers Tailored solvers for embedded linear and nonlinear MPC Offline Based For linear and piecewise linear systems: explicit MPC Mixing pre-computation and online-optimization Suboptimal MPC Fast Online Implementation on embedded FPGA, GPU Analog MPC: microsecond sampling time

Major effort over the past 20 years for enlarging MPC application domain

A very biased story

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 15

Iterative Learning Model Predictive Control

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 16

Three Forms of Learning 1 - Skill acquisition

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 17

Three Forms of Learning 2 - Performance Improvement

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 18

Three Forms of Learning 3 - Computation Load Reduction

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 19

Three Forms of Learning. Practice in order to:

Learning from demonstration Transfer learning Learning from simulations Iterative Learning Computational load reduction of control policy Acquire a Skill Improve Performance Reduce Computational Load

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 20

Three Forms of Learning. Practice in order to:

Learning from demonstration Transfer learning Learning from simulations Iterative Learning Computational reduction

  • f control policy

Acquire a Skill Improve Performance Reduce Computational Load

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 21

Learning MPC Applied to Robo-Cars

(instead of robo-soccer players..)

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 22

Autonomous Cars @MPC Lab

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 23

Autonomous Vehicles- Motion Control Through:

Acceleration, Braking, Steering Also:

4 braking torques Gear Ratio Engine torque + front and rear distribution 4 dampers for active suspensions

Hyundai Genesis G70

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 24

Static Nonlinearities: Tires Inequality Constraints: Safety region Uncertain Tire Model, Road Friction, Obstacles Nonlinear Dynamical System

Useful Model Abstraction

and

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 25

Tires and Road

Simplified Nonlinear Model

Slip Angle Lateral Force Dry Asphalt Snow Ice

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 28

Berkeley Autonomous 1/10 Race Car Project www.barc-project.com

Complete Open Source Ubuntu, RoS, OpenCV, Julia, IPOPT Camera, IMU, Ultrasounds, LIDAR Cloud-Based

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 30

Three Forms of Learning 1 - Skill acquisition

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 31

Three Forms of Learning 2 - Performance Improvement

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 32

Three Forms of Learning 3 - Computation Load Reduction

Average CPU Load at each iteration Lap Time at each iteration

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 34

Three Forms of Learning

How we do this?

Model Predictive Control A Simple Idea (which exploits the iterative nature of the tasks) Important Design Steps

Acquire a Skill Improve Performance Reduce Computational Load

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 35

Iterative Learning Model Predictive Control

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 36

One task execution referred to as “iteration” or “episode” Same initial and terminal state at each iteration Notation:

Iterative Tasks - Problem Setup

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 37

Iterative Tasks - Problem Setup

One task execution referred to as “iteration” or “episode” Same initial and terminal state at each iteration Notation:

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 38

Iterative Tasks - Problem Setup

One task execution referred to as “iteration” or “episode” Same initial and terminal state at each iteration Notation:

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 39

Goal

Safety guarantees: Constraint satisfaction at iteration j → satisfaction at iteration j+1 Performance improvement guarantees: Closed loop cost at iteration j+1 ≤cost at iteration j

Learned from data

Iterative Learning MPC Incorporating data in advanced model based controller

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 40

Learned from data

Learning MPC Incorporating data in advance model based controller

Simplification (general case later)

Known/nominal model Infinite Horizon Task Uncertainty and model adaptation later (and at this conference)

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 43

Learning Model Predictive Control (LMPC)

  • Recursive feasibility
  • Iterative feasibility
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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 44

Iteration 0

Assume at iteration 0 the closed-loop trajectory is feasible

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 45

Iteration 0

Assume at iteration 0 the closed-loop trajectory is feasible

Fact

is a control invariant

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 46

Iteration 1, Step 0

Use SS0 as terminal set at Iteration 1

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 47

Iteration 1, Step 0

Use SS0 as terminal set at Iteration 1

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 48

Iteration 1, Step 1

Use SS0 as terminal set at Iteration 1

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 49

Iteration 1, Step 1

Use SS0 as terminal set at Iteration 1

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 50

Iteration 1, Step 2

Use SS0 as terminal set at Iteration 1

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 51

Iteration 1, Step 2

Use SS0 as terminal set at Iteration 1

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 52

Iteration 1, Step 3

Use SS0 as terminal set at Iteration 1

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 53

Iteration 1, Step 4

Use SS0 as terminal set at Iteration 1

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 54

Iteration 2 Safe Set

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 55

Constructing the terminal set

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 56

Terminal Set : Convex all of Sample Safe Set

for Constrained Linear Dynamical Systems is a Control Invariant Set

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 57

Learning Model Predictive Control (LMPC)

  • Convergence
  • Performance

improvement

  • Local optimality
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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 58

Terminal Cost at Iteration 0

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 59

Terminal Cost at Iteration 0 A control Lyapunov “function”

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 60

Terminal Cost at the j-th iteration

≡

Define Compute terminal cost as

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 61

Terminal Cost: Barycentric Approximation of Q()

Control Lyapunov Function

(for Constrained Linear Dynamical Systems)

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 62

ILMPC Summary

MPC strategy:

Optimize over inputs and lambdas For constrained linear systems

Safety guarantees:

  • Constraint satisfaction at iteration j=> satisfaction at iteration j+1

Performance improvement guarantees:

  • Closed loop cost at iteration j >= cost at iteration j+1

Convergence to global optimal solution Constraint qualification conditions required for cost decrease

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 63

Performance Improvement Proof

Conjecture Notation Closed-loop state and input trajectory at iteration j

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 64

Step 1:

Performance Improvement Proof

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 65

Performance Improvement Proof

Step 1:

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 66

Performance Improvement Proof

Step 1:

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 67

Performance Improvement Proof

Step 1:

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 68

Performance Improvement Proof

Step 1: Step 2:

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 69

Performance Improvement Proof

Step 1: Step 2:

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 70

Performance Improvement Proof

Step 1: Step 2:

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 71

Performance Improvement Proof

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 72

Iterative Learning MPC

Optimize over inputs and lambdas Simple proofs For constrained linear systems

Safety and Performance improvement guarantees Convergence to global optimal solution (for linear Constraint qualification conditions required for cost decrease

If full column rank, improvement cannot be

  • btained
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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 73 Control objective System dynamics System constraints Starting Position

Constrained LQR Example

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 74 Control objective System dynamics System constraints Terminal Constraint Initial Condition

Iterative LMPC with horizon N=2

Will not work! Will work if one sets N=3

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 75

Comparison with R.L.??

RL term too broad Two good references: Bertsekas paper connecting MPC and ADP* Lewis and Vrabile survey on CSM** Recht survey (section 6): https://arxiv.org/abs/1806.09460 ILMPC highlights Continuous state formulation Constraints satisfaction and Sampled Safe Sets Q-function constructed (learned) locally based on cost/model driven exploration and past trails Q-function at stored state is “exact” and lowerbounds property at intermediate points (for convex problems)

*Dynamic Programming and Suboptimal Control: A Survey from ADP to MPC **Reinforcement Learning and Adaptive Dynamic Programming for Feedback Control

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 76

About Model Learning in Racing

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 77

Autonomous Racing Control Problem

Start & end position System dynamics System constraints Obstacle avoidance Control objective

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 78

Learning Model Predictive Control (LMPC)

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 80

Learning Process

The lap time decreases until the LMPC converges to a set of trajectories

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 82

Learning Model Predictive Control (LMPC)

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 83

Nonlinear Dynamical System

Useful Vehicle Model Abstraction

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 84

Nonlinear Dynamical System

Useful Vehicle Model Abstraction

Kinematic Equations

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 85

Nonlinear Dynamical System

Useful Vehicle Model Abstraction

Kinematic Equations

Identifying the Dynamical System

Linearization around predicted trajectory

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 86

Nonlinear Dynamical System

Useful Vehicle Model Abstraction

Kinematic Equations Dynamic Equations

Identifying the Dynamical System

Local Linear Regression Linearization around predicted trajectory

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 87

Useful Vehicle Model Abstraction

Identifying the Dynamical System Important Design Steps

1.

Compute trajectory to linearize around uses previous optimal inputs and inputs in the safe set

2.

Enforce model-based sparsity in local linear regression

Local Linear Regression Linearization around predicted trajectory

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 88

Nonlinear Dynamical System

Useful Vehicle Model Abstraction

The velocity update is not affected by Position and Acceleration command

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 89

Useful Vehicle Model Abstraction

Identifying the Dynamical System Important Design Steps

1.

Compute trajectory to linearize aroundpusing previous optimal inputs and inputs in the safe set

2.

Enforce model-based sparsity in local linear regression

3.

Use data close to current state trajectory for parameter ID

4.

Use kernel K() to weight differently data as a function of distance to linearized trajectory

Local Linear Regression Linearization around predicted trajectory

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 90

Accelerations

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 91

Results

Gain from steering to lateral velocity

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 95

About Model Learning Ball in Cup

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 96

Ball in a Cup System with MuJoCo

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 97

Ball in a Cup Control Problem

Start & end position System dynamics System constraints Obstacle avoidance Control objective

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 98

Learning Model Predictive Control (LMPC)

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 99

Useful Mujoco Model Abstraction

Identifying the Dynamical System

Local Linear Regression Linearization around predicted trajectory

Important Design Steps

1.

Compute trajectory to linearize aroundpusing previous optimal inputs and inputs in the safe set

2.

Enforce model-based sparsity in local linear regression

3.

Use data close to current state trajectory for parameter ID

4.

Use kernel K() to weight differently data as a function of distance to linearized trajectory

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 100

Ball in a Cup System

At iteration 0 find a sequence by sampling parametrized inputs profiles (takes 5mins) Use ILMPC: At iteration 1, time reduced of 10%, cup height movement reduced of 35%

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 101

Back to our main chart..

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 102

Three Forms of Learning

Skill acquisition Performance improvement

How we do this? Model Predictive Control + A Simple Idea + Good Practices

Reduce load for Routine Execution

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 103

Offline π(∙) and Online π(x) Computation

Option 1 (Offline Based): “Complex” Offline, “Simple” Online π(∙) often Piecewise Constant (except special classes) Dynamic Programing is one choice Basic rule: n>5 impossible Option 2 (Online Based): “Simple” Offline, “Complex” Online Compute on-line π(x) with a “sophisticated” algorithm Interior point method solver is one choice Basic Rule: avoid use `home-made’ solvers

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 104

One Simple Way: Data-Based Policy for π(∙)

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 105

One Simple Way: Data-Based Policy for π(∙) Historical data

  • f converged iterations
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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 107

Three Forms of Learning 3 - Computation Load Reduction

Average CPU Load at each iteration Lap Time at each iteration

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 108

Experimental Results

Factor of 10

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 109

Data Based Policy: Alternatives

Nearest Neighbor Train ReLU Neural Network Local Explicit MPC

All Continuous Piecewise Affine Policies

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 110

Learned from data

Learning MPC Incorporating data in advance model based controller

In Practice Noise and model uncertainty: Robust case

What about noise and model uncertainty?

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 111

ILMPC – Robust and Adaptive design

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 112

At Iteration 0

Linear System Terminal Goal Set Successful Iteration

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 113

At Iteration 1

CVX hull is not a robust invariant!

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 114

ILMPC – Robust and Adaptive design

Robust invariants “Robustify” Q-function (and dualize for computational efficiency)

Chance constraints

See my group papers at this conference if interested..

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 115

For Iterative tasks I discussed By using Iterative learning MPC, i.e.

Model Predictive Control A Simple Idea

(which exploits the iterative nature of the tasks)

A Few Important Design Steps

How to obtain performance improvement and reduced computational load while satisfying constraints

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Borrelli (UC Berkeley) Iterative Learning MPC 2018 CDC– Slide 116

The End