Linear MPC Robert Platt Northeastern University Linear Model - - PowerPoint PPT Presentation

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Mar 26, 2024 245 likes •484 views

Linear MPC Robert Platt Northeastern University Linear Model Predictive Control Drawbacks to LQR: hard to encode constraints suppose you have a hard goal constraint? suppose you have piecewise linear state and action constraints?

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Linear MPC

Robert Platt Northeastern University

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Linear Model Predictive Control

Drawbacks to LQR: hard to encode constraints – suppose you have a hard goal constraint? – suppose you have piecewise linear state and action constraints? Answer: – solve control as a new optimization problem on every time step

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Linear Model Predictive Control

Given:

System: Cost function: where:

Calculate:

Initial state: U that minimizes J(X,U)

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Linear Model Predictive Control

Given:

System: Cost function: where:

Calculate:

Initial state: U that minimizes J(X,U)

We're going to solve this problem by expressing it explicitly as a quadratic program

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Quadratic program

Minimize: Subject to:

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Quadratic program

Minimize: Subject to: Constants are part of problem statement: x is the variable Problem: find the value of x that minimizes the objective subject to the constraints

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Quadratic program

Quadratic objective function Linear inequality constraints Linear equality constraints Minimize: Subject to:

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Quadratic program

Minimize: Subject to:

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Quadratic program

Minimize: Subject to:

Why?

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Quadratic program

Quadratic objective function

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Quadratic program

Quadratic objective function Inequality constraints

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Quadratic program

Quadratic objective function equality constraints

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QP versus Unconstrained Optimization

Minimize: Subject to: Original QP

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QP versus Unconstrained Optimization

Minimize: Unconstrained version of original QP Subject to:

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QP versus Unconstrained Optimization

Minimize: Unconstrained version of original QP How do we minimize this expression?

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QP versus Unconstrained Optimization

Minimize: Unconstrained version of original QP How do we minimize this expression?

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Linear Model Predictive Control

Minimize: Subject to:

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Linear Model Predictive Control

Minimize: Subject to:

What are the variables?

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Linear Model Predictive Control

Minimize: Subject to: What other constraints might we want add?

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Linear Model Predictive Control

Minimize: Subject to:

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Linear Model Predictive Control

Minimize: Subject to: Can't express these constraints in standard LQR

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Linear MPC Receding Horizon Control

Minimize: Subject to: Re-solve the quadratic program on each time step: – always plan another T time steps into the future