Designing model predictive controllers with prioritised constraints and objectives

1
Designing model predictive controllers with
prioritised constraints and objectives

• Eric Kerrigan
• Jan Maciejowski
• Cambridge University Engineering Department

2
Overview

• Motivation
• Prioritised, multi-objective optimisation
• Lexicographic programming
• Classes of objectives to be considered
• Costs
• Constraints
• Application to model predictive control
• Conclusions

3
Motivation

• When designing controllers, difficult to express
  all objectives as single cost
• Different types of objectives
• Costs, e.g. minimise fuel
• Constraints, e.g. safety, performance
• Objective hierarchy
• Some objectives more important than others
• Model predictive control
• optimise cost subject to constraints

4
Prioritised multi-objective optimisation problem
Given The multi-objective optimisation problem
(MOP)
where the objectives have been prioritised from
most to least important.
The lexicographic minimum is given by
5
Properties of a prioritised MOP

• A lexicographic minimiser exists
• The lexicographic minimum is unique
• If each objective function is convex, then need
  only solve a finite number of
• convex,
• constrained,
• single-objective optimisation problems (SOPs)
• If a single objective function is strictly convex
• then the minimiser is unique

6
Classes of objective functions

• Quadratic cost function
• Largest constraint violation
• Weighted sum of constraint violations
• Largest element in index set of violated
  constraints

7
Model predictive control (MPC)

• Linear, discrete-time system
• Constraints on inputs and outputs
• Set of input- and state-dependent objectives
• Minimise costs (quadratic, linear)
• Satisfy constraints (quadratic, linear)
• Some objectives more important than others
• For the current measured state, find an optimal,
  finite sequence of inputs

8
Example

• Two outputs, linear constraints performance,
  safety
• Minimise, in order of importance
• Duration of constraint violations for 1st output
• Duration of constraint violations for 2nd output
• Largest constraint violation for 1st output
• L1 norm of constraint violations for 2nd output
• Quadratic norm of deviations of outputs from
  reference
• Quadratic norm of deviations of inputs from 0
• Each objective translates into an objective
  function that has been considered here
• Need solve a sequence of convex, constrained SOPs
• LPs,1 LCQP,1 QCQP (SDP)

9
Conclusions

• Often objectives in an MOP can be prioritised
• A minimiser exists and the minimum is unique
• If all the objectives are convex, then solve a
  finite number of
• convex,
• constrained, single-objective problems
• A linear model, linear and/or quadratic
  constraints and costs, then one can
• set up a flexible, prioritised MOP that can be
• solved efficiently using convex programming
• LP,QP,SDP, etc.