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.