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 1 st output Duration of constraint violations for 2 nd output Largest constraint violation for 1 st output L1 norm of constraint violations for 2 nd 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.