### Project UT-C: Model Predictive Control

**Manager:** Jan Maciejowski

**Investigators:** David Mayne and Jan Maciejowski

**Research Staff:** Marco Gallieri (Research Student, Cambridge appointed 01.10.2010)

**Collaborators:** Dr Andrea Lecchini Visintini (University of Leicester), Prof. K.V. Ling (NTU Singapore), Prof. John Lygeros (ETH Zurich) and Dr A.G. Richards (University of Bristol)

**Sponsors:** Honeywell, Wind Technologies

**Start date:** 01/10/2010

**Linked Projects:** PS-B, PS-E, EET-A and EET-B

**Summary.** Model Predictive Control (MPC) is an optimisation-based controller design methodology, which has found wide-spread application in the process industries, because it can take account of plant nonlinearities and of constraints on variables in a simple, direct manner. This project aims to extend the MPC approach in several directions, matched to the requirements of the applied projects PS-B, EET-A and EET-B in the areas of wind energy extraction, engine management and air traffic control. The new control algorithms will cover a variety of system descriptions, and will also provide controllers that conform to a required decentralised structure (‘Decentralised MPC’). For both classes of algorithms, a theoretical framework will be provided, giving precise conditions under which they will achieve their proposed control objectives. Algorithm performance will be assessed in case studies centred on wind energy extraction and air vehicle collision avoidance.

**Current Status.**

At Imperial College:

Model predictive control is the most widely applied, by far, of modern control techniques in process industry, giving rise to a multitude of different approaches. This is a result of several factors including its ability to deal with constraints. The most relevant approaches are discussed and compared in [RM09, M13, M11] pointing out their weaknesses and strengths and paving the road to the current research results and future challenges. Opportunities and challenges open in the application of advanced control to telecommunication systems, electricity networks, mining, minerals processing, agriculture, power electronics and drive have been investigated in [GMCCMQ10, GMCCMQ10a].

The research reported in [MKWF11] is the first successful extension of tube-based model predictive control to the control of uncertain nonlinear systems. Examples studied to date support its potential for this class of problems. Further research aimed at quantifying the advantages of this method of control and developing the preliminary results in [MKF11]. In particular, the proposed scheme presents an online complexity comparable to that encountered in conventional MPC and is “generic”, in the sense that do not depend heavily of the special features of the plant to be controlled, and therefore have the potential for widespread application.

The problem of obtaining robust model predictive control of nonlinear systems with un-modelled dynamics has received relatively little attention in the literature. In [FM11, FM13] we show how to convert the problem of obtaining robustness against unstructured uncertainty into the easier problem of achieving robustness against a bounded disturbance while satisfying an additional output constraint. The bound on the disturbance and the output constraint depend on the magnitude of the uncertainty. As a result of this conversion, the powerful tube-based model predictive control may be employed, including in its formulation mixed constraints and generic disturbances as well.

The tube model predictive control of uncertain non-linear systems proposed in [MKWF11, FM11, FM13] has been successfully applied to the control of a high dimensional solar collector plant [FM11, FM13].

The research previously described has originated the development of an open-source (BSD) Matlab toolbox with code allowing users to define and solve optimal control problems with general path and boundary constraints and free or fixed final time [WPK10]. The toolbox includes the nonlinear optimisation code IPOPT and the SUNDIALS ODE solver CVODES for sensitivity analysis. The toolbox is currently used in several universities and it has been employed in many applications of practical interest.

An important control problem is tracking of a reference that may vary with time. It is, however, usual in the literature to consider the case when the reference tends to be a constant because this simplifies analysis. However in electromechanical, power electronics and telecommunication problems, it is no longer true that the reference signal can be considered constant, or even piecewise constant.

In [GCMSS11] the question of reference tracking in MPC is re-examined and a novel strategy is formulated aimed at problems where the reference is not constant and the system is affected by unmodelled dynamics and unmeasured disturbances. The strategy combines both preview and feed-forward of the reference signal to seek improvements of the nominal MPC tracking performance.

In [FM12, FM13a, F14] Model Predictive Control strategies are proposed when the reference is affected by a randomly varying component. These kind of problems arise for instance in the control of wind power systems. The performance of the proposed algorithms are analysed and the set to which the tracking error converges is characterised.

In [F14] the convergence property of the strategy proposed in [FM12] is discussed under more general assumptions while a simple and computationally efficient model predictive control scheme for linear systems is introduced in [FM13a].

When the reference signal is periodic and generated by a finite dimensional exosystem model predictive control may be employed to solve the global constrained tracking problem [FM13b] and thereby avoid the difficult task of determining the control invariant manifold and its associated control law.

In ongoing research, in collaboration with Professors J. B. Rawlings (University of Wisconsin, Madison) and Professor G. Pannocchia, a fast method for online computation of model predictive control has been developed. To obtain robustness and other advantages, the optimal control problem solved online has an infinite horizon; methods for solving this problem efficiently are proposed in [PRMM09, PRMM10, PMR13, PRMM13].

The methods in [PRMM09, PRMM10] use a non-uniform discretisation and a piece-wise linear parameterisation of the input. The discretisation is refined in a novel manner to improve accuracy and efficiency.

Novel first and second order lower bounds on the optimal cost are employed enabling the computation to be terminated when a pre-assigned accuracy is achieved [PMR13, PRMM13]. Efficiency is achieved by employing exact determination of state and control trajectories and of the cost using pre-computed quantities. The solution methods are independent of sample time and actuator hardware.

In collaborative research with Professor G. C. Goodwin, University of Newcastle, Australia the networked control problem when the communication channel is bit-rate limited is studied [GMCCMQ10a]. The main issue arising in networked control is related to the choice on how to implement a controller over a channel that supports a certain fixed bit rate. A restricted architecture in which linear filters are used for the encoder/decoder is considered and a quantizer with linear feedback is deployed. Subject to these restrictions, a procedure for designing the controller and associated filters is presented in [GCFM14, CGFM14]. These filters are then deployed to choose the best number of bits per sample. It is shown, subject to the above restrictions, that it is generally best to use one bit per sample, in which case, the sampling frequency is equal to the bit rate.

Distributed algorithms for Cooperative Economic Model Predictive Control of Linear and Nonlinear Systems are being investigated. The aim is to design distributed algorithms to optimise economic performance and, at the same time, ensure fulfillment of operational constraints when the computational burden is shared by multiple controllers. In [LA12, LA11a] control algorithms for this type of task are derived assuming all-to-all communication and their convergence properties, recursive feasibility and guaranteed performance bounds are investigated. This line of investigation has attracted considerable attention from the community and invited sessions on Economic Model Predictive Control as well as invited tutorial sessions on the topic were organised in recent conferences, such as MTNS and IEEE CDC.

Our latest efforts have focused on nonlinear systems and non-convex objective cost functionals [LA13]. Lack of convexity prevents the possibility of a parallel update of the current solution computed by individual agents and a hierarchical approach based on a bipartition of the agents set has been proposed to overcome these difficulties and still retain a certain degree of parallelism. Simulations studies have been carried out on models of chemical reactors and a journal paper is in preparation illustrating these recent findings.

At Cambridge:

The notion of *rubber horizon* was invented for use with model predictive control, in finite-duration problems in which some kind of completion is required (*e.g.* of a manoeuvre), but the duration is too long, and the sampling interval is too high, to be practical with a single conventional prediction horizon. This is replaced by two horizons, which are joined by a ‘rubber’ horizon of unspecified length.

The concept of *multiplexed model predictive control (MMPC)* has been further elaborated. Both the nominal and the robust case have been treated. The idea here is to obtain simpler (hence faster) MPC problems by optimising for only one input at a time (in a multi-input system). This is particularly useful for certain multi-agent problems [SM11a]. A paper on MMPC was published by *Automatica* [LMRW10]. The benefits of MMPC have been demonstrated by application (using a high-fidelity nonlinear simulation) to the control of a large civil airliner in both a nominal configuration and after sustaining heavy damage [HML12]. There are nearly 30 independent control actuators, which are grouped into seven clusters, each containing four controls. The use of MMPC results in a computational speed-up of about 50 (very close to the theoretical prediction) with no noticeable loss of performance, when compared with conventional MPC. A further development of MMPC has been proposed, called ‘channel-hopping MPC’ in which the individual control actuators are updated in a sequence, which depends on the benefit to be obtained, rather than in a fixed periodic sequence [LMGS11].

Work has begun on the use of sum-of-norms regularisation in MPC. This results in ‘sparse’ controls with attendant advantages such as reduced fuel use, reduced actuator wear and tear, preferential use of one actuator in a multi-input system, etc. [GM12]. The potential of this approach for spatial and temporal actuator sparseness has been demonstrated on the course-keeping and roll-damping of a vessel with rudder and stabiliser fins. The algorithm performs a thresholding on the actuator signals, which results in a beneficial reduction of the rudder activity. Stability results of different kinds have been established for non-linear systems governed by differentiable vector fields. A conventional dual-mode approach yields a stability guarantee easily, but results in the sparseness benefits of Lasso-MPC being lost in the terminal set.

Lasso MPC with quadratic terminal cost (LASSO MPC version 1) provides ultimate boundedness of the system trajectory, under mild conditions. In this case each actuator has a specific dead-zone. A second approach has been addressed, which considers a completion constraint and contractive sets. This formulation is indeed stabilising, but it does not, in general, allow for temporal sparsity. Spatial sparsity is, on the other hand, possible, for systems that can be stabilised with a smaller actuator set. The approach is able to select a preferred actuator set, to be used all of the time, while maintaining a ‘dead-zone’ for the remaining actuators (to act as a backup, for example). The performance of Lasso-MPC with quadratic terminal cost has been compared with that of quadratic, L1 and L-infinity MPC. While the last two can give very poor performance, Lasso MPC tends to give the good closed-loop behaviour of quadratic MPC, with the benefit of sparse input signals.

Another approach, (LASSO MPC VERSION 2) based more closely on a standard MPC paradigm, has also been investigated [GM13a]. A terminal cost is used to obtain guaranteed stability. A terminal constraint set and a candidate terminal controller are also used, providing an extra degree of freedom for the designer. A technique is proposed for computation of the terminal controller, so that an approximation of the feasible region is maximised. Spatial sparsity is retained, as above.

In order to simplify the computation of the terminal ingredients for closed-loop asymptotic stability, in [GM13] the terminal set for LASSO MPC version 2 is a lambda-contractive set, and the terminal cost is a scaled version of its Minkowski function.

The geometry of the LASSO and elastic-net cost functions have been studied, showing that lasso-MPC has a smooth and unique solution, as for quadratic MPC, if a positive definite quadratic input penalty is included in the cost. At the same time, the solution of lasso-MPC can be sparse as for MPC based on Lp-norm costs. This is obtained for lasso-MPC without the drawbacks of LP-MPCs, such as actuator chattering.

High fidelity simulations have been performed, for the problem of roll reduction of an ocean vessel by means of rudder and stabiliser fins. The problem requires the reduction of the high frequency wave induced roll motion, preferably by means of the fins, as well as the regulation of the low frequency yaw motion by means of the rudder. In particular, lasso-MPC version 1 has been shown to outperform the more common LQ, L1 and L∞-MPC in terms of the reduction of roll variance and low frequency yaw as well as for the minimisation of the rudder activity.

A tuning procedure has been developed, for LTI system, that allows the *a priori* selection of a group of preferred actuators. This will be reported in Gallieri’s forthcoming PhD thesis . The remaining ones will be idle in a particular neighbourhood of the state-space origin. This region has been characterised explicitly, and can be a design parameter. Limitations on the attainable region depend on the horizon length and on the constraints. The proposed procedure is based on exact penalty functions and requires the solution of a multi-parametric QP.

Lasso-MPC for prioritised actuators has been demonstrated to be effective for the minimisation of spoiler use for the roll control of the linearised lateral dynamics of a Boeing 747, at different flight points, as well as for the control of an abstracted distribution network with prioritised links. This will be reported in forthcoming PhD thesis of Gallieri.

Lasso-MPC has also been applied to a problem of spacecraft rendezvous, where inputs are subject to a non-convex minimum thrust constraint [HGM13]. LASSO is used to approximate the non-convex constraint. The approach can reduce the amount of input that falls below the minimum thrust threshold, and it provides better results than quadratic and L-1 MPC.

A robust LASSO-MPC for LTI tracking of piecewise-constant references has been formulated, which makes use of soft-constraints [GM13]. Sufficient conditions have been given to obtain a bound on the ∞-norm of the allowable additive disturbances, for which the MPC is recursively feasible. The feasible region is a robustly positively invariant set, and the closed-loop system is Input-to-State Stable. The proposed conditions are based on soft constraints, which can be inevitable in the presence of uncertainty, and on the introduction of a novel terminal set for tracking. This new set is formulated so that its projection on the state-error space is a λ-contractive set. Conditions are also given for the existence and finite time (offline) computation of the required ingredients. The proposed controller is capable of providing a much larger region of attraction as well as to track a much larger set of references than is achieved by a standard robust MPC based on constraint tightening.

**Publications**

**[F14]**
P. Falugi,
*Model predictive control for tracking randomly varying references*,
International Journal of Control ,
submitted 2014

**[F14a]**
P. Falugi,
*Model predictive control: a passive scheme*,
19th IFAC World Congress,
August 2014

**[CGFM14]**
Mauricio G. Cea and G. C. Goodwin and Arie Feuer and David Q. Mayne,
*On the Control Rate versus Quantizer-Resolution Trade Off in Networked Control*,
19th IFAC world congress, August 2014,
August 2014

**[GCFM14]**
Graham C. Goodwin, Mauricio G. Cea, Arie Feuer and D. Q. Mayne,
*On the Use of One Bit Quantizers in Networked Control*,
Automatica online,
Feb. 2014

**[FM13]**
P. Falugi and D. Q. Mayne,
*Getting robustness against unstructured uncertainty: a tube-based MPC approach*,
IEEE Transactions on Automatic Control, Issue 99,
2013

**[LA13]**
J. Lee and D. Angeli,
*Cooperative economic model predictive control for linear systems with convex objectives*,
European Journal of Control,
Volume 20, Issue 3, pp 141–151, May 2014

**[GM13]**
M. Gallieri and J.M. Maciejowski,
*Soft-constrained LASSO MPC for Robust LTI : Enlarged feasible region and an ISS gain estimate*,
Proc. Decision and Control Conference, Firenze,
Dec. 2013

**[GM13a]**
M. Gallieri and J.M. Maciejowski,
*Stabilising Terminal Cost and Terminal Controller for lasso-MPC: Enhanced Optimality and Region of Attraction*,
Proc. European Control Conference, Zurich,
Jul. 2013

**[HGM13]**
E.N. Hartley, M. Gallieri and J.M. Maciejowski,
*Terminal spacecraft rendezvous and capture with Lasso MPC*,
International Journal of Control, vol.86, no.11, pp.2104-2113,
2013

**[FM13a]**
P. Falugi and D. Q. Mayne,
*Model predictive control for tracking random references*,
Proc. of European Control Conference, Zurich,
Jul. 2013

**[FM13b]**
P. Falugi and D. Q. Mayne,
*Tracking a periodic reference using nonlinear model predictive control*,
Proc. of 52nd IEEE Conference on Decision and Control Model,
2013

**[M13]**
D. Q. Mayne,
*An apologia for stabilising conditions in model predictive control*,
International Journal of Control, vol. 86, no. 11, pp. 2090-2095,
2013

**[PMR13]**
Gabriele Pannocchia, David Q. Mayne and James B. Rawlings,
*A parsimonious algorithm for the solution of continuous-time constrained LQR problems with guaranteed convergence*,
Proc. of European Control Conference, Zurich,
Jul. 2013

**[PRMM13]**
Gabriele Pannocchia, James B. Rawlings, David Q. Mayne and Giulio M. Mancuso,
*Whither discrete time model predictive control?*,
IEEE Transactions on Automatic Control,
DOI: 10.1109/TAC.2014.2324131, Volume PP, Issue 99, 2014

**[HML12]**
E.N.Hartley, J.M.Maciejowski and K.V.Ling,
*Performance evaluation of multiplexed model predictive control for a large airliner in nominal and contingency scenarios*,
American Control Conference, Montreal,
Jun. 2012

**[GM12]**
M. Gallieri and J.M. Maciejowski,
*The lasso-MPC: Smart regulation of over-actuated systems*,
American Control Conference, Montreal,
Jul. 2012

**[GM12a]**
M. Gallieri and J.M. Maciejowski,
*A tuning procedure for lasso-MPC for over-actuated systems*,
Proc IEEE Conference on Decision and Control, Hawaii,
Dec. 2012

**[FM12]**
P. Falugi and D. Q. Mayne,
*Tracking performance of model reference control*,
Proc. 50th IEEE Conference on Decision and Control, Hawaii, USA,
Dec. 2012

**[LA12]**
J. Lee and David Angeli,
*Distributed cooperative nonlinear economic MPC*,
Proc. Conference on Mathematical Theory of Networks and Systems (MTNS),
2012

**[MKWF11]**
D.Q.Mayne, E.C.Kerrigan, E.J.van Wyk and P.Falugi,
*Tube based robust nonlinear model predictive control*,
International Journal of Robust and Nonlinear Control, Vol 21, no 11, pp 1341-1353,
2011

**[GCMSS11]**
G.C.Goodwin, D.S.Carrasco, D.Q.Mayne, M.E.Salgado and M.M.Saron,
*Preview and feedforward in model predictive control: conceptual and design issues*,
Proceedings of the 18th IFAC World Congress, Milan, Italy,
Aug-Sept 2011

**[MKF11]**
D. Q. Mayne, C. Kerrigan E, and P. Falugi,
*Robust nonlinear model predictive control: advantages and disadvantages of tube-based methods*,
Proc. 18th IFAC World Congress, Milano, Italy,
2011

**[FM11]**
P. Falugi and D. Q. Mayne,
*Tube-based model predictive control for nonlinear systems with unstructured uncertainty*,
Proc. 50th IEEE Conference on Decision and Control, pages 2656–2661, Orlando, Florida, USA,
Dec. 2011

Warning: Can't find topic Publications.LA11a

**[LMGS11]**
K.V.Ling, J.M.Maciejowski, J.Guo and E.Siva,
*Channel-hopping model predictive control*,
Proceedings of the 18th IFAC World Congress, Milan, Italy,
Aug-Sept 2011

**[SM11a]**
E.Siva and J.M.Maciejowski,
*Robust multiplexed model predictive control for distributed multi-agent systems*,
Proceedings of the 18th IFAC World Congress, Milan, Italy,
Aug-Sept 2011

**[M11]**
D. Q. Mayne,
*The role of model predictive control*,
Second Monterey Workshop: Computational Issues in Nonlinear Control, Monterey,
2011

**[LMRW10]**
K.V. Ling, J.M. Maciejowski, A.G. Richards and B.F. Wu,
*Multiplexed Model Predictive Control*,
Automatica, vol.48, no.2, 396-401,
Feb. 2012

**[WPK10]**
E. J. V. Wyk, P. Falugi, and E. C. Kerrigan,
*Imperial College London Optimal Control Software (ICLOCS) (Solves nonlinear optimal control problems with constraints.)*,
http://www.ee.ic.ac.uk/ICLOCS/,
2010

**[GMCCMQ10]**
G.C.Goodwin, D.Q.Mayne, K-Y.Chen, C.Coates, G.Mirazaeva and D.E.Quevedo,
*An introduction to the control of switching electronic systems*,
Annual Reviews in Control, Vol 34, 2, pp 209-220,
Dec., 2010

**[PRMM10]**
G.Pannocchia, J.B.Rawlings, D.Q.Mayne and W.Marquadt,
*On computing solutions to the continuous time constrained linear quadratic regulator*,
IEEE Transactions on Automatic Control, Vol 55, 9, pp 2192-2198,
Sep., 2010

**[GMCCMQ10a]**
Graham C. Goodwin, David Q. Mayne, Keng-Yuan Chen, Colin Coates, Galina Mirzaeva, and Daniel E. Quevedo,
*Opportunities and challenges in the application of advanced control to power electronics and drives*,
IEEE-ICIT2010 (Industrial Conference on Industrial Technology),
2010

**[LMRW10a]**
K.V.Ling, J.M.Maciejowski, A.G.Richards and B.F.Wu,
*Multiplexed model predictive control*,
Technical Report CUED/F-INFENG/TR.657, Cambridge University Engineering Department,
2010

**[RM09]**
J.B.Rawlings and D.Q.Mayne,
*Model Predictive Control: Theory and Design*,
Nob Hill Publishing, Madison Wisconsin, 533pp ISBN 978-0975937709,
Aug., 2009

**[PRMM09]**
G. Pannocchia, J. B. Rawlings, D. Q. Mayne, and W. Marquardt,
*Computation of the infinite horizon continuous time constrained linear quadratic regular*,
In S. Engell and Y. Arkun, Editors, Proc of ADCHEM 2009 (IFAC Symposium of Advanced Control of Chemical Processes), Vol. 1, pp.243-248,
2009

**[MRFA09]**
D.Q.Mayne, S.V.Rakovic, R.Findeisen and F.Allgower,
*Robust output feedback model predictive control of constrained linear systems: time varying case*,
Automatica, 45, 9, pp 2082-2087,
Sep., 2009

**[SKMJ09]**
J.Spitvold, E.C.Kerrigan, D.Q.Mayne and T.A.Johansen,
*Inf-sup control of discontinuous piece-wise affine systems*,
Int Journal of Robust and Nonlinear Control, 19, 13, pp 1471-1492,
Sep., 2009