Model-based predictive control (MPC), arguably the most effective control methodology for constrained systems, has seen rapid growth over the last few decades. The theory of classical MPC is well established by now, and robust MPC (RMPC) that deals with uncertainty (either in the form of additive disturbance or imprecise and/or time-varying knowledge of the system parameters) is itself reaching a state of maturity. There have been a number of new developments reported in the area of stochastic MPC (SMPC), which deals with the case where uncertainty is random and some or all of the constraints are probabilistic. The present paper surveys these developments, setting the scene by first discussing the key ingredients of classical MPC, then highlighting some major contributions in RMPC, and finally, describing recent results in SMPC. The discussion of the latter is restricted to uncertainty with bounded support, which is consistent with practice and provides the basis for the establishment of control theoretic properties, such as recurrent feasibility, stability, and convergence.

References

References
1.
Kailath
,
T.
,
1980
,
Linear Systems
,
Prentice-Hall
,
New York
.
2.
Mayne
,
D. Q.
,
Rawlings
,
J. B.
,
Rao
,
C. V.
, and
Scokaert
,
P. O. M.
,
2000
, “
Constrained Model Predictive Control: Stability and Optimality
,”
Automatica
,
36
(
6
), pp. 
789
814
.10.1016/S0005-1098(99)00214-9
3.
Gilbert
,
E. G.
, and
Tan
,
K. T.
,
1991
, “
Linear Systems With State and Control Constraints: The Theory and Application of Maximal Output Admissible Sets
,”
IEEE Trans. Autom. Control
,
36
(
9
), pp. 
1008
1020
.10.1109/9.83532
4.
Kouvaritakis
,
B.
,
Rossiter
,
J. A.
, and
Chang
,
A. O. T.
,
1992
, “
Stable Generalised Predictive Control: An Algorithm With Guaranteed Stability
,”
Proc. IEE Pt-D
,
139
(
4
), pp. 
349
362
.
5.
Bertsekas
,
D. P.
, and
Rhodes
,
I. B.
,
1973
, “
Sufficiently Informative Functions and the Minimax Feedback Control of Uncertain Dynamic Systems
,”
IEEE Trans. Autom. Control
,
18
(
2
), pp. 
117
124
.10.1109/TAC.1973.1100241
6.
Scokaert
,
P. O. M.
, and
Mayne
,
D. Q.
,
1998
, “
Min-Max Feedback Model Predictive Control for Constrained Linear Systems
,”
IEEE Trans. Autom. Control
,
43
(
8
), pp. 
1136
1142
.10.1109/9.704989
7.
van Hessem
,
D. H.
, and
Bosgra
,
O. H.
,
2002
, “
A Conic Reformulation of Model Predictive Control Including Bounded and Stochastic Disturbances Under State and Input Constraints
,”
Proceedings of the 41st IEEE Conference on Decision and Control
,
Las Vegas, NV
,
IEEE
, pp. 
4643
4648
.
8.
Löfberg
,
J.
,
2003
, “
Approximations of Closed-Loop MPC
,”
Proceedings of the 42nd IEEE Conference on Decision and Control
,
Maui, Hawaii
,
IEEE
, pp. 
1438
1442
.
9.
Goulart
,
P. J.
,
Kerrigan
,
E. C.
, and
Maciejowski
,
J. M.
,
2006
, “
Optimization Over State Feedback Policies for Robust Control with Constraints
,”
Automatica
,
42
(
4
), pp. 
523
533
.10.1016/j.automatica.2005.08.023
10.
Rakovic
,
S. V.
,
Kouvaritakis
,
B.
,
Cannon
,
M.
,
Panos
,
C.
, and
Findeisen
,
R.
,
2012
, “
Parameterized Tube Model Predictive Control
,”
IEEE Trans. Autom. Control
,
57
(
11
), pp. 
2746
2761
.10.1109/TAC.2012.2191174
11.
Rakovic
,
S. V.
,
Kouvaritakis
,
B.
,
Cannon
,
M.
, and
Panos
,
C.
,
2012
, “
Fully Parameterized Tube Model Predictive Control
,”
Int. J. Robust Nonlinear Control
,
22
(
12
), pp. 
1330
1361
.10.1002/rnc.2825
12.
Munoz-Carpintero
,
D.
,
Kouvaritakis
,
B.
, and
Cannon
,
M.
,
2014
, “
Striped Parameterized Tube Model Predictive Control
,”
Proceedings of the 19th IFAC World Congress
,
Cape Town, South Africa
,
International Federation of Automatic Control
, pp. 
11998
12003
.
13.
Lee
,
Y. I.
, and
Kouvaritakis
,
B.
,
1999
, “
Constrained Receding Horizon Predictive Control for Systems With Disturbances
,”
Int. J. Control
,
72
(
11
), pp. 
1027
1032
.
14.
Lee
,
Y. I.
, and
Kouvaritakis
,
B.
,
2000
, “
Linear Matrix Inequalities and Polyhedral Invariant Sets in Constrained Robust Predictive Control
,”
Int. J. Robust Nonlinear Control
,
10
(
13
), pp. 
1079
1090
.10.1002/(ISSN)1099-1239
15.
Lee
,
Y. I.
,
Kouvaritakis
,
B.
, and
Cannon
,
M.
,
2002
, “
Constrained Receding Horizon Predictive Control for Nonlinear Systems
,”
Automatica
,
38
(
12
), pp. 
2093
2102
.10.1016/S0005-1098(02)00133-4
16.
Evans
,
M. A.
,
Cannon
,
M.
, and
Kouvaritakis
,
B.
,
2012
, “
Robust MPC for Linear Systems With Bounded Multiplicative Uncertainty
,”
Proceedings of the 51st IEEE Conference on Decision and Control
,
Maui, Hawaii
,
IEEE
, pp. 
248
253
.
17.
Fleming
,
J.
,
Kouvaritakis
,
B.
, and
Cannon
,
M.
,
2014
, “
Robust Tube MPC for Linear Systems With Multiplicative Uncertainty
,”
IEEE Trans. Autom. Control
(in press), 10.1109/TAC.2014.2336358.
18.
Kouvaritakis
,
B.
,
Cannon
,
M.
, and
Tsachouridis
,
V.
,
2004
, “
Recent Developments in Stochastic MPC and Sustainable Development
,”
Ann. Rev. Control
,
28
(
1
), pp. 
23
35
.
19.
Kouvaritakis
,
B.
,
Cannon
,
M.
, and
Couchman
,
P.
,
2006
, “
MPC as a Tool for Sustainable Development Integrated Policy Assessment
,”
IEEE Trans. Autom. Control
,
51
(
1
), pp. 
145
149
.10.1109/TAC.2005.861702
20.
Evans
,
M. A.
,
Cannon
,
M.
, and
Kouvaritakis
,
B.
,
2014
, “
Robust MPC Tower Damping for Variable Speed Wind Turbines
,”
IEEE Trans. Control Sys. Techno.
,
23
(
1
), pp.
290
296
.
21.
Kouvaritakis
,
B.
,
Cannon
,
M.
,
Rakovic
,
S. V.
, and
Cheng
,
Q.
,
2010
, “
Explicit Use of Probabilistic Distributions in Linear Predictive Control
,”
Automatica
,
46
(
10
), pp. 
1719
1724
.10.1016/j.automatica.2010.06.034
22.
Cannon
,
M.
,
Kouvaritakis
,
B.
,
Rakovic
,
S. V.
, and
Cheng
,
Q.
,
2011
, “
Stochastic Tubes in Model Predictive Control With Probabilistic Constraints
,”
IEEE Trans. Autom. Control
,
56
(
1
), pp. 
194
200
.10.1109/TAC.2010.2086553
23.
Fleming
,
J.
,
Cannon
,
M.
, and
Kouvaritakis
,
B.
,
2014
, “
Stochastic Tube MPC for LPV Systems With Probabilistic Set Inclusion Conditions
,”
Proceedings of the 53rd IEEE Conference on Decision and Control
,
Los Angeles
,
IEEE
, pp.
4783
4788
.
24.
Boyd
,
S.
,
El Ghaoui
,
L.
,
Feron
,
E.
, and
Balakrishnan
,
V.
,
1994
,
Linear Matrix Inequalities in System and Control Theory
(Studies in Applied Mathematics, Vol. 15),
SIAM
,
Philadelphia
.
25.
Rossiter
,
J. A.
,
Rice
,
M. J.
, and
Kouvaritakis
,
B.
,
1998
, “
A Numerically Robust State-Space Approach to Stable Predictive Control Strategies
,”
Automatica
,
34
(
1
), pp. 
65
73
.10.1016/S0005-1098(97)00171-4
26.
Kouvaritakis
,
B.
,
Rossiter
,
J. A.
, and
Schuurmans
,
J.
,
2000
, “
Efficient Robust Predictive Control
,”
IEEE Trans. Autom. Control
,
45
(
8
), pp. 
1545
1549
.10.1109/9.871769
27.
Cannon
,
M.
, and
Kouvaritakis
,
B.
,
2005
, “
Optimizing Prediction Dynamics for Robust MPC
,”
IEEE Trans. Autom. Control
,
50
(
11
), pp. 
1892
1897
.10.1109/TAC.2005.858679
28.
Lee
,
Y. I.
, and
Kouvaritakis
,
B.
,
2000
, “
Receding Horizon H-Infinity Predictive Control for Systems With Input Saturation
,”
Proc. IEE Pt-D
,
147
(
2
), pp. 
153
158
.
29.
Kouvaritakis
,
B.
,
Cannon
,
M.
, and
Rossiter
,
J. A.
,
2002
, “
Who Needs QP for Linear System MPC Anyway?
Automatica
,
38
(
5
), pp. 
879
884
.10.1016/S0005-1098(01)00263-1
30.
Rakovic
,
S. V.
,
Kerrigan
,
E.
,
Kouramas
,
K.
, and
Mayne
,
D.
,
2005
, “
Invariant Approximations of the Minimal Robustly Positively Invariant Set
,”
IEEE Trans. Autom. Control
,
50
(
3
), pp. 
406
410
.10.1109/TAC.2005.843854
31.
Mayne
,
D. Q.
,
Seron
,
M. M.
, and
Rakovic
,
S. V.
,
2005
, “
Robust Model Predictive Control of Constrained Linear Systems With Bounded Disturbances
,”
Automatica
,
41
(
2
), pp. 
219
224
.10.1016/j.automatica.2004.08.019
32.
Rakovic
,
S. V.
,
Kouvaritakis
,
B.
,
Findeisen
,
R.
, and
Cannon
,
M.
,
2012
, “
Homothetic Tube Model Predictive Control
,”
Automatica
,
48
(
8
), pp. 
1631
1638
.10.1016/j.automatica.2012.05.003
33.
Kothare
,
M. V.
,
Balakrishnan
,
V.
, and
Morari
,
M.
,
1996
, “
Robust Constrained Model Predictive Control Using Linear Matrix Inequalities
,”
Automatica
,
32
(
10
), pp. 
1361
1379
.10.1016/0005-1098(96)00063-5
34.
Schuurmanns
,
J.
, and
Rossiter
,
J. A.
,
2000
, “
Robust Piecewise Linear Control for Polytopic Systems With Input Constraints
,”
IEE Proc. CTA
,
147
(
1
), pp. 
13
18
.
35.
Casavola
,
A.
,
Gianneli
,
M.
, and
Mosca
,
E.
,
2000
, “
Min-Max Predictive Control Strategies for Input-Saturated Polytopic Systems
,”
Automatica
,
36
(
1
), pp. 
125
133
.10.1016/S0005-1098(99)00112-0
36.
Blanchini
,
F.
, and
Miani
,
S.
,
2007
,
Set-Theoretic Methods in Control
,
Springer
,
New York
.
37.
Gautam
,
A.
,
Chu
,
Y.-C.
, and
Soh
,
Y. C.
,
2012
, “
Optimized Dynamic Policy for Receding Horizon Control of Linear Time-Varying Systems With Bounded Disturbances
,”
IEEE Trans. Autom. Control
,
57
(
4
), pp. 
973
988
.10.1109/TAC.2011.2170109
38.
Munoz-Carpintero
,
D.
,
Cannon
,
M.
, and
Kouvaritakis
,
B.
,
2013
, “
Recursively Feasible Robust MPC for Linear Systems With Additive and Multiplicative Uncertainty Using Optimized Polytopic Dynamics
,”
Proceedings of the 52nd IEEE Conference on Decision and Control
,
Florence, Italy
,
IEEE
, pp. 
1101
1106
.
39.
Astrom
,
K. J.
, and
Wittenmark
,
B.
,
1973
, “
On Self Tuning Regulators
,”
Automatica
,
9
(
2
), pp. 
185
199
.10.1016/0005-1098(73)90073-3
40.
Clarke
,
D. W.
, and
Gawthrop
,
P. J.
,
1975
, “
Self-Tuning Controller
,”
Proc. Inst. Electr. Eng.
,
122
(
9
), pp. 
929
934
.10.1049/piee.1975.0252
41.
Schwarm
,
A. T.
, and
Nikolaou
,
M.
,
1999
, “
Chance-Constrained Model Predictive Control
,”
AIChE J.
,
45
(
8
), pp. 
1743
1752
.10.1002/(ISSN)1547-5905
42.
Stoorvogel
,
A. A.
,
Weiland
,
S.
, and
Batina
,
I.
,
2007
,
Model Predictive Control by Randomized Algorithms for Systems With Constrained Inputs and Stochastic Disturbances
, wwwhome.math.utwente.nl/∼stoorvogelaa/subm01.pdf.
43.
Calafiore
,
G. C.
, and
Fagiano
,
L.
,
2013
, “
Stochastic Model Predictive Control of LPV Systems Via Scenario Optimization
,”
Automatica
,
49
(
6
), pp. 
1861
1866
.10.1016/j.automatica.2013.02.060
44.
Cannon
,
M.
,
2008
, “
Stochastic Model Predictive Control: State Space Methods
,”
Tutorial Workshop on Stochastic MPC, IFAC World Congress
,
Seoul, Korea
, http://users.ox.ac.uk/~engs0169/pdf/cannon_ifac08c.pdf.
45.
Kouvaritakis
,
B.
,
Cannon
,
M.
, and
Munoz-Carpintero
,
D.
,
2013
, “
Efficient Prediction Strategies for Disturbance Compensation in Stochastic MPC
,”
Int. J. Sys. Sci.
,
44
(
7
), pp. 
1344
1353
.
46.
Campi
,
M. C.
, and
Garatti
,
S.
,
2011
, “
A Sampling-and-Discarding Approach to Chance-Constrained Optimization: Feasibility and Optimality
,”
J. Optim. Theory Appl.
,
148
(
2
), pp. 
257
280
.10.1007/s10957-010-9754-6
47.
Cannon
,
M.
,
Kouvaritakis
,
B.
, and
Wu
,
X.
,
2009
, “
Model Predictive Control for Systems With Stochastic Multiplicative Uncertainty and Probabilistic Constraints
,”
Automatica
,
45
(
1
), pp. 
167
172
.10.1016/j.automatica.2008.06.017
48.
Cannon
,
M.
,
Kouvaritakis
,
B.
, and
Wu
,
X.
,
2009
, “
Probabilistic Constrained MPC for Multiplicative and Additive Stochastic Uncertainty
,”
IEEE Trans. Autom. Control
,
54
(
7
), pp. 
1626
1632
.10.1109/TAC.2009.2017970
49.
Cannon
,
M.
,
Kouvaritakis
,
B.
, and
Ng
,
D.
,
2009
, “
Probabilistic Tubes in Linear Stochastic Model Predictive Control
,”
Sys. Control Lett.
,
58
(
10
), pp. 
747
753
.
50.
Korda
,
M.
,
Gondhalekar
,
R.
,
Oldewurtel
,
F.
, and
Jones
,
C. N.
,
2014
, “
Stochastic MPC Framework for Controlling the Average Constraint Violation
,”
IEEE Trans. Autom. Control
,
59
(
7
), pp. 
1706
1721
.10.1109/TAC.2014.2310066
51.
Cheng
,
Q.
,
Cannon
,
M.
,
Kouvaritakis
,
B.
, and
Evans
,
M.
,
2014
, “
Stochastic MPC for Systems with Both Multiplicative and Additive Disturbances
,”
19th IFAC World Congress
,
Cape Town, South Africa
,
International Federation of Automatic Control
, pp. 
2291
2296
.
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