Decentralized and distributed control - Decentralized and...

Decentralized and distributed controlDecentralized and distributed control for constrained discrete-time

systems

M. Farina1 G. Ferrari Trecate2

1Dipartimento di Elettronica, Informazione e Bioingegneria (DEIB)Politecnico di Milano, Italy

[email protected]

2Dipartimento di Ingegneria Industriale e dell’Informazione (DIII)Universita degli Studi di Pavia, [email protected]

EECI-HYCON2 Graduate School on Control 2015Supelec, France

Farina, Ferrari Trecate () Decentralized and distributed control EECI-HYCON2 School 2015 1 / 101

[email protected]

[email protected]

Outline

1 Models

2 MPC decentralized and distributed algorithms

3 Analysis of prototypical algorithms

4 ExamplesTemperature controlThree-tank system

5 Conclusions

6 Suggested readings


Outline

1 Models




5 Conclusions



Models

The design of decentralized and distributed MPC control systemsrequires the availability of partitioned models of large-scale systemsinto a number M of sub-models Si .

Here we deal with discrete-time systems.

Unstructured discrete-time modelxo(k +1) = Aoxo(k)+Bou(k)y(k) = Coxo(k)

Partitioning methodsIn this part we focus on:

non-overlapping decompositions;completely overlapping decompositions.


ModelsNon-overlapping decomposition

Interaction-oriented model

{xi(k +1) = Aiixi(k)+Biiui(k)+∑

Mj 6=i Aijxj(k)+∑

Mj 6=i Bijuj(k)

yi(k) = Cixi(k)

where xi ∈ Rni , ui ∈ Rmi , and yi ∈ Rpi .

Recall that, possibly under a permutation:

xo =

x1...

xM

, u =

u1...

uM

, y =

y1...

yM

i.e., the input/state/output vectors are completely partitioned.


ModelsNon-overlapping decomposition

Features of non-overlapping decompositionThe main objectives are:

to obtain small scale sub-models (minimality of therepresentation),to identify cascade configurations (simplicity of the controlstructure),to identify subsystems that are weakly connected together (allowdecentralized/distributed controllers),to minimize the communication/interaction links (minimization ofthe communication).


ModelsCompletely overlapping decomposition

Interaction-oriented model{xi(k +1) = Aiixi(k)+∑

Mj=1 Bijuj(k)

yi(k) = Cixi(k)

i.e., the dynamically-decoupled form. where xi ∈ Rni , ui ∈ Rmi , andyi ∈ Rpi .

Recall that, possibly under a permutation:

u =

u1...

uM

, y =

y1...

yM

i.e., the input/output vectors are completely partitioned, while ni isgenerally greater that n.


ModelsCompletely overlapping decomposition

Features of completely overlapping decompositionThe main objectives are:

to obtain sub-models with local (and low-order) input and outputvariables,to avoid the dependance of the local equations upon the statevariables of neighboring subsystems.


ModelsMain requirements

Many features of the distributed MPC algorithms that will be laterpresented strongly depend on the type and the characteristics of theused partition.

Minimality of the representationneeded to limit the algorithm computational burden,the number of involved state variables (for each subsystem)generally affects the computational load of the algorithm (at leastwhen state constraints are accounted for).


Models

Reduction of the communication burdenReducing the communication burden reduces the probability oftransmission delays, overloads and package losses.

It is strongly dependent onI) the number of neighbors for each subsystem: this is mostly

dependent on the type of decomposition;II) the required information to be transmitted on-line: this mostly

depends on the selected distributed control algorithm.


ModelsReduction of the required information to be stored androbustness

Minimal non-overlapping representations require that only theknowledge of local dynamics (and of how the neighboringvariables affect them) is available to each local controller

I scalable memory load;I robustness with respect to uncertainties on the dynamics of other

subsystems;overlapping decompositions imply that possibly the dynamicalmodel of the overall system be stored by local control units

I this requires non-scalable memory load (it increases as the order ofthe system increases);

I this makes the local control system affected by uncertainties andperturbations.

Remark that globally optimal performances can be generallyobtained with non-minimal representations, such as the completelyoverlapping ones.


Outline

1 Models




5 Conclusions



MPC decentralized and distributed algorithms

In the following we consider the case where the system is partitionedusing a non-specified interaction-oriented decomposition and M = 2:


MPC decentralized and distributed algorithmsInteracting subsystems:

S1 =

{x1(k +1) = A11x1(k)+B11u1(k)+A12x2(k)+B12u2(k)y1(k) = C1x1(k)

S2 =

{x2(k +1) = A22x2(k)+B22u2(k)+A21x1(k)+B21u1(k)y2(k) = C2x2(k)

Collecting together the interaction oriented models, we obtain a (possiblynon-minimal) collective model:{

x(k +1) = Ax(k)+Bu(k)y(k) = Cx(k)

where

x(k) =[x1(k)x2(k)

], y(k) =

[y1(k)y2(k)

], u(k) =

[u1(k)u2(k)

]and

A =

[A11 A12A21 A22

], B =

[B11 B12B21 B22

], C =

[C1 00 C2

]Remark: A, B, and C are different, in general, from the minimal ones Ao, Bo, and Co.


MPC decentralized and distributed algorithms

Cost functionThe cost function is

V =t+N−1

∑k=t

12{‖x(k)‖2Q +‖u(k)‖2R}+

12‖x(t +N)‖2P

Separability of the cost functionThe cost function is assumed to be formally separable, i.e.,

V = ρ1V1 +ρ2V2


MPC decentralized and distributed algorithmsSeparability of the cost function is obtained by setting block diagonal weightingmatrices

Q =

[ρ1Q1 0

0 ρ2Q2

], R =

[ρ1R1 0

0 ρ2R2

], P =

[ρ1P1 0

0 ρ2P2

]in such a way that

V = ∑t+N−1k=t

12{[xT

1 (k) xT2 (k)

][ρ1Q1 00 ρ2Q2

][x1(k)x2(k)

]+

+[uT

1 (k) uT2 (k)

][ρ1R1 00 ρ2R2

][u1(k)u2(k)

]}+

12[xT

1 (t +N) xT2 (t +N)

][ρ1P1 00 ρ2P2

][x1(t +N)x2(t +N)

]= ρ1V1 +ρ2V2

whereV1 = ∑

t+N−1k=t

12{‖x1(k)‖2Q1

+‖u1(k)‖2R1}+ 1

2‖x1(t +N)‖2P1

V2 = ∑t+N−1k=t

12{‖x2(k)‖2Q2

+‖u2(k)‖2R2}+ 1

2‖x2(t +N)‖2P2


MPC decentralized and distributed algorithmsCentralized MPC (cMPC)

Minimization problem

minu1(t :t+N−1),u2(t :t+N−1)

ρ1V1 +ρ2V2

System model

S : x(k +1) = Ax(k)+Bu(k)


MPC decentralized and distributed algorithmsDecentralized MPC (dMPC)

Decentralized controllocal regulators designed to control xi (k)using input ui (k) independently of xj (k) anduj (k), j 6= i ;

mutual interactions neglected (model error);

design of local regulators trivial (low orderproblems);

if interactions between S1 and S2 are“sufficiently” weak (i.e., matrices A12, B12,A21, and B21 are small), then closed loopstability;

strong interactions may prevent stabilityand/or acceptable performances;

stability analysis of the closed loop system nottrivial.



Regulator R1Minimization problem:

minu1(t :t+N−1)

V1

Model of S1 (”wrong”)

x1(k +1) = A11x1(k)+B11u1(k)


minu2(t :t+N−1)

V2

Model of S2 (”wrong”)

x2(k +1) = A22x2(k)+B22u2(k)



Suggested readings on decentralized MPCL. Magni and R. Scattolini. Stabilizing decentralized model predictive control of nonlinearsystems. Automatica, 42(7):1231 - 1236, 2006.

D. M. Raimondo, L. Magni, and R. Scattolini. Decentralized MPC of nonlinear system: Aninput-to-state stability approach. International Journal of Robust and Nonlinear Control,17(5):1651 - 1667, 2007.

A. Alessio and A. Bemporad. Stability conditions for decentralized model predictive controlunder packet drop communication. In Proc. of ACC: 3577 - 3582, 2008.

D. Barcelli and A. Bemporad. Decentralized model predictive control ofdynamically-coupled linear systems: Tracking under packet loss. Proc. IFAC Workshop onEstimation and Control of Networked Systems, 2009.

A. Alessio, D. Barcelli, and A. Bemporad. Decentralized model predictive control ofdynamically-coupled linear systems. Journal of Process Control, 21(5):705 – 714, 2011.

S. Riverso, M. Farina, and G. Ferrari Trecate. Plug-and-Play Decentralized ModelPredictive Control for Linear Systems. IEEE Transactions on Automatic Control,58(10):2608-2614, 2013.

S. Riverso, M. Farina, G. Ferrari Trecate. Plug-and-play model predictive control based onrobust control invariant sets. Automatica, 50(8): 2179-2186, 2014.


MPC decentralized and distributed algorithmsDistributed MPC (DMPC)

Distributed controlinformation is transmitted among localregulators, e.g., future predicted state andcontrol variables computed locally, to predictthe interaction effects;

transmitted data is strictly is related to theemployed model decomposition. E.g.,

I. non-overlapping partition: S1 needs thepredictions of both x2(k) and u2(k) tocompute predictions of the evolution of x1(k);

II. completely overlapping partitions: S1 needsu2(k) to compute predictions of the evolutionof x2(k).



Classification of DMPC algorithms:

Topology of the transmission networkfully connected transmission networks (all-to-allcommunication): information is transmitted from anylocal regulator to the others;

partially connected transmission networks(neighbor-to-neighbor communication): informationis transmitted among the local regulators ofsubsystems only with direct dynamic/input coupling(it allows to significantly reduce the transmissionload especially with non-overlapping - sparse -partitions).




Exchange of informationiterative algorithms:information is transmitted (andreceived) more than once within each samplingtime (e.g., optimality is obtained when the algorithmsteps reach convergence);

non-iterative algorithms:information is transmitted(and received) once within each sampling time.




Rationale of the algorithmThe distributed MPC problems described in this sectioncan all be cast as dynamic non-cooperative games:

Nash solution of a non-cooperative game wherethe utility functions of the players differ from eachother (independent DMPC, iDMPC);

robust solution of a non-cooperative gamewhere the utility functions of the players differ fromeach other (robust DMPC, rDMPC);

solution of a ”cooperative game” where the utilityfunctions are the same for all players (cooperativeDMPC, cDMPC).


MPC decentralized and distributed algorithmsIndependent DMPC (iDMPC)


minu1(t :t+N−1)

V1

Model of S1

x1(k +1) = A11x1(k)+B11u1(k)+A12x∗2(k)+B12u∗2(k)

where predictions u∗2(t : t +N−1), x∗2(t : t +N−1) areavailable




minu2(t :t+N−1)

V2

Model of S2

x2(k +1) = A22x2(k)+B22u2(k)+A21x∗1(k)+B21u∗1(k)

where predictions u∗1(t : t +N−1), x∗1(t : t +N−1) areavailable



Remarks:

iDMPC methods can be either iterative ornon-iterative: predicted trajectories can becomputed locally and transmitted iteratively duringeach sampling time to obtain more reliablepredictions;

the needed transmission network is partiallyconnected.


MPC decentralized and distributed algorithmsCooperative DMPC (cDMPC)

Note that, cooperative solutions have been proposed for dynamically decoupled models, i.e., incase A12 = 0 and A21 = 0.


minu1(t :t+N−1)

ρ1V1 +ρ2V2

Model of S

x1(k +1) = A11x1(k)+B11u1(k)+B12u∗2(k)x2(k +1) = A22x2(k)+B21u1(k)+B22u∗2(k)

where predictions u∗2(t : t +N−1) are available.



Note that, cooperative solutions have been proposed for dynamically decoupled models, i.e., incase A12 = 0 and A21 = 0.


minu2(t :t+N−1)

ρ1V1 +ρ2V2

Model of S

x1(k +1) = A11x1(k)+B12u2(k)+B11u∗1(k)x2(k +1) = A22x2(k)+B22u2(k)+B21u∗1(k)

where predictions u∗1(t : t +N−1) are available.



Remarks:

cDMPC methods are iterative: predicted trajectoriesmust be computed locally and transmitted iterativelyduring each sampling time to obtain more reliablepredictions;

the needed transmission network is fully connected:predictions of the overall system response to ui (k)must be computed;

some available algorithms guarantee stability evenin the non-iterative formulation, while globaloptimality is guaranteed only when iterationsconverge.


MPC decentralized and distributed algorithmsRobust DMPC (rDMPC)

Main ideaAt time t , S1

sends to S2 the nominal predicted referencetrajectories u∗1(t : t +N−1) and x∗1(t : t +N−1),

guarantees (through suitable constraints in theoptimization problem) that the actual trajectoriesu1(t : t +N−1) and x1(t : t +N−1) lie in suitableneighborhoods of u∗1(t : t +N−1) andx∗1(t : t +N−1), respectively, i.e.,

u1(k) ∈ u∗1(k)⊕U1x1(k) ∈ x∗1(k)⊕Z1

In this way, the regulator R2 has the following model ofS2:

x2(k +1) = A22x2(k)+B22u2(k)+A21x1(k)+B21u1(k)= A22x2(k)+B22u2(k)+A21x∗1(k)+B21u∗1(k)++A21(x1(k)−x∗1(k))+B21(u1(k)−u∗1(k))= A22x2(k)+B22u2(k)+A21x∗1(k)+B21u∗1(k)+w2(k)



Main ideaTherefore, at time t , the regulator R2 has thefollowing model of S2:

x2(k +1) = A22x2(k)+B22u2(k)+A21x∗1(k)++B21u∗1(k)+w2(k)

The disturbance w2(k) is bounded: a robust(tube-based) approach is used for control of thissystem.




minu1(t :t+N−1)

V1

Model of S1

x1(k +1) = A11x1(k)+B11u1(k)++A12x∗2(k)+B12u∗2(k)+w1(k)

where predictions (reference trajectories) u∗2(t : t +N−1),x∗2(t : t +N−1) are available, and w1(k) is unknown butbounded.Further constraints:

u1(k) ∈ u∗1(k)⊕U1x1(k) ∈ x∗1(k)⊕Z1




minu2(t :t+N−1)

V2

Model of S2

x2(k +1) = A22x2(k)+B22u2(k)++A21x∗1(k)+B21u∗1(k)+w2(k)

where predictions (reference trajectories) u∗1(t : t +N−1),x∗1(t : t +N−1) are available, and w2(k) is unknown butbounded.Further constraints:

u2(k) ∈ u∗2(k)⊕U2x2(k) ∈ x∗2(k)⊕Z2



Remarks:

rDMPC methods are non-iterative: predictedtrajectories must be computed locally andtransmitted once during each sampling time;

the needed transmission network is partiallyconnected.


MPC decentralized and distributed algorithmsDMPC

Suggested readings on DMPCW. B. Dunbar. Distributed receding horizon control of dynamically coupled nonlinearsystems. IEEE Transactions on Automatic Control, 52(7):1249 - 1263, 2007.

J. Liu, D. Munoz de la Pena, and P. D. Christofides. Distributed model predictive control ofnonlinear process systems. AIChE Journal, 55(9):1171 - 1184, 2009.

J. Liu, D. Munoz de la Pena, and P. D. Christofides. Distributed model predictive control ofnonlinear systems subject to asynchronous and delayed measurements. Automatica,46:52 - 61, 2010.

M. Farina and R. Scattolini. An output feedback distributed predictive control algorithm.Proceedings of the 50th IEEE Conference on Decision and Control: 8139 - 8144, 2011.

M. Farina and R. Scattolini. Distributed predictive control: a non-cooperative algorithm withneighbor-to-neighbor communication for linear systems. Automatica, 48(6):1088-1096,2012.

M. Farina, L. Giulioni, G. Betti, and R. Scattolini. An approach to distributed predictivecontrol for tracking - theory and applications. IEEE Transactions on Control SystemTechnology, 22(4):1558-1566, 2014.

G. Betti, M. Farina, R. Scattolini. Realization issues, tuning, and testing of a distributedpredictive control algorithm. Journal of Process Control, 24(4):424-434, 2014.

M. Farina, G. Betti, R. Scattolini. Distributed Predictive Control of continuous-time systems.Systems and Control Letters, 74: 32-40, 2014.



Suggested readings on DMPCJ. M. Maestre, D. Munoz de la Pena, E. F. Camacho, and T. Alamo. Distributed modelpredictive control based on agent negotiation. J. Process Control, 21(5):685 - 697, 2011.

A. N. Venkat, I. A. Hiskens, J. B. Rawlings, and S.J. Wright. Distributed MPC strategieswith application to power system automatic generation control. IEEE Transactions onControl Systems Technology, 16(6):1192 - 1206, 2008.

B. T. Stewart, A. N. Venkat, J. B. Rawlings, S. J. Wright, and G. Pannocchia. Cooperativedistributed model predictive control. System and Control Letters, 59:460 - 469, 2010.

P. Trodden, A. Richards. Cooperative distributed MPC of linear systems with coupledconstraints. Automatica, 49(2):479-487, 2013.

A. Ferramosca, D. Limon, I. Alvarado, E.F. Camacho. Cooperative distributed MPC fortracking. Automatica, 49(4): 906-914, 2013.

M. A. Muller, M. Reble, and F. Allgower. Cooperative control of dynamically decoupledsystems via distributed model predictive control. Int. J. Robust Nonlinear Control,22(12):1376-1397, 2012.

M. A. Muller, and F. Allgower. Distributed economic MPC: a framework for cooperativecontrol problems. Proc. IFAC WC, 2014: 1029-1034.

M. Lopes de Lima, E. Camponogara, D. Limon, D. Munoz de la Pena. Distributedsatisficing MPC. IEEE Transactions on Control Systems Technology, 23(1):305 - 312,2015.



Recent applications to power networks, smart grids, (building)energy management (I)

S. Roshany-Yamchi, M. Cychowski, R.R. Negenborn, B. De Schutter, K. Delaney, J.Connell. Kalman Filter-Based Distributed Predictive Control of Large-Scale Multi-RateSystems: Application to Power Networks. IEEE Transactions on Control SystemsTechnology, 21(1): 27-39, 2013.

D. Wang, M. Glavic, L. Wehenkel. Comparison of centralized, distributed and hierarchicalmodel predictive control schemes for electromechanical oscillations damping in large-scalepower systems. International Journal of Electrical Power & Energy Systems, 58:32-41,2014.

A. J. del Real, A. Arce, C. Bordons. Combined environmental and economic dispatch ofsmart grids using distributed model predictive control. International Journal of ElectricalPower & Energy Systems, 54: 65-76, 2014.

Miaomiao Ma, Hong Chen, Xiangjie Liu, Frank Allgower. Distributed model predictive loadfrequency control of multi-area interconnected power system. International Journal ofElectrical Power & Energy Systems, 62: 289-298, 2014.



Recent applications to power networks, smart grids, (building)energy management (II)

G.K.H. Larsen, N.D. van Foreest, J.M.A. Scherpen. Distributed MPC Applied to a Networkof Households With Micro-CHP and Heat Storage. IEEE Transactions on Smart Grid, 5(4):2106-2114, 2014.

H.F. Scherer, M. Pasamontes, J.L. Guzman, J.D. Alvarez, E. Camponogara, J.E.Normey-Rico . Efficient building energy management using distributed model predictivecontrol. Journal of Process Control, 24(6): 740-749, 2014.

A.J. del Real, A. Arce, C. Bordons. An Integrated Framework for Distributed ModelPredictive Control of Large-Scale Power Networks. IEEE Transactions on IndustrialInformatics, 10(1): 197-209, 2014.



Recent applications to multi-vehicle systemsM. Zhu, S. Martinez. On distributed constrained formation control in operator-vehicleadversarial networks. Automatica, 49(12): 3571-3582, 2013.

P. Wang, B. Ding. A synthesis approach of distributed model predictive control forhomogeneous multi-agent system with collision avoidance. International Journal ofControl, 87(1):52-63, 2014.

P. Wang, B. Ding. Distributed RHC for Tracking and Formation of NonholonomicMulti-Vehicle Systems. IEEE Transactions on Automatic Control, 59(6): 1439-1453, 2014.



Other recent applicationsF. Fele, J. M. Maestre, S. M. Hashemy, D. Munoz de la Pena, E. F. Camacho. Coalitionalmodel predictive control of an irrigation canal. Journal of Process Control, 24(4): 314-325,2014.

G. Li, M. R. Belmont. Model predictive control of sea wave energy converters - Part II: Thecase of an array of devices. Renewable Energy, 68: 540-549, 2014.

V. Kirubakaran, T.K. Radhakrishnan, N. Sivakumaran. Distributed multiparametric modelpredictive control design for a quadruple tank process. Measurement, 47: 841-854.

A. Ferrara, A. Nai Oleari, S. Sacone, S. Siri. Freeways as Systems of Systems: ADistributed Model Predictive Control Scheme. IEEE Systems Journal, to appear.


MPC decentralized and distributed algorithmsDistributed solution of centralized MPC optimization (DcMPC)

Distributed solution of a centralizedMPC problem

from a control-theoretical perspective, it isequivalent to solve cMPC;

from an optimization perspective, theone-step MPC problem is:

I R1 and R2 solve small-scaleoptimization problems;

I R1 and R2 exchange information;I this procedure is iterated until

convergence;

there exist both hierarchical and distributedarchitectures.



Minimization problem

minu1(t :t+N−1),u2(t :t+N−1)

ρ1V1 +ρ2V2

System model

S : x(k +1) = Ax(k)+Bu(k)



Remark:

price coordination, primal and dualdecomposition methods are widespreadDcMPC algorithms;

optimality and/or feasibility are generallyattained only when the convergence of thedecomposition algorithm (within eachsampling time) is achieved.


MPC decentralized and distributed algorithmsDistributed solutions of centralized MPC optimization (DcMPC)

Suggested readings on distributed solutions of centralized MPCH. Everett. Generalized Lagrange multiplier method for solving problems of optimumallocation of resources. Operations Research, 11:399417, 1963.

M. D. Mesarovic, D. Macko, and Y. Takahara. Theory of Hierarchical, Multilevel, Systems.Academic Press, New York, 1970.

D. A. Wismer. Optimization methods for large-scale systems with applications.McGraw-Hill New York, 1971.

C. Savorgnan and M. Diehl. Report of literature survey, analysis, and comparison of on-lineoptimization methods for hierarchical and distributed MPC. HD-MPC Deliverable D4.1.1.Available at http://www.ict-hd-mpc.eu/deliverables/hd_mpc_D_4_1_1.pdf.

C. Conte, M. N. Zeilinger, M. Morari, and C. N. Jones. Cooperative distributed trackingMPC for constrained linear systems: theory and synthesis. Proc. of CDC, 38123817, 2013.

P. Giselsson, M.D. Doan, T. Keviczky, B. De Schutter, A. Rantzer. Accelerated gradientmethods and dual decomposition in distributed model predictive control. Automatica,49(3):829-833, 2013.

P. Giselsson, A. Rantzer. On Feasibility, Stability and Performance in Distributed ModelPredictive Control. IEEE Transactions on Automatic Control, 59(4):1031-1036, 2014.


http://www.ict-hd-mpc.eu/deliverables/hd_mpc_D_4_1_1.pdf

MPC decentralized and distributed algorithmsSummary

ALGO cMPC dMPC iDMPC cDMPC rDMPC DcMPC#CSs 1 M M M M MModel S Si Si Si Si Si

Mod.Type any any any dyn.dec any anyCost fcn V Vi Vi V Vi V

tNW CS to all no n2n all-to-all n2n all-to-allCS load high small small small small small

iters no no yes/no yes no yesopt yes no no yes no yes


Outline

1 Models




5 Conclusions



Analysis of prototypical algorithmsMAIN ASSUMPTIONS

the system is partitioned in M = 2 subsystems;

we have dynamically decoupled subsystems

S1 =

{x1(k +1) = A11x1(k)+B11u1(k)+B12u2(k)y1(k) = C1x1(k)

S2 =

{x2(k +1) = A22x2(k)+B22u2(k)+B21u1(k)y2(k) = C2x2(k)

A11 and A22 are asymptotically stable;

The cost functions are:

V1 = ∑t+N−1k=t

12{‖y1(k)‖2Q∗1 +‖u1(k)‖2R1

}+ 12‖x1(t +N)‖2P1

= ∑t+N−1k=t

12{‖x1(k)‖2Q1

+‖u1(k)‖2R1}+ 1

2‖x1(t +N)‖2P1

V2 = ∑t+N−1k=t

12{‖y2(k)‖2Q∗2 +‖u2(k)‖2R2

}+ 12‖x2(t +N)‖2P2

= ∑t+N−1k=t

12{‖x2(k)‖2Q2

+‖u2(k)‖2R2}+ 1

2‖x2(t +N)‖2P2

where Q1 = CT1 Q∗1C1 ≥ 0 and Q2 = CT

2 Q∗2C2 ≥ 0;


Analysis of prototypical algorithmsMAIN ASSUMPTIONS

constraints on input and state variables are neglected;

being A11 and A22 asymptotically stable, the decentralized ”stabilizing” auxiliarycontrol law is u1 = 0 and u2 = 0 (i.e., K1 = 0 and K2 = 0);

The final cost matrices P1 and P2 are chosen in such a way that

AT11P1A11−P1 = −Q1

AT22P2A22−P2 = −Q2

i.e., 12‖x1‖2P1

and 12‖x2‖2P2

are the infinite-horizon costs-to-go under zero control;

the global cost function isV = ρ1V1 +ρ2V2

where ρ1 +ρ2 = 1.


Analysis of prototypical algorithmsCentralized control (cMPC)

cMPC problem

minu

V

subject tox(k +1) = Ax(k)+Bu(k)

where

V =t+N−1

∑k=t

12{‖x(k)‖2Q +‖u(k)‖2R}+

12‖x(t +N)‖2P

where

A =

[A11 00 A22

], B =

[B11 B12B21 B22

]and

Q =

[ρ1Q1 0

0 ρ2Q2

], R =

[ρ1R1 0

0 ρ2R2

], P =

[ρ1P1 0

0 ρ2P2

]


Analysis of prototypical algorithmsCentralized control (cMPC)

Since:

A is assumed to be asymptotically stable,

u(k) = 0 (with K = 0) is a suitable stabilizing auxiliary control law,

AT PA−P =−Q,

the pair (A,Q) is detectable,

then, according to the theory of MPC

stability of cMPC is guaranteed!


Analysis of prototypical algorithmsDecentralized control (dMPC)

dMPC problemsThe unconstrained minimization problems solved by regulators R1 and R2 are

minu1(t :t+N−1)V1 minu2(t :t+N−1)V2subject to subject tox1(k +1) = A11x1(k)+B11u1(k) x2(k +1) = A22x2(k)+B22u2(k)

The explicit solutions these problems can be found, since state and input constraintsare not present:

uo1 (k) = K d

1 x1(k)uo

2 (k) = K d2 x2(k)


Analysis of prototypical algorithmsDecentralized control (dMPC)In fact, consider for example the model of S1

x1(t)x1(t +1)

...x1(t +N)

︸︷︷︸

=

A0

11A1

11...

AN11

︸︷︷︸

x1(t)+

0 0 . . . 0

B11 0 . . . 0...

.... . .

...AN−1

11 B11 AN−211 B11 . . . B11

︸︷︷︸

u1(t)

u1(t +1)...

u1(t +N−1)

︸︷︷︸

x1(t : t +N) A11 B11 u1(t : t +N−1)

Denoting Q1 =diag(Q1, . . . ,Q1,P1) and R1 =diag(R1, . . . ,R1), V1 can be written as

V1 =12‖A11x1(t)+B11u1(t : t +N−1)‖2Q1

+12‖u1(t : t +N−1)‖2R1

The problem minu1(t :t+N−1) V1 leads to the solution

uo1 (t : t +N−1) =−(R1 +BT

11Q1B11)−1BT

11Q1A11x1(t)

Denoting Ei (i = 1,2) the matrix that selects from the vector uoi (t : t +N−1) the subvector uo

i (t),i.e.,

Ei =[I 0 . . . 0

]we obtain that the gain K d

1 is

K d1 =−E1 (R1 +BT

11Q1B11)−1BT

11Q1A11


Analysis of prototypical algorithmsDecentralized control (dMPC)

ThereforeK d

1 = −E1 (R1 +BT11Q1B11)

−1BT11Q1A11

K d2 = −E1 (R2 +BT

22Q2B22)−1BT

22Q2A22

Since K d1 and K d

2 result from separated ”well-posed” MPC problems:

A11 +B11K d1 and A22 +B22K d

2 are asymptotically stable.

However, the collective close-loop system results in[x1(t +1)x2(t +1)

]=

[A11 +B11K d

1 B12K d2

B21K d1 A22 +B22K 2

2

] [x1(t)x2(t)

]


Analysis of prototypical algorithmsDecentralized control (dMPC)[

x1(t +1)x2(t +1)

]=

[A11 +B11K d

1 B12K d2

B21K d1 A22 +B22K 2

2

] [x1(t)x2(t)

]

RemarksThe stability of the system is not guaranteed, due to the nonblock-diagonal elements B12K d

2 and B21K d1 .

B12 = 0 and/or B21 = 0 if

I the two subsystems are non-interacting (when both B12 = 0 andB21 = 0)

I the two subsystems are arranged in a cascaded setting (when onlyone of the two matrices is non-zero).

For continuity arguments, under the assumption of small interaction(“small” B12 and/or B21) the stability of the closed loop system ispreserved.


Analysis of prototypical algorithmsDistributed control with independent agents (iDMPC)

iDMPC problemsGiven predictions Given predictionsu(p−1)

2 (t : t +N−1) u(p−1)1 (t : t +N−1)

Model of S1: Model of S2:x1(k +1) = A11x1(k)+B11u1(k)++B12u(p−1)

2 (k)x2(k +1) = A22x2(k)+B22u2(k)+

+B21u(p−1)1 (k)

Minimization problem for R1: Minimization problem for R2:minu1(t :t+N−1) V1 minu2(t :t+N−1) V2

During each sampling period, at iteration p ≥ 0 of the ”negotiation algorithm”:

R1 knows the predicted trajectory u(p−1)2 (t : t +N−1), generated by R2 (at step p−1),

the result of the optimization carried out by R1 is u∗1(t : t +N−1|t),

u(p)1 (t : t +N−1) is generated as

u(p)1 (t : t +N−1) = (1−w1)u

(p−1)1 (t : t +N−1)+w1u∗1(t : t +N−1|t)

where w1 ∈ [0,1].



Focusing for example on R1

x1(t : t +N) = A11x1(t)+B11u1(t : t +N−1)+

+

0 0 . . . 0

B12 0 . . . 0...

.... . .

...AN−1

11 B12 AN−211 B12 . . . B12

︸︷︷︸

u(p−1)

2 (t)u(p−1)

2 (t +1)...

u(p−1)2 (t +N−1)

︸︷︷︸

B12 u(p−1)2 (t : t +N−1)

one obtains that

u∗1(t : t +N−1|t) =−(R1 +BT11Q1B11)

−1BT11Q1(A11x1(t)+B12u(p−1)

2 (t : t +N−1))

We denoteK i ,N

1 =−(R1 +BT11Q1B11)

−1BT11Q1A11

andLi ,N

1 =−(R1 +BT11Q1B11)

−1BT11Q1B12



Collectively we can write

[u∗1(t : t +N−1|t)u∗2(t : t +N−1|t)

]=

[K i ,N

1 00 K i ,N

2

][x1(t)x2(t)

]+

[0 Li ,N

1Li ,N

2 0

][u(p−1)

1 (t : t +N−1)u(p−1)

2 (t : t +N−1)

]

and

[u(p)

1 (t : t +N−1)

u(p)2 (t : t +N−1)

]=

[w1K i ,N

1 00 w2K i ,N

2

]︸︷︷︸

[x1(t)x2(t)

]+

[(1−w1)I w1Li ,N

1w2Li ,N

2 (1−w2)I

]︸︷︷︸

[u(p−1)

1 (t : t +N−1)

u(p−1)2 (t : t +N−1)

]︸︷︷︸

Ki ,N Li ,N u(p−1)(t : t +N−1)

leading to the iterative equation

u(p)(t : t +N−1) = Ki ,Nx(k)+Li ,Nu(p−1)(t : t +N−1)



u(p)(t : t +N−1) = Ki ,Nx(k)+Li ,Nu(p−1)(t : t +N−1)

Convergence of the inter-sampling iterationsOnly if the matrix Li ,N has all the eigenvalues inside the unit circle, then

u(p)(t : t +N−1)p→+∞−→ u(t : t +N−1) = (I−Li ,N)−1Ki ,Nx(k)

If such a solution exists KD = E(I−Li ,N)−1Ki ,N is the resulting (Nash solution) controlgain, being E =diag(E1,E2).The collective close-loop system results in

x(k +1) = (A+BKD)x(k)

However

there is no guarantee that A+BKD is asymptotically stable!



Different casesI) the Nash equilibrium does not exist (i.e., Li ,N is an unstable matrix): no control

law can be found (at least after an infinite number of negotiation steps);

II) the Nash equilibrium exists (i.e., Li ,N is an asymptotically stable matrix), butthe closed loop is unstable: a control law is found after an infinite number ofnegotiation steps, but it leads to instability of the closed loop system;

III) the Nash equilibrium is stable (i.e., Li ,N is an asymptotically stable matrix),and the closed loop is stable (A+BKD is stable): a stabilizing control law can befound after an infinite number of negotiation steps. However, global optimality ofthe controlled system is generally not obtained.


Analysis of prototypical algorithmsDistributed control with cooperative agents (cDMPC)

cDMPC problemsGiven predictions Given predictionsu(p−1)

2 (t : t +N−1) u(p−1)1 (t : t +N−1)

minu1(t :t+N−1) ρ1V1 +ρ2V2 minu2(t :t+N−1) ρ1V1 +ρ2V2

subject to subject tox1(k+1)=A11x1(k)+B11u1(k)+B12u(p−1)

2 (k) x2(k+1)=A22x2(k)+B22u2(k)+B21u(p−1)1 (k)

x2(k+1)=A22x2(k)+B21u1(k)+B22u(p−1)2 (k) x1(k+1)=A11x1(k)+B12u2(k)+B11u(p−1)

1 (k)

During each sampling period, at iteration p ≥ 0 of the ”negotiation algorithm”:

R1 knows the predicted trajectory u(p−1)2 (t : t +N−1), generated by R2 (at step p−1),

the result of the optimization carried out by R1 is u∗1(t : t +N−1|t),

u(p)1 (t : t +N−1) is generated as

u(p)1 (t : t +N−1) = (1−w1)u

(p−1)1 (t : t +N−1)+w1u∗1(t : t +N−1|t)

where w1 ∈ [0,1].


Analysis of prototypical algorithmsDistributed control with cooperative agents (cDMPC)Define

V(x(t),u∗1(t : t +N−1),u∗2(t : t +N−1))

as the value assumed by V subject to system

x1(k +1) = A11x1(k)+B11u1(k)+B12u2(k)x2(k +1) = A22x1(k)+B21u1(k)+B22u2(k)

where the inputs of the previous system are set as the argument of the functions V,i.e.,

u1(k) = u∗1(k) , k = t , . . . , t +N−1u2(k) = u∗2(k) , k = t , . . . , t +N−1

and with initial condition (x1(t),x2(t)) = x(t).

ResultsIt is possible to prove that

I) the optimal cost function V decreases from the time instant t−1 to the timeinstant t , if no negotiation steps are performed;

II) the optimal cost function V decreases at each negotiation step.


Analysis of prototypical algorithmsDistributed control with cooperative agents (cDMPC)

Main resultFrom I) and II), the decrease of the cost function V is proved.Stability of cDMPC is guaranteed for an arbitrary number of iterations(i.e., negotiation steps) of the algorithm within each sampling interval,and that (see in particular item II)), for p→+∞

u(p)i (t : t +N−1)→ u(opt)

i (t : t +N−1)

for i = 1,2. That is, cDMPC leads to the optimal solution obtainablewith centralized MPC.


Analysis of prototypical algorithmsDistributed control with robust agents (rDMPC)Assume that reference (predicted) trajectories u∗1(t) and u∗2(t) are defined for u1(t)and u2(t), respectively.

x1(t +1) = A11x1(t)+B11u1(t)+B12u∗2(t)+w1(t)x2(t +1) = A22x2(t)+B22u2(t)+B21u∗1(t)+w2(t)

Remarksthrough suitable constraints in the optimization problem we guarantee

u1(t) ∈ u∗1(t)⊕U1u2(t) ∈ u∗2(t)⊕U2

then the termsw1(t) = B12(u2(t)−u∗2(t))w2(t) = B21(u1(t)−u∗1(t))

can be treated as bounded unknown disturbances for subsystems S1 and S2,specifically

w1(t) ∈ W1 = B12U2w2(t) ∈ W2 = B21U1


Analysis of prototypical algorithmsDistributed control with robust agents (rDMPC)

Considering the equations:

x1(t +1) = A11x1(t)+B11u1(t)+B12u∗2(t)+w1(t)x2(t +1) = A22x2(t)+B22u2(t)+B21u∗1(t)+w2(t)

Exogenous signals and noise

the effect of w1(t) and w2(t) must be rejected by the respectiveregulators;

u∗1(k) and u∗2(k) are known exogenous input, whose presence is to becompensated (i.e., explicitly taken into account in the control law).

We use tube-based robust MPC to reject w1(t) and w2(t).



Perturbed modelsx1(t +1) = A11x1(t)+B11u1(t)+B12u∗2(t)+w1(t)x2(t +1) = A22x2(t)+B22u2(t)+B21u∗1(t)+w2(t)

Nominal modelsx1(t +1) = A11x1(t)+B11u1(t)+B12u∗2(t)x2(t +1) = A22x2(t)+B22u2(t)+B21u∗1(t)

whereu1(t) = u1(t)+K1(x1(t)− x1(t))u2(t) = u2(t)+K2(x2(t)− x2(t))

and Ki is defined in such a way that Aii +BiiKi is asymptotically stable for i = 1,2.



rDMPC problemsGiven u∗2(t : t +N−1) Given u∗1(t : t +N−1)

minu1(t :t+N−1)V1 minu2(t :t+N−1)V2subject to subject to

x1(k +1) = A11x1(k)+B11u1(k)+ x2(k +1) = A22x2(k)+B22u2(k)+

+B12u∗2(k) +B21u∗1(k)

u1(k)−u∗1(k) ∈ΔU1, k = t , . . . , t +N−1 u2(k)−u∗2(k) ∈ΔU1, k = t , . . . , t +N−1



Denoting z1(t) = x1(t)− x1(t) and z2(t) = x2(t)− x2(t):

z1(t +1) = (A11 +B11K1)z1(t)+w1(t)z2(t +1) = (A22 +B22K2)z2(t)+w2(t)

Robust positively invariant setsDefine Z1 and Z2 as an robust positively invariant sets for z1 and z2,respectively, such that

xi(0)− xi(0) ∈ Zi , wi ∈Wi

guarantees that

xi(t)− xi(t) ∈ Zi and ui(t)− ui(t) ∈ KiZi =ΔUi

for all t ≥ 0.



Additional constraints in the optimization problemsIn the MPC problems, the following constraints are present:

u1(k)−u∗1(k) ∈ ΔU1u2(k)−u∗2(k) ∈ ΔU1

for k = t , . . . , t +N−1.

Boundedness of the disturbanceThe robust positive invariant set and the additional constraints guarantee theboundedness of w1 and w2. In fact

ui(t)−u∗i (t) = ui(t)− ui(t)︸︷︷︸ + ui(t)−u∗i (t)︸︷︷︸ ∈Ui

∈ KiZi ⊕ ΔUi ⊆Ui



Main resultConvergence of rDMPC is guaranteed if

I) for all i = 1,2, Aii +BiiKi is stable,

II) for all i = 1,2, there exist Ui and Zi such that KiZi ⊂Ui . This is needed toguarantee the existence of ΔUi = Ui KiZi ;

III) the gains Ki must stabilize the system in a decentralized fashion. In fact, denotingK =diag(K1,K2), A+BK is stable;

IV) the auxiliary control must guarantee that

‖x(k +1)‖2P ≤ ‖x(k)‖2P− (‖x(k)‖2Q +‖u(k)‖2R)

where P, Q, and R are block-diagonal matrices and where the state is updatedusing the decentralized auxiliary control law K: u(k) = Kx(k) andx(k +1) = (A+BK)x(k).



Remark on II)For all i = 1,2, there exist Ui and Zi such that KiZi ⊂Ui .

WHY?

Equations


where

w1(t) = B12(u2(t)−u∗2(t))w2(t) = B21(u1(t)−u∗1(t))

1. Given U2, then W1 = B12U2 is defined.Given U1, then W2 = B21U1 is defined.

2. Given W1, then Z1 is defined.Given W1, then Z1 is defined.

3. u1(t)− u1(t) ∈ K1Z1.u2(t)− u2(t) ∈ K2Z2.

4. Only if K1Z1 ⊂U1 we can defineΔU1 = U1K1Z1 to constrain u1(t)− u1(t)in the MPC problem.Only if K2Z2 ⊂U2 we can defineΔU2 = U2K2Z2 to constrain u2(t)− u2(t)in the MPC problem.



Remark on II)For all i = 1,2, there exist Ui and Zi such that KiZi ⊂Ui .

IT IS A SMALL GAIN CONDITION!

Equations


where

w1(t) = B12(u2(t)−u∗2(t))w2(t) = B21(u1(t)−u∗1(t))



Application of the main result to the simple caseUnder the given assumptions, the choice K1 = 0, K2 = 0 is sufficient to guarantee thatall the assumptions (I-IV) are satisfied.

In fact

I) for all i = 1,2, Aii +BiiKi = Aii is stable;

II) define arbitrary neighborhoods of the origin U1 and U2. In turn, this definesW1 = B12U2 and W2 = B21U1: Since A11 +B11K1 and A22 +B22K2 are stable,RPI sets Z1 and Z2 can be automatically defined. Since K1 = 0 and K2 = 0, then

K1Z1 ⊂U1 and K2Z2 ⊂U2

III) A+BK = A is stable;

IV) matrix P =diag(P1,P2) satisfies

AT PA−P =−Q

This is a very conservative solution: better performances can be obtained by selectingdifferent values of K1 and K2.


Analysis of prototypical algorithmsSummary

Among the presented algorithms:

Stability can be guaranteed a-priori for

I centralized MPC (cMPC);I distributed control with cooperative agents (cDMPC);I distributed control with robust agents (rDMPC).

Optimality is guaranteed by:

I centralized MPC (cMPC);I distributed control with cooperative agents (cDMPC), when

inter-sampling iterations reach the convergence;

cDMPC extensions: output-feedback, tracking, nonlinear and unstablesystems;

rDMPC extensions: output-feedback, tracking, unstable systems;


Analysis of prototypical algorithmsA simple example

The systemsx1(t +1) = a1x1(t)+b11u1(t)+b12u2(t)x2(t +1) = a2x2(t)+b22u2(t)+b12u1(t)

Model parameters a1 = 0.99 and a2 = 0.9.

MPC parameters: N = 1, R1 = R2 = 0.1, Q1 = Q2 = 1 and Pi = Qi/(1−ai ),i = 1,2.

iDMPC and cDMPC parameters: w1 = w2 = 0.5 and ρ1 = ρ2 = 0.5.

Three cases:

I) b11 = 1, b12 = .5, b21 = .2, and b22 = 1;

II) b11 = 0.001, b12 =−1.1, b21 =−0.9, and b22 = 10;

III) b11 = 1, b12 = 5, b21 = 2, and b22 = 1.



Case IcMPC: the spectral radius of the controlled system is 0.009.

dMPC: b12 and b21 are sufficiently small to preserve stability of the closed-loopsystem. Spectral radius: 0.297.

iDMPC: The matrix Li ,N is stable, then there exists a Nash equilibrium. If p→ ∞

the obtained controlled system has a spectral radius 0.008.

cDMPC: If p→ ∞ the obtained controlled system has a spectral radius 0.009(i.e., it is equal to the cMPC).



Case ISolutions of the optimization problems using different distributed control approaches.



Case ISimulations



Case IIcMPC: the spectral radius of the controlled system is 0.103.

dMPC: b12 and b21 are not sufficiently small to preserve stability, and theclosed-loop system is unstable (spectral radius 1.071).

iDMPC: The matrix Li ,N is stable, then there exists a Nash equilibrium. If p→ ∞

the obtained controlled system has a spectral radius 1.098.




Case IISolutions of the optimization problems using different distributed control approaches.



Case IISimulations



Case IIIcMPC: the spectral radius of the controlled system is 0.003.

dMPC: b12 and b21 are not sufficiently small to preserve stability of theclosed-loop system. Spectral radius: 2.974.

iDMPC: The matrix Li ,N is unstable (spectral radius equal to 2.073). No control.




Case IIISolutions of the optimization problems using different distributed control approaches.



Case IIISimulations


Outline

1 Models




5 Conclusions



ExamplesTemperature control

Consider the problem of controlling the temperature of the building



Dynamically decoupled submodelsThe M = 2 dynamically decoupled sub-models are:

δT (1)

A (t +1)δT (1)

B (t +1)δT (1)

C (t +1)δT (1)

D (t +1)

= A11

δT (1)

A (t)δT (1)

B (t)δT (1)

C (t)δT (1)

D (t)

+B11

[δqA(t)δqC(t)

]+B12

[δqB(t)δqD(t)

]

δT (2)

A (t +1)δT (2)

B (t +1)δT (2)

C (t +1)δT (2)

D (t +1)

= A22

δT (2)

A (t)δT (2)

B (t)δT (2)

C (t)δT (2)

D (t)

+B22

[δqB(t)δqD(t)

]+B21

[δqA(t)δqC(t)

]

y1(t) = C1

δT (1)

A (t)δT (1)

B (t)δT (1)

C (t)δT (1)

D (t)

y2(t) = C2

δT (2)

A (t)δT (2)

B (t)δT (2)

C (t)δT (2)

D (t)



Dynamically decoupled submodels (forward Euler discretization)

Aii=

1− γol τ γ2τ γ1τ 0

γ2τ 1− γol τ 0 γ1τ

γ1τ 0 1− γol τ γ2τ

0 γ1τ γ2τ 1− γol τ

,

B11=B21=

1 00 00 10 0

τ,B12=B22=

0 01 00 00 1

τ

C1 = 1τBT

11, and C2 = 1τBT

22.

Parametersconstraints on inputs ∈ [−2 ·10−3,2 ·10−3],

ρ1 = ρ2 = w1 = w2 = 0.5, Q1 =diag(1,0,1,0), Q2 =diag(0,1,0,1), R1 = R2 = I2,

N = 7.



ResultsDynamics of the aggregate state variable

(14(δT 2

A +δT 2B +δT 2

C +δTD))1/2



ResultsDynamics of the aggregate input variable

14(δq2

A +δq2B +δq2

C +δqD))1/2


Outline

1 Models




5 Conclusions



ExamplesThree-tank system

Consider the system illustrated in the following Figure, consisting in acascade interconnection of three tanks.



Dynamically decoupled submodels (forward Euler discretization)The M = 2 dynamically decoupled sub-models are:

δx (1)2 (t +1) = (1− τk2)δx (1)

2 (t)+ τδu2(t)δx (2)2 (t +1)

δx (2)1 (t +1)

δx (2)3 (t +1)

= A22

δx (2)2 (t)

δx (2)1 (t)

δx (2)3 (t)

+B22δu1 +B21

where

A22=I3+

−1 0 00 −1 11 0 −1

τ,B22=

001

,B21=

100

,C2=[0 1 0

]

Parametersconstraints on inputs ∈ [−0.3,0.3],

ρ1 = ρ2 = w1 = w2 = 0.5, Q1 = 1, Q2 =diag(0,1,0), R1 = R2 = 1,

N = 7.



ResultsDynamics of the aggregate state variable

(13(δx2

1 +δx22 +δx2

3 )1/2



ResultsDynamics of the aggregate input variable

(12(δu2

1 +δu22))

1/2


Outline

1 Models




5 Conclusions



ConclusionsTake-home messages:

there are many different methods for distributed and decentralizedMPC, with different performances and features;for all methods there are pros and cons. E.g.,

I to obtain global optimality, relevant communication load is required,I to reduce as much as possible both communication and

computational load, robustness-based methods are often required:loss of optimality,

for some of the proposed methods convergence can be a-prioriguaranteed.


Outline

1 Models




5 Conclusions



Suggested readings

Generic referencesE. Camponogara, D. Jia, B. H. Krogh, and S. Talukdar. Distributed model predictivecontrol. IEEE Control Systems Magazine, 1:44 - 52, February 2002.

J. B. Rawlings and B. T. Stewart. Coordinating multiple optimization-based controllers: newopportunities and challenges. Journal of Process Control, 18: 839 – 845, 2008.

J. B. Rawlings and D. Q. Mayne. Model Predictive Control: Theory and Design. Nob HillPublishing, Madison, WI, 2009.

R. Scattolini. Architectures for distributed and hierarchical Model Predictive Control – Areview. Journal of Process Control, 19(5):723 – 731, May 2009.

P. D. Christofides, R. Scattolini, D. Munoz de la Pena, J. Liu. Distributed model predictivecontrol: A tutorial review and future research directions, Computers & ChemicalEngineering, 51:21-41, 2013.

J. M. Maestre, R.R. Negenborn (editors). Distributed predictive control made easy.Springer, 2014.

R.R. Negenborn, J.M. Maestre. Distributed model predictive control - An overview androadmap of future research opportunities. IEEE Control Systems Magazine, 34(4):87-97,2014.


Decentralized and distributed control - Decentralized and...

Documents

Transcript of Decentralized and distributed control - Decentralized and...