Hybrid MPC

Hybrid Systems: Model Predictive Control

Alberto Bemporad - “Model Predictive Control” course - Academic year 2016/17

MPC of hybrid systems

Useahybriddynamicalmodeloftheprocesstopredictitsfutureevolutionandchoosethe“best”controlaction

processmodel-basedoptimizer

reference outputcontrolinput

measurements

r(t) u(t) y(t)

xk+1 = Axk +B1uk +B2k +B3zk +B5yk = Cxk +D1uk +D2k +D3zk +D5

E2k + E3zk E4xk + E1uk + E5

wheretheequilibriumconditionisobtainedbysolvingthefollowingmixed-integerfeasibilityproblem

•Attimetsolvethefinite-horizonoptimalcontrolproblemw.r.t. U ,

64u0...

minu0,...,uN1

k=0kyk rk2Q + kuk urk2R

+kk rk2 + kzk zrk2 + kxk xrk2

subject to MLD equations

x0 = x(t)

xN = xr

kvk2Q , v0Qv

xr = Axr +B1ur +B2r +B3zrr = Cxr +D1ur +D2r +D3zr

E2r + E3zr E4xr + E1ur + E5

Q,R 0, > 0

HybridModelPredictiveControlMPC for hybrid systems

•Applyonlyu(t)=u0*(discardtheremainingoptimalinputs)•Attimet+1:getnewmeasurements,repeatoptimization

(xr, ur, r, zr)

notation:

y0kQyk + u0kRuk

20H + x

0(t)F +1

2x0(t)Y x(t)

s.t. G W + Sx(t)

= [u0, . . . , uN1, 0, . . . , N1, z0, . . . , zN1]

u Rmc 0,1mb

0,1rbz Rrc

R(mc+rc)N 0,1(mb+rb)N

MIQPFormulationofMPCMIQP formulation of MPC

MixedIntegerQuadraticProgram(MIQP)

(Bemporad,Morari,1999)

•Optimizationvector:

vectorξhasbothrealandbinaryvalues

8Q,R 0, 8 > 0

limt!1

y(t) = r

limt!1

u(t) = ur

limt!1

x(t) = xr

limt!1

(t) = r

limt!1

z(t) = zr

Closed-loopConvergenceClosed-loop convergence

Proof:EasilyfollowsfromstandardLyapunovarguments(Bemporad,Morari1999)

TheoremLetbetheequilibriumvaluescorrespondingtosetpointr.Assumex(0)issuchthattheMPCproblemisfeasibleattimet=0.

Thentheclosed-loophybridMPCloopconvergesasymptotically

andallconstraintsarefulfilledateachtimet≥0.

(xr, ur, r, zr)

(Lazar,Heemels,Weiland,Bemporad,2006)

Lyapunovasymptoticstabilityandexponentialstabilitycanbeguaranteedbychoosingaproperterminalcostandconstraintset

• Byconstructionisfeasible@t+1,asitsatisfiesallMLDconstraintsandterminalconstraint

• SinceR,Q>0,,andallothervariablesconverge.

• ismonotonicallydecreasingand≥0,so

• Thecostofis

• Letbetheoptimalsequence@t

V (x(t)) 9 limt!1

V (x(t)) , V1

V (x(t)) ky(t) rk2Q ku urk2R

k(t) rk2 + kz(t) zrk2 + kx(t) xrk2

V (x(t+1))

k(t) rk2, kz(t) zrk2, kx(t) xrk2 ! 0ky(t) rk2Q, ku(t) urk2R

limt!1

y(t) = r, limt!1

u(t) = ur

t = [ut0, . . . , utN1,

t0, . . . ,

tN1, z

t0, . . . , z

t+1 = [ut1, . . . , u

tN1, ur,

t1, . . . ,

tN1, r, z

t1, . . . , z

tN1, zr]

Convergence proof

Globaloptimumisnotneededtoproveconvergence!

Mainidea:UsevaluefunctionV*(x(t))asaLyapunovfunction

• Henceand

kQykk1 + kRukk1

minx, s.t. x, x

yk ±Qiyk i = 1, . . . ,my k = 0, . . . , N 1uk ±Riuk i = 1, . . . ,mu k = 0, . . . , N 1

yk kQykk1uk kRukk1

Q 2 Rmyny

R 2 Rmunu

= [y0, . . . , yN1,

u0, . . . ,

uN1, u0, . . . , uN1, 0, . . . , N1, z0, . . . , zN1]

MIQPFormulationofMPCMILP formulation of MPC(Bemporad,Borrelli,Morari,2000)

MixedIntegerLinearProgram(MILP)

•Optimizationvector:

•Introduceslackvariables:

yk + uk

s.t. G W + Sx(t)

vectorξ hasbothrealandbinaryvalues

HybridMPC-ExampleHybrid MPC example

open-loopsimulation

PWAsystem:

constraints:

gotodemo/demos/hybrid/bm99sim.m

/demos/hybrid/bm99.hys

/* 2x2 PWA system - Example from the paper A. Bemporad and M. Morari, ``Control of systems integrating logic, dynamics, and constraints,'' Automatica, vol. 35, no. 3, pp. 407-427, 1999. (C) 2003 by A. Bemporad, 2003 */

SYSTEM pwa INTERFACE STATE REAL x1 [-10,10]; REAL x2 [-10,10];

INPUT REAL u [-1.1,1.1];

OUTPUT REAL y;

PARAMETER REAL alpha = 1.0472; /* 60 deg in radiants */ REAL C = cos(alpha); REAL S = sin(alpha);

IMPLEMENTATION AUX REAL z1,z2; BOOL sign; AD sign = x1<=0;

DA z1 = IF sign THEN 0.8*(C*x1+S*x2) ELSE 0.8*(C*x1-S*x2) ; z2 = IF sign THEN 0.8*(-S*x1+C*x2) ELSE 0.8*(S*x1+C*x2) ; CONTINUOUS x1 = z1; x2 = z2+u;

OUTPUT y = x2;

HYSDELmodel

k=1|yk r(t)|

closed-loop:

performanceindex:

output ref

output & ref

Hybrid MLD SystemHybrid Controller

k=0kx2k r(t)k1

(x1k 25, k = 1,2MLD model

Hybrid MPC – Temperature control

>>[XX,UU,DD,ZZ,TT]=sim(C,S,r,x0,Tstop);

>>C=hybcon(S,Q,N,limits,refs);

>>refs.x=2; % just weight state #2>>Q.x=1; % unit weight on state #2>>Q.rho=Inf; % hard constraints>>Q.norm=Inf; % infinity norms>>N=2; % prediction horizon>>limits.xmin=[25;-Inf];

Hybrid controller based on MLD model S <heatcoolmodel.hys> [Inf-norm]

2 state measurement(s) 0 output reference(s) 0 input reference(s) 1 state reference(s) 0 reference(s) on auxiliary continuous z-variables 20 optimization variable(s) (8 continuous, 12 binary) 46 mixed-integer linear inequalitiessampling time = 0.5, MILP solver = 'glpk' Type "struct(C)" for more details.>>

Hybrid MPC – Temperature control

Mixed-IntegerProgramSolversMixed-Integer Program (MIP) solvers

•Generalpurposebranch&bound/branch&cutsolversavailableforMILP

andMIQP(CPLEX,GLPK,Xpress-MP,CBC,Gurobi,...)

•Noneedtoreachglobaloptimum(seeproofofthetheorem),althoughperformancedeteriorates

http://plato.la.asu.edu/bench.htmlMoresolversandbenchmarks:

•Mixed-IntegerProgrammingisNP-complete

•QPalgorithmbasedonNNLSisextendedheretohandle- equalityconstraints- bilateralconstraints- warm-starts

V (z) , 1

2z0Qz + c0z

s.t. ` Az u

Gz = g

Aiz 2 ¯i, ui, i = 1, . . . , q

Q = Q0 0

¯i Aiz ui

¯i = 0, ui = 1, Ai = [0 . . .0 1 0 . . .0]

Branch & Bound for MIQP (Using NNLS solver for QP)•WeconsideraMIQPproblemofthefollowingform

•Binaryconstraintsonzareaspecialcase:

sotosolveMIQPrelaxations()veryefficiently

(Bemporad,NMPC,2015)

Aiz 2 ¯i, ui8i = 1, . . . , q

Branch & Bound for MIQP (Using NNLS solver for QP)

•Branchandboundscheme:

V (z) , 1

2z0Qz + c0z

s.t. ` Az u

Gz = g

¯ Az u

QPinfeasible

no yesMIQPinfeasible

integerfeasible

MIQPsolutionfound

startbranching...

(lucky case)(lucky case)

(typical case)

QP relaxation

V (z) , 1

2z0Qz + c0z

s.t. ` Az u

Gz = g

Aiz = ¯i¯j Ajz uj, j 6= i min

zV (z) , 1

2z0Qz + c0z

s.t. ` Az u

Gz = g

Aiz = ui¯j Ajz uj, j 6= i

Aiz¯i+ui

2 ¯i ui

•Branching:pickupindexisuchthatisclosestto

V (z) , 1

2z0Qz + c0z

s.t. ` Az u

Gz = g

¯ Az u

QP1 QP2

•SolvetwonewQPproblems:

WarmstartfromprevioussolutionofQP0heavilyexploitedinsolvingQP1,QP2!

updateupperbound

onMIQPsolution

QP1 QP2

QPinfeasible

integerfeasible

keepbranching...

stopbranchingonsubtree

upperbound

onMIQPisavailable

QP1 QP2

optimum>=V0 ?

keepbranching...

stopbranching!(addingfurtherequality

constraintscanonlyincreasetheoptimalcost)

• LowerboundVonoptimalsolutionofQPrelaxationareimmediatelyavailableduringtheiterationsoftheQPalgorithm(dualcost)

This saves a lot of QP active set iterations !

•Whennofurtherbranchingispossible,eithertheMIQPproblemisrecognizedinfeasibleortheoptimalsolutionhasbeenfound

V V0stopsolvingQPrelaxation(andfurtherbranching)if

MIQP via NNLS: Numerical results

QPalgorithmincompiledEmbeddedMATLABcode,B&BininterpretedMATLABcode.CPUtimemeasuredonthisMac

(NNLSLDL in the table) of

MP`d`P`

MPu duPuN f

recursively as described in (Bemporad, 2015b) when the

dimension of vectorh

iis smaller or equal than n, and,

for improved numerical robustness, QR factorization incase more than n elements must be optimized in Prob-lem (4). As an alternative, we purely updated the QRfactorization of the same matrix (NNLSQR in the table)recursively as described in (Lawson and Hanson, 1974,Chap. 24, Method 1).

n m q NNLSLDL NNLSQR GUROBI CPLEX10 5 2 2.3 1.2 1.4 8.010 100 2 5.7 3.3 6.1 31.450 25 5 4.2 6.1 14.1 30.150 200 10 68.8 104.4 114.6 294.1

100 50 2 4.6 10.2 37.2 69.2100 200 15 137.5 365.7 259.8 547.8150 100 5 15.6 49.2 157.2 260.1150 300 20 1174.4 3970.4 1296.1 2123.9

Table 1. Worst-case CPU time (ms) on randomMIQP problems over 20 instances for each

combination of n, m, q.

It is apparent that on such a set of random MIQPproblems, Algorithm 2 performs comparably well withrespect to the commercial solvers GUROBI and CPLEX,especially when the number q of binary constraints is smallcompared to n and m, probably due to the pure B&Bnature of Algorithm 2.

The results shown in Table 2 are obtained, under the sameconditions, on purely binary quadratic programs (n = q,m = 5n). When turning the presolver on, in GUROBI andCPLEX the results remain rather similar.

n m q NNLSLDL NNLSQR GUROBI CPLEX2 10 2 5.1 4.0 0.7 8.44 20 4 8.9 4.3 4.5 16.78 40 8 19.2 18.0 37.1 14.7

12 60 12 59.7 57.8 82.3 47.920 100 20 483.5 457.7 566.8 99.625 250 25 110.4 93.3 1054.4 169.430 150 30 1645.4 1415.8 2156.2 184.5

Table 2. Worst-case CPU time (ms) for randombinary QP problems with n variables and 5mconstraints, over 20 instances for each value of

n and the corresponding m, q.

5.2 Hybrid MPC problem

In order to test Algorithm 2 in a hybrid MPC problem (2)–(3), we consider the hybrid control problem describedin (Bemporad and Morari, 1999, Example 5.1) with allzero weights except a unit weight on the output of thesystem (these are the settings of the demo bm99sim.m inthe Hybrid Toolbox for MATLAB (Bemporad, 2003)) anda prediction horizon T between 2 and 10.

The regularization term 104I was added on the resultingHessian matrix Q to make the resulting MIQP’s positivedefinite. This induces a small di↵erence in the input and

N NNLSLDL NNLSQR GUROBI CPLEX2 2.2 2.3 1.2 3.03 3.4 3.9 2.0 6.54 5.0 6.5 2.6 8.15 7.6 9.8 3.7 9.06 12.3 17.7 4.3 11.07 20.5 30.5 5.8 13.18 28.9 47.1 7.3 17.39 38.8 62.5 9.5 18.9

10 55.4 98.2 10.9 22.4

Table 3. Hybrid MPC problem: CPU time(ms) per sampling step for di↵erent prediction

horizons N

output trajectories, however the norm of the di↵erencebetween the entire trajectories smaller than 0.001. Wecompare Algorithm 2 with preconditioning (16) againstGUROBI and CPLEX with presolvers enabled. The resultsare reported in Table 3. For T = 10, the MIQP problemhas n = 40, q = 10 (i.e., 30 continuous variables and10 binary variables) and m = 160 linear inequalities.We observed that disabling presolvers in GUROBI andCPLEX sometimes speeds up sometimes slows down thesolver.

6. CONCLUSIONS

In this paper we have proposed a new MIQP solver basedon B&B that is tailored to embedded hybrid MPC appli-cations. The approach extends an active set method re-cently developed by the author to solve QP relaxations asnonnegative least-squares problems. While the presentedapproach was shown e↵ective in simulations compared toreference commercial solvers, current research is devotedto combine hybrid models and MIQP solution methods forreaching even higher degrees of e↵ectiveness.

REFERENCES

Axehill, D. and Hansson, A. (2006). A mixed integer dualquadratic programming algorithm tailored for MPC. InProc. 45th IEEE Conference on Decision and Control,5693–5698. San Diego, CA, USA.

Bemporad, A. (2003). Hybrid Toolbox – User’sGuide. http://cse.lab.imtlucca.it/~bemporad/hybrid/toolbox.

Bemporad, A. (2015a). A multiparametric quadraticprogramming algorithm with polyhedral computationsbased on nonnegative least squares. IEEE Trans. Auto-matic Control. In press.

Bemporad, A. (2015b). A quadratic programming algo-rithm based on nonnegative least squares with appli-cations to embedded model predictive control. IEEETrans. Automatic Control. Conditionally accepted forpublication.

Bemporad, A. and Morari, M. (1999). Control of systemsintegrating logic, dynamics, and constraints. Automat-ica, 35(3), 407–427.

Bertsekas, D. (2009). Convex Optimization Theory.Athena Scientific.

Bierlaire, M., Toint, P., and Tuyttens, D. (1991). Oniterative algorithms for linear ls problems with boundconstraints. Linear Algebra and Its Applications, 143,111–143.

n=#variables,m=#inequalityconstraints,noequalities,q=#binaryconstraints

•Worst-caseCPUtimeonrandomMIQPproblems:

NNLSLDL=recursiveLDLfactorizationusedtosolveleast-squareproblemsinQPsolverNNLSQR=recursiveQRfactorizationusedinstead(numericallymorerobust)

(Bemporad,NMPC2015)

n = 40, m = 160, q = 10

Numerical results

•Worst-caseCPUtime(ms)onrandompurelybinaryQPproblems:

MP`d`P`

MPu duPuN f

100 50 2 4.6 10.2 37.2 69.2100 200 15 137.5 365.7 259.8 547.8150 100 5 15.6 49.2 157.2 260.1150 300 20 1174.4 3970.4 1296.1 2123.9

12 60 12 59.7 57.8 82.3 47.920 100 20 483.5 457.7 566.8 99.625 250 25 110.4 93.3 1054.4 169.430 150 30 1645.4 1415.8 2156.2 184.5

10 55.4 98.2 10.9 22.4

horizons N

6. CONCLUSIONS

REFERENCES

•Worst-caseCPUtime(ms)onahybridMPCproblem (Bemporad,Morari,1999)

MP`d`P`

MPu duPuN f

100 50 2 4.6 10.2 37.2 69.2100 200 15 137.5 365.7 259.8 547.8150 100 5 15.6 49.2 157.2 260.1150 300 20 1174.4 3970.4 1296.1 2123.9

12 60 12 59.7 57.8 82.3 47.920 100 20 483.5 457.7 566.8 99.625 250 25 110.4 93.3 1054.4 169.430 150 30 1645.4 1415.8 2156.2 184.5

10 55.4 98.2 10.9 22.4

horizons N

6. CONCLUSIONS

REFERENCES

N=predictionhorizon

MIQPregularizedtomakeQstrictly>0(-->solutiondifferenceisnegligible)

V (z) , 1

2z0Qz + c0z

s.t. ` Az u

Aeqz = beq

Aiz 2 ¯i, ui, i = 1, . . . , p

¯i Aiz ui

Fast gradient projection for MIQP•MIQPproblem

•Usebranch&bound,relaxbinaryconstraintsto

special case:binary constraints zi∊0,1

constraint is relaxed

constraint is fixed

constraint is ignored

(Naik,Bemporad,2016)

Aiz ui ! yik+1 = maxnyik + sik,0

Aiz = ui ! yik+1 = yik + sik

Aiz = ¯i ! yk+1i = 0

•OnlyprojectionchangesfromoneQPrelaxationtoanother:

yi≥0

yi≷0

Fast gradient projection for MIQP

•Numericalresults(timeinms):

First we consider the hybrid vehicle example describedin (Takapoui et al., 2016), that consists of the combinationof a battery, an electric motor/generator, and an engine.For a given power demand P

dest at time t = 0, . . . , T 1,

the objective is to plan the battery power P battt and engine

power P engt for the time interval t = 0, . . . , T1, such that

Pbattt + P

engt P

dest . Let Et be the energy of the battery

at time t, Et+1 = Et Pbattt , where is the sample time.

The hybrid vehicle control problem is

min (ET Emax)2 +

f(P engt , zt) + (zt zt1)+

s.t. Et+1 = Et Pbattt

Pbattt + P

engt P

zt 2 0, 1, t = 0, . . . , T 1

where Pbattt , P

engt , zt (engine on/o↵) and Et are the

optimization variables. The term (zt zt1)+ penalizesthe engine going from the o↵ to the on state. By choosingT = 72 steps, the resulting MIQP problem has n = 862optimization variables, p = 72 binary variables, m = 503inequality constraints, q = 575 equality constraints.

The solver GUROBI (Gurobi Optimization, Inc., 2014)computes the optimal cost V

= 135.9 in 21.05 s withdefault options. The cost calculated by the ADMM-basedheuristic approach of Takapoui et al. (2016), denotedas miqpADMM, is 138.1 1 and is obtained in 0.40 s forpreconditioning + 3.55 s for solving the problem.

The performance of Algorithm 1+4+2 implemented ininterpreted MATLAB code, denoted as miqpGPAD-H, isreported in Table 1 for di↵erent values of the feasibil-ity tolerance A and optimality tolerance V . Whenevercondition (9) is satisfied at a given iteration k, ratherthan restarting the values of k we just assign wk yk,weq,k k in Step (5) for that iteration. We also intro-duce the regularization term 103

I in the cost function tomake the primal Hessian matrix Q positive definite. Thetolerance value used in Algorithm 2 is I = 1e 2.

V , A Cost Precond., Solving Constr. violation

1e-2, 1e-2 131.5 1.19, 3.81 s 3.62e-1

1e-2, 1e-3 135.9 1.12, 8.62 s 8.09e-3

1e-3, 1e-3 135.9 1.07, 8.67 s 7.98e-3

1e-3, 1e-4 136.0 1.14, 13.37 s 2.59e-3

1e-4, 1e-3 136.1 1.10, 15.87 s 1.08e-3

Table 1. Performance comparison with di↵er-ent values of V , A for miqpGPAD-H

Figure 1 shows the trajectories obtained with miqpGPAD-

H for V = 1e2, A = 1e3 and compare them with theones obtained by GUROBI and miqpADMM. It is apparentthat the proposed heuristic approach is computationallyfaster and very simple to implement in an embeddedcontrol platform, and keeps the quality of the solution overa sucient level for the practical application at hand.

Next, we test the B&B method miqpGPAD on randomlygenerated MIQP problems with n variables, m inequality1 https://github.com/cvxgrp/miqp_admm/tree/master/matlab/vehicle.m

0 10 20 30 40 50 60 70

t0 10 20 30 40 50 60 70

miqpADMM

GUROBI

miqpGPAD

Fig. 1. Engine power, battery power, battery energy, andengine on/o↵ signals versus time.(V = 1e 2, A =1e 3 case for miqpGPAD), cost=135.9

constraints, p binary constraints, q equality constraints,condition number = 10 of the primal Hessian Q

2 .Algorithm 3 is implemented in interpreted MATLAB codeand Algorithms 1, 2, 4 are implemented in EmbeddedMATLAB and complied. The tolerance values used in

n m p q miqpGPAD GUROBI

10 100 2 2 15.6 6.56

50 25 5 3 3.44 8.74

50 150 10 5 63.22 46.25

100 50 2 5 6.22 26.24

100 200 15 5 164.06 188.42

150 100 5 5 31.26 88.13

150 200 20 5 258.80 274.06

200 50 15 6 35.08 144.38

Table 2. Average CPU time (ms) on randomMIQP problems over 50 instances for each

combination of n, m, p, q.

Algorithm 1 are V , A = 1e 5, in Algorithm 2 I =1e 2. The CPU time reported for solving feasible MIQPproblems, averaged over 50 executions is listed in Table 2.The results show that the proposed scheme performs wellas compared to the commercial GUROBI solver, but hasthe advantage of a very simple coding, therefore making itvery suitable for embedded control applications.

6. CONCLUSION

In this paper we have presented an exact and a heuristicapproach to solve MIQPs based on accelerated gradientprojection methods applied on the dual QP relaxations. Inspite of their simplicity of code, the proposed approaches2

The entries of matrix A are generated from the normal distribution

N (0, 0.0025), `, u from the uniform distribution U(0, 100), c from

N (0, 1); matrix Q = UV 0, where U, V are orthogonal matrices

generated by QR decomposition of random n n matrices, and

is diagonal with nonzero entries having logarithms equally spaced

between ± log()/4 (Bierlaire et al., 1991).

•SamedualQPmatrices,preconditioningonlycomputedatrootnode

•Warm-startexploited,dualcostusedtostopQPrelaxationsearlier

•CriterionbasedonFarkaslemmatodetectQPinfeasibility

(Naik,Bemporad,2016)

min 12x

0Qx+ q0xs.t. ` Ax u

Aix 2 `i, ui, i 2 I

quantization

Heuristic ADMM method for (Suboptimal) MIQP(Takapoui,Moehle,Boyd,Bemporad,ACC’16)

•MIQPproblem:

• ADMMiterations:

• Iterationsconvergetoa(local)solution

xk+1 = (Q+ ATA)1(AT (yk zk) + q)

zk+1 = minmaxAxk+1 + yk, `, u

zk+1i =

:`i if zk+1

i < `i+ui2

ui if zk+1i `i+ui

2 , i 2 I

yk+1 = yk +Axk+1 zk+1

(Naik,Bemporad,2016)• Similarideaalsoapplicabletofastgradientmethods

ADMM method for MIQP(Takapoui,Moehle,Boyd,Bemporad,ACC’16)

• Example:parallelHybridElectricVehiclecontrolproblem

0 10 20 30 40 50 60 700

0 10 20 30 40 50 60 70-20

0 10 20 30 40 50 60 700

0 10 20 30 40 50 60 70-20

0 10 20 30 40 50 60 700

Fig. 1: Engine power, battery power, battery energy, and engine on/off signals versus time. Left: the globalsolution. Right: the solution found using ADMM (Algorithm 1).

+−utVdc

Fig. 2: Converter circuit model.

Figure 1, we see that qualitatively, the optimal trajectoryand the trajectory generated by ADMM are very similar.

C. Power converter control

We consider control of the switched-mode powerconverter shown in Figure 2. The circuit dynamics are

ξt+1 = Gξt +Hut, t = 0, 1, . . . , T − 1,

where ξt = (i1,t, v1,t, i2,t, v2,t) is the system state atepoch t, consisting of all inductor currents and capacitorvoltages, and ut ∈ −1, 0, 1 is the control input. Thedynamics matrices G ∈ R4×4 and H ∈ R4×1 areobtained by discretizing the dynamics of the circuit inFigure 2.

We would like to control the switch configurations sothat v2 tracks a desired sinusoidal waveform. This canbe done by solving

minimize!T

t=0(v2,t − vdes)2 + λ|ut − ut−1|subject to ξt+1 = Gξt +Hut

ξ0 = ξTu0 = uT

ut ∈ −1, 0, 1,

where λ ≥ 0 is a tradeoff parameter between output volt-age regulation and switching frequency. The variablesare ξt for t = 0, . . . , T and ut for t = 0, . . . , T − 1.

Note that if we take λ = 0, and take the inputvoltage ut to be unconstrained (i.e., allow ut to take anyvalues in R), (6) can be solved as a convex quadraticminimization problem, with solution ξlst . Returning to

our original problem, we can penalize deviation fromthis ideal waveform by including a regularization termµ∥ξ − ξlst ∥

2 to (6), where µ > 0 is a positive weightingparameter. We solved this regularized version of (6), withL1 = 10 µH, C1 = 1 µF, L2 = 10 µH, C2 = 10 µF,R = 1Ω, Vdc = 10V, T = 100 (with a discretizationinterval of 0.5 µs), λ = 1.5V2, and µ = 0.1. Werun algorithm 1 with ρ = 2.7 and 500 iterations forthree different initializations. An approximate solution isfound via our heuristic in less than 2 seconds, whereasit takes MOSEK more than 4 hours to find the globalsolution. Figure 3 compares the approximate solutionderived by the heuristic with the global solution.

D. Signal decoding

We consider maximum-likelihood decoding of a mes-sage passed through a linear multiple-input and multiple-output (MIMO) channel. In particular, we have

y = Hx+ v,

where y ∈ Rp is the message received, H ∈ Rp×n isthe channel matrix, x ∈ Rn is the message sent, and theelements of the noise vector v ∈ Rp are independent,identically distributed Gaussian random variables. Wefurther assume that the elements of x belong to the signalconstellation −3,−1, 1, 3. The maximum likelihoodestimate of x is given by the solution to the problem

minimize ∥Hx− y∥2

subject to xi ∈ − 3,−1, 1, 3, i = 1, . . . , n,(7)

where x ∈ Rn is the variable.We generate 1000 random problem instances with

H ∈ R2000×400 chosen from a standard normal distri-bution. The uncorrupted signal x is chosen uniformlyrandomly and the additive noise is Gaussian such thatthe signal to noise ratio (SNR) is 8 dB. For sucha problem in embedded application, branch-and-bound

CONFIDENTIAL. Limited circulation. For review only.

Preprint submitted to 2016 American Control Conference.

Received September 27, 2015.

0 10 20 30 40 50 60 700

0 10 20 30 40 50 60 70-20

0 10 20 30 40 50 60 700

0 10 20 30 40 50 60 70-20

0 10 20 30 40 50 60 700

+−utVdc

ξt+1 = Gξt +Hut, t = 0, 1, . . . , T − 1,

minimize!T

ξ0 = ξTu0 = uT

ut ∈ −1, 0, 1,

D. Signal decoding

y = Hx+ v,

subject to xi ∈ − 3,−1, 1, 3, i = 1, . . . , n,(7)

optimalsolution ADMMsolution

energy stored in battery

engine on/off

engine power

electrical power

ADMM method for (Suboptimal) MIQP(Takapoui,Moehle,Boyd,Bemporad,ACC’16)

• Example:powerconvertercontrol

optimalsolution ADMMsolution

output voltagev2

input voltage sign ut

0 10 20 30 40 50 60 700

0 10 20 30 40 50 60 70-20

0 10 20 30 40 50 60 700

0 10 20 30 40 50 60 70-20

0 10 20 30 40 50 60 700

+−utVdc

ξt+1 = Gξt +Hut, t = 0, 1, . . . , T − 1,

minimize!T

ξ0 = ξTu0 = uT

ut ∈ −1, 0, 1,

D. Signal decoding

y = Hx+ v,

subject to xi ∈ − 3,−1, 1, 3, i = 1, . . . , n,(7)

0 10 20 30 40 50 60 700

0 10 20 30 40 50 60 70-20

0 10 20 30 40 50 60 700

0 10 20 30 40 50 60 70-20

0 10 20 30 40 50 60 700

+−utVdc

ξt+1 = Gξt +Hut, t = 0, 1, . . . , T − 1,

minimize!T

ξ0 = ξTu0 = uT

ut ∈ −1, 0, 1,

D. Signal decoding

y = Hx+ v,

subject to xi ∈ − 3,−1, 1, 3, i = 1, . . . , n,(7)

0 10 20 30 40 50 60 70 80 90 100−1

−0.5

0 10 20 30 40 50 60 70 80 90 100−1

−0.5

0 10 20 30 40 50 60 70 80 90 100-1

Fig. 3: The switch configuration and the output voltage. Left: the global solution. Right: the solution usingADMM (Algorithm 1).

methods are not desirable due to their worst-case timecomplexity. We run the heuristic with only one initial-ization, with 10 iterations to find xadmm. The averageruntime for each problem (including preprocessing) is 80milliseconds, which is substantially faster than branch-and-bound based methods. We compare the performanceof the points xadmm with the points found by relax-and-round technique xrlx. In Figure 4 we have plotted thehistogram of the difference between the objective valuesevaluated at xadmm and xrlx. Depicted in Figure 4, we seethat in 95% of the cases, the bit error rate (BER) usingour heuristic was at least as good as the bit error rateusing relax and round.

IV. CONCLUSIONS

In this paper, we introduced an effective heuristicfor finding approximate solutions to convex quadraticminimization problems over the intersection of affine andnonconvex sets. Our heuristic is significantly faster thanbranch-and-bound algorithms and has shown effective ina variety of embedded problems including hybrid vehiclecontrol, power converter control, and signal decoding.

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