Model Predictive Control with a Relaxed Cost Function for...

Research ArticleModel Predictive Control with a Relaxed Cost Function forConstrained Linear Systems

DavidSoteloAntonioFavela-Contreras ViacheslavVKalashnikov andCarlosSotelo

Tecnologico de Monterrey Escuela de Ingenieri a y Ciencias Monterrey Mexico

Correspondence should be addressed to Antonio Favela-Contreras antoniofavelatecmx

Received 5 September 2019 Accepted 4 March 2020 Published 31 March 2020

Academic Editor Marek Lefik

Copyright copy 2020 David Sotelo et al -is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

-e Model Predictive Control technique is widely used for optimizing the performance of constrained multi-input multi-outputprocesses However due to its mathematical complexity and heavy computation effort it is mainly suitable in processes with slowdynamics Based on the Exact Penalization -eorem this paper presents a discrete-time state-space Model Predictive Controlstrategy with a relaxed performance index where the constraints are implicitly defined in the weighting matrices computed ateach sampling time-e performance validation for theModel Predictive Control strategy with the proposed relaxed cost functionuses the simulation of a tape transport system and a jet transport aircraft during cruise flight Without affecting the trackingperformance numerical results show that the execution time is notably decreased compared with two well-known discrete-timestate-space Model Predictive Control strategies -is makes the proposed Model Predictive Control mainly suitable for con-strained multivariable processes with fast dynamics

1 Introduction

Model Predictive Control (MPC) for linear systems is now awell-established discipline providing stability feasibilityand robustness [1ndash6] Due to its inherent ability to take intoaccount constraints and deal with multi-input multi-outputvariables [7ndash10] it has been applied in a wide range ofapplications including chemical processes industrial sys-tems energy health environment and aerospace [11ndash16]In [17] a robust MPC strategy is presented to handle thetrajectory tracking problem for an underactuated two-wheeled inverted pendulum vehicle Moreover based on anMPC scheme in [18] a control strategy is designed to anunmanned aerial vehicle for its automatic carrier landingsystem Nevertheless the computation complexity makesthe multivariable MPC ineffectual for high speed applica-tions where the controller must execute in a fewmilliseconds[19ndash23] Moreover the problem becomes much morecomplicated solving such an online constrained optimiza-tion problem by computing a numerical solver [24ndash26]Several MPC techniques are used to overcome theseproblems For instance in [27] an explicit model predictive

control moves major part of computation offline whichmakes it enable to be implemented in real time for widerange of fast systems Also in [28] to reduce the onlinecomputational time all the state trajectories are included inthe optimal control problem as the constraints in the pre-diction horizon then only a quadratic programmingproblem is solved In [29] based on a mixed integer qua-dratic programming problem the control input is calculatedat each discrete time In contrast to common MPC ap-proaches where an optimization toolbox is required thiswork presents a relaxed performance index in which theweighting matrices are computed online using the conceptof Taylor series expansion and standard inverse distanceweighting (IDW) functions -en tracking performanceunder input-output constraints is well obtained lightercomputation load is achieved and execution time to solve aQuadratic Program (QP) is reduced -us a computa-tionally efficient constrained MPC for discrete-time state-space multivariable systems is obtained

-e paper is organized as follows Section 2 gives thepreliminaries of the proposed MPC strategy Section 3 de-scribes the proposed relaxed cost function Section 4

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 7485865 10 pageshttpsdoiorg10115520207485865

presents a tape transport system and a jet transport aircraftas study cases Simulation results show the performance ofthe proposed MPC strategy and the execution time im-provement compared with two well-known MPC strategiesFinally Section 5 discusses the conclusions Acknowledg-ments and the list of references finish the paper

2 Model Predictive Control Based on Discrete-Time State Space Model

-is section presents a brief review of MPC based on dis-crete-time state-space model -e original controller isproposed by Alamir in [7] In this previous work consid-ering the predictions of the states the control action isobtained through the solution of a constrained optimizationproblem by using a cost function with constant weightingmatrices At each sampling time an optimal control problemis solved whose results are computationally expensive -esystem dynamics is denoted by the Linear Time Invariant(LTI) State-Space Model taking the following structure

x(k + 1) Ax(k) + Bu(k)

yr(k) Crx(k)(1)

where x isinRn is the state vector u isinRnu is the controlledinput vector yr isinR

nr is the output vector A isinRntimesn is thestate matrix B isinRntimesnu stands for the input matrixCr isinR

nrtimesn is the output matrix and k isin N denotes thesampling instant number Hence as in [7] from (1) the statepredictions for the consecutive sampling instants are

x(k + 1) Ax(k) + Bu(k)

x(k + 2) A2x(k) + ABu(k) + Bu(k + 1)

⋮

x(k + i) Ai1113960 1113961x(k) + Aiminus 1B AB B1113858 1113859

u(k)

⋮

u(k + i minus 2)

u(k + i minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


⨿ nuN( )1

⋮

⨿ nuN( )iminus 1

⨿ nuN( )i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957u(k)

x(k + i) Φix(k) + Ψi1113957u(k) forall i isin 1 N

(2)

where Φi isinRntimesn and Ψi isinR

ntimes(Nnu) are used to representthe N-step-ahead prediction map for Linear Time Invariant(LTI) systems in a compactness form (2) -us system (2)can be reformulated using the following vector-matrixnotation

1113957x(k) Φx(k) + Ψ1113957u(k) (3)

where

Φ ≔

Φ1

Φ2

⋮

ΦN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ψ ≔

Ψ1

Ψ2

⋮

ΨN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4)

and 1113957x(k) isinRNmiddotn is the whole state trajectory ofx x1 x2 xn1113858 1113859

T 1113957u(k) isinRNmiddotnu concatenates thecomputed sequence of u u1 u2 unu

1113960 1113961T and

1113957yr(k) isinRNmiddotnr is the output trajectory ofyr y1 y2 ynr

1113960 1113961T

1113957x(k) ≔

x(k + 1)

x(k + 2)

⋮x(k + N)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957u(k) ≔

u(k)

u(k + 1)

⋮u(k + N minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957yr(k) ≔

yr(k + 1)

yr(k + 2)

⋮yr(k + N)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(5)

Now any candidate sequence of actions 1113957u(k) has thecorresponding future behavior of the system contained inthe state trajectory 1113957x(k) and consequently the output tra-jectory 1113957yr(k)

Defining the projection matrix ⨿(nN)i

⨿(nN)i ≔

0ntimesn 0ntimesn1113980radicradicradicradicradic11139791113978radicradicradicradicradic1113981(iminus 1)terms

Intimesn 0ntimesn 0ntimesn1113980radicradicradicradicradic11139791113978radicradicradicradicradic1113981(Nminus i)terms

1113890 1113891 (6)

where n is the vector length N corresponds to the predictionhorizon and i stands for the desired term then the statecontrol and output vectors at a specific instant k + i can beobtained as follows

x(k + i) ⨿(nN)i middot 1113957x(k)

u(k + i minus 1) UnuN( )

i middot 1113957u(k)

yr(k + i) U(nN)i middot 1113957yr(k)

(7)

3 Proposed Relaxed Cost Function

31 Development To find the best sequence of control ac-tion in the present work the value of the cost functionJ(1113957u | x(k) 1113957wr(k) up) is defined over the prediction horizon[k k + N] as follows

2 Mathematical Problems in Engineering

J ≔ 1113944N

i1⨿ nrN( )

i middot 1113957yr minus ⨿ nrN( )i middot 1113957wr

2

Q+ ⨿ nuN( )

1 middot 1113957u minus up

2

R

+ 1113944N

i2⨿ nuN( )

i minus ⨿ nuN( )iminus 11113876 11138771113957u

2

R

(8)where considering a past control action value up isinRnu thelast two terms are added to penalize the rate excursion of thecontrol vector by the weighting matrix R isinRnutimesnu More-over in the first term the pondering matrix Q isinRnrtimesnr

penalizes the error between the output trajectory1113957yr(k) isinRNmiddotnr and the reference 1113957wr(k) isinRNmiddotnr of the vectorwr w1 w2 wnr

1113960 1113961T which is expressed as follows

1113957wr(k) ≔

wr(k + 1)

wr(k + 2)

⋮

wr(k + N)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(9)

Using notation (7) and (8)

J ≔ 1113944N

i1yr(k + i) minus wr(k + i)

2Q

+ ⨿ nuN( )1 middot 1113957u minus up

2

R

+ 1113944N

i2⨿ nuN( )


2

R

(10)Moreover substituting (1) and (2) in the first element of (10)

the cost function is expressed in state-space representation

J ≔ 1113944N

i1CrΨi1113957u + CrΦix(k) minus wr(k + i)

2Q


2

R+ 1113944

N

i2⨿ nuN( )


2

R

(11)-us to obtain the control law the performance index

(11) is rewritten in the following compact form

J ≔12

1113957uTH1113957u + F1x(k) + F2 1113957wr + F3up

1113858 1113859T

1113957u + Cst (12)

where

H ≔ 21113944N

i1ΨT

i CTr QCrΨi1113960 1113961 + ⨿ nuN( )

11113874 1113875TR ⨿ nuN( )

11113874 1113875

+ 1113944N

i2⨿ nuN( )

i minus ⨿ nu N( )iminus 11113874 1113875

TR ⨿ nuN( )

i minus ⨿ nuN( )iminus 11113874 11138751113890 1113891

F1 ≔ 21113944N

i1ΨT

i CTr QCrΦi1113960 1113961

F2 ≔ minus 21113944N

i1ΨT

i CTr Q⨿

nrN( )i1113876 1113877

F3 ≔ minus 2 ⨿ nuN( )11113874 1113875

TR

Cst ≔ 1113944N

i1CrΦix(k) minus wr(k + i)

2Q + up

2R

(13)

-en the control law is obtained from (12) as follows

1113957uopt(x(k)) minus Hminus 1 F1x(k) + F2 1113957wr + F3up

1113858 1113859 (14)

and applying the first action of the best sequence controlaction 1113957uopt(x(k)) the MPC state feedback is

KMPC(x(k)) ≔ ⨿ nuN( )1 middot 1113957uopt x(k) 1113957wr(k) up

( 1113857 (15)

-e MPC optimization problem (16) is solved at eachsampling instant in which the measurements of the outputsand state variables are updated continuously

P(x(k)) min1113957uisinRNnu

J 1113957u | x(k) 1113957wr(k) up( 1113857 (16)

32 Definition of the Constraints -e systematic handling ofconstraints provided by predictive control strategies allowsfor significant improvements in performance over con-ventional control methodologies [30] As in [7 31] thesequence of the future actions 1113957u cannot generally be freelychosen in RNmiddotnu At least saturation constraints on theactuators have to be taken into account giving the followingoptimization problem P(x(k))


J(1113957u | x(k)) st g(1113957u | x(k))le 0 (17)

-e cost function J(1113957u | x(k)) is minimized consideringthe constraints arranged in g(1113957u | x(k)) Nevertheless solvinggeneral optimization problems involving a high number ofdecision variables and high number of constraints is gen-erally a very difficult task For this reason in the presentwork based on the Inverse Distance Weighting (IDW)method [24 32ndash36] the cost function (12) input and outputconstraints are included in Q and R matrices where toreduce computational complexity Taylor series expansion isused to obtain the predicted manipulation 1113954up and predictedoutput 1113954yp [23 37ndash41]

1113954up(k + 1) u(k) +k minus (k minus 1)

1

u(k) minus u(k minus 1)

k minus (k minus 1)1113896 1113897 + middot middot middot

(18)

1113954yp(k + 1) y(k) +k minus (k minus 1)

1

y(k) minus y(k minus 1)


(19)

-e penalization at the output Q isinRnrtimesnr is defined

Q

q1 middot middot middot 0

⋮ ⋱ 0

0 0 qp

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (20)

where qp eσ middot cforallp isin 1 nr1113864 1113865 α is the tuning parameterset for a desired closed-loop performanceσ α + |1113954yp minus wp|min(|1113954yp minus ymax

p | |1113954yp minus yminp |) and

c 1(min(|1113954up minus umaxp | |1113954up minus umin

p |)) which corresponds toan IDW function

Mathematical Problems in Engineering 3

-e deviation between the predicted output 1113954yp and thereference wp is penalized by a function of distance |1113954yp minus wp|Furthermore to avoid that the predicted process variable 1113954yp

remains close to the lower bound yminp or the upper bound

ymaxp the min(|1113954yp minus ymax

p | |1113954yp minus yminp |) term is added Fi-

nally c term is included in order to increase the penalizationwhile the predicted manipulation variable 1113954up is close to theupper or lower bounds umax

p and uminp respectively

-e penalization at the input R isinRnutimesnu is defined

R

r1 middot middot middot 0

⋮ ⋱ 0

0 0 rq

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (21)

where rq c + |δ1113954uq minus δupq |forallq isin 1 nu1113864 1113865

-e first term is included to avoid that the predictedmanipulated variable 1113954up remains close to the upper or lowerbounds umax

q and uminq while the second term is added to

reduce the large excursion between the increment of thepredicted manipulated variable δ1113954uq and the increment of theprevious manipulated variable δu

pq

-e resulting optimization problem P(x(k)) expressedas the cost function J(1113957u | x(k) 1113957wr(k) up) is minimized ateach instant k by a sequence of future actions 1113957uopt(x(k))

respecting all the constraints As it is described Q and R aredefined as functionals of constraints depending on thecurrent state x(k) the desired trajectory 1113957wr(k) and the pastcontrolled input up -us the matrices H Fi i 1 2 3 arecomputed online looking for relaxing the optimizationproblem

33 Algorithm for the ProposedMPC Strategy -e algorithmused for the proposed discrete-time state-space constrainedMPC strategy consists of

(1) Define the linear time invariant mathematical modelof the physical system in state-space representation(equation (1))

(2) Compute the matrices Φi and Ψiforalli isin 1 N

used to represent the N-step-ahead prediction mapfor LTI systems (equation (4))

(3) Estimate the manipulation 1113954up (equation (18)) andthe output 1113954yp (equation (19)) at the next samplinginstant using Taylor series expansion

(4) Compute the weighting matrices Q (equation (20))and R (equation (19)) based on the proposedfunctions

(5) Compute the online matrices H Fi i 1 2 3 of theproposed performance index (equation (12))

(6) Obtain the control law 1113957uopt(x(k)) (equation (14))and apply to the system the first action of the bestsequence control action KMPC(x(k)) (equation (15))

(7) At the next sampling instant (k ≔ k + 1 ) the ref-erence wr the bounds ymin ymax umin and umax themeasurements of the outputs yr and the state var-iables x(k) are updated and the MPC optimization

problem is solved again (equation (17)) the algo-rithm goes to step 3

4 Simulation and Results

-e present work and the MPC strategies in [7 31] aredeveloped using the same computational platform for itsevaluation Hence to solve the optimization problem in[7 31] quadprog toolbox from MATLABreg is usedMATLABreg codes for the following study cases are availableMPC Matlab Files_MPE

41 Example 1 TapeTransport System -e tape drive systemconsists of two reels to supply and file data Here the datatransfer rate is proportional to the tape transport speed-us the tape drive mechanism must be able to rapidlytransport a fragile tape with an accurate tension regulationFigure 1 shows the schematic of a tape transport systemwhere its components and variables involved are the tapestiffness and the damping denoted by K and D the reel radiiand the inertia represented as r and J the motor torqueconstant Kt and the viscous friction coefficient denoted byβ

Assuming there is no force loss across the head the tapetension T T1 T2 [42] Although the physical model ofthe process contains nonlinearities in (22) a simplifiedstate-space model is presented in continuous time [42ndash44]

_x(t) Acx(t) + Bcu(t)

y(t) Ccx(t)

Ac

minus Dr21J1

+r22J2

1113888 1113889 Dβr1J1

minus Kr1 Kr2 minus Dβr2J2

r1

J1minusβJ1

0

minusr2

J20 minus

βJ2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Bc

minus DKt

r1

J1DKt

r2

J2

Kt

J10

0Kt

J2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cc

0r12

r22

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)

-emodel has three states x T ω1 ω21113858 1113859T where T is

the tension tape in N meanwhile ω1 and ω2 represent the


supply and take-up reel in rads respectively Moreover thesystem has two inputs u u1 u21113858 1113859

T that represent thevoltages applied to the reel motors in volts and two outputsy y1 y21113858 1113859

T which stand for the tape speed vrw at the read-write head in ms and the tape tension T respectively -econtrol strategy described in the present work is simulatedusing parameters from the tested tape system described in[42 45 46] whose parameters are summarized in Table 1Considering that the motors are nominally identical forboth motors it is used as the same motor torque constant Kt

and viscous friction coefficient β [45 46]Discretizing (23) with a sampling time τ 01 s and

considering a prediction horizon N 10 with a tuning pa-rameter α 37 the simulation results are shown in Figure 2 Itis divided in two main parts the first part corresponds to theoutputs and the inputs of the system using the MPC with therelaxed cost function while the second part shows the coeffi-cient values of the weighting matrices Q and R As it is showncomputing the coefficient values q1 and q2 the constrained tapespeedy1 and tape tensiony2 present a good performanceHerey1 do not present overshoot and has a maximum settling timeof 01 seconds while y2 has a maximum overshoot of 3 and asettling time of 03 seconds On the contrary themotor voltagesu1 for the supply reel and u2 for the take-up reel remains insidetheir lower and upper bounds by using the computed coefficientvalues r1 and r2 Here the values of r1 are lower than the valuesof r2 due to the rates of the manipulations

In order to see the closed-loop stability of the system thestability indicator SN is defined as in [7]

SN ≔ maxjisin 1n

λj A minus BKN( 111385711138681113868111386811138681113868

11138681113868111386811138681113868 (23)

where λj stands for the eigenvalues of the system forj isin 1 n and KN ≔ ⨿

(nuN)1 middot Hminus 1[F1x(k)] is the

closed-loop gain Figure 3 shows that the system remainsstable during the test Finally Table 2 is presented tocompare the execution time between the present work andprevious works [7 31]

As it can be seen the total execution time is reduced bytaking advantage of the relaxed cost function Here thecomputation of 1113957u(k) is 1924 seconds and 2148 secondsfaster than the computation of the manipulations using[7 31] MPCs strategies Henceforth the percentage con-sumption of time to obtain the control actions is notablydecreased

42 Example 2 Jet Transport Aircraft -e Jet TransportAircraft Boeing 747 in high-lift configuration addressescomplex geometries and physical phenomena that make thecontroller design a difficult process Figure 4 illustrates theJet Transport Aircraft with its components and variablesinvolved such as the angles β and φ and the angular velocitiesψ and θ

Although the physical model of the Boeing 747 islengthy in (24) the simplified state-space model duringcruise flight at MATCH 08 and H 40 000 ft is pre-sented in continuous time [47]


y(t) Ccx(t)

Ac

minus 00558 minus 09968 00802 00415

0598 minus 0115 minus 00318 0

minus 305 0388 minus 0465 0

0 00805 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Bc

00729 0

minus 475 0775

0153 143

0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cc 0 1 0 0

0 0 0 1⎡⎢⎣ ⎤⎥⎦

(24)

-e model has four states x β ψ θ φ1113858 1113859T where β is

the sideslip angle φ stands for the bank angle and mean-while ψ and θ represent the yaw and roll rate respectivelyHerein all the angles are in rad and the angular velocities inrads -e system has two inputs u u1 u21113858 1113859

T the rudderand the aileron deflections and two outputs y y1 y21113858 1113859

Tthe yaw rate ψ and the bank angle φ

Using a sampling time τ 02 s system (24) is dis-cretized -en a set of changes in the output reference and aseries of variations in the constraints are used to test thepresent work Considering a prediction horizon N 20 anda tuning parameter α 45 simulation results are shown inFigure 5 Here using the computed coefficient values q1 and

r1

ω1

ω2

J1 β1 Kt1 J2 β2 Kt2

T1 T2

D

K

r2

Supply reel Take-up reel

Figure 1 Tape transport system

Table 1 Tape transport system parameters

Symbol Parameter ValueK Tape stiffness 2times103NmD Damping 2N sm2

r1 Radius of supply reel 212times10minus 3mr2 Radius of take-up reel 975times10minus 3m

J1Moment of inertia of the supply

reel 142times10minus 6 Kgm2

J2Moment of inertia of the take-up


Kt Motor torque constant 248times10minus 3NmV

β Viscous friction coefficient 103times10minus 4Nm srad


q2 the yaw rate y1 has a maximum overshoot of 29 with amaximum settling time of 1 second while the bank angle y2has a maximum overshoot of 66 with a maximum settlingtime of 12 seconds Moreover with the computed coeffi-cient values r1 and r2 the rudder deflection u1 and theaileron deflection u2 remains inside their bounds withoutsaturation Here the values of r2 are greater than the valuesof r1 this is because the rate of the manipulation u2 is greater

than the rate of the manipulation u1 On the contraryFigure 6 shows the stability behavior during the test As it isshown the system remains stable

Finally the execution time comparison between thepresent work and previous works [7 31] is shown inTable 3 As it can be seen the total execution time isconsiderably reduced due to the relaxed cost functionHere the computation of 1113957u(k) is 6074 seconds and 3155

4

3

2

1

0Tape

spee

d (m

s)

0 2 4 6 8 10 12Time (s)

w1y1y1

minmax

(a)

0 2 4 6 8 10 12Time (s)

w2y2y2

minmax

108060402

0Tape

tens

ion

(N)

(b)

0 2 4 6 8 10 12Time (s)

2

1

0

ndash1Mot

or v

olta

ge (V

)

u1u1

minmax

(c)

0 2 4 6 8 10 12Time (s)

Mot

or v

olta

ge (V

)u2u2

minmax

0504030201

0ndash01

(d)

0 2 4 6 8 10 12Time (s)

600

400

200

0

Coe

ffici

ent v

alue

q1

(e)

0 2 4 6 8 10 12Time (s)

Coe

ffici

ent v

alue

q2

1000800600400200

0

(f)

Coe

ffici

ent v

alue

0 2 4 6 8 10 12Time (s)

4

3

2

1

0

r1

(g)

00

2 4 6 8 10 12Time (s)

20

15

10

5

Coe

ffici

ent v

alue

r2

(h)

Figure 2 Simulation results for tape transport system under the MPC strategy with the relaxed cost function


012

010

008

006

004

002

00 2 4 6 8 10 12

Time

Stab

ility

indi

cato

rs (S

N)

Figure 3 Stability behavior

Table 2 Execution time comparison with previous works

MPC Total (s) Computation of 1113957u(k) (s) Percentage consumption ()[5] 2919 1936 663[22] 2907 2160 743-is work 0892 0012 13

Aileron

Longitudinalaxis

Top view

Rudder

Front view

X X

Y Z

φ

ψ

β

+ +θ

Figure 4 Jet transport aircraft (a) Top view (b) Front view

ndash05

0

05

Yaw

rate

(rad

s)

5 403520 30 450 1510 5025Time (s)

w1y1y1

minmax

(a)

ndash1

ndash05

0

05

1

Bank

angl

e (ra

d)

5 403520 30 450 1510 5025Time (s)

w2y2y2

minmax

(b)

Figure 5 Continued


ndash2ndash1

012

Rudd

er d

efle

ctio

n (r

ad)

1510 25 505 30 35 40 45200Time (s)

u1minmax

u1

(c)

u2minmax

u2

5 403520 30 450 1510 5025Time (s)

ndash2ndash1

012

Aile

ron

defle

ctio

n (r

ad)

(d)

0

50

100

150

200

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

q1

(e)

0

200

400

600

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

q2

(f )

0

1

2

3

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

r1

(g)

0

2

4

6

8

Coe

ffici

ent v

alue

5 403520 30 450 1510 5025Time (s)

r2

(h)

Figure 5 Simulation results for jet transport aircraft under the MPC strategy with the relaxed cost function

1010

1005

1

0995

0990

0985

0980

Stab

ility

indi

cato

r (S N

)

0 5 10 15 20 25 30 35 40 45 50Time





seconds faster than the computation of the manipulationsusing [7 31] MPCs strategies -en as in example 1 thepercentage consumption of time to obtain the controlactions is notably decreased

5 Conclusions

-is paper presents a discrete-time state-space MPCapproach for multivariable systems Based on the IDWmethod and the concept of Taylor series expansion arelaxed performance index with constraints defined in theonline weighting matrices is proposed to compute thecontrol action -us as in study cases the proposed MPCstrategy is used to control a tape transport system and a jettransport aircraft during cruise flight

Simulation results show that the proposed MPC strategywith the relaxed cost function has a good performance nomatter abrupt changes of set-points and constraints occureven at the same time Additionally compared with twowell-known discrete-time state-space MPC strategies thereis a significant improvement on the execution time withoutaffecting the tracking performance -e percentage con-sumption of time to compute the best sequence of controlactions 1113957u(k) is 13 for the tape transport system and 17for the jet transport aircraft Henceforth it takes almost 01milliseconds for the tape transport system and 012 milli-seconds for the jet transport aircraft to obtain the manip-ulation u(k) that minimizes the proposed cost functionwhile respecting the constraints -us the proposed MPCstrategy with the relaxed cost function is mainly suitable forconstrained multivariable real processes with fast dynamics

Data Availability

-e data of the conducted experiments and simulations areavailable upon requirement

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-e research activities of the third co-author were par-tially supported by the Mexico National Council forScience and Technologies (CONACYT) with Grants CB-2013-01-221676 and FC-2016-01-1938 -e authors alsothank the Research Groups of Sensors and Devices and ofOptimization and Data Science of the School of Engi-neering and Sciences for their support of the developmentof this work and MSc Arturo Pinto for his fruitfuldiscussions

References

[1] R Heydari andM Farrokhi ldquoRobust model predictive controlof biped robots with adaptive on-line gait generationrdquo In-ternational Journal of Control Automation and Systemsvol 15 no 1 pp 329ndash344 2017

[2] S Riverso M Farina and G Ferrari-Trecate ldquoPlug-and-playdecentralized model predictive control for linear systemsrdquoIEEE Transactions on Automatic Control vol 58 no 10pp 2608ndash2614 2013

[3] R Zhang J Lu H Qu and F Gao ldquoState space modelpredictive fault-tolerant control for batch processes withpartial actuator failurerdquo Journal of Process Control vol 24no 5 pp 613ndash620 2014

[4] M Zhao C-C Jiang and M-H She ldquoRobust contractiveeconomic MPC for nonlinear systems with additive distur-bancerdquo International Journal of Control Automation andSystems vol 16 no 5 pp 2253ndash2263 2018

[5] H Shi P Li J Cao C Su and J Yu ldquoRobust fuzzy predictivecontrol for discrete-time systems with interval time-varyingdelays and unknown disturbancesrdquo IEEE Transactions onFuzzy Systems 2019

[6] H Shi P Li C Su Y Wang J Yu and J Cao ldquoRobustconstrained model predictive fault-tolerant control for in-dustrial processes with partial actuator failures and intervaltime-varying delaysrdquo Journal of Process Control vol 75pp 187ndash203 2019

[7] M Alamir A Pragmatic Story of Model Predictive ControlSelf-Contained Algorithms and Case-Studies CNRS-Univer-sity of Grenoble Grenoble France 2013

[8] I Chang and J Bentsman ldquoConstrained discrete-time state-dependent Riccati equation technique a model predictivecontrol approachrdquo in Proceedings of the of 52nd IEEE Con-ference on Decision and Control pp 5125ndash5130 FlorenceItaly December 2013

[9] B Zhu H Tazvinga and X Xia ldquoSwitched model predictivecontrol for energy dispatching of a photovoltaic-diesel-batteryhybrid power systemrdquo IEEE Transactions on Control SystemsTechnology vol 23 no 3 pp 1229ndash1236 2015

[10] H-y Shi C-l Su J-t Cao P Li Y-l Song and N-b LildquoIncremental multivariable predictive functional control andits application in a gas fractionation unitrdquo Journal of CentralSouth University vol 22 no 12 pp 4653ndash4668 2015

[11] Y Wang and S Boyd ldquoFast model predictive control usingonline optimizationrdquo IEEE Transactions on Control SystemsTechnology vol 18 no 2 pp 267ndash278 2010

[12] S-K Kim D-K Choi K-B Lee and Y I Lee ldquoOffset-freemodel predictive control for the power control of three-phaseACDC convertersrdquo IEEE Transactions on Industrial Elec-tronics vol 62 no 11 pp 7114ndash7126 2015

[13] R Zhang A Xue and F Gao ldquoTemperature control of in-dustrial coke furnace using novel state space model predictivecontrolrdquo IEEE Transactions on Industrial Informatics vol 10no 4 pp 2084ndash2092 2014

[14] M Preindl and S Bolognani ldquoModel predictive direct speed controlwith finite control set of PMSM drive systemsrdquo IEEE Transactionson Power Electronics vol 28 no 2 pp 1007ndash1015 2013

[15] B Hredzak V G Agelidis and M Minsoo Jang ldquoA modelpredictive control system for a hybrid battery-ultracapacitorpower sourcerdquo IEEE Transactions on Power Electronicsvol 29 no 3 pp 1469ndash1479 2014

[16] H Shi C Su J Cao P Li J Liang and G Zhong ldquoNonlinearadaptive predictive functional control based on the Takagi-Sugeno model for average cracking outlet temperature of theethylene cracking furnacerdquo Industrial amp Engineering Chem-istry Research vol 54 no 6 pp 1849ndash1860 2015

[17] M Yue C An and J-Z Sun ldquoAn efficient model predictivecontrol for trajectory tracking of wheeled inverted pendulumvehicles with various physical constraintsrdquo International


Journal of Control Automation and Systems vol 16 no 1pp 265ndash274 2018

[18] S Koo S Kim J Suk Y Kim and J Shin ldquoImprovement ofshipboard landing performance of fixed-wing UAV usingmodel predictive controlrdquo International Journal of ControlAutomation and Systems vol 18 no 1 pp 265ndash274 2018

[19] R Zhang S Wu and F Gao ldquoState space model predictivecontrol for advanced process operation a review of recentdevelopment new results and insightrdquo Industrial amp Engi-neering Chemistry Research vol 56 no 18 pp 5360ndash53942017

[20] X Yu-Geng L De-Wei and L Shu ldquoModel predictivecontrolmdashstatus and challengesrdquo Acta Automatica Sinicavol 39 no 3 pp 222ndash236 2013

[21] S J Qin and T A Badgwell ldquoA survey of industrial modelpredictive control technologyrdquo Control Engineering Practicevol 11 no 7 pp 733ndash764 2003

[22] P D Christofides R Scattolini D Muntildeoz de la Pentildea andJ Liu ldquoDistributed model predictive control a tutorial reviewand future research directionsrdquo Computers amp Chemical En-gineering vol 51 pp 21ndash41 2013

[23] J YangW X Zheng S Li B Wu andM Cheng ldquoDesign of aprediction-accuracy-enhanced continuous-time MPC fordisturbed systems via a disturbance observerrdquo IEEE Trans-actions on Industrial Electronics vol 62 no 9 pp 5807ndash58162015

[24] A Bemporad ldquoGlobal optimization via inverse distanceweightingrdquo 2019 httpsarxivorgabs190606498

[25] A Bemporad ldquoModel predictive control design new trendsand toolsrdquo in Proceedings of the 45th IEEE Conference onDecision and Control pp 6678ndash6683 San Diego CA USADecember 2006

[26] A Bemporad ldquoA multiparametric quadratic programmingalgorithm with polyhedral computations based on nonneg-ative least squaresrdquo IEEE Transactions on Automatic Controlvol 60 no 11 pp 2892ndash2903 2015

[27] M A Mohammadkhani F Bayat and A A Jalali ldquoDesign ofexplicit model predictive control for constrained linear sys-tems with disturbancesrdquo International Journal of ControlAutomation and Systems vol 12 no 2 pp 294ndash301 2014

[28] M A Mousavi B Moshiri and Z Heshmati ldquoA new pre-dictive motion control of a planar vehicle under uncertaintyvia convex optimizationrdquo International Journal of ControlAutomation and Systems vol 15 no 1 pp 129ndash137 2017

[29] D Satoh K Kobayashi and Y Yamashita ldquoMPC-based co-design of control and routing for wireless sensor and actuatornetworksrdquo International Journal of Control Automation andSystems vol 16 no 3 pp 953ndash960 2018

[30] G Serale M Fiorentini A Capozzoli D Bernardini andA Bemporad ldquoModel predictive control (MPC) for en-hancing building and HVAC system energy efficiencyproblem formulation applications and opportunitiesrdquo En-ergies vol 11 no 3 p 631 2018

[31] L Wang Model Predictive Control System Design andImplementation Using MATLABreg Springer Science amp Busi-ness Media Berlin Germany 2009

[32] Z Li K Wang H Ma and Y Wu ldquoAn adjusted inversedistance weighted spatial interpolation methodrdquo in Pro-ceedings of the 2018 3rd International Conference on Com-munications Information Management and Network Security(CIMNS 2018) November 2018

[33] Z Fan J Li and M Deng ldquoAn adaptive inverse-distanceweighting spatial interpolationmethod with the consideration

of multiple factorsrdquo Geomatics and Information Science ofWuhan University vol 6 p 20 2016

[34] D Shepard ldquoA two-dimensional interpolation function forirregularly-spaced datardquo in Proceedings of the of the 23rd ACMNational Conference pp 517ndash524 New York NY USA 1968

[35] G Y Lu and D W Wong ldquoAn adaptive inverse-distanceweighting spatial interpolation techniquerdquo Computers ampGeosciences vol 34 no 9 pp 1044ndash1055 2008

[36] DWWang L N Li C Hu Q Li X Chen and PW HuangldquoA modified inverse distance weighting method for inter-polation in open public places based on wi-fi probe datardquoJournal of Advanced Transportation Article ID 760279211 pages 2019

[37] M Ławrynczuk ldquoNonlinear predictive control of a boiler-turbine unit a state-space approach with successive on-linemodel linearisation and quadratic optimisationrdquo ISA Trans-actions vol 67 pp 476ndash495 2017

[38] M Ławrynczuk ldquoNonlinear statendashspace predictive controlwith onndashline linearisation and state estimationrdquo InternationalJournal of AppliedMathematics and Computer Science vol 25no 4 pp 833ndash847 2015

[39] W H Chen ldquoPredictive control of general nonlinear systemsusing approximationrdquo IEE ProceedingsmdashControl Deory andApplications vol 151 no 2 pp 137ndash144 2004

[40] R Errouissi A Al-Durra and S M Muyeen ldquoA robustcontinuous-timeMPC of a DC-DC boost converter interfacedwith a grid-connected photovoltaic systemrdquo IEEE Journal ofPhotovoltaics vol 6 no 6 pp 1619ndash1629 2016

[41] C Sotelo A Favela-Contreras F Beltran-Carbajal G Dieck-Assad P Rodrıguez-Cantildeedo and D Sotelo ldquoA novel discrete-time nonlinear model predictive control based on state spacemodelrdquo International Journal of Control Automation andSystems vol 16 no 6 pp 2688ndash2696 2018

[42] P D Mathur and W C Messner ldquoController developmentfor a prototype high-speed low-tension tape transportrdquo IEEETransactions on Control Systems Technology vol 6 no 4pp 534ndash542 1998

[43] Y LU and W C Messner ldquoRobust servo design for tapetransportrdquo in Proceedings of the 2001 IEEE InternationalConference on Control Applications (CCArsquo01) (Cat No01CH37204) pp 1014ndash1019 Mexico City Mexico September2001

[44] D Tenne and T Singh ldquoRobust feed-forwardfeedback designfor tape transportrdquo in Proceedings of the of the AIAA Guid-ance Navigation and Control Conference and Exhibit p 5119Providence RI USA August 2004

[45] M D Baumgart and L Y Pao ldquoRobust control of nonlineartape transport systems with and without tension sensorsrdquoJournal of Dynamic Systems Measurement and Controlvol 129 no 1 pp 41ndash55 2007

[46] M D Baumgart and L Y Pao ldquoRobust control of tapetransport systems with no tension sensorrdquo in Proceedings ofthe 2004 43rd IEEE Conference on Decision and Control (CDC)(IEEE Cat No 04CH37601) pp 4342ndash4349 Nassau Baha-mas December 2004

[47] S Singh and T R Murthy ldquoSimulation of sensor failureaccommodation in flight control system of transport aircrafta modular Approachrdquo World Journal of Modelling andSimulation vol 11 no 1 pp 55ndash68 2015


presents a tape transport system and a jet transport aircraftas study cases Simulation results show the performance ofthe proposed MPC strategy and the execution time im-provement compared with two well-known MPC strategiesFinally Section 5 discusses the conclusions Acknowledg-ments and the list of references finish the paper

2 Model Predictive Control Based on Discrete-Time State Space Model

-is section presents a brief review of MPC based on dis-crete-time state-space model -e original controller isproposed by Alamir in [7] In this previous work consid-ering the predictions of the states the control action isobtained through the solution of a constrained optimizationproblem by using a cost function with constant weightingmatrices At each sampling time an optimal control problemis solved whose results are computationally expensive -esystem dynamics is denoted by the Linear Time Invariant(LTI) State-Space Model taking the following structure

x(k + 1) Ax(k) + Bu(k)

yr(k) Crx(k)(1)

where x isinRn is the state vector u isinRnu is the controlledinput vector yr isinR

nr is the output vector A isinRntimesn is thestate matrix B isinRntimesnu stands for the input matrixCr isinR

nrtimesn is the output matrix and k isin N denotes thesampling instant number Hence as in [7] from (1) the statepredictions for the consecutive sampling instants are

x(k + 1) Ax(k) + Bu(k)

x(k + 2) A2x(k) + ABu(k) + Bu(k + 1)

⋮


u(k)

⋮

u(k + i minus 2)

u(k + i minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦


⨿ nuN( )1

⋮

⨿ nuN( )iminus 1

⨿ nuN( )i

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957u(k)

x(k + i) Φix(k) + Ψi1113957u(k) forall i isin 1 N

(2)

where Φi isinRntimesn and Ψi isinR

ntimes(Nnu) are used to representthe N-step-ahead prediction map for Linear Time Invariant(LTI) systems in a compactness form (2) -us system (2)can be reformulated using the following vector-matrixnotation

1113957x(k) Φx(k) + Ψ1113957u(k) (3)

where

Φ ≔

Φ1

Φ2

⋮

ΦN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ψ ≔

Ψ1

Ψ2

⋮

ΨN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4)

and 1113957x(k) isinRNmiddotn is the whole state trajectory ofx x1 x2 xn1113858 1113859

T 1113957u(k) isinRNmiddotnu concatenates thecomputed sequence of u u1 u2 unu

1113960 1113961T and

1113957yr(k) isinRNmiddotnr is the output trajectory ofyr y1 y2 ynr

1113960 1113961T

1113957x(k) ≔

x(k + 1)

x(k + 2)

⋮x(k + N)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957u(k) ≔

u(k)

u(k + 1)

⋮u(k + N minus 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957yr(k) ≔

yr(k + 1)

yr(k + 2)

⋮yr(k + N)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(5)

Now any candidate sequence of actions 1113957u(k) has thecorresponding future behavior of the system contained inthe state trajectory 1113957x(k) and consequently the output tra-jectory 1113957yr(k)

Defining the projection matrix ⨿(nN)i

⨿(nN)i ≔

0ntimesn 0ntimesn1113980radicradicradicradicradic11139791113978radicradicradicradicradic1113981(iminus 1)terms

Intimesn 0ntimesn 0ntimesn1113980radicradicradicradicradic11139791113978radicradicradicradicradic1113981(Nminus i)terms

1113890 1113891 (6)

where n is the vector length N corresponds to the predictionhorizon and i stands for the desired term then the statecontrol and output vectors at a specific instant k + i can beobtained as follows

x(k + i) ⨿(nN)i middot 1113957x(k)

u(k + i minus 1) UnuN( )

i middot 1113957u(k)

yr(k + i) U(nN)i middot 1113957yr(k)

(7)

3 Proposed Relaxed Cost Function

31 Development To find the best sequence of control ac-tion in the present work the value of the cost functionJ(1113957u | x(k) 1113957wr(k) up) is defined over the prediction horizon[k k + N] as follows


J ≔ 1113944N

i1⨿ nrN( )


2

Q+ ⨿ nuN( )


2

R

+ 1113944N

i2⨿ nuN( )


2

R




1113957wr(k) ≔

wr(k + 1)

wr(k + 2)

⋮

wr(k + N)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(9)


J ≔ 1113944N


2Q


2

R

+ 1113944N

i2⨿ nuN( )


2

R



J ≔ 1113944N


2Q


2

R+ 1113944

N

i2⨿ nuN( )


2

R



J ≔12

1113957uTH1113957u + F1x(k) + F2 1113957wr + F3up

1113858 1113859T

1113957u + Cst (12)

where

H ≔ 21113944N

i1ΨT

i CTr QCrΨi1113960 1113961 + ⨿ nuN( )

11113874 1113875TR ⨿ nuN( )

11113874 1113875

+ 1113944N

i2⨿ nuN( )


TR ⨿ nuN( )


F1 ≔ 21113944N

i1ΨT

i CTr QCrΦi1113960 1113961

F2 ≔ minus 21113944N

i1ΨT

i CTr Q⨿

nrN( )i1113876 1113877

F3 ≔ minus 2 ⨿ nuN( )11113874 1113875

TR

Cst ≔ 1113944N


2Q + up

2R

(13)



1113858 1113859 (14)



( 1113857 (15)



J 1113957u | x(k) 1113957wr(k) up( 1113857 (16)



J(1113957u | x(k)) st g(1113957u | x(k))le 0 (17)



1



(18)


1



(19)


Q


⋮ ⋱ 0

0 0 qp

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (20)













R


⋮ ⋱ 0

0 0 rq

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (21)





pq


















y(t) Ccx(t)

Ac

minus Dr21J1

+r22J2

1113888 1113889 Dβr1J1


r1

J1minusβJ1

0

minusr2

J20 minus

βJ2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Bc

minus DKt

r1

J1DKt

r2

J2

Kt

J10

0Kt

J2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cc

0r12

r22

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)










SN ≔ maxjisin 1n

λj A minus BKN( 111385711138681113868111386811138681113868

11138681113868111386811138681113868 (23)








y(t) Ccx(t)

Ac

minus 00558 minus 09968 00802 00415

0598 minus 0115 minus 00318 0

minus 305 0388 minus 0465 0

0 00805 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Bc

00729 0

minus 475 0775

0153 143

0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cc 0 1 0 0

0 0 0 1⎡⎢⎣ ⎤⎥⎦

(24)






r1

ω1

ω2


T1 T2

D

K

r2
















4

3

2

1

0Tape

spee

d (m

s)

0 2 4 6 8 10 12Time (s)

w1y1y1

minmax

(a)

0 2 4 6 8 10 12Time (s)

w2y2y2

minmax

108060402

0Tape

tens

ion

(N)

(b)

0 2 4 6 8 10 12Time (s)

2

1

0

ndash1Mot

or v

olta

ge (V

)

u1u1

minmax

(c)

0 2 4 6 8 10 12Time (s)

Mot

or v

olta

ge (V

)u2u2

minmax

0504030201

0ndash01

(d)

0 2 4 6 8 10 12Time (s)

600

400

200

0

Coe

ffici

ent v

alue

q1

(e)

0 2 4 6 8 10 12Time (s)

Coe

ffici

ent v

alue

q2

1000800600400200

0

(f)

Coe

ffici

ent v

alue

0 2 4 6 8 10 12Time (s)

4

3

2

1

0

r1

(g)

00

2 4 6 8 10 12Time (s)

20

15

10

5

Coe

ffici

ent v

alue

r2

(h)



012

010

008

006

004

002

00 2 4 6 8 10 12

Time

Stab

ility

indi

cato

rs (S

N)




Aileron

Longitudinalaxis

Top view

Rudder

Front view

X X

Y Z

φ

ψ

β

+ +θ


ndash05

0

05

Yaw

rate

(rad

s)

5 403520 30 450 1510 5025Time (s)

w1y1y1

minmax

(a)

ndash1

ndash05

0

05

1

Bank

angl

e (ra

d)

5 403520 30 450 1510 5025Time (s)

w2y2y2

minmax

(b)

Figure 5 Continued


ndash2ndash1

012

Rudd

er d

efle

ctio

n (r

ad)

1510 25 505 30 35 40 45200Time (s)

u1minmax

u1

(c)

u2minmax

u2

5 403520 30 450 1510 5025Time (s)

ndash2ndash1

012

Aile

ron

defle

ctio

n (r

ad)

(d)

0

50

100

150

200

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

q1

(e)

0

200

400

600

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

q2

(f )

0

1

2

3

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

r1

(g)

0

2

4

6

8

Coe

ffici

ent v

alue

5 403520 30 450 1510 5025Time (s)

r2

(h)


1010

1005

1

0995

0990

0985

0980

Stab

ility

indi

cato

r (S N

)

0 5 10 15 20 25 30 35 40 45 50Time






5 Conclusions



Data Availability




Acknowledgments


References




















































J ≔ 1113944N

i1⨿ nrN( )


2

Q+ ⨿ nuN( )


2

R

+ 1113944N

i2⨿ nuN( )


2

R




1113957wr(k) ≔

wr(k + 1)

wr(k + 2)

⋮

wr(k + N)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(9)


J ≔ 1113944N


2Q


2

R

+ 1113944N

i2⨿ nuN( )


2

R



J ≔ 1113944N


2Q


2

R+ 1113944

N

i2⨿ nuN( )


2

R



J ≔12

1113957uTH1113957u + F1x(k) + F2 1113957wr + F3up

1113858 1113859T

1113957u + Cst (12)

where

H ≔ 21113944N

i1ΨT

i CTr QCrΨi1113960 1113961 + ⨿ nuN( )

11113874 1113875TR ⨿ nuN( )

11113874 1113875

+ 1113944N

i2⨿ nuN( )


TR ⨿ nuN( )


F1 ≔ 21113944N

i1ΨT

i CTr QCrΦi1113960 1113961

F2 ≔ minus 21113944N

i1ΨT

i CTr Q⨿

nrN( )i1113876 1113877

F3 ≔ minus 2 ⨿ nuN( )11113874 1113875

TR

Cst ≔ 1113944N


2Q + up

2R

(13)



1113858 1113859 (14)



( 1113857 (15)



J 1113957u | x(k) 1113957wr(k) up( 1113857 (16)



J(1113957u | x(k)) st g(1113957u | x(k))le 0 (17)



1



(18)


1



(19)


Q


⋮ ⋱ 0

0 0 qp

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (20)













R


⋮ ⋱ 0

0 0 rq

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (21)





pq


















y(t) Ccx(t)

Ac

minus Dr21J1

+r22J2

1113888 1113889 Dβr1J1


r1

J1minusβJ1

0

minusr2

J20 minus

βJ2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Bc

minus DKt

r1

J1DKt

r2

J2

Kt

J10

0Kt

J2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cc

0r12

r22

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)










SN ≔ maxjisin 1n

λj A minus BKN( 111385711138681113868111386811138681113868

11138681113868111386811138681113868 (23)








y(t) Ccx(t)

Ac

minus 00558 minus 09968 00802 00415

0598 minus 0115 minus 00318 0

minus 305 0388 minus 0465 0

0 00805 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Bc

00729 0

minus 475 0775

0153 143

0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cc 0 1 0 0

0 0 0 1⎡⎢⎣ ⎤⎥⎦

(24)






r1

ω1

ω2


T1 T2

D

K

r2
















4

3

2

1

0Tape

spee

d (m

s)

0 2 4 6 8 10 12Time (s)

w1y1y1

minmax

(a)

0 2 4 6 8 10 12Time (s)

w2y2y2

minmax

108060402

0Tape

tens

ion

(N)

(b)

0 2 4 6 8 10 12Time (s)

2

1

0

ndash1Mot

or v

olta

ge (V

)

u1u1

minmax

(c)

0 2 4 6 8 10 12Time (s)

Mot

or v

olta

ge (V

)u2u2

minmax

0504030201

0ndash01

(d)

0 2 4 6 8 10 12Time (s)

600

400

200

0

Coe

ffici

ent v

alue

q1

(e)

0 2 4 6 8 10 12Time (s)

Coe

ffici

ent v

alue

q2

1000800600400200

0

(f)

Coe

ffici

ent v

alue

0 2 4 6 8 10 12Time (s)

4

3

2

1

0

r1

(g)

00

2 4 6 8 10 12Time (s)

20

15

10

5

Coe

ffici

ent v

alue

r2

(h)



012

010

008

006

004

002

00 2 4 6 8 10 12

Time

Stab

ility

indi

cato

rs (S

N)




Aileron

Longitudinalaxis

Top view

Rudder

Front view

X X

Y Z

φ

ψ

β

+ +θ


ndash05

0

05

Yaw

rate

(rad

s)

5 403520 30 450 1510 5025Time (s)

w1y1y1

minmax

(a)

ndash1

ndash05

0

05

1

Bank

angl

e (ra

d)

5 403520 30 450 1510 5025Time (s)

w2y2y2

minmax

(b)

Figure 5 Continued


ndash2ndash1

012

Rudd

er d

efle

ctio

n (r

ad)

1510 25 505 30 35 40 45200Time (s)

u1minmax

u1

(c)

u2minmax

u2

5 403520 30 450 1510 5025Time (s)

ndash2ndash1

012

Aile

ron

defle

ctio

n (r

ad)

(d)

0

50

100

150

200

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

q1

(e)

0

200

400

600

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

q2

(f )

0

1

2

3

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

r1

(g)

0

2

4

6

8

Coe

ffici

ent v

alue

5 403520 30 450 1510 5025Time (s)

r2

(h)


1010

1005

1

0995

0990

0985

0980

Stab

ility

indi

cato

r (S N

)

0 5 10 15 20 25 30 35 40 45 50Time






5 Conclusions



Data Availability




Acknowledgments


References



























































R


⋮ ⋱ 0

0 0 rq

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (21)





pq


















y(t) Ccx(t)

Ac

minus Dr21J1

+r22J2

1113888 1113889 Dβr1J1


r1

J1minusβJ1

0

minusr2

J20 minus

βJ2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Bc

minus DKt

r1

J1DKt

r2

J2

Kt

J10

0Kt

J2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cc

0r12

r22

1 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)










SN ≔ maxjisin 1n

λj A minus BKN( 111385711138681113868111386811138681113868

11138681113868111386811138681113868 (23)








y(t) Ccx(t)

Ac

minus 00558 minus 09968 00802 00415

0598 minus 0115 minus 00318 0

minus 305 0388 minus 0465 0

0 00805 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Bc

00729 0

minus 475 0775

0153 143

0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cc 0 1 0 0

0 0 0 1⎡⎢⎣ ⎤⎥⎦

(24)






r1

ω1

ω2


T1 T2

D

K

r2
















4

3

2

1

0Tape

spee

d (m

s)

0 2 4 6 8 10 12Time (s)

w1y1y1

minmax

(a)

0 2 4 6 8 10 12Time (s)

w2y2y2

minmax

108060402

0Tape

tens

ion

(N)

(b)

0 2 4 6 8 10 12Time (s)

2

1

0

ndash1Mot

or v

olta

ge (V

)

u1u1

minmax

(c)

0 2 4 6 8 10 12Time (s)

Mot

or v

olta

ge (V

)u2u2

minmax

0504030201

0ndash01

(d)

0 2 4 6 8 10 12Time (s)

600

400

200

0

Coe

ffici

ent v

alue

q1

(e)

0 2 4 6 8 10 12Time (s)

Coe

ffici

ent v

alue

q2

1000800600400200

0

(f)

Coe

ffici

ent v

alue

0 2 4 6 8 10 12Time (s)

4

3

2

1

0

r1

(g)

00

2 4 6 8 10 12Time (s)

20

15

10

5

Coe

ffici

ent v

alue

r2

(h)



012

010

008

006

004

002

00 2 4 6 8 10 12

Time

Stab

ility

indi

cato

rs (S

N)




Aileron

Longitudinalaxis

Top view

Rudder

Front view

X X

Y Z

φ

ψ

β

+ +θ


ndash05

0

05

Yaw

rate

(rad

s)

5 403520 30 450 1510 5025Time (s)

w1y1y1

minmax

(a)

ndash1

ndash05

0

05

1

Bank

angl

e (ra

d)

5 403520 30 450 1510 5025Time (s)

w2y2y2

minmax

(b)

Figure 5 Continued


ndash2ndash1

012

Rudd

er d

efle

ctio

n (r

ad)

1510 25 505 30 35 40 45200Time (s)

u1minmax

u1

(c)

u2minmax

u2

5 403520 30 450 1510 5025Time (s)

ndash2ndash1

012

Aile

ron

defle

ctio

n (r

ad)

(d)

0

50

100

150

200

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

q1

(e)

0

200

400

600

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

q2

(f )

0

1

2

3

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

r1

(g)

0

2

4

6

8

Coe

ffici

ent v

alue

5 403520 30 450 1510 5025Time (s)

r2

(h)


1010

1005

1

0995

0990

0985

0980

Stab

ility

indi

cato

r (S N

)

0 5 10 15 20 25 30 35 40 45 50Time






5 Conclusions



Data Availability




Acknowledgments


References


























































SN ≔ maxjisin 1n

λj A minus BKN( 111385711138681113868111386811138681113868

11138681113868111386811138681113868 (23)








y(t) Ccx(t)

Ac

minus 00558 minus 09968 00802 00415

0598 minus 0115 minus 00318 0

minus 305 0388 minus 0465 0

0 00805 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Bc

00729 0

minus 475 0775

0153 143

0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cc 0 1 0 0

0 0 0 1⎡⎢⎣ ⎤⎥⎦

(24)






r1

ω1

ω2


T1 T2

D

K

r2
















4

3

2

1

0Tape

spee

d (m

s)

0 2 4 6 8 10 12Time (s)

w1y1y1

minmax

(a)

0 2 4 6 8 10 12Time (s)

w2y2y2

minmax

108060402

0Tape

tens

ion

(N)

(b)

0 2 4 6 8 10 12Time (s)

2

1

0

ndash1Mot

or v

olta

ge (V

)

u1u1

minmax

(c)

0 2 4 6 8 10 12Time (s)

Mot

or v

olta

ge (V

)u2u2

minmax

0504030201

0ndash01

(d)

0 2 4 6 8 10 12Time (s)

600

400

200

0

Coe

ffici

ent v

alue

q1

(e)

0 2 4 6 8 10 12Time (s)

Coe

ffici

ent v

alue

q2

1000800600400200

0

(f)

Coe

ffici

ent v

alue

0 2 4 6 8 10 12Time (s)

4

3

2

1

0

r1

(g)

00

2 4 6 8 10 12Time (s)

20

15

10

5

Coe

ffici

ent v

alue

r2

(h)



012

010

008

006

004

002

00 2 4 6 8 10 12

Time

Stab

ility

indi

cato

rs (S

N)




Aileron

Longitudinalaxis

Top view

Rudder

Front view

X X

Y Z

φ

ψ

β

+ +θ


ndash05

0

05

Yaw

rate

(rad

s)

5 403520 30 450 1510 5025Time (s)

w1y1y1

minmax

(a)

ndash1

ndash05

0

05

1

Bank

angl

e (ra

d)

5 403520 30 450 1510 5025Time (s)

w2y2y2

minmax

(b)

Figure 5 Continued


ndash2ndash1

012

Rudd

er d

efle

ctio

n (r

ad)

1510 25 505 30 35 40 45200Time (s)

u1minmax

u1

(c)

u2minmax

u2

5 403520 30 450 1510 5025Time (s)

ndash2ndash1

012

Aile

ron

defle

ctio

n (r

ad)

(d)

0

50

100

150

200

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

q1

(e)

0

200

400

600

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

q2

(f )

0

1

2

3

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

r1

(g)

0

2

4

6

8

Coe

ffici

ent v

alue

5 403520 30 450 1510 5025Time (s)

r2

(h)


1010

1005

1

0995

0990

0985

0980

Stab

ility

indi

cato

r (S N

)

0 5 10 15 20 25 30 35 40 45 50Time






5 Conclusions



Data Availability




Acknowledgments


References























































4

3

2

1

0Tape

spee

d (m

s)

0 2 4 6 8 10 12Time (s)

w1y1y1

minmax

(a)

0 2 4 6 8 10 12Time (s)

w2y2y2

minmax

108060402

0Tape

tens

ion

(N)

(b)

0 2 4 6 8 10 12Time (s)

2

1

0

ndash1Mot

or v

olta

ge (V

)

u1u1

minmax

(c)

0 2 4 6 8 10 12Time (s)

Mot

or v

olta

ge (V

)u2u2

minmax

0504030201

0ndash01

(d)

0 2 4 6 8 10 12Time (s)

600

400

200

0

Coe

ffici

ent v

alue

q1

(e)

0 2 4 6 8 10 12Time (s)

Coe

ffici

ent v

alue

q2

1000800600400200

0

(f)

Coe

ffici

ent v

alue

0 2 4 6 8 10 12Time (s)

4

3

2

1

0

r1

(g)

00

2 4 6 8 10 12Time (s)

20

15

10

5

Coe

ffici

ent v

alue

r2

(h)



012

010

008

006

004

002

00 2 4 6 8 10 12

Time

Stab

ility

indi

cato

rs (S

N)




Aileron

Longitudinalaxis

Top view

Rudder

Front view

X X

Y Z

φ

ψ

β

+ +θ


ndash05

0

05

Yaw

rate

(rad

s)

5 403520 30 450 1510 5025Time (s)

w1y1y1

minmax

(a)

ndash1

ndash05

0

05

1

Bank

angl

e (ra

d)

5 403520 30 450 1510 5025Time (s)

w2y2y2

minmax

(b)

Figure 5 Continued


ndash2ndash1

012

Rudd

er d

efle

ctio

n (r

ad)

1510 25 505 30 35 40 45200Time (s)

u1minmax

u1

(c)

u2minmax

u2

5 403520 30 450 1510 5025Time (s)

ndash2ndash1

012

Aile

ron

defle

ctio

n (r

ad)

(d)

0

50

100

150

200

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

q1

(e)

0

200

400

600

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

q2

(f )

0

1

2

3

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

r1

(g)

0

2

4

6

8

Coe

ffici

ent v

alue

5 403520 30 450 1510 5025Time (s)

r2

(h)


1010

1005

1

0995

0990

0985

0980

Stab

ility

indi

cato

r (S N

)

0 5 10 15 20 25 30 35 40 45 50Time






5 Conclusions



Data Availability




Acknowledgments


References




















































012

010

008

006

004

002

00 2 4 6 8 10 12

Time

Stab

ility

indi

cato

rs (S

N)




Aileron

Longitudinalaxis

Top view

Rudder

Front view

X X

Y Z

φ

ψ

β

+ +θ


ndash05

0

05

Yaw

rate

(rad

s)

5 403520 30 450 1510 5025Time (s)

w1y1y1

minmax

(a)

ndash1

ndash05

0

05

1

Bank

angl

e (ra

d)

5 403520 30 450 1510 5025Time (s)

w2y2y2

minmax

(b)

Figure 5 Continued


ndash2ndash1

012

Rudd

er d

efle

ctio

n (r

ad)

1510 25 505 30 35 40 45200Time (s)

u1minmax

u1

(c)

u2minmax

u2

5 403520 30 450 1510 5025Time (s)

ndash2ndash1

012

Aile

ron

defle

ctio

n (r

ad)

(d)

0

50

100

150

200

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

q1

(e)

0

200

400

600

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

q2

(f )

0

1

2

3

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

r1

(g)

0

2

4

6

8

Coe

ffici

ent v

alue

5 403520 30 450 1510 5025Time (s)

r2

(h)


1010

1005

1

0995

0990

0985

0980

Stab

ility

indi

cato

r (S N

)

0 5 10 15 20 25 30 35 40 45 50Time






5 Conclusions



Data Availability




Acknowledgments


References




















































ndash2ndash1

012

Rudd

er d

efle

ctio

n (r

ad)

1510 25 505 30 35 40 45200Time (s)

u1minmax

u1

(c)

u2minmax

u2

5 403520 30 450 1510 5025Time (s)

ndash2ndash1

012

Aile

ron

defle

ctio

n (r

ad)

(d)

0

50

100

150

200

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

q1

(e)

0

200

400

600

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

q2

(f )

0

1

2

3

Coe

ffici

ent v

alue

5 10 15 20 25 30 35 40 45 500Time (s)

r1

(g)

0

2

4

6

8

Coe

ffici

ent v

alue

5 403520 30 450 1510 5025Time (s)

r2

(h)


1010

1005

1

0995

0990

0985

0980

Stab

ility

indi

cato

r (S N

)

0 5 10 15 20 25 30 35 40 45 50Time






5 Conclusions



Data Availability




Acknowledgments


References





















































5 Conclusions



Data Availability




Acknowledgments


References




















































Model Predictive Control with a Relaxed Cost Function for...

Documents

Transcript of Model Predictive Control with a Relaxed Cost Function for...