Nonlinear modeling and feedback control ofboom barrier automation

Daniel Cunico, Angelo Cenedese, Luca Zaccarian and Mauro Borgo

Abstract— We address modeling and control of a gate accessautomation system. A model of the mechatronic system isderived and identified. Then an approximate explicit feedbacklinearization scheme is proposed, which ensures almost linearresponse between the external input and the delivered torque.A nonlinear optimization problem is solved offline to generatea feasible trajectory associated with a feedforward action anda low level feedback controller is designed to track it. Thefeedback gains can be conveniently tuned by solving a set ofconvex linear matrix inequalities, performing a multi-objectivetrade-off between disturbances attenuation and closed-loopperformance. Finally, the proposed control strategy is testedon the real system and experimental results show that it caneffectively meet the requirements in terms of robustness, loaddisturbance rejection and tracking performance.

I. INTRODUCTION

Due to the increasing global industrial competition forextreme performance and reliability, research on mechatronicsystems is becoming highly multidisciplinary, with an ever-increasing integration of mechanical, electronic, and infor-mation disciplines [1]. Nonlinear actuator phenomena, suchas saturations, dead-zones, backlash or sampling/quantizationeffects, appear frequently in mechatronic systems [2]–[4]and have proved to be a source of performance degradationand closed-loop instability. The efficiency highly dependson the control architecture and its ability to consider thelimitations and constraints in order to optimize the function-ing and to avoid dangerous working conditions. Moreover,robust control design techniques [5] are key for taking intoaccount uncertain model parameters within a broad rangeof real situations, e.g. due to changes in the environmentalconditions or wear of the mechanical components.

Access automation systems are used in several residentialor public areas to prevent unwanted access or to regulatetraffic flow [6], [7]. An active industrial research area dealswith the performance and quality improvement for suchsystems, while lowering the manufacturing costs and thepower consumption.

The standard control techniques are those typical ofelectromechanical motion systems [8]–[11] comprising twohierarchical levels: a trajectory planner generates the desired

D. Cunico, A. Cenedese are with the Department of Informa-tion Engineering, University of Padua, Padova, Italy, 35131, Email:(daniel.cunico@studenti.unipd.it; angelo.cenedese@unipd.it).

D. Cunico, M. Borgo are with BFT SpA, Schio (VI), Italy, 36015, Email:(daniel.cunico@bft-automation.com; mauro.borgo@bft-automation.com).

L. Zaccarian is with the Department of Industrial Engineering, Univer-sity of Trento, Italy, and LAAS-CNRS, Universite de Toulouse, CNRS,Toulouse, France, Email: zaccarian@laas.fr

o x

y

θ

Fig. 1. The boom barrier experimental system.

reference, taking into account the nonlinear dynamics and theconstraints; a linear error feedback reduces the deviation ofthe actual trajectory from the desired reference. The currentindustrial practice for parameter tuning is based on runningseveral experimental tests and adjusting certain PID gainsvia trial and error, until acceptable results are obtained, thusrequiring much time and man power from the technicaldepartment.

In this paper, we focus on the modeling and control ofa road automatic barrier represented in Fig. 1. The originalcontributions of this work are highlighted next. 1) First wederive and experimentally validate a nonlinear mathematicalmodel of the underlying unidirectional power converter, theelectrical motor and the mechanical transmission moving theload by well representing the interplay of mechanical andelectrical components (the mechatronic device). 2) Secondly,we propose an approximate feedback inversion scheme,whose effectiveness is proven by relying on formally certifiedinterval arithmetic combined with formal Taylor expansion(thanks to the Coq Interval tactic [12]): through this scheme,we include a feedback linearizing pre-compensator, preciselycharacterizing the state-dependent saturation values of thevirtual input proportional to the exerted voltage. 3) Thirdly,based on this feedback linearizing structure we propose afeedforward/feedback architecture, whose feedforward termis generated through the minimization of a nonlinear func-tional cost under constraints, and the feedback term isconveniently tuned via a linear matrix inequality (LMI)formulation [13]. The LMI constraints allow us to optimizea disturbance rejection performance under uncertain modelparameters, while constraining the closed-loop poles in asuitable region of the left half-plane [14] to induce a suitable

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transient response. 4) Fourth, rigorous statements certifythe effectiveness of our scheme in terms of stabilizationof the error dynamics and feasibility of our LMIs. 5)Lastly, and most importantly, experimental results on theindustrial device confirm the effectiveness of the proposedstrategy, which induces regular (no oscillations) and fastbarrier opening, despite the system nonlinear dynamics anduncertain parameters. Beyond the performance improvementin the specifically considered application, the approach is ofgeneral interest and it can be easily extended to many similarapplications that use the same control electronics. Sometechnological details are omitted and all the units of measureare normalized in the experimental results for reasons ofconfidentiality. However, the proposed design strategy isfully parametric and has been tested successfully with manydifferent parameter selections. The paper is organized asfollows. In Section II the experimental setup is describedand the closed-loop goals are clarified. In Section III themathematical modeling of the road barrier gate is derived,considering all the mechatronic components. In Section IVthe augmented plant model and the parameter identificationprocedure are illustrated. Section V describes the electricaldrive and the feedback linearization method. In Section VIthe control architecture is presented and an LMI based tuningprocedure is proposed. Experimental tests are discussed inSection VII. Concluding remarks are reported in SectionVIII.

II. SYSTEM DESCRIPTION AND GOALS

The considered mechatronic system can be represented assketched in Fig. 2. The electronic parts are the power sourcecircuit and the driver of the motor. A DC motor convertselectrical energy into mechanical energy and produces thetorque required to move the load with the desired outputangular speed. The torque is transmitted through a gearboxto the mechanical system. Two main elements compose themechanics of the automatic road barrier: a bar that rotatesabout one of its ends, and a spring-damper system used tocompensate for the weight of the bar. The sensor devicesrepresent the part related to the data acquisition system, i.e.the group of sensors and transducers with their conditioningcircuits. In the present case study, the acquired measurementsare the motor speed ωm and the motor current ia. Theelectronic, gearmotor and mechanical subsystems togetherwith the sensors form the so-called augmented plant. Finally,the embedded control software produces the duty cycle δ ofa PWM signal to control the actuator with precise timing.

The main problems and goals regarding the control of thisapplication can be summarized in the following points:

1) (Limitations) The low-cost electronic board does notallow exerting a motor torque/current in the brakingdirection. Therefore, the braking phase is often slowand the system only decelerates due to the action offriction.

2) (Safety) The gate opening maneuver must end with asufficient low speed at the mechanical stop in order toavoid damaging the device.

ControlSoftware

ElectronicSubsystem

GearmotorSubsytem

MechanicalSubsystem

Sensors

δ

ωm, ia

Augmented Plant

Fig. 2. Blocks diagram of a common mechatronic system.

3) (Performance) The gate opening should be regular(without oscillations) and fast.

In addition to the aims defined above, the controller shouldbe robust with respect to possible slow unmodeled dynamics,small delays in the loop, quantization effects, variationsrelated to environmental conditions and aging. Finally, theproposed control strategy must be easy enough to be imple-mented in the micro-controller unit of the industrial device,which has limited computational capacity.

III. MODELING

A. Electric motor

A brushed DC electric motor exerts the torque on themechanical subsystem. The DC motor dynamic model is wellknown in the literature [15]. The electrical equation is:

ua(t) = Raia(t) + Ladia(t)

dt+ ea(t), (1)

where ua(t) is the terminal voltage, ia(t) is the armaturecurrent, Ra is the armature winding resistance, La is thephase inductance and ea(t) is the back electromotive force(BEMF). The BEMF and the torque exerted at the motorshaft correspond to

ea(t) = ktωm(t), (2)τm(t) = ktia(t), (3)

where kt is the torque constant and ωm is the mechanicalspeed of the motor. Note that the two constants in eq. (2)and eq. (3) coincide because of the balance between the inputelectrical power and the output mechanical power.

B. Mechanical system

The mechanical subsystem of the automatic road barrier,represented in Fig. 3, is composed by two main elements:1) a bar rotating about one of its ends (the point O), assumedto be an ideal rod of length la and mass ma, whose angularposition with respect to the x-axis in Fig. 3 is described bythe angle θ.2) a spring-damper of natural length ls,0 and spring constantks, with one of its ends connected to the bar through a leverof length l`. The lever element is fixed to the bar in O, thusforming with it a constant angle ϕ. The damper elementproduces a force proportional to the velocity, according tothe viscous coefficient bs, allowing for the stabilization ofthe entire mechanical system.

O x

yma, la

l`

d

bsks, ls

θϕ

αβ

s

Fig. 3. Mechanical subsystem of the automatic barrier.

0 /8 /4 3 /8 /2

theta [rad]

0

0.5

1

torq

ue [norm

aliz

ed] bar torque

spring torque

overall torque r( )

Fig. 4. Main components of the torque τr(θ) in (4b). The spring is pre-compressed of a length s0 so that θe = π/4.

Furthermore, it is possible to pre-compress the spring of alength s0 in order to calibrate the resulting force. Typicallys0 is tuned in such a way that the entire system be at theequilibrium when θ = θe = π/4. We assume that the massof the spring and of the lever are negligible. From geometricconsiderations, we obtain the following expressions for theangle α(θ) of the lever w.r.t. the x-axis, the length ls(θ) ofthe spring and the compression s(θ) of the spring:

α(θ) = π − ϕ− θ,

ls(θ) =√d2 + l2` − 2d l`cos (β + α(θ)),

s(θ) = ls,0 − ls(θ) + s0.

Following the notation used in Fig. 3, the inertia and thefriction of the mechanical load are

Ja =1

3mal

2a, b(θ) = bs

l`d

ls(θ)sin (β + α(θ)) , (4a)

and the reaction torque exerted by the rod at the hingecorresponds to

τr(θ) = −kss(θ)l`d

ls(θ)sin (β + α(θ))︸ ︷︷ ︸

spring torque

+g

2malacos(θ)︸ ︷︷ ︸

bar torque

. (4b)

Fig. 4 shows the evolution of the overall external torque τras a function of the angle θ. The first and the second termat the right-hand side of equation (4b) are respectively thespring and bar contributions to the torque, tuned to generatean equilibrium point at θ = θe.

C. Mechanical transmission

The mechanical transmission consists of a gear trainsystem. The gearbox is modelled by means of the classicalmechanical approach assuming rigid coupling [16]. In anideal transmission, i.e. under the assumption of losslesspower transfer, denoting by ω(t) = θ(t) the speed at theoutput of the gear, we have that:

ω =r1

r2ωm = Ngωm, θ = Ngθm (5)

where r1 and r2 are the gear wheels radii and Ng is thetransmission gear ratio. A better description is achieved byconsidering an efficiency η < 1 of the transmission gear, andcharacterizing load torque τ` as

τ`(θm(t), ωm(t)) = τr(Ngθm(t))Ng

η+ τcsign(ωm(t)), (6)

where τr is defined in eq. (4b) and τc represents the Coulombfriction torque [17]. The resulting mechanical equation of thesystem is

τm(t) = Jtotdωm(t)

dt+ btotωm(t) + τ`(θm(t), ωm(t)) (7)

Jtot = Jmg + Ja

N2g

η, btot = bmg + b(θm)

N2g

η. (8)

where Jmg and bmg are the gearmotor inertia and friction.

IV. AUGMENTED PLANT AND IDENTIFICATION

A. Augmented Plant modeling

Combining (1), (2), (3), (7) and recalling that θ = Ngθm,a state-space model of the augmented plant can be obtained.Denoting by ua the voltage applied to the motor terminals, byx1 = ia the motor current, by x2 = θm the motor positionand by x3 = ωm the motor velocity, we have, with x =[x1, x2, x3]

>,

x = f(x, ua) =

−Ra

Lax1 −

kt

Lax3 +

1

Laua

x3

kt

Jtotx1 −

btot(x2)

Jtotx3 −

τ`(x2, x3)

Jtot(9)

where τ`(x2, x3) = τr(Ngx2)Ng

η + τcsign(x3) according to(6). Table I reports all the relevant quantities appearing in(8), (9), and their definitions.

B. System identification

The focus is now on the identification of the model pa-rameters and their experimental validation. To this aim, well-established System Identification methods [18] are used. Thenominal model parameters are provided from the literatureand the technical data-sheet, while other parameter valuescan be estimated from the experimental data, to accuratelydescribe the system response. Following [19], for the elec-trical and mechanical parameters of the motor and gearsubsystem, a set of experiments has been performed withan independent laboratory acquisition system to measure the

0 10 20 30 40 50 600

0.5

1

Vo

lta

ge

0 10 20 30 40 50 60-1

0

1

Cu

rre

nt

0 10 20 30 40 50 60

time [s]

0

0.5

1

Sp

ee

d0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.25

0.5 input duty cycle

0 0.5 1 1.5 2 2.5 3 3.5 40

0.25

0.5 measured motor speed

simulated motor speed

0 0.5 1 1.5 2 2.5 3 3.5 4

time [s]

0

0.25

0.5 measured motor current

simulated motor current

Fig. 5. (Left) Example of data acquired for the identification of the gear-motor parameters: from above to below, respectively, voltage supplied to themotor, motor current and motor speed. (Right) Experimental response to a staircase input duty cycle δ, compared to the identified model simulation. Theoutput signals are the motor speed ωm and the motor current ia, while ωm,M and ia,M are normalization factors.

TABLE IPARAMETERS OF MODEL (8), (9).

Symbol Name Defined in

Ra Armature resistance [Ω] eq. (1)La Armature inductance [H] eq. (1)kt Torque constant [Nm/A] eq. (3)Ng Gear ratio eq. (5)η Gear efficiency eq. (6)bmg Gearmotor viscous friction [Nms] eq. (7)Jmg Gearmotor inertia [Kgm2] eq. (7)Ja Rod inertia [Kgm2] eq. (4a)b Nonlinear spring damping [Nms] eq. (4a)τr Reaction torque [Nm] eq. (4b)τc Coulomb friction torque [Nm] eq. (9)

responses to canonical input signals. For identification pur-poses, the sampling frequency is 20 times higher than that ofthe industrial product and a 16-bit ADC resolution is used. ANational Instruments DAQ board (USB-6216) has been usedto acquire the data and the motor has been equipped witha 12-bit resolution encoder (Eltra ER38F). Voltage, currentand speed are continuous-time signals acquired through asampling that produces two discrete-time datasets of lengthn.

For the estimation of the electrical parameters Ra, La ofeq. (1), the following equation has been identified:

dia(t)

dt= −Ra

Laia(t) +

1

Laua(t), ya(t) = ia(t−∆) (10)

where ∆ is the delay due to the laboratory acquisitionsystem. Note that, as compared with (1), ea is not present in(10) since the identification phase is performed under lockedrotor condition, that is, from (2), ea is zero. Notably, thedelay should not be identified since the industrial devicedoes not contain the acquisition system used to performthe parameter identification experiments. Since input ua isconstant during each sampling period, the sampled measure-ments ya(k) collected from (10) depend on two values ofthe discretized input as follows

ya(k + 1) = Φya(k) + Γ0ua(k + 1) + Γ1ua(k), (11)

Φ = e−RaLats ,

Γ0 =1

Ra

(1− e−Ra

La(ts−∆)

),

Γ1 =1

Ra

(e−

RaLa

(ts−∆) − e−RaLats),

where ua(k) is the voltage supplied to the motor at the k-th sampling time, tk = kts, k = 1, 2, . . . , n, and ts is thesampling time. From eq. (11), with a least-squares estimateof the parameters of the model, we can obtain the value ofthe electrical parameters as:

Ra =1− Φ

Γ0 + Γ1, La = −Ra ts

lnΦ.

In a similar way, for the estimation of the mechanicalparameters bmg, Jmg of eq. (8), we consider the discretizeddynamics between the applied voltage ua and the delayedspeed measurement ωm(t−∆), giving the following equation

ωm(k + 1)− Φωm(k) = Γ0ua(k + 1) + Γ1ua(k), (12)

Φ = e− bmgRa+k2t

JmgRats ,

Γ0 =kt

bmgRa + k2t

(1− e−

bmgRa+k2tJmgRa

(ts−∆)),

Γ1 =kt

bmgRa + k2t

(e− bmgRa+k2t

JmgRa(ts−∆) − e−

bmgRa+k2tJmgRa

ts

).

Then, similar to before, the mechanical parameters are com-puted as

bmg =1

Ra

(kt(1− Φ)

Γ0 + Γ1− k2

t

), Jmg = − (bmgRa + k2

t )tsRalnΦ

.

Fig. 5(left) shows an example of data acquisition forthe identification of the gearmotor parameters. The parame-ter values of the mechanical system are generally known,however to obtain a suitable plant representation, and toimprove the prediction capability, certain model parameters(the mechanical Coulomb friction τc, the damping of thespring b) have been adjusted around their nominal values.Hence, a calibration procedure has been carried out bycomparing the acquired and the simulated data and theparameter set that minimizes the root mean square error

Vac v+Ra La

eaua

ia +

Motor

Mosfet

Driver

δ ∈ [0, 1] −→

VD

1

Fig. 6. The electrical scheme of the driver and the equivalent motor circuit.

0 T 2T 3T 4T 5T 6T 7T 8T 9T 10T

time

0

0.5

1

am

plit

ude (

norm

aliz

ed)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Fig. 7. (Top) Time evolution of the signals related to the specific dynamicsof the drive. (Bottom) Illustration of eq. (14) for constant BEMF values,where ea,M and ua,M are normalization factors. The blue and red dotsrepresent, respectively, the minimum and maximum points discussed insection V-B.

has been selected. Using all the identified parameters, il-lustrated in Table I, model (9) has been validated on a setof independent experiments where the following quantitieshave been acquired via the serial communication device ofthe mechatronic system under analysis: the duty cycle δ, themotor speed ωm and the motor current ia. Fig. 5(right) showsa sample outcome of the model validation results, obtainedby comparing the identified model simulation outputs (motorcurrent ia and angular velocity ωm) with the correspondingsignals measured from the physical system, when the sameinput signal (duty cycle δ) is used. Specifically, we show theresponses to a staircase input. The qualitative trend of thesimulated signals is close to the experimental measurements.

V. ELECTRICAL DRIVE AND FEEDBACK LINEARIZATION

A. Electric motor drive

The motor driver, illustrated in Fig. 6, supplies the voltageto the electric motor, based on the reference signal δ ∈ [0, 1],

provided by the control law. With reference to Fig. 6, thealternating voltage source with effective value Vac is rectifiedby means of a Graetz bridge. The motor is controlled bychopping the non-negative semi-sinusoids v+, thus modify-ing the average voltage depending on the on-off time of theMosfet switching. A flyback diode, in parallel to the motor,forms a circulating path of the inductive load current. To easethe mathematical modeling, several approximations havebeen carried out. The Mosfet is modelled as an ideal switchand all diodes are considered as a voltage generator whenconducting current, whose voltage VD is set to the valuespecified in the diode data-sheet. The circuit is completedby a relay, which allows reversing the polarity of the motorwhen switching between opening and closing maneuvers.The reference voltage δ ∈ [0, 1] coming from the controlsoftware governs the driver operation. Considering a singleperiod T of the rectified semi-sinusoid v+, given the dutycycle δ ∈ [0, 1] during the “Mosfet off” portion of the dutycycle lasting toff = (1− δ)T seconds, the voltage ua acrossthe motor terminals is:

t ∈ [0, (1− δ)T ]⇒ ua(t) =

−VD if ia(t) > 0

ea(t) if ia(t) = 0(13a)

while during the remaining “Mosfet on” portion of the dutycycle lasting ton = δT seconds, we have:

t ∈ [(1− δ)T, T ]⇒ ua(t) =

v+(t) if ia(t) > 0

ea(t) if ia(t) = 0.(13b)

An example of the corresponding signals is reported at thetop of Fig. 7, where it can be observed that the load ismainly resistive, since the PWM switching period is high,as compared to the electrical time constant of the armaturewindings. Therefore, to simplify eq. (13) we can assumeia = 0 during the toff phase and ia > 0 during the ton

phase. Since the controller has a sampling period T , we aimto determine the average voltage ua of the waveform in theinterval t ∈ [0, T ]. This can be computed as

ua(δ, ea)=1

T

[∫ (1−δ)T

0

eadt+

∫ T

(1−δ)T

√2Vacsin

( πTt)dt

],

where the first term is the weighted contribution of ua(δ, ea)during the toff phase, while the weighted contribution on theton phase is given by the semi-sinusoid v+ with amplitude√

2Vac and period T . It results that ua is equal to

ua(δ, ea) = ea(1− δ) +

√2Vac

π(1− cos(πδ)). (14)

The bottom of Fig. 7 shows eq. (14) as a function of δ, for0 ≤ δ ≤ 1 and for constant values of ea, 0 ≤ ea ≤ ea,max.

B. Feedback linearization of the motor drive dynamics

Using a feedback linearization approach [20], if we man-age to locally invert function (14), it is possible to selectthe duty cycle δ so that the plant seen by the controller islinear. Inspecting the blue and red dots in the lower plot ofFig. 7, it is clear that the local inversion needs to rely on

0

0.5

1

0 0.2 0.4 0.6 0.8 1

-0.01

0

0.01

Fig. 8. Function ua in (17c) and the mismatch function Ψ(δ, ea) definedin (18), represented for various values of the BEMF ea.

the values of δ providing the minimum and maximum of ua.Indeed, eq. (14) is invertible only in a range depending onea. In particular, differentiating (14) with respect to δ, we candetermine the minimum point at δ = δm and the maximumone at δ = δM (represented by the blue and red dots of Fig.7), as a function of ea ∈ [0,

√2Vac), as follows:

δm(ea) =1

πarcsin

(ea√2Vac

)∈[0,

1

2

), (15a)

δM(ea) = 1− δm(ea) ∈(

1

2, 1

]. (15b)

Note that the values of δm and δM are well-defined since wehave always ea <

√2Vac. The corresponding extreme values

ua,m(ea) = ua(δm(ea), ea) and ua,M(ea) = ua(δM(ea), ea)of ua can be conveniently expressed as a function of δm,omitting the dependence on ea at the right-hand side, forcompact notation:

ua,m(ea) =√

2Vac

[(1− δm)sin(πδm) +

1

π− cos(πδm)

π

],

(16a)

ua,M(ea) =√

2Vac

[δmsin(πδm) +

1

π+

cos(πδm)

π

].

(16b)

To suitably invert (14) based on the quantities above, definethe normalized input δ and output ua as follows

δ(δ, ea) =δ − δm(ea)

δM(ea)− δm(ea), (17a)

δ = γ(δ, ea) = δm(ea) + (1− 2δm(ea))δ, (17b)

ua(δ, ea) =ua(γ(δ, ea), ea)− ua,m(ea)

ua,M(ea)− ua,m(ea), (17c)

both of them taking values in [0, 1].In the special case ea = 0, the expression of ua in

(17c) simplifies to ua(δ, 0) = 1−cos(πδ)2 = sin2

(πδ2

). More

generally, we may decompose

ua(δ, ea) =1

2

(1− cos(πδ)

)+ Ψ(δ, ea), (18)

where the mismatch function Ψ(δ, ea) is small, thereforeneglectable, as visible from Fig. 8 and as characterized in

the next lemma, whose proof is given in Section V-C toavoid breaking the flow of the exposition.

Lemma 1. For any δ ∈ [0, 1] and any ea ∈ [0,√

2Vac), itholds that |Ψ(δ, ea)| < 0.01001.

Based on the above, we can now state our main resultabout the inversion of function (14).

Proposition 1. For any u ∈ [ua,m(ea), ua,M(ea)], selecting

δ = δm(ea)+1−2δm(ea)

πarccos

(1− 2(u− ua,m(ea))

ua,M(ea)− ua,m(ea)

)(19)

the resulting average input obtained from (14) is

ua(δ, ea) = u+ψ with |ψ| ≤ 0.01001(ua,M(ea)−ua,m(ea)).

Namely, the mismatch between the requested input u and theapplied input ua is about one percent of the input range.

Proof. Substituing expression (19) in (17a) we obtainδ(δ, ea) = 1

πarccos(1− 2(u−ua,m(ea))

ua,M(ea)−ua,m(ea)

). Substituting

this last quantity in (18) and using Lemma 1, weobtain ua(δ, ea) =

u−ua,m(ea)ua,M(ea)−ua,m(ea) + Ψ(δ, ea), with

|Ψ(δ, ea)| < 0.01001. The result then immediately followsfrom (17c).

C. Proof of Lemma 1

First, we state below a result of independent interestabout a polynomial approximation of the sine function. Forthe proof of the result, we adopt formally certified intervalarithmetic combined with formal Taylor expansion, thanksto the Coq Interval tactic [12].

Proposition 2. For any α ∈ [0, 1] the following bound holds:| sin

(π2α)− 1

2 (3α−α3)| < 0.02002. Namely, the polynomialfunction 1

2 (3α − α3) approximates sin(π2α)

with an errorof about 2%.

Proof. The proof is carried out using formally certifiedinterval arithmetic software [12]. In particular, denotingξ(α) := sin

(π2α)− 1

2 (3α − α3), it is readily certifiedthat ξ(α) ≤ 0.02002 for all α ∈ [0, 1]. To prove thatξ(α) ≥ 0, a certificate of positivity is immediate in theinterval α ∈ [0, 0.99], while proving non-negativity in theremaining interval [0.99, 1] requires proving that, in thisinterval, ξ′′(α) ≥ 0 (so that ξ′(α) is non-drecreasing). Sinceξ′(1) = 0 and ξ′(0.99) < 0, the above monotonicity propertyproves ξ′(α) ≤ 0 for α ∈ [0.99, 1], which means the ξ istherein non-increasing. Since ξ(0.99) > 0 and ξ(1) = 0,this means ξ(α) ≥ 0 for α ∈ [0.99, 1], thus completing theproof.

To the end of proving Lemma 1, denoting with σ = π(1−2δm(ea)), after some simplifications, we obtain from (17c),

ua(δ, ea) =tan(πδm)(sin(σδ)− σδ) + 1− cos(σδ)

2− σtan(πδm). (20)

Then, using ua(δ, 0) in (18), we obtain the expression ofΨ(δ, ea) = ua(δ, ea)− ua(δ, 0) as

Ψ=tan(πδm)(sin(σδ)+σsin2(πδ2 )−σδ)+cos(πδ)−cos(σδ)

2− σtan(πδm),

(21)which clearly satisfies (see also Fig. 8),

Ψ(0, ea) = 0, Ψ(1, ea) = 0, Ψ(δ, 0) = 0. (22)

Moreover, the following symmetry is also visible from thelower representation in Fig. 8.

Lemma 2. For any α ∈ [0, 1] and any ea ∈ [0,√

2Vac), itholds that Ψ

(1+α

2 , ea

)= −Ψ

(1−α

2 , ea

).

Proof. From eq. (21) consider tan(πδm)(σsin2(πδ2 ) −σδ) + cos(πδ). Substituting sin2(πδ2 ) = 1

2 − 12 cos(πδ)

and noting that cos(π2 + πα2 ) = −sin(πα2 ) it

follows that tan(πδm)( 12 sin(πα2 ) + α

2 ) − sin(πα2 ) =−[tan(πδm)(− 1

2 sin(πα2 )− α2 ) + sin(πα2 )

]. Recalling that

σ = π − 2πδm, the proof is completed for the remainingterms, applying the trigonometric addition formulas andnoting that tan(πδm)sin(σ2 )cos(σα2 ) − cos(σ2 )cos(σα2 ) = 0since tan(πδm)sin(σ2 ) = cos(σ2 ) = sin(πδm).

Based on Lemma 2, for proving the bound in Lemma 1,we may focus on its values in the range δ ∈ [0, 0.5], whichcan be parametrized by δ = 1−α

2 , α ∈ [0, 1], and ea =√2Vac

(1− 2

π s), s ∈ (0, π2 ]. This provides

supδ∈[0,1],

ea∈[0,√

2Vac)

|Ψ(δ, ea)| = supα∈[0,1],s∈(0,π2 ]

|Ψ(α, s)|, (23)

where Ψ(α, s) := Ψ(

1−α2 ,√

2Vac

(1− 2

π s))

. Function Ψcan be expressed as follows, after some simplifications,

Ψ =1

2

(sin(π

2α)− α− α(1− α)

Φ(s)− Φ(αs)

(1− α)s

1

Φ′(s)

),

where Φ(s) = sin(s)s and Φ′(s) = s cos(s)−sin(s)

s2 denotesits derivative. In particular, due to the positivity of Φ(s)and non-positivity of both Φ′(s) and Φ′′(s), it holds thatΦ(s)−Φ(αs)

(1−α)s1

Φ′(s) is non-decreasing for s ∈(0, π2

]. Hence,

due to positivity of α(1−α), for each α ∈ [0, 1], the functions 7→ Ψ(α, s) is nonincreasing in

(0, π2

]. Since the function

is zero for s = π2 , then the function is everywhere positive

and its supremum is given by

ΨM(α) = lims→0+

Ψ(α, s) =1

2

(sin(π

2α)− 1

2(3α− α3)

),

where the right-hand side expression has been computed byapplying L’Hopital’s rule three times. Finally, applying (23)and Proposition 2, we obtain an upper bound on |Ψ(δ, ea)|equal to 1

20.02002, thus completing the proof of Lemma 1.

Ref.PD

controllerFeedback

linearizationAugmented

Plant

Feed-Forward

r e ufb u δ y

uffLinearized Plant

Fig. 9. Blocks diagram of the control architecture.

VI. CONTROLLER DESIGN

In this section we describe the design of a closed-loopspeed controller, optimizing the performance during theopening and closing operations. The algorithm brings thebarrier angular position from θ0 = 0 at rest to θf = π/2at rest (in the opening phase) or vice-versa (in the closingphase), while satisfying a number of operating constraints.While the presented algorithm is generic, we will focus onthe opening task, which is more critical due to the stringentopening time requirements.

The proposed control architecture is depicted in Fig. 9. Theaugmented plant model (9) has been defined and identifiedin Section IV. An optimization problem is solved offline togenerate a feedforward input uff and a reference trajectory r,specifying the desired motor angular velocity ωm, which issubsequently used as reference to be tracked by the feedbackcontroller. The feedback block in Fig. 9 consists of a PDcontroller, operating at 100Hz to be compatible with theembedded software implementation. The overall control lawu, used to compute δ from the feedback linearizing law (19),is given by

u(t) = uff(t) + ufb(t), (24)

where uff is an optimized feedforward input associated withthe reference motion r and ufb is an error feedback stabilizerexploiting the plant measurement y. Due to Proposition 1,the dynamics from u to ua is almost an identity (with a1% error) if u ∈ [ua,m(ea), ua,M(ea)]. The design paradigmfor the feedforward and the feedback blocks of Fig. 9 isexplained in the next sections and can be extended to similaraccess automation systems.

A. Reference and feedforward generation

The reference r and the feedforward term uff are obtainedby solving a constrained nonlinear optimization problem.The constraints associated to the physical limits of the systemare

ua,m(ea) ≤ ua ≤ ua,M(ea), 0 ≤ ia ≤ ia,M, (25)

where ua,m(ea) and ua,M(ea) are defined in (16) and ia,Mis the maximum current. To leave some input margin for thefeedback action, we define 5% tighter constraints than theactual ones. To avoid feasibility issues, the formulation witha soft constraint is introduced by adding a time-varying slackvariable ε and incorporating it into the cost functional. Theconstraints for the optimization problem become

ua,m(ea) + µ(ea) ≤ u ≤ ua,M(ea)− µ(ea),

0.05 ia,M − ε ≤ x1 ≤ 0.5 ia,M,(26)

where µ(ea) = 0.05 (ua,M(ea) − ua,m(ea)) and ε ∈[0, 0.05 ia,M]. Note that the motor current can only flowin one direction, which is reversed by a relay when togglingbetween the opening and closing phases. In the following wefocus on the opening phase, the closing one being similar.Moreover, discontinuities of the input u are not feasible,so we enforce continuity of u by using as input the timederivative v = u of u. The optimal control problem (OCP)is formulated as follows

minx(·),u(·),v(·),ε(·)

∫ tf

t0

‖h(x(t), v(t), ε(t))‖2W dt+ ‖hf(x(tf))‖2Wf

(27a)subject to:

x(t) = f(x(t), u(t)), u(t) = v(t), ∀t ∈ [t0, tf ], (27b)ψ(x(t), u(t), ε(t)) ≤ 0, ∀t ∈ [t0, tf ], (27c)

where t0 (initial time), tf (terminal time) are fixed and thecost functions h and hf are defined as

h(x, v, ε) = [ia, θ − θf , v, ε]>,

hf(x) = [ia, θ − θf ]>.

(28)

weighted by diagonal matrices

W = diag([

10−1, 102, 10−3, 107]),

Wf = diag([

10−1, 102]),

(29)

to equalize the range of the corresponding variables. Theequality constraints (27b) represent the dynamics of thesystem, where f(x, u) is defined in (9), while the inequality(27c) comprises the constraints (26) with

ψ(x, u, ε) =

ua,m(ea) + µ(ea)− uu− ua,M(ea) + µ(ea)

0.05ia,M − ε− x1

x1 − 0.5ia,M−ε

ε− 0.05ia,M

. (30)

The penalty related to the position error θ − θf in (28) isneeded to optimize the barrier opening time, while the onerelated to the current ia penalizes high currents. The cost onthe input v ensures a sufficiently smooth control action, whilethe slack variable ε is needed to implement the soft constraintgiven in (26). The OCP (27) is solved offline in MATLABusing the software package MATMPC [21], [22], [23], anopen-source tool to solve nonlinear programming (NLP). InMATMPC, a NLP problem is formulated by discretizing theOCP using multiple shooting [24] over the prediction horizontf , which is divided into N shooting intervals [t0, t1, . . . , tN ].

MATMPC has been set up with a 4th order Runge-Kuttaintegrator and qpOASES as QP solver [25]. The systemdynamics is discretized with a sampling time Ts = 0.01sand a total number of N = 500 shooting intervals, enabling aprediction length of 5s. Given the ensuing optimal solutionsx∗(·), u∗(·), we use the resulting profile r(·) = x∗3(·) asreference for ωm and the optimal u∗(·) as feedforward term,corresponding respectively to the dashed curve in Fig. 11(a)and the grey line in Fig. 11(c).

B. PD feedback controller

Based on the optimized solutions of (9), computed in (27)of the previous section, corresponding to trajectory x∗, andinput uff = u∗, we may obtain the dynamics of the mismatchstate x = x∗ − x to be stabilized by input ufb:

La˙x1 = −Rax1 − ktx3 − ufb

˙x2 = x3

˙x3 =kt

Jtotx1 −

btot

Jtotx3 +

1

Jtotw,

(31)

where the exogenous signal w represents the nonlinearmismatch terms comprising• the viscous friction term btot(x

∗2)− btot(x2);

• the external torque τ`(x∗2, x∗3)− τ`(x2, x3).

Model (31) can be reduced to a lower-order system withthe objective of simplifying the feedback tuning procedure.Since the inductance La is small, we can reduce (31) byignoring the (fast) electrical time constant. Fixing La = 0,the first equation in (31) becomes

0 = −Rax1 − ktx3 + ufb, (32)

which provides x1 = ufb−ktx3

Ra. This can be replaced in eq.

(31) to obtaineθ = eω

eω = −(k2

t − btotRa

RaJtot

)eω −

kt

RaJtotufb +

1

Jtotw,

(33)where eθ and eω represent the angular position and velocityerrors, respectively. Considering the reduced model (33) anddefining the error dynamics e = [eθ, eω] ∈ R2, dynamics(33) can be written as

e =Ae+Bufb + Ew (34)

:=

[0 1

0 −k2t +btotRa

RaJtot

]e+

[0

− ktRaJtot

]ufb +

[01Jtot

]w.

Lemma 3. The pair (A,B) is controllable.

Proof. This property is verified simply by noting that thecontrollability matrix C = [B|AB] has full rank.

Motivated by Lemma 3, the goal is to tune the parametersof a PD feedback control law

ufb = Ke =[kp kd

] [eθeω

]= kpeθ + kdeω, (35)

for the closed loop system (34) ensuring desirable closed-loop dynamic performance.

C. PD gains tuning

In this section we propose an LMI-based technique to tunethe feedback controller parameters K = [kp, kd] in (35).The objectives of the tuning are to guarantee the stability,to optimize the rejection of disturbance w, and to shapethe transient performance by constraining the closed-loopeigenvalues in the shaded region of Fig. 10 (left).

Specifically, fixing parameters α ≥ 0, ρ > α, ϑ ∈ [0, π/2]and choosing a matrix C characterizing a performance outputz = Ce, consider the following optimization problem:

minW∈R2×2,X∈R1×2,γ∈R

γ subject to:

W = W> > 0 (36a)

M +M> + 2αW < 0 (36b)[(M +M>)sin(ϑ) (M −M>)cos(ϑ)(M> −M)cos(ϑ) (M +M>)sin(ϑ)

]≤ 0 (36c)[

−ρW M>

M −ρW

]≤ 0 (36d)M +M> E WC>

E> −γI 0CW 0 −γI

< 0, (36e)

where M := AW + BX , ϑ ∈ [0, π/2] and I is the identitymatrix of proper dimensions. Constraints (36b), (36c), (36d)force the closed-loop poles to lie in the shaded region ofthe complex plane, shown at the left of Fig. 10, whichis the intersection of three elementary LMI regions: anα-stability region, a disk of radius ρ, and a conic sectordetermined by ϑ. The shape of this region can be adjustedusing these parameters, modifying the dynamical propertiesof the system.

Proposition 3. Under Lemma 3, for any value of α ≥ 0, ϑ ∈[0, π/2] and ρ > α LMI (36) is feasible. Moreover, for anyfeasible solution to (36), selecting K = XW−1 the followingproperties hold: i) the closed-loop matrix (A + BK) haseigenvalues with absolute value less than ρ, ii) the dampingfactor of the poles is larger than cos(ϑ), iii) (A+BK) haseigenvalues with real part smaller than −α, iv) the L2 gainfrom w to z = Ce for (34) with ufb = Ke is smaller thanγ.

Proof. Feasibility of (36) comes from the fact that thecontrollability property in Lemma 3 implies a matrix Kthat places the eigenvalues of the closed-loop system on theregion of the complex semiplane defined by ρ, ϑ and α.i-ii) The eigenvalues of (A+BK) having an absolute valuesmaller than ρ and the damping factor larger than cos(ϑ) area direct application of the results in [14, Equations (10) and(13)].iii) This follows from noticing that (36b) implies (A+BK+αI)W + W (A + BK + αI)> ≤ 0, which holds positivedefinite W only if A+BK has convergence abscissa smallerthan −α.iv) The proof is a standard application of the boundedreal lemma and the use of quadratic Lyapunov functions.Defining V (e) = e>We, W = W> > 0 by constraint(36a), performing a Schur complement on (36e), left-rightmultiplying by [e, w]

> we obtain that ∀ [e, w] 6= (0, 0):

e>(M +M>)e+ 2e>Ew + 1γ z>z − γw>w < 0.

Substituting M = AW + BKW we get 〈∇V (e), e〉 +1γ z>z < γw>w. Integrating both sides, we obtain the desired

Re(s)

Im(s)

α

ϑ

ρ

0 2 4 6 8 10 12

103

104

Trade-off curves between and

Fig. 10. (Left) The shaded region where the closed-loop eigenvaluesare constrained by (36b)-(36d). (Right) Trade-off curves between α andγ obtained by solving the optimization problem (36) for model (34),considering increasing values of α and for two different values of ϑ. Thecolored dots correspond to the operating points chosen for the experimentalresults illustrated in Fig. 11.

bound on the L2 gain from w to z, i.e. ‖z‖2 ≤ γ‖w‖2 (orequivalently on the H∞ norm).

Remark 1. In the presence of input saturation, the feedbacksystem has the form considered in [26, eqs. (1),(2)], withparameters a1 = 0, a2 =

k2t +btotRa

RaJtot, k = kt

RaJtotand the

function β(e) = [kp, kd] e (using the notation of [26]).Applying [26, Thm 1], we may conclude that even with asaturated feedback ufb, the origin of the error system (34)remains globally asymptotically stable and locally exponen-tially stable.

The LMI-based design approach (36) is an effective toolfor performing the design of K. For our application, to theend of reducing as much as possible the oscillations, weselect

z = Ce =[0 1

]e = eω. (37)

The suggested use of (36) is to fix parameters ρ and ϑto ensure a maximum natural frequency and a minimumdamping ratio. Then, for a fast decay rate, several differentvalues of α can be tested to generate the corresponding trade-off curve, as reported in the right of Fig. 10, where we showthe curves for ϑ = π

6 and ϑ = π18 . The trade-off between

γ and α is easily seen from the resulting curves, wherethe solution of the optimization problem (36) provides theoptimal gain γ∗(α) for each value of the parameter α. Theoperating points highlighted with colored dots correspond tothe feedback gains used in the experiments reported in Fig.11 of the next section.

Remark 2. To certify stability of the error dynamics (34)in the presence of uncertain model parameters, suppose Aand B correspond to a nominal model and that the actualmatrices are not precisely known, but belong to a polytopicdomain D. Any matrix inside the domain D can be writtenas a convex combination of the vertices Aj and Bj of theuncertainty polytope. We can then augment (36) with thefollowing LMIs

AjW +BjX +WA>j +X>B>j < 0, j = 1, . . . p (38)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

reference

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

uff

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

Fig. 11. System responses for different values of the tuning parameters αand ϑ. (a) reference speed r and velocities ωm; (b) motor currents ia; (c)control input u delivered by the controller and (d) duty cycle δ obtainedfrom the feedback linearization. Constants ωm,M, ia,M, uM and tM arenormalization factors.

where p is the number of vertices of the polytope to ensurerobust exponential stability of the error dynamics for anyparameter in the polytope.

VII. EXPERIMENTAL RESULTS

All the experiments have been conducted on an indus-trial automatic barrier whose model has been identified asdescribed in Section IV-B. The controller has been im-plemented on a 8-bit microcontroller (Microchip PIC18F)mounted on a control board provided by the companydeveloping the boom barrier. The firmware has been writtenin C language through MPLAB X IDE. The reference andfeedforward terms have been determined offline as describedin Section VI-A, while the PD controller and the feedbacklinearization have been implemented directly on the micro-controller.

Fig. 11 shows the results of the proposed control strategyand illustrates the main variables involved in the controlloop. Following Remark 2, rather than the nominal designin (36), given the structure of A and B in (34), we consideran uncertainty of ±20% on the a2,2 element of matrix Aand the b2 element of B. This provides four vertices thathave been taken into account in our robust design for eachone of the considered gain selections. The response fordifferent values of the LMI tuning parameters α and ϑ showa reasonable trade-off between disturbance rejection and

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

proposed controller

production controller

reference

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

Fig. 12. Output responses (a-b) and the control input responses (c) withthe proposed controller and the production standard controller. ConstantsθM, ωm,M and tM are normalization factors.

closed-loop performance. We select ϑ = π6 for lower values

of α and ϑ = π18 for higher values, since large values of

α may lead to optimal closed-loop gains inducing undesiredoscillations, especially in the accelerating phase, probablydue the mechanical backlash of the gearbox. We successfullyremove the oscillations by reducing ϑ and consequentlyincreasing the closed-loop damping ratio (see Fig. 10). Whenα is too low, the tracking performance degenerates, and anundesirably large time is needed to complete the openingmaneuver, due to the imperfect tracking of the referencesignal.

Specifically, in Fig. 11(a) the dashed line shows thereference x∗3 obtained by solving the optimization problem(27) in Section VI-A. Fig. 11(b) shows the evolution of thecurrent x1, while Fig. 11(c) and 11(d) illustrate, respectively,the control input u, which is the sum of the feedforwardterm uff and the feedback signal ufb, and the duty cycleδ obtained from the feedback linearization map described inSection V-B. The controller allows tracking the reference sig-nal, compensating for disturbances and model uncertainties,resulting in a desirably small tracking error. Fig. 12 shows acomparison between the opening maneuver responses of theproposed solution and the production standard controller. Tonumerically quantify the gap between the two controllersin terms of tracking performance, we consider the NRMSEmetric

NRMSE =||ri − yi||

||ri −mean(r)|| i = 1, . . . , n : ri 6= 0,

(39)where ri = x∗3,i and yi = ωm,i , i = 1, . . . , n are thesamples of the reference (dashed line in Fig. 12b) andthe output (solid lines in the same figure). The computedNRMSE correspond to 0.1861 for the production controller

and 0.0719 for the proposed showing a reduction to morethan one half, revealing a substantial tracking accuracyimprovement. Multiple experimental tests have been carriedout for a large variety of working conditions, providingexcellent results, thus confirming the desirable features ofthe proposed solution.

VIII. CONCLUSION

We addressed modeling and control of a nonlinearmotion system. A feedback linearization-based feedfor-ward/feedback architecture was derived, associated with rig-orous optimized performance and feasibility guarantees. Thisstrategy is novel, it is general enough to be applicable toalternative systems sharing similar mechatronic structure,and solves the cumbersome manual gain tuning currentlyemployed at the industrial level.

In light of the satisfactory experimental results on theconsidered industrial device, future work may include veri-fying the effectiveness of the control algorithm to explicitlyaccount for saturation for optimizing saturated performance.We will also test the proposed strategy on alternative similaraccess automation systems (such as horizontal automaticgates). The robust results highlighted in Remark 2 will alsobe better investigated. Finally, a limitation of the proposedapproach is the inability to account for systematic trajectorytracking errors. As this type of application is subject torepetitive maneuvers, it would be interesting to developadaptive control techniques, despite the fact that they couldbe more demanding from a computational viewpoint.

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