Modeling, Identification and Control, Vol. 30, No. 2, 2009, pp. 57–70, ISSN 1890–1328

Constrained Control Design for DynamicPositioning of Marine Vehicles with Control

Allocation

Tristan Perez 1 ,2 Alejandro Donaire 1

1Centre for Complex Dynamic Systems and Control (CDSC), The University of Newcastle, Callaghan, NSW-2308,Australia. E-mail: Tristan.Perez/Alejandro.Donaire@newcastle.edu.au.

2Centre for Ships and Ocean Structures (CeSOS) Norwegian University of Science and Technology, N-7491 Trond-heim, Norway. E-mail: Tristan.Perez@ntnu.no.

Abstract

In this paper, we address the control design problem of positioning of over-actuated marine vehicleswith control allocation. The proposed design is based on a combined position and velocity loops ina multi-variable anti-windup implementation together with a control allocation mapping. The vehiclemodelling is considered with appropriate simplifications related to low-speed manoeuvring hydrodynamicsand vehicle symmetry. The control design is considered together with a control allocation mapping. Wederive analytical tuning rules based on requirements of closed-loop stability and performance. The anti-windup implementation of the controller is obtained by mapping the actuator-force constraint set into aconstraint set for the generalized forces. This approach ensures that actuation capacity is not violatedby constraining the generalized control forces; thus, the control allocation is simplified since it can beformulated as an unconstrained problem. The mapping can also be modified on-line based on actuatoravailability to provide actuator-failure accommodation. We provide a proof of the closed-loop stabilityand illustrate the performance using simulation scenarios for an open-frame underwater vehicle.

Keywords: Marine control systems, dynamic positioning, control allocation, anti-windup, fault accom-modation.

1. Introduction

Dynamic positioning of marine vehicles refers to the useof the propulsion system to regulate the vessel motionin the horizontal plane: North-position, East-position,and heading. In the case of underwater vehicles, thiscan be extended with the additional requirement ofdepth regulation.

In order to increase reliability, vehicles that oper-ate in dynamic positioning are usually equipped withmore actuators than the minimum required. Then, theforces used to control the motion can be produced bydifferent combinations of the forces produced by the ac-

tuators. This practice enables the vehicle to continueoperation or to start a safe shut-down in case of an ac-tuator failure by reconfiguring the forces produced bythe remaining actuators. Within this framework, theship motion control system is separated into two maincomponents:

• motion controller

• control allocation mapping.

The motion controller generates demands for general-ized forces (forces and moments) in the degrees of free-dom in which the vehicle is controlled, namely, surge,

doi:10.4173/mic.2009.2.2 c© 2009 Norwegian Society of Automatic Control

Modeling, Identification and Control

sway, and yaw. Since the motion controller operatesin terms of generalized forces, the control design andtuning can be done independently of the actuator con-figuration to certain extend. The control allocationmapping then transforms the controller demands intoindividual actuator commands such that the demandedgeneralized forces are implemented. This separationbetween motion control and actuator force allocationprovides a modular design and makes the vessel motioncontrol independent, to a certain degree, of the numberof actuators available (Fossen, 2002).

The vessel actuator configuration is designed suchthat the control allocation problem has multiple solu-tions. This problem is normally posed as a constrainedoptimization problem, and it is solved on-line to findthe actuator commands that implement the desiredgeneralized forces while satisfying power, magnitude,rate, and other constraints—see, for example, Fossenand Johansen (2006), Peterson and Bodson (2006),Fossen et al. (2008), and references therein.

Due to the characteristics of the ship motion controlproblem, the controller is required to provide integralaction to reject low-frequency disturbances due to cur-rents, mean wave drift and wind forces (Perez, 2005).It is well known that the combination of integral actionand limited actuation capacity can result in degradedperformance and even instability of the closed-loop sys-tem if saturation is not handled appropriately (Good-win et al., 2005). Matters become worse in the caseof actuator failure since the likelihood of saturation in-creases. In a recent work, Perez (2009) combined thepositioning control and control allocation problem us-ing a particular anti-windup implementation, in whichthe actuator-force constraint set is mapped into a con-straint set for the generalized forces.

In this paper, we extend the work of Perez (2009),by deriving analytical tuning rules that ensure stabil-ity and desired performance. These tuning rules can beused as a basis of auto-tuning control. Auto-tuning cansimplify the commissioning of the control system, andfor underwater vehicles, it provides a mean for con-trol re-tuning if significant changes in mass distribu-tion and hydrodynamic characteristics take place dueto mission-dependent equipment configurations. Here,we also give a theoretical proof of the stability of theclosed-loop. In addition, we show how the constraintscan be adjusted online to accommodate actuator fail-ures provided the vehicle remains fully-actuated afterthe failure. The performance of the proposed controlsystem is demonstrated based on a simulation scenarioof horizontal-plane positioning of a general open-framevehicle with four actuators. We consider both nominaland actuator faulty cases.

2. Horizontal-plane positioningcontrol

In this paper, we concentrate on the positioning prob-lem in the horizontal plane, that is, surge, sway, andyaw. For the positioning problem and degrees of free-dom being considered, the vehicle dynamics can be rep-resented as

η = R(η)ν (1)

Mν + C(ν,νc) + D(ν,νc) = τ . (2)

The generalized position, velocity and force vectorsare given by

η ,

neψ

ν ,uvr

τ ,XYN

, (3)

where n and e denote the North and East positionsrelative to a local geographical frame with a coordi-nate system convention of North-East-Down, and ψdenotes the heading or yaw angle relative to the Northdirection. The velocities are expressed in the vehicle’sframe of reference (body-fixed frame) with a forward-starboard-down coordinate-system convention. Thesurge and sway velocities are denoted by u and v re-spectively, and the yaw rate is denoted by r. The gen-eralized velocity vector νc represents the velocity ofthe current expressed in body-fixed coordinates. Thesurge and sway forces along the body-axis are repre-sented by X and Y , and the yaw torque about thez-axis is represented by N .

Equation (1) is a kinematic transformation thattakes the body-fixed generalized velocity vector intothe time derivative of the generalized position vector.The matrix R(η) is single axis rotation matrix aboutthe vehicles vertical axis, that is R(ψ). Equation (2)is a vector equation of motion. The parameter Mis the total mass and inertia matrix, which includesrigid-body and hydrodynamic added mass components.The second term in (2) represents Coriolis-centripetalforces, which appear as a result of expressing the equa-tions of motion in body-fixed coordinates. This termhas a component due to rigid body and another dueto hydrodynamics (potential component of the currentforces). The third term represents damping forces,which reflect the work done on the fluid due to themotion of the vehicle. We assume that the vehicle isneutrally buoyant, and therefore we do not considergravity and buoyancy in (2). For further details onmodels of marine vehicles see Fossen (2002).

For vehicles manoeuvring at very-low speed, a fewsimplifications can be made to the model (2). First,

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Perez and Donaire, “Constrained Control for Positioning of Marine Vehicles with Control Allocation”

the Coriolis-centripetal terms become small since theseterms have a quadratic dependency on the velocities.Second, the nonlinear damping term in (2) is domi-nated by a linear damping at low speeds; and there-fore, we can separate the effect of the current forces andconsider them as a slowly varying disturbance (Fossen,2002). These lead to the following model:

η = R(η)ν (4)

Mν + Dν = τc + τctrl. (5)

In addition, we assume that the hydrodynamic cou-pling between yaw and sway is negligible. This assump-tion is motivated by the usual fore-aft quasi-symmetryof most surface vessel hulls, which also applies to someunderwater vehicles (Smallwood and Whitcomb, 2003).The latter assumption leads to diagonal matrices in (5),and thus, to a fully decoupled kinetic model.

3. Positioning control with controlallocation

3.1. Positioning controller architecture

Figure 1 shows a block diagram describing the mainfunctionality of a marine vehicle motion control sys-tem. The navigation system uses various sensors toprovide reliable signals related to the vehicles velocityand position. The controller processes information re-lated to desired position and the actual position and ve-locity generated by the navigation system. This infor-mation is used to produce generalized force commandsthat correct the motion of the vehicle. The allocationsystem maps the generalized force commands into ap-propriate force commands for the various actuators onthe vehicle. The separation between generalized forcecontrol and actuator-force allocation makes the controltuning independent, to a certain extent, of the charac-teristics of the force actuators and provides the basisfor fault accommodation in the presence of actuatorfailure.

Figure 2 shows a block diagram of the proposedposition-regulation controller. This controller consistsof an outer position-feedback loop with a proportionalcontroller and an inner velocity-feedback loop with aproportional-integral controller. This architecture pro-vides integral action in both position and velocity; thatis, it regulates position despite constant force distur-bances. In the proposed controller, the measured posi-tions are transformed by the inverse of the kinematictransformation (1). This transformation results in alinearization of the kinematics provided that the vehi-cle heading rate is low (Fossen, 2002):

ηp ≈ ν. (6)

The advantage of using the rotated variables (non-linear controller), is that the control tuning can be ad-dressed within a linear framework.

For surface vehicles, the generalized position and ve-locity vectors are wave-filtered before being used bythe controller shown in Figure 2. The topic of wavefiltering for positioning control of surface vessels goesbeyond the scope of this paper. The interested readershould see, for example, Fossen and Perez (2009).

3.2. Unconstrained control allocation

The force vectors produced by the actuators can be de-composed into rectangular coordinates (along the lon-gitudinal and transverse direction on the vessel) andcombined into a single vector u,

u =[TX1 TY 1 TX2 TY 2 · · ·TXN TY N

]T, (7)

where N is the number of actuators. This vector ismapped into the generalized forces via the actuatorconfiguration matrix B, which depends only on the lo-cation of the actuators on the vehicle:

τ = Bu. (8)

That is, we assume that the angles of the azimuth ac-tuators, if any, are fixed.

Since u has more components than τ , there are dif-ferent vectors u satisfying (8) for a given value of τ . Inorder to limit the number of solutions, one can pose theproblem as an optimization problem, like for example,

u∗ = arg minu

(uTWu)

subject to τ d = Bu.(9)

The objective function uTWu is representative of thetotal energy or control effort, where W is a positivedefinite matrix weighting the actuator cost. Thus, thecontrol allocation seeks the solution that implementsthe desired generalized force τ d whilst minimizing thecontrol effort. As shown in Fossen (2002), the solutionof the above problem is given by

u∗ = B†τ d, B† = W−1BT (BW−1BT )−1. (10)

Note that since B depends only on the location of theactuators on the vehicle, the right inverse B† can bepre-computed.

The optimization problem (9) does not take into ac-count the fact that the vector u must belong to a con-straint set due to the maximum force that the var-ious actuators can produce. Adding this constraintto (9) requires on-line numerical optimization, andthe solution (10) is no longer the optimal solution.

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Modeling, Identification and Control

Controller Allocation

Navigation

Environment

Generalised Force Commands

Actuator Force Commands

Desired Position

Generalised Position

Generalised Velocity

Figure 1: Uninhabited underwater vehicle motion control system with control allocation.

P PI Generalised Force Commands

Desired Position

Generalised Position

Generalised Velocity

Proportional Position Controller

Proportional-Integral Velocity Controller

Desired Velocity

!

!d

!

RT (!)

!p

!dp

! d

Figure 2: Details of motion controller proportional (P) position controller and proportional-integral (PI) velocitycontroller. For surface vessels, the generalized position and velocity vectors are wave-filtered beforebeing used by the controller in this figure.

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Perez and Donaire, “Constrained Control for Positioning of Marine Vehicles with Control Allocation”

Constrained numerical optimization brings out time-feasibility issues since the optimization algorithm mustalways provide a feasible solution within the requiredsampling period (Fossen and Johansen, 2006; Petersonand Bodson, 2006). An alternative to this approachis to constrain the desired generalized force τ d suchthat the constraints on u are always satisfied (Perez,2009). By so doing, the force controller is also in-formed about reaching constraints, which prevents per-formance degradation due to the combination actuatorsaturation and integral action. This control method isdescribed in the next section.

3.3. Constrained control via input scaling

One of the key issues in control design for systems thatrequire integral action and present a potential for ac-tuator saturation is that of integrator windup. Thatis, if the actuators saturate and the integral controlleris not informed about the saturation, the integratorscontinue integrating the error signals but the controlaction is not seen by the system. This often producesa degradation of performance in terms of undesirableoscillations and even instability. Control schemes thatdeal with this effect are called anti-windup schemes(Bernstein and Michel, 1995).

If a linear controller C(s) is minimum phase and bi-proper (as in the case of a PI and PID controller), thenanti-windup can be achieved simply by implementation(Goodwin et al., 2001). This implementation is shownin Figure 3. In this figure, Lim represents a saturation

c!e

yC(s)!1 ! c!1

"

!uc! dLim

Figure 3: Anti-windup implementation of a minimum-phase bi-proper controller.

(magnitude, rate, or a combination of both), and thegain

c∞ = lims→∞

C(s). (11)

Note that if the limitation is not active, the loop ofFigure 3, reduces to the controller:

C(s) =[I + c∞

(C(s)−1 − c−1∞

)]−1c∞. (12)

When the limitation becomes active, it prevents thecontrol signal τ d from exceeding its limits, and the

constrained signal drives the states of the controller,which are all on the feedback path. This anti-windupstrategy is simply a particular implementation of theoriginal controller. That is, it does not require anyadditional design.

The anti-windup scheme described above can be ap-plied to the velocity PI controllers of the velocity loopsof the vehicle positioning controller:

Cvel(s) =

Cu(s) 0 00 Cv(s) 00 0 Cr(s)

, (13)

where for i = u, v, r,

Ci(s) = Kip

T iIs+ 1

T iIs. (14)

In order to constrain τ d we need to construct a setsuch that

τ d ∈ T ⇔ u ∈ U . (15)

One way of enforcing these constraint is by comput-ing an unconstrained control τuc and then scale it downif it is outside the constraint set, that is

τ d =

{τuc if τuc ∈ Tατuc if τuc /∈ T , α < 1 : ατuc ∈ ∂T

(16)

By scaling the vector τuc, we preserve its direction.In order to implement (16) we need to compute τucand then obtain α. This implementation leads to thecomplete control scheme depicted in Figure 4.

The explicit computation of the set T is not al-ways easy. To simplify this computation we canconsider an approximating polyhedron subset U? ⊆ Uof the actuator force components (Ruth et al., 2009):

U? = {u : L < u < U}, (17)

or alternatively,

U? = {u : Fu ≤ f}, F =

[I−I

], f =

[U−L

]. (18)

Using the set U?, the following algorithm determinesthe scaling factor (Perez, 2009):

1. Compute the unconstrained control, and evaluatefuc = FB†τuc,

2. Set α = 1,

3. If fuc(i) > f(i) and f(i)/fuc(i) < α. Then, re-setα = f(i)/fuc(i). Do this for all the componentsi = 1, ..., 4N .

Once the scaling factor is computed the control can beimplemented as

τ d = ατuc. (19)

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Modeling, Identification and Control

P Generalised Force Commands

Desired Position

Generalised Position

Generalised Velocity

Proportional Position Controller

Desired Velocity

!

!d

!

RT (!)

!p

!dp

! dc! !I

Cvel(s)!1 ! c!1

"

PI Velocity Controller in Anti-wind-up form with scaling

Figure 4: Vehicle positioning controller with velocity loop in anti-windup form with scaling.

3.4. Stability and control tuning

The proposed controller ensures asymptotic stabilityfor position regulation provided that the following con-ditions hold for the parameters of the proportional-integral velocity controllers (14) and the proportionalposition controllers:

Kip > 0, T iI > 0, and T iI 6= T iv, (20)

0 < P i < 1/T iv, (21)

where T iv is the open-loop time-constant of the vehi-cle velocity response. The condition (20) ensures thatthere is no zero-pole cancelation between the velocitycontroller and the velocity response of the vehicle. Thecondition (21) establishes that the position control loopmust be slower than the open-loop velocity response.

The stability proof follows from re-arranging theclosed-loop system with control allocation into a Lure’ssystem and application of Lyapunov and passivity the-ories (Khalil, 2000; Brogliato et al., 2007). Details ofthe stability proof are given in Appendix B. The ful-filment of both conditions ensure of passivity of thedynamic linear part of the Lure’s system.

Based on the above conditions on the controller pa-rameters, practical tuning rules can be obtained using,for example, the root-locus technique. The main ideais to create a faster velocity loop response and thenset the proportional gain to obtain a position loop re-sponse that is slower than the open-loop velocity re-sponse. Therefore, we propose the following set of tun-ing rules.

Given the velocity response models,

Giv(s) =Kiv

(T ivs+ 1), (22)

then,

1. set the zero of the velocity PI at a higher frequencythan the pole of the velocity response (22), thatis,

T iI = βiT iv, βi = 0.2, (23)

2. determine the proportional gain of the velocity PIcontroller, Ki

p, such that the velocity closed looppoles are real and above the zero of the controller.These conditions are satisfied when Ki

p is greater

than the critical value Kipc given by

Kipc =

[(2βi − 1

)+

√(2βi − 1

)2− 1

]Kiv

, (24)

3. set the gain of the proportional position controllerbased on the closed-loop stability constraint (21).Faster response results when P i is selected closerto 1/T iv.

0 < P i < 1/T iv. (25)

The above tuning rules are practical and simple to fol-low. Note that the only data necessary for the tuningare the parameters of the simplified vehicle open-loopvelocity response models (22). These parameters canbe estimated from data of zig-zag tests. By combiningthe estimation with the above tuning rules, we obtainthe basis of an auto-tuning controller.

3.5. Actuator-fault accommodation

The proposed controller offers a possibility of vehiclefault accommodation in the case of actuator failure pro-vided that the vehicle remains fully-actuated after thefailure. Should this be the case, one can simply changethe actuator configuration matrix B and its pseudo-inverse B†. Then, constraint set U? and thus the scal-ing factor α are automatically re-computed with the

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Perez and Donaire, “Constrained Control for Positioning of Marine Vehicles with Control Allocation”

algorithm proposed in Section 3.3. Note that the newactuator configuration and pseudo inverse matrices forthe faulty case can be pre-computed. Therefore, oncethe fault diagnosis system identifies the failed actuator,this actuator can be switched off and the correspond-ing configuration and pseudo inverse matrices be usedto compute the scaling factor of the anti-windup im-plementation.

If the actuator failure is such that vehicle becomesunder-actuated after the failure, then one needs to con-sider control reconfiguration (change the set of input-output variables used to implement the controller).Control reconfiguration goes beyond the scope of thispaper.

4. Simulation case study

In this section, we present numerical simulations thatboth verify and illustrate the performance of the pro-posed control system. We consider a model of an open-frame underwater vehicle—like the one shown in Fig-ure 5. This vehicle has four thrusters in an x-typeconfiguration, namely, bow-starboard, bow-port, stern-port, and stern-starboard. Further details of the vehi-cle model are given in Appendix A.

Figure 5: Example of open frame underwater vehiclewith x-type thruster configuration. Picturecourtesy of Saab Seaeye Ltd, UK.

Figure 7 shows the results of the first experiment,in which the vehicle is in position hold (regulation) inthe presence of a constant ocean current and then af-ter 20 seconds we set a reference set-point change insurge, sway and yaw simultaneously. This figure showsthe demanded and actual surge, sway and yaw posi-tions, velocities and generalized forces. As we can seefrom the demanded and actual generalized forces, theanti-windup scheme works such that the demands arefeasible. Due to the saturation of the actuators, how-ever, the velocity demands cannot be followed. Figure8 shows the corresponding forces of the four actuators.As we can see from the latter figure, the forces remainwithin the constraints on the maximum force magni-tude. The control system presents a good performance

despite the actuators reaching the saturation levels.Figures 9 and 10 show the results of a second ex-

periment with the same environmental conditions andposition set-point change as in the first experiment, butin this case the anti-windup is inactive (that is α = 1).In this case the controller is unaware of the limitationof the actuators, and this, as expected, produces a se-vere degradation in the positioning performance sincethe force demands are not feasible and the integratorswindup. Although the vehicle finally goes to the de-sired position (in an inadmissible way), the stability ofthe closed-loop cannot not be guaranteed by the the-ory.

In the third experiment the ocean current and thereference steps are the same as the previous experi-ments, but the stern-port thruster is turned off to em-ulate a failure. The controller parameters remain un-altered and the matrix B adopts a new value given bythe new actuator configurations, while the anti-windupstrategy is active. The time evolution of demandedand actual positions, velocities and generalized forcesin surge, sway and yaw are shown in Figure 11. Thissimulation evidences that the performance of the con-trol system is still good despite one thruster is unavail-able, only a slight difference in the time response is ap-preciated between the controller with three or four ac-tuators, as can be intuitively thought. Figure 12 showsthe actuator forces, the stern-port force is compensatedby the other actuators to produce the demanded gen-eralized forces. The last experiment is the same as thethird experiment but in this case we disable the anti-windup algorithm. Figures 13 and 14 show that underthese conditions the closed loop becomes unstable.

5. Conclusions

In this paper, we have considered the design of a po-sitioning controller in conjunction with control alloca-tion for over-actuated marine vehicles. The proposedcontrol architecture is based on a velocity loop imple-mented with a proportional-integral controller in anti-windup form and a position loop implemented with aproportional controller. This control architecture pro-vides integral action for position regulation.

The use of control allocation allows the control de-sign to be focussed on the generalized forces related tothe degrees of freedom of concern rather than on ac-tuator forces. In the proposed control system, we mapthe constraint set of actuator forces due to the lim-ited authority of the actuators into a constraint set forthe generalized forces. This approach results in a con-strained control design with a simple, unconstrained,control allocation problem.

We provide an algorithm for an on-line implementa-

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Modeling, Identification and Control

tion of the mapping, and we prove closed-loop asymp-totic stability using Lyapunov and passivity theory.The use of control allocation also provides the basisfor actuator fault accommodation. The proposed con-troller offers the possibility of actuator fault accommo-dation provided that the vehicle remains fully-actuatedafter the failure.

The tuning of the proposed control system is basedon simple practical rules that follow from closed-loopstability considerations and performance. The key as-sumptions is that the hydrodynamic coupling betweenthe different degrees of freedom is negligible. Thisassumption is motivated by the usual fore-aft quasi-symmetry of most surface vessel hulls and some un-derwater vehicles. The tuning rules use only the pa-rameters of the open-loop velocity response. These pa-rameters can be estimated from data of zig-zag tests.By combining the estimation with the tuning rules, weobtain the basis of an auto-tuning controller.

The performance of the proposed controller is illus-trated based on simulation case scenarios for a genericopen frame underwater vehicle with four actuators innominal and faulty modes.

References

Bernstein, D. and Michel, A. A chronological bibliogra-phy on saturating actuators. International Journalof Robust and Nonlinear Control, 1995. 5(5):375–380. doi:10.1002/rnc.4590050502.

Brogliato, B., Lozano, R., Maschke, B., and Ege-land, O. Dissipative System Analysis and Control.Springer-Verlag, London, 2007.

Fossen, T. Marine Control Systems. Marine Cybernet-ics AS, Trondheim, 2002.

Fossen, T. and Johansen, T. A survey of control al-location methods for ships and underwater vehicles.In Proc. of the 14th IEEE Mediterranean Conferenceon Control and Automation. Ancona, Italy, 2006 .

Fossen, T., Johansen, T., and Perez, T. A Survey ofControl Allocation Methods for Underwater Vehicles,chapter 7, pages 109–128. In-Tech, Vienna, Austria,2008.

Fossen, T. and Perez, T. Kalman filtering for position-ing and heading control of ships and offshore rigs.IEEE Control Systems Magazine, 2009. 29(6):32–46.doi:10.1109/MCS.2009.934408.

Goodwin, G., Graebe, S., and Salgado, M. ControlSystem Design. Prentice-Hall Inc., New Jersey, 2001.

Goodwin, G., Seron, M., and DeDona, J. ConstrainedControl and Estimation: An Optimisation Approach.Communications and Control Engineering. Springer,London, 2005.

Khalil, H. Nonlinear Systems. Prentice-Hall, New Jer-sey, 2000.

Perez, T. Ship Motion Control. Advances in IndustrialControl. Springer-Verlag, London, 2005.

Perez, T. Anti-windup designs for ship dynamic posi-tioning with control allocation. In Proc. of the 8thIFAC International Conference on Manoeuvring andControl of Marine Craft. Guaruja, Brazil, 2009 .

Peterson, J. and Bodson, M. Constrained quadraticprogramming techniques for control allocation.IEEE Transaction on Control System Technology,2006. 14(1):91–98. doi:10.1109/TCST.2005.860516.

Ruth, E., Smogeli, O., Perez, T., and Sorensen, A.Antispin thrust allocation for marine vessels. IEEETransactions on Control Systems Technology, 2009.17(6):1257 – 1269. doi:10.1109/TCST.2008.2006187.

Smallwood, D. and Whitcomb, L. Adaptiveidentification of dynamically positioned under-water vehicles. IEEE Transaction on Con-trol System Technology, 2003. 11(4):505–515.doi:10.1109/TCST.2003.813377.

A. Open-frame underwater vehiclemodel parameters

We consider a vehicle with a mass of 10,000 Kg, andit is of a similar form as the vehicle shown in Figure 5.The vehicle has four thrusters in an x-type configu-ration, namely, bow-starboard, bow-port, stern-port,and stern-starboard. The maximum force produced byeach thruster is 150N, and for simplicity, we considera symmetric force characteristic. The thruster config-uration matrix is given by

BT =

√2

2

1 −1 (lBP1 − lBP2 )1 1 (−lBS1 + lBS2 )−1 −1 (−lSP1 + lSP2 )−1 1 (lSS1 − lSS2 )

, (26)

and the distances that determine the horizontal loca-tion of the thrusters in the vehicle are l1 = 0.6m andl2 = 0.4m for all four actuators. These distances aregiven with respect to the point at the centre of thebox that encloses the vehicle. The matrix B for the

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Perez and Donaire, “Constrained Control for Positioning of Marine Vehicles with Control Allocation”

faulty case is obtained from equation (26) taking offthe row corresponding with the faulty actuator. Forreason of symmetry, the weighted matrix W is chosenas the identity matrix.

The vector equation of motion is formulated at thecentre of the box that encloses the vehicle. Table 1gives the parameters of the velocity response dynamicmodel and those of the controller.

Table 1: Model and controller parameters

Model ControllerSurge Tv = 0.0105s Kp = 1718

Kv = 3.0384N.s/m TI = 0.6077P = 0.3258

Sway Tv = 0.0083s Kp = 2160Kv = 3.5383N.s/m TI = 0.7077

P = 0.2798Yaw Tv = 0.0095s Kp = 1894

Kv = 1.2564N.s/m TI = 0.2513P = 0.7879

B. Stability proof of the closed loop

In this appendix, we prove that the closed-loop sys-tem is asymptotic stable using Lyapunov and passivitytheory (Khalil, 2000; Brogliato et al., 2007). Underthe modeling hypothesis considered in Section 2, theclosed-loop dynamics on each degree of freedom canbe described as a feedback interconnection of a linearsystem and a memoryless function as follows

xi1xi2xi3

=

− dli

mi 0 01mi 0 00 0 − 1

T iI

xi1xi2xi3

+

10− 1Ki

p

τ i

yi =[Ki

p

mi KipP

i Kip

T iI

]xi1xi2xi3

(27)

τ = −Limi(t, yi), (28)

where i indicates the degree of freedom surge, swayand yaw. The states xi1 and xi2 represent generalizedmomenta (muu, mvv and mrr) and positions (np, epand ψp) respectively, xi3 is the state of the velocitycontrollers, τ i is the generalized force, mi and dli arethe diagonal entries of the mass and damping matricesM and D of the model (5).

After the loop transformation shown in Figure 6, thenonlinearity in the feedback loop belongs to the sector

[0,∞]. In this form, the asymptotic stability of theclosed loop origin is equivalent to the absolute stabilityof the Lure’s problem (Khalil, 2000).

Theorem 1 (Asymptotic stablity) The closed -loop system obtained by the feedback interconnectionof the linear system (27) and the memoryless function(28) with controller parameters satisfying

0 <1

TI6= dl

m,

0 <P <dl

m, and

0 ≤Kp

(29)

has an asymptotic equilibrium point at the origin.

Proof First, we prove that the transfer functionGT (s) = 1+C(sI−A)−1B is a positive real, where A,B and C are the matrices that characterize the linearsystem (27):

• the poles of GT (s) are{

0;− dlm ;− 1

TI

}∈ Re[s] ≤ 0,

• for all real ω for which jw is not a pole of GT (s),GT (jω) + GTT (−jω) ≥ 0 if and only if 0 < TI ,0 < P < dl/m and Kp > 0, and

• lims→j0

(s− j0)GT (s) =PKp

dl ≥ 0.

Under these conditions GT (s) is positive real (Khalil,2000). Given that the pairs (A,B) and (A,C) arecontrollable and observable respectively if 1/TI 6= dl/mand P 6= dl/m, and GT (s) is positive real, then thereexist matrices P = PT > 0, L, and W such that

PA + ATP = −LTLPB = CT − LTW

WTW = D + DT

(30)

This is the well-known Positive Real Lemma (Khalil,2000; Brogliato et al., 2007), which proves that thelinear system is passive with supply rate τy and storagefunction V (x) = 1

2xTPx.

In order to prove stability of the closed loop, we con-sider V (x) as a Lyapunov function candidate. Then,using (30), the orbital derivative of V (x) results

65

Modeling, Identification and Control

1s

1s

1m

Lim(t,.)

dl

x1 x2τ

Kp1T

1I

P

Kp

x3

y

G (s)

yT

T

Lim (t,.)T

s

Figure 6: Loop transformation of the control system

V =1

2xTPx +

1

2xTPx

=1

2xT (PA + APT )x + xTPBτ

= −1

2xTLTLx + xT (CT − LTW)τ

= −1

2xTLTLx + (Cx + Dτ)T τ − τTDτ − xTLTWτ

= −1

2(Lx + Wu)T (Lx + Wu) + yTT τ

V ≤yTT τ (which proves passivity)

V ≤− yTT LimT (t, yT ) ≤ 0

This proves Lyapunov stability of the closed-loopsystem, then asymptotic stability follows by applyingthe invariance theorem (Khalil, 2000), that is,

V (x) = 0⇒ yT = 0⇒ Cx = 0⇒

x1 = −dl

mx1

x2 =1

mx1

x3 = − 1

TIx3

The above equations establish that there is no invari-ant subset of the closed-loop system state space whereV (x) = 0, unless the subset is the origin. Then, theorigin is an asymptotically stable equilibrium point. �

66

Perez and Donaire, “Constrained Control for Positioning of Marine Vehicles with Control Allocation”

0 20 40 60−5

0

5

10

15

20

25Surge

Time [s]

Sur

ge p

ositi

on [m

]

0 20 40 60−5

0

5

10

15

20

25Sway

Time [s]

Sw

ay p

ositi

on [m

]

0 20 40 60−100

−80

−60

−40

−20

0

20Yaw

Time [s]

Yaw

ang

le [ °]

0 20 40 60−4

−2

0

2

4

6

8

Time [s]

Sur

ge v

eloc

ity [m

/s]

demandedactual

0 20 40 60−2

0

2

4

6

8

Time [s]

Sw

ay v

eloc

ity [m

/s]

demandedactual

0 20 40 60−80

−60

−40

−20

0

20

Time [s]

Yaw

ang

le [ °/

s]

demandedactual

0 20 40 60−200

−100

0

100

200

Time [s]

Sur

ge c

ontr

ol fo

rce

[N]

actualdemanded

0 20 40 600

100

200

300

400

Time [s]

Sw

ay c

ontr

ol fo

rce

[N]

actualdemanded

0 20 40 60−40

−30

−20

−10

0

10

Time [s]

Yaw

con

trol

mom

ent [

N.m

]

actualdemanded

Figure 7: Performance of vehicle position regulation controller in current and position set point change.

0 10 20 30 40 50 60−200

0

200Actuator Forces

Time[s]

Bow

−P

ort [

N]

0 10 20 30 40 50 60−200

0

200

Time[s]

Bow

−S

tarb

oard

[N]

0 10 20 30 40 50 60−200

0

200

Time[s]

Ste

rn−

Por

t [N

]

0 10 20 30 40 50 60−200

0

200

Time[s]

Ste

rn−

Sta

rboa

rd [N

]

Figure 8: Actuator forces for a vehicle change in position while in current.

67

Modeling, Identification and Control

0 100 200 300 400−40

−20

0

20

40

60Surge

Time [s]

Sur

ge p

ositi

on [m

]

0 100 200 300 400−20

0

20

40

60Sway

Time [s]

Sw

ay p

ositi

on [m

]

0 100 200 300 400−200

−150

−100

−50

0

50Yaw

Time [s]

Yaw

ang

le [ °]

0 100 200 300 400−15

−10

−5

0

5

10

15

Time [s]

Sur

ge v

eloc

ity [m

/s]

demandedactual

0 100 200 300 400−15

−10

−5

0

5

10

15

Time [s]

Sw

ay v

eloc

ity [m

/s]

demandedactual

0 100 200 300 400−100

−50

0

50

100

Time [s]

Yaw

ang

le [ °/

s]

demandedactual

0 100 200 300 400−6

−4

−2

0

2x 10

5

Time [s]

Sur

ge c

ontr

ol fo

rce

[N]

actualdemanded

0 100 200 300 400−2

0

2

4

6x 10

5

Time [s]

Sw

ay c

ontr

ol fo

rce

[N]

actualdemanded

0 100 200 300 400−10

−5

0

5x 10

4

Time [s]Y

aw c

ontr

ol m

omen

t [N

.m]

actualdemanded

Figure 9: Performance of vehicle position regulation controller in current and position set point change, for thecase where the anti-windup is not active.

0 50 100 150 200 250 300 350 400−200

0

200Actuator Forces

Time[s]

Bow

−P

ort [

N]

0 50 100 150 200 250 300 350 400−200

0

200

Time[s]

Bow

−S

tarb

oard

[N]

0 50 100 150 200 250 300 350 400−200

0

200

Time[s]

Ste

rn−

Por

t [N

]

0 50 100 150 200 250 300 350 400−200

0

200

Time[s]

Ste

rn−

Sta

rboa

rd [N

]

Figure 10: Actuator forces for a vehicle change in position while in current for the case where the anti-windupis not active.

68

Perez and Donaire, “Constrained Control for Positioning of Marine Vehicles with Control Allocation”

0 50 100−5

0

5

10

15

20

25Surge

Time [s]

Sur

ge p

ositi

on [m

]

0 50 100−5

0

5

10

15

20

25Sway

Time [s]

Sw

ay p

ositi

on [m

]

0 50 100−100

−80

−60

−40

−20

0

20Yaw

Time [s]

Yaw

ang

le [ °]

0 50 100−4

−2

0

2

4

6

8

Time [s]

Sur

ge v

eloc

ity [m

/s]

demandedactual

0 50 100−2

0

2

4

6

8

Time [s]

Sw

ay v

eloc

ity [m

/s]

demandedactual

0 50 100−80

−60

−40

−20

0

20

Time [s]

Yaw

ang

le [ °/

s]

demandedactual

0 50 100−150

−100

−50

0

50

100

150

Time [s]

Sur

ge c

ontr

ol fo

rce

[N]

actualdemanded

0 50 1000

100

200

300

Time [s]

Sw

ay c

ontr

ol fo

rce

[N]

actualdemanded

0 50 100−25

−20

−15

−10

−5

0

5

Time [s]Y

aw c

ontr

ol m

omen

t [N

.m]

actualdemanded

Figure 11: Performance of vehicle position regulation controller in current and position set point change underan actuator fault.

0 10 20 30 40 50 60 70 80 90 100−200

0

200Actuator Forces

Time[s]

Bow

−P

ort [

N]

0 10 20 30 40 50 60 70 80 90 100−200

0

200

Time[s]

Bow

−S

tarb

oard

[N]

0 10 20 30 40 50 60 70 80 90 100−200

0

200

Time[s]

Ste

rn−

Por

t [N

]

0 10 20 30 40 50 60 70 80 90 100−200

0

200

Time[s]

Ste

rn−

Sta

rboa

rd [N

]

Figure 12: Actuator forces for a vehicle change in position while in current with an actuator fault.

69

Modeling, Identification and Control

0 50 100 150 200−150

−100

−50

0

50Surge

Time [s]

Sur

ge p

ositi

on [m

]

0 50 100 150 200−20

−10

0

10

20

30Sway

Time [s]

Sw

ay p

ositi

on [m

]

0 50 100 150 200−400

−200

0

200Yaw

Time [s]

Yaw

ang

le [ °]

0 50 100 150 200−40

−20

0

20

40

Time [s]

Sur

ge v

eloc

ity [m

/s]

demandedactual

0 50 100 150 200−40

−20

0

20

40

Time [s]

Sw

ay v

eloc

ity [m

/s]

demandedactual

0 50 100 150 200−300

−200

−100

0

100

200

Time [s]

Yaw

ang

le [ °/

s]

demandedactual

0 50 100 150 200−1

−0.5

0

0.5

1x 10

6

Time [s]

Sur

ge c

ontr

ol fo

rce

[N]

actualdemanded

0 50 100 150 200−2.5

−2

−1.5

−1

−0.5

0

0.5x 10

6

Time [s]

Sw

ay c

ontr

ol fo

rce

[N]

actualdemanded

0 50 100 150 200−4

−2

0

2x 10

5

Time [s]

Yaw

con

trol

mom

ent [

N.m

]

actualdemanded

Figure 13: Performance of vehicle position regulation controller in current and position set point change underan actuator fault, for the case where the anti-windup is not active.

0 100 200 300 400 500 600 700 800 900 1000−200

0

200Actuator Forces

Time[s]

Bow

−P

ort [

N]

0 100 200 300 400 500 600 700 800 900 1000−200

0

200

Time[s]

Bow

−S

tarb

oard

[N]

0 100 200 300 400 500 600 700 800 900 1000−200

0

200

Time[s]

Ste

rn−

Por

t [N

]

0 100 200 300 400 500 600 700 800 900 1000−200

0

200

Time[s]

Ste

rn−

Sta

rboa

rd [N

]

Figure 14: Actuator forces for a vehicle change in position while in current with an actuator fault, for the casewhere the anti-windup is not active.

70