Nonlinear Control of an Active HeaveCompensation System

K. D. Do∗ and J. PanSchool of Mechanical Engineering, The University of Western Australia

35 Stirling Highway, Crawley, WA 6009, Australia∗Corresponding author. Tel: +61864883125, Fax: +61864881024,

Email: duc@mech.uwa.edu.au (K. D. Do)

Abstract: Heave motion of a vessel or a rig has an adverse impact on the response of adrill-string or a riser. To compensate for heave motion, passive and active devices are usuallyused. Active heave compensators, in which a control system is an essential part, allow conductingoperations under more extreme weather

ljkhij conditions than passive ones. This paper presents a constructive method to design anonlinear controller for an active heave compensation system using an electro-hydraulic systemdriven by a double rod actuator. The control system reduces the effect of the heave motionof the vessel on the response of the riser by regulating the distance from the upper end of theriser to the seabed. The control development is based on Lyapunov’s direct method and dis-turbance observers. The paper also includes a method to select the control and disturbanceobserver gains such that actuator saturations are avoided. Stability of the closed loop system iscarefully examined. Simulation results illustrate the effectiveness of the proposed control system.

Keywords: Heave compensation, Nonlinear control, Riser.

1 Introduction

Figure 1: Schematic of an active heave compensationsystem (http://www.oceandrilling.org).

Offshore oilfield development has movedto a deeper, and more severe environ-ment for new oil sources. Moreover,offshore oil-rigs have become small andlight weight. Therefore, waves, windand ocean currents easily cause therig to move away from their desiredposition both horizontally and verti-cally. This in turn has adverse ef-fects on the riser(s) connecting be-tween the rig and the well at the seabed, (Sarpkaya, 1981). The horizon-tal motion of the rig (hence the up-per end of the risers) is often con-trolled by a dynamic positioning sys-tem in deep-water applications. Basedon linear and nonlinear control theo-ries, a number of control techniques fordynamic positioning systems has beenproposed in literature, see for exam-ple (Grimble et al., 1980), (Fossen and

1

Strand, 2001), (Do et al., 2002), (Soresen, 2005). However, in these papers the vertical (heave)motion of the rig, which directly changes the tension in the riser, is completely ignored. Toreduce the effect of the heave motion of the rig, an active heave compensation system is oftenused in combination with a passive riser compensator to provide a stable position of the crownblock referred to seabed. This system is based on a hydraulic cylinder installed on top of thepassive crown block compensator. The cylinder is suspended by specially designed sphericalbearings. The cylinder is operated in such a way that it applies force to the crown block toovercome frictions and the force acting between the riser and the active heave compensationunit to maintain a constant distance from the upper end of the riser to the seabed. The resultis that the crown block position with respect to seabed is constant within (0.1 − 0.5)m with righeave up to (4 − 5)m, (Aker, 1998). The cylinder is controlled by means of input from a heavemotion sensor (e.g. 3 axis accelerometer) and from a cylinder position sensor. A schematic of anactive heave compensation system is depicted in Figure 1. Acting cylinders are attached directlybetween the crown block and the structure of the derrick.

It is surprising that while controlling dynamic positioning systems receives a lot of attention,control of active heave compensation systems has received much less attention from researchers.In (Korde, 1998), a control system based on linear control techniques is proposed for an activeheave compensation system on drill-ships. Recently, active control of heave compensated cranesor module handling systems during the water entry phase of a subsea installation was studied in(Johansen et al., 2003). In this work, a concept referred to as wave synchronization is introduced,where the main idea is to utilize a wave amplitude measurement in order to compensate directlyfor the water motion due to waves inside the moonpool. In (Skaare and Egeland, 2006), theauthors proposed a parallel force/position controller for the control of loads through the wavezone in marine operations. The controller structure has similarities to the parallel force/positioncontrol scheme used in robotics. Their controller achieved a significant improvement of theminimum value of the wire tension over the wave synchronization approach which was used in(Johansen et al., 2003). In (Marconi et al., 2002), an autopilot for the autonomous landing of avertical take off and landing vehicle on a ship whose deck oscillates in the vertical direction dueto high sea states is proposed. The heave motion of the deck is compensated using an internal-model-based error feedback dynamic regulator. Designing a high performance controller for anactive heave compensation system is a challenging task. This is partially due to the fact that theforce acting between the riser/drill-string and the active heave compensation unit is difficult tomodel accurately. Moreover, due to complex response of the riser/drill-string it is complicatedto determine the upper bound of this force as a function of the hydraulic piston motion. Thisupper bound is usually required in robust and adaptive control design methods, see (Krstic etal., 1995) and (Khalil, 2002) for example.

This paper focuses on an active heave compensation objective, which is defined as minimiza-tion of the effect of the heave motion of the vessel on the response of the riser by regulating thedistance from the upper end of the riser to the seabed. We present a constructive method todesign a nonlinear controller for an active heave compensation system using an electro-hydraulicsystem driven by a double rod actuator. A disturbance observer is also developed to estimate theforce acting on the piston of the hydraulic system, and the heave acceleration of the vessel. Thisdisturbance observer is then implemented in the control design to result in a high performancecontrol system. The control development and stability analysis are based on Lyapunov’s directmethod.

2

2 Problem formulation and preliminaries

2.1 Problem formulation

We consider an active heave compensation system depicted in Figure 2. It consists of an electro-hydraulic system driven by a double rod actuator. The riser connects to the piston of thehydraulic system via a ball joint. Hence there is no bending moment. The hydraulic system’shouse is fixed to the vessel/rig. The heave motion of the vessel/rig is denoted by z(t), which iscoordinated in the earth-fixed frame, OEXEYEZE . Let xH , which is coordinated in the vesselbody-fixed frame, be the displacement of the cylinder rod (piston) of the hydraulic system, i.e.the motion of the piston with respect to the vessel in the vertical direction. The mathematicalformulation and control design in this paper is generic. Hence they also work for one or morehydraulic systems.

Ship/Rig

1P

2P

,1 1P Q

2 2,P Q

Hi

HSP0HR

1 2

1 2

2

H

H

P P PQ QQ

Hx

z

Water level

(,,

,,

)H

Htx

zx

z

ball joint

Riser

wind

waves

currents

ball jointseabed

buoyancy module

well head

EO

EZ

EX

EY

Figure 2: Representative structure of an active heavecompensation system

To regulate the distance from theupper end of the riser to the seabed,it is obvious that we want to keep thesum |xH(t)+ z(t)−L| as small as pos-sible, where L is a constant, which ispreset to achieve the desired preten-sion of the riser. This is done in thecalibration process when the riser sys-tem is installed. Let iH be the currentinput to the hydraulic system. Thenthe control objective can be describedas designing the control input iH todrive the cylinder rod in such a waythat |xH(t)+ z(t)−L| is kept as smallas possible. In sequel, we will developa mathematical model describing dy-namics of the active heave compensa-tion system to formulate the controlobjective. The dynamics of the pis-ton/rod can be described as

mH xH = AHPH − bH xH

+Δ(t, xH , z, xH , z) (1)

where mH is the mass of the rod ofthe hydraulic system, PH = P1 − P2 isthe load pressure of the cylinder withP1 and P2 being the pressures in theupper and lower compartments of thecylinder, see Figure 2, AH is the ram area of the cylinder, bH represents the combined coefficientof the modeled damping and viscous friction forces on the cylinder rod, and Δ(t, xH , z, xH , z)denotes the force acting on the rod from the riser or drill-string. It is noted that there wouldbe a force from the riser acting on the cylinder rod in the horizontal plane. This force willaffect the motions in sway and surge. However, we do not consider the sway and surge motions,which are considered in a dynamic positioning system, in this paper. It is complicated to derive

3

an explicit formula for the force Δ(t, xH , z, xH , z). This force would depend on many factorssuch as xH , z, xH , z, initial tension in the riser, weight of the riser, current profile and currentmagnitude acting on the riser, and shape of the riser. In addition, the term Δ(t, xH , z, xH , z) caninclude unmodeled friction force between the piston and the cylinder of the hydraulic system.This friction force is also difficult to accurately model. Hence, in this paper we consider the forceΔ(t, xH , z, xH , z) as a disturbance. We do not ignore this disturbance but will later develop adisturbance observer to estimate Δ(t, xH , z, xH , z) for the control design purpose. Neglecting theleakage flows in the cylinder and the servovalve, the actuator or the cylinder dynamics is writtenas (Merritt, 1967)

VH

4βHe

PH = −AH xH − CHT PH + QH (2)

where VH is the total volume of the cylinder and the hoses between the cylinder and the ser-vovalve, βHe is the effective bulk modulus, CHT is the coefficient of the total internal leakageof the cylinder due to pressure, QH is the load flow. The load flow QH is related to the spooldisplacement of the servovalve, xHv, by (Merritt, 1967)

QH = CHDWHxHv

√PHS − sgn(xHv)PH

ρH(3)

where CHD is the discharge coefficient, WH is the spool valve area gradient, PHS is the supplypressure of the fluid, sgn denotes the standard signum function, and ρH is density of the oil. It isnoted that since the supply pressure PHS is always higher than the load pressure PH , i.e. thereexists a strictly positive constant ε1 such that PHS − sgn(xHv)PH ≥ ε1. Hence the equation (3)is well-defined all xHv ∈ R. The servovalve dynamics can be described as

τHvxHv = −xHv + kHviH (4)

where τHv and kHv are the time constant and gain of the servovalve, respectively, iH is the currentinput to the hydraulic system. Since PH is usually very large and τHv is usually very small, wescale the pressure PH and the spool displacement xHv as PH = PH

CH3and xHv = xHv

CH4where CH3

and CH4 are constants, to avoid numerical error and facilitating the control gain tuning process.Now the entire system consisting of (1), (2) and (4) can be rewritten as

xH =AHCH3

mHPH − bH

mHxH +

1

mHΔ(t, xH , z, xH , z),

˙PH = −4βHeAH

VHCH3xH − 4βHeCHT

VHPH +

4βHeCHDWHCH4xHv

VH√

ρH

√CH3

√PHS − sgn(xHv)PH ,

˙xHv = − 1

τHv

xHv +kHv

τHvCH4

iH (5)

where PHS = PHS

CH3. For stabilization/tracking control of the hydraulic system (5) without using

disturbance observers but the ”concept of dominating the size of disturbances”, the reader is re-ferred to (Yao et al., 2001) and references therein. We now restate the control objective as follows:

Control objective. Under the assumption that z(t) and Δ(t, xH , z, xH , z) and their deriva-tives are bounded, design the control input iH to regulate the distance from the upper end ofthe riser to seabed, i.e. to keep the sum |z(t) + xH(t) − L| as small as possible, where L is a

4

constant which is preset to achieve a desired tension in the riser.

Remark 1. A heave compensation system is usually calibrated in such a way that at thezero vessel/rig heave, the riser tension is at its desired value, i.e. the constant L is availablefor the control design. Since the dynamics of the vessel/rig and the riser in large motion modeare slow (i.e. we do not consider vibration mode of the riser in this paper), we can assumeboundedness of z(t) and Δ(t, xH , z, xH , z) and their derivatives in the control design. Therefore,assumptions made in the above control objective are reasonable in practice.

2.2 Preliminaries

Since we assume that Δ(t, xH , z, xH , z) and z(t) are not available for the control design, wepresent in this subsection a disturbance observer, which will be used to estimate Δ(t, xH , z, xH , z)and z(t) in the control design in the next section. In addition, discontinunity of the term√

PHS − sgn(xHv)PH in (5) causes difficulty when taking derivatives of this term with respectto time in the control design. Therefore, we also present a p-times differentiable signum functionto approximate the signum function in (5).

2.2.1 Disturbance observer

Consider the following systemx = f(x) + u + d(t, x) (6)

where x ∈ Rn, f(x) is a vector of known functions of x, u is the control input vector, and d(t, x)

is a vector of unknown disturbances. We assume that there exists a nonnegative constant Cd

such that ‖d(t, x)‖ ≤ Cd. Now we want to design the control input u to stabilize the system(6) at the origin. It is obvious that if we can design a disturbance observer, d(t, x), that es-timates d(t, x) sufficiently accurately, then the control input u is straightforwardly designed asu = −κx− f(x)− d(t, x) with κ a positive definite matrix. The disturbance observer is given inthe following lemma.

Lemma 1. Consider the following disturbance observer

d(t, x) = ξ + ρ(x)

ξ = −K(x)ξ − K(x)(f(x) + u + ρ(x)) (7)

where K(x) = ∂ρ(x)∂x

, ρ(x) is chosen such that the matrix K(x) is positive definite for allx ∈ R

n. The disturbance observer (7) guarantees that the disturbance observer error de(t, x) =d(t, x) − d(t, x) exponentially converges to a ball centered at the origin. The radius of this ballcan be made arbitrarily small by adjusting the function ρ(x). In the case Cd = 0, the disturbanceobserver error de(t, x) exponentially converges to zero.

Proof. See Appendix A. The disturbance observer (7) is a dynamical system. The variable ξis generated by the second equation of (7), which is an ordinary differential equation, with someinitial value ξ(t0), where t0 is the initial time. The choice of the function ρ(x), which results inthe matrix K(x) directly affects performance of the disturbance observer. The larger eigenvaluesof the matrix K(x) are the faster the response of the disturbance observer is with a trade-off ofa large overshoot of the observer, and vice-versa. The initial value ξ(t0) is ideally chosen to be

5

d(t0, x(t0)) − ρ(x(t0)), where d(t0, x(t0)) and ρ(x(t0)) are the initial values of d(t, x) and ρ(x),respectively. However, d(t0, x(t0)) is unavailable since the disturbance d(t, x) is unknown, weusually take ξ(t0) = 0 in practice. Of course, one can choose ξ(t0) to be any constant but thetransient response of the disturbance observer might be bad because as d(t0, x(t0)) is unknown,the chosen constant ξ(t0) can make |d(t0, x(t0)) − ρ(x(t0))| large.

To illustrate the effectiveness of the disturbance observer (7), we perform some numericalsimulations. In the simulations, we consider a scalar system in the form of (6) with f(x) =arctan(x + x2) and d(t, x) =

∑5i=1

(sin(it) + sin(x) sin( it

2)). The function ρ(x) is taken as

ρ(x) = 20(x+ x3

3). This choice gives K(x) = 20(1+x2), which is positive for all x ∈ R. The initial

conditions are x(0) = 1 and ξ(0) = 0. The control law is designed as u = −κx − f(x) − d(t, x)with κ = 5. We run two simulations. In the first one, the disturbance d(t, x) is ignored in thecontrol design, i.e. we set d(t, x) = 0 in the above control law. Simulation results are presentedin the top two sub-figures A) and B) of Figure 3. In the second simulation, we include d(t, x)in the control law. Simulation results are plotted in the bottom two sub-figures C) and D) ofFigure 3. It is seen from the sub-figures A) and C) that in the case where the disturbanced(t, x) is ignored in the control design, the state x converges to a much larger ball than the casewhere the disturbance observer is used in the control design. In the sub-figures B) and D), thedisturbance d(t, x) is plotted in the solid line while the disturbance estimate d(t, x) is plotted inthe dash-dotted line. From the sub-figure D), we can see that the disturbance observer providesan excellent estimate of the time and state dependent disturbance d(t, x). Finally, it is notedthat the value of disturbance d(t, x) plotted in the sub-figure B) is slightly different from that ofthe one plotted in the sub-figure D) because the disturbance d(t, x) depends on the state x.

0 5 10 15−2

−1

0

1

2

Time [s]

x

0 5 10 15−20

−10

0

10

20

30

Time [s]

d, d ha

t

0 5 10 15−1.5

−1

−0.5

0

0.5

1

1.5

Time [s]

x

0 5 10 15

−10

−5

0

5

10

Time [s]

d, d ha

t

A) B)

C) D)

Figure 3: Effectiveness of the proposed disturbance observer.

2.2.2 p-times differentiable signum function

6

−0.4 −0.2 0 0.2 0.4−1

−0.5

0

0.5

1

x

h(x,

−0.1

,0.1

)

Figure 4: A twice differentiable signum function.

Definition 1. A scalar function h(x, a, b) iscalled a p-times differentiable signum functionif it enjoys the following properties

1) h(x, a, b) = −1 if −∞ < x ≤ a,

2) h(x, a, b) = 1 if x ≥ b,

3) − 1 < h(x, a, b) < 1 if a < x < b, (8)

4) h(x, a, b) is p times differentiable

with respect to x

where p is a positive integer, x ∈ R+, and aand b are constants such that a < 0 < b. More-over, if the function h(x, a, b) is infinite times differentiable with respect to x, then it is called asmooth signum function.

Lemma 2. Let the scalar function h(x, a, b) be defined as

h(x, a, b) = 2

∫ x

af(τ − a)f(b − τ)dτ∫ b

af(τ − a)f(b − τ)dτ

− 1 (9)

where the function f(y) is defined as follows

f(y) = 0 if y ≤ 0 and f(y) = yp if y > 0 (10)

with p being a positive integer. Then the function h(x, a, b) is a p times differentiable signumfunction. Moreover, if f(y) = yp in (10) is replaced by f(y) = e−1/y then property 4) is replacedby 4’) h(x, a, b) is a smooth signum function.

Proof. See Appendix B. An illustration of a twice differentiable signum function (a =−0.1, b = 0.1) is given in Fig. 4.

Remark 2. As specified in Lemma 2, the p-times differentiable signum functions can beobtained explicitly, i.e. the integral (9) can be solved analytically. The smooth signum functionscannot obtained explicitly, i.e. the integral (9) cannot be solved analytically, thus a numericalprocedure is needed to solve the integral (9). A simpler version of the p-times differentiablesignum function was also given in (Do, 2007).

With materials presented in this subsection, we rewrite the entire system of the heave com-pensation dynamics (5) in a standard state space form for the purpose of control design in thenext section as follows:

x1 = x2,

x2 = θ21x3 − θ22x2 + Δ(t, x1, x2, z, z),

x3 = −θ31x2 − θ32x3 + θ33g3(x3, x4),

x4 = −θ41x4 + θ42iH (11)

7

with x1 = xH , x2 = xH , x3 = PH , x4 = xHv, and

θ21 =AHCH3

mH, θ22 =

bH

mH, Δ(t, x1, x2, z, z) =

1

mHΔ(t, xH , z, xH , z),

θ31 =4βHeAH

VHCH3, θ32 =

4βHeCHT

VH, θ33 =

4βHeCHDWHCH4

VH√

ρH

√CH3

, g3(x3, x4) = x4

√PHS − h(x4, a, b)x3,

θ41 =1

τHv, θ42 =

kHv

τHvCH4(12)

where we have used the p-times differentiable signum function h(x4, a, b) to replace the signumfunction sgn(x4).

Remark 3. The use of the p-times differentiable signum function h(x4, a, b) instead of thesignum function sgn(x4) in (11) not only makes the the function g3(x3, x4) differentiable withrespect to x3 and x4 but also represents the actual dynamics of the spool dynamics. This isbecause there is always certain inaccuracy in manufacturing the servovalve, i.e. the flow in theservovalve does not change its direction immediately.

3 Control design

A close look at the entire system (11) shows that the system is of a strick-feedback form, (Krsticet al., 1995). Therefore, we will use the backstepping technique to design the control input iHto achieve the control objective stated in the previous section. The control design consists of 4steps. At steps 1 and 2, we will use the disturbance observer presented in Subsection 2.2.1 todesign an estimate for z(t) and Δ(t, x1, x2, z, z).

3.1 Step 1

Based on the backstepping technique, at this step we consider the first equation of the entiresystem (11) where the state x2 is considered as a control. As such, we define

x1e = x1 + z(t) − L,

x2e = x2 − α1 (13)

where α1 is a virtual control of x2. Differentiating both sides of the first equation of (13) alongthe solutions of the second equation of (13) and the first equation of (11) gives

x1e = α1 + x2e + w (14)

where w := z(t). The goal is to design the virtual control α1 to stabilize x1e at the origin whiletreating w as a disturbance. Also at this step, we do not need to consider x2e. Regulating x2e

to a small neighborhood of the origin will be done at the next step. Ideally, we would regulatex2e to the origin but it is impossible since (14) contains a time varying disturbance w and thedisturbance observer can only estimate the time varying disturbance w approximately. It is seenthat (14) is exactly the same form as (6) if we ignore x2e. Hence, from (14) and the disturbanceobserver design proposed in Subsection 2.2.1, we design the virtual control α1 and the estimate

8

w of w as follows:

α1 = −k11x1e − w,

w = ξ1 + k12x1e,

ξ1 = −k12ξ1 − k12(x2e + α1 + k12x1e) (15)

where k11 and k12 are positive constants. It is noted that the above disturbance observer is anapplication from Subsection 2.2.1 where we took ρ(x) = k12x. Using (14) and (15), we have thefollowing first closed loop sub-system

x1e = −k11x1e + x2e + we,

we = −k12we + w (16)

where we = w−w. The dynamics (16) will be used in stability analysis of the closed loop system.It is of interest to note that the virtual control α1 is a smooth function of x1e and ξ1.

3.2 Step 2

Our goal at this step is to regulate x2e to a small neighborhood of the origin by considering thesecond equation of the entire system (11). Similarly to Step 1, the state x3 is considered as acontrol. As such, we define

x3e = x3 − α2 (17)

where α2 is a virtual control of x3. Now differentiating both sides of the second equation of (13)along the solutions of the second equation of (11), (15) and (16) gives

x2e = θ21(x3e + α2) − θ22(x2e + α1) + Δ(t, x1, x2, z, z) − ∂α1

∂x1e(−k11x1e + x2e + we) − ∂α1

∂ξ1ξ1.

(18)

Again at this step we do not consider x3e, of which regulation to a small neighborhood ofthe origin will be done at the next step. Clearly, it is seen that (18) is the same form as(6) in Subsection 2.2.1 if we ignore x3e. Therefore, from (18) and the disturbance observerdesign proposed in Subsection 2.2.1, we choose the virtual control α2 and an estimate Δ of thedisturbance Δ(t, x1, x2, z, z) as follows

α2 =1

θ21

(− x1e − k21x2e + θ22(x2e + α1) +

∂α1

∂x1e(−k11x1e + x2e) +

∂α1

∂ξ1ξ1 − Δ

),

Δ = ξ2 + k22x2e,

ξ2 = −k22ξ2 − k22

(θ21(x3e + α2) − θ22(x2e + α1) − ∂α1

∂x1e(−k11x1e + x2e) − ∂α1

∂ξ1ξ1

)(19)

where k21 and k22 are positive constants. It is again noted that the above disturbance observeris an application from Subsection 2.2.1 where we took ρ(x) = k22x. Using (18) and (19), we havethe following second closed loop sub-system

x2e = −x1e − k21x2e + θ21x3e − ∂α1

∂x1e

we + Δe,

Δe = −k22Δe + k22∂α1

∂x1ewe + Δ (20)

where Δe = Δ − Δ. The dynamics (20) will be used in stability analysis of the closed loopsystem. It is noted that the virtual control α2 is a smooth function of x1e, ξ1, x2e and ξ2.

9

3.3 Step 3

The objective of this step is to regulate x3e to a small neighborhood of the origin by consideringthe third equation of the entire system (11). To simplify the design process, we consider g3(x3, x4)a control instead of x4. As such, we define

x4e = g3(x3, x4) − α3 (21)

where α3 is a virtual control of g3(x3, x4). Now differentiating both sides of (17) along thesolutions of the third equation of (11), (15), (16) and (20) gives

x3e = −θ31x2 − θ32x3 + θ33(α3 + x4e) − ∂α2

∂x1e(−k11x1e + x2e + we) − ∂α2

∂ξ1ξ1 −

∂α2

∂x2e

(− x1e − k21x2e + θ21x3e − ∂α1

∂x1e

we + Δe

)− ∂α2

∂ξ2

ξ2. (22)

Again at this step we do not consider x4e, of which regulation to a small neighborhood of theorigin will be done at the next step. From (22), we choose the virtual control α3 as

α3 =1

θ33

(− θ21x2e − k31x3e + θ31x2 + θ32x3 +

∂α2

∂x1e

(−k11x1e + x2e) +∂α2

∂ξ1

ξ1 +

∂α2

∂x2e(−x1e − k21x2e + θ21x3e) +

∂α2

∂ξ2ξ2

)(23)

where k31 is a positive constant. Now substituting (23) into (22) gives the following third closedloop sub-system

x3e = −θ21x2e − k31x3e + θ33x4e −(

∂α2

∂x1e

− ∂α2

∂x2e

∂α1

∂x1e

)we − ∂α2

∂x2e

Δe (24)

which will be used in stability analysis of the closed loop system. It is noted that the virtualcontrol α3 is a smooth function of x1e, ξ1, x2e, ξ2, x3e since x2 = x2e + α1 and x3 = x3e + α2.

3.4 Step 4

This is the final step. The actual control input iH will be designed to regulate x4e to a smallneighborhood of the origin. To do so, differentiating both sides of (21) along the solutions of thelast equation of the entire system (11) gives

x4e =∂g3

∂x3(−θ31x2 − θ32x3 + θ33g3(x3, x4)) +

∂g3

∂x4(−θ41x4 + θ42iH) − ∂α3

∂ξ1ξ1 − ∂α3

∂ξ2ξ2 −

∂α3

∂x1e

(−k11x1e + x2e + we) − ∂α3

∂x2e

(− x1e − k21x2e + θ21x3e − ∂α1

∂x1e

we + Δe

)−

∂α3

∂x3e

(− θ21x2e − k31x3e + θ33x4e −

(∂α2

∂x1e− ∂α2

∂x2e

∂α1

∂x1e

)we − ∂α2

∂x2eΔe

). (25)

10

From (25), we choose the actual control iH as follows

iH =1

θ42∂g3

∂x4

(− θ33x3e − k41x4e + θ41x4 − ∂g3

∂x3(−θ31x2 − θ32x3 + θ33g3(x3, x4)) +

∂α3

∂ξ1

ξ1 +∂α3

∂ξ2

ξ2 +∂α3

∂x1e

(−k11x1e + x2e) +∂α3

∂x2e

(−x1e − k21x2e + θ21x3e) +

∂α3

∂x3e(−θ21x2e − k31x3e + θ33x4e)

)(26)

where k41 is a positive constant. It should be noted that since PHS−|PH | = CH3(PHs−|x3|) ≥ ε1,see Subsection 2.1 just below equation (3), and that

∂g3

∂x4=

√PHS − x3h(x4, a, b) − x3x4

∂h(x4,a,b)∂x4

2√

PHS − x3h(x4a, b). (27)

From properties of the p-times differentiable signum function h(x4, a, b), we can see that thereexists a strictly positive constant ε2 such that | ∂g3

∂x4| ≥ ε2. This means that the control iH given

in (26) is well-defined. Substituting (26) into (25) results in the fourth closed loop sub-system

x4e = −θ33x3e − k41x4e −(

∂α3

∂x1e− ∂α3

∂x2e

∂α1

∂x1e− ∂α3

∂x3e

∂α2

∂x1e+

∂α3

∂x3e

∂α2

∂x2e

∂α1

∂x1e

)we −(

∂α3

∂x2e

− ∂α3

∂x3e

∂α3

∂x2e

)Δe. (28)

The control design has been completed. Stability analysis of the closed loop system consistingof four closed loop sub-systems (16), (20), (24) and (28) is given in Appendix C. An appropriatechoice of the control and observer gains k11, k12, k21, k22, k31, and k41 to avoid actuator satura-tions is given in the next section. For convenience of the reader, we summarize all the formulaethat need to implement the proposed controller in Table 1. We now state the main result of thepaper in the following theorem.

Theorem 1. Under the assumption that z(t) and Δ(t, xH , z, xH , z) and their derivatives arebounded, the control input iH given in (26) and the disturbance observers given in (15) and (19)solve the control objective. In particular, the closed loop system consisting of (16), (20), (24)and (28) is forward complete, and the active heave compensation error x1e(t) = x1(t) + z(t)−Lexponentially converges to a small value. This value can be made arbitrarily small by choosingsufficiently large control and observer gains k11, k12, k21, k22, k31, and k41. If the derivatives ofz(t) and Δ(t, xH , z, xH , z) are zero, the active heave compensation error x1e(t) = x1(t)+ z(t)−Lexponentially converges to zero.

Proof. See Appendix C.

11

Step 1:

x1e = x1 + z(t) − L,

x2e = x2 − α1,

α1 = −k11x1e − w,

w = ξ1 + k12x1e,

ξ1 = −k12ξ1 − k12(x2e + α1 + k12x1e).

Step 2:

x3e = x3 − α2,

α2 =1

θ21

(− x1e − k21x2e + θ22(x2e + α1) +

∂α1

∂x1e(−k11x1e + x2e) +

∂α1

∂ξ1ξ1 − Δ

),

Δ = ξ2 + k22x2e,

ξ2 = −k22ξ2 − k22

(θ21(x3e + α2) − θ22(x2e + α1) − ∂α1

∂x1e(−k11x1e + x2e) − ∂α1

∂ξ1ξ1

).

Step 3:

x4e = g3(x3, x4) − α3,

α3 =1

θ33

(− θ21x2e − k31x3e + θ31x2 + θ32x3 +

∂α2

∂x1e(−k11x1e + x2e) +

∂α2

∂ξ1ξ1 +

∂α2

∂x2e

(−x1e − k21x2e + θ21x3e) +∂α2

∂ξ2

ξ2

).

Step 4:

iH =1

θ42∂g3

∂x4

(− θ33x3e − k41x4e + θ41x4 − ∂g3

∂x3(−θ31x2 − θ32x3 + θ33g3(x3, x4)) +

∂α3

∂ξ1

ξ1 +∂α3

∂ξ2

ξ2 +∂α3

∂x1e

(−k11x1e + x2e) +∂α3

∂x2e

(−x1e − k21x2e + θ21x3e) +

∂α3

∂x3e(−θ21x2e − k31x3e + θ33x4e)

).

Table 1. Summary of formulae for control implementation.

4 Determination of control and observer gains

The control and observer gains k11, k12, k21, k22, k31, and k41 directly affect performance of theclosed loop system. The larger these gains are, the faster the response of the closed loop systemis with the trade-off of a larger control effort at the transient response time, and vice-versa.Unfortunately, the current input iH cannot be arbitrarily large for any hydraulic systems used inan active compensation system in practice. This means that we need to choose the above controland observer gains such that they result in an as fast as possible response of the closed loopsystem, and avoidance of control saturation, i.e. the current input iH must lie in a certain range,say |iH | ≤ imax

H where imaxH is the maximum current inputting to the hydraulic system with an

assumption of a symmetric current input hydraulic system. For a nonsymmetric current inputhydraulic system, the following discussion can be carried out similarly. Moreover, each hydraulic

12

system has a limited bandwidth. This means that if we force a hydraulic system to responsetoo fast, the hydraulic system will not properly response to the commanded input, rather it justoscillates/vibrates.

In general, actuator saturation deteriorates the overall performance of the closed loop system,and can even destabilize the closed loop system. These problems are difficult to deal with butare important to take care of. In many cases, global stabilization and/or tracking result cannotbe achieved when actuators are subject to saturation. For example, consider the scalar system

x = x2 + u (29)

with x the state and u the control. For this system, no bounded control u can globally exponen-tially/asymptotically stabilize the system at the origin. In fact, any bounded control u can resultin a finite escape time. To illustrate the idea of dealing with control saturation, we consider aproblem of stabilizing the scalar system (29). First, we design a control as

u = −kx − x2 (30)

where the control gain k is a positive constant. This control results in a closed loop system

x = −kx (31)

which is globally exponentially stable at the origin. Now assume that the control u must lie ina certain range due to the actuator saturation, say |u(t)| ≤ umax, ∀t ≥ t0 ≥ 0 where umax isthe maximum magnitude of the control input u. Given the initial value x(t0) and umax with afeasible assumption that umax is strictly larger than x2(t0), we need to find the control gain ksuch that |u(t)| ≤ umax, ∀t ≥ t0 ≥ 0. From (31), we have x(t) = x(t0)e

−k(t−t0), which impliesthat

|x(t)| ≤ |x(t0)|, ∀t ≥ t0 ≥ 0. (32)

Therefore, from (30) and (32), we have

|u(t)| ≤ k|x(t)| + x2(t) ≤ k|x(t0)| + x2(t0), ∀t ≥ t0 ≥ 0. (33)

Since we want |u(t)| ≤ umax, ∀t ≥ t0 ≥ 0, from (34) it is sufficient to choose the control gain k as

k|x(t0)| + x2(t0) ≤ umax ⇒ k ≤ umax − x2(t0)

|x(t0)| , ∀t ≥ t0 ≥ 0. (34)

Since the feasible condition (i.e. umax is strictly larger than x2(t0)) implies that umax − x2(t0)is strictly positive, we can always find a positive constant k such that the condition (34) holds.In practice, we need to provide the maximum value of |x(t0)| satisfying the feasible condition todetermine a range of the control gain k must be lie in to stabilize the system (29) at the origin,and to avoid the control saturation. The condition (34) also implies that the control u withthe control gain k satisfying (34) can only provide a local (not global) exponential stabilizationsolution. In literature, only restrictive tracking control and stabilization can be achieved formany systems including one consisting of a simple integrator chain (Teel, 1992).

We now discuss how to use the proposed controller given in Section 3 when both actuatormagnitude and rate saturations present in the active heave compensation in question. We firstdeal with the case where the actuator magnitude saturation presents then move to the casewhere the actuator rate saturation is imposed.

13

4.1 Actuator magnitude saturation case

Motivated by the aforementioned example, we will first set |iH | ≤ imaxH , then find restrictions

on the control and observer gains, and the initial conditions such that the condition |iH | ≤ imaxH

holds. From (26), the condition |iH | ≤ imaxH is equivalent to∣∣∣ 1

θ42∂g3

∂x4

(− θ33x3e − k41x4e + θ41x4 − ∂g3

∂x3(−θ31x2 − θ32x3 + θ33g3(x3, x4)) +

∂α3

∂ξ1ξ1 +

∂α3

∂ξ2ξ2 +

∂α3

∂x1e(−k11x1e + x2e) +

∂α3

∂x2e(−x1e − k21x2e + θ21x3e) +

∂α3

∂x3e

(−θ21x2e − k31x3e + θ33x4e)

)∣∣∣ ≤ imaxH . (35)

To find a range that the control and observer gains should lie in, with a note that | ∂g3

∂x4| ≥ ε2,

see Subsection 3.4, we need to calculate upper bounds of all the terms in the left-hand side of(35). These bounds depend on the control and observer gains, and the initial conditions. Afterthe bounds are substituted into the left-hand side of (35), we will be able to find a feasiblerange of the control and observer gains. We first find the upper bounds of x1e, x2e, x3e, x4e,we and Δe. Based on these values, we are then be able to find upper bounds of all the termsin the left-hand side of (35). As such, integrating both sides of (56), see Appendix C, we haveV (t) ≤ (V (t0) − λ

c)e−c(t−t0) + λ

c, ∀t ≥ t0 ≥ 0, which implies that V (t) ≤ V (t0) + λ

c. From the

definition of V (t), see (51) in Appendix C, we have(x2

1e(t) + x22e(t) + x2

3e(t) + x24e(t) + δ1w

2e(t) + δ2Δ

2e(t)

) ≤(x2

1e(t0) + x22e(t0) + x2

3e(t0) + x24e(t0) + δ1w

2e(t0) + δ2Δ

2e(t0)

)+

λ

c:= Ω0. (36)

From (36) and the definition of x1e, x2e, x3e, x4e, we and Δe, we first calculate the followingimmediate upper bounds of α1, x2, ξ1, ξ1, α2, x3, ξ2, ξ2, α3 and x4 as follows:

|x1e| ≤ Ω0, |x2e| ≤ Ω0, |x3e| ≤ Ω0, |x4e| ≤ Ω0, |we| ≤ Ω0

δ1

, |Δe| ≤ Ω0

δ2

,

|α1| ≤ k11|x1e| + |w| ≤ k11|x1e| + |we| + |w| ≤ k11Ω0 +Ω0

δ1

+ Cw := αM1 ,

|x2| ≤ |x2e| + |α1| ≤ Ω0 + k11Ω0 +Ω0

δ1

+ Cw := xM2 ,

|ξ1| ≤ |w| + k12|x1e| ≤ Ω0

δ1

+ Cw + k12Ω0 := ξM1 ,

|ξ1| ≤ k12|ξ1| + k12(|x2e| + |α1| + k12|x1e|) := ξM1

|Δ| ≤ |Δ| + |Δe| ≤ CΔ +Ω0

δ2:= ΔM ,

|α2| ≤ 1

|θ21|(|x1e| + k21|x2e| + |θ22|(|x2e| + |α1|) +

∣∣∣ ∂α1

∂x1e

∣∣∣(k11|x1e| + |x2e|) +

∣∣∣∂α1

∂ξ1

∣∣∣|ξ1| + |Δ|)

:= αM2 ,

|x3| ≤ |x3e| + |α2| := xM3 ,

|ξ2| ≤ |Δ| + k22|x2e| := ξM2 ,

14

|ξ2| ≤ k22|ξ2| + k22

(|θ21|(|x3e| + |α2|) + |θ22|(|x2e| + |α1|) +

∣∣∣ ∂α1

∂x1e

∣∣∣(k11|x1e| + |x2e|) +∣∣∣∂α1

∂ξ1

∣∣∣|ξ1|)

:= ξM2 ,

|α3| ≤ 1

|θ33|(|θ21||x2e| + k31|x3e| + |θ31||x2| + |θ32||x3| +

∣∣∣ ∂α2

∂x1e

∣∣∣(k11|x1e| + |x2e|) +

∣∣∣∂α2

∂ξ1

∣∣∣|ξ1| +∣∣∣ ∂α2

∂x2e

∣∣∣(|x1e| + k21|x2e| + |θ21||x3e|) +∣∣∣∂α2

∂ξ2

∣∣∣|ξ2|)

:= αM3 ,

|x4| ≤ |x4e| + |g3(x3, x4)| := xM4 (37)

where Cw and ΔM is the maximum value of |w| and |Δ|, respectively. It is noted that all theterms involving the partial derivatives can be easily upper bounded based on (37). Now thecondition (35) can be written as

1

|θ42|| ∂g3

∂x4|

(|θ33||x3e| + k41|x4e| + |θ41||x4| +

∣∣∣∂g3

∂x3

∣∣∣(|θ31||x2| + |θ32||x3| + |θ33||g3(x3, x4)|) +

∣∣∣∂α3

∂ξ1

∣∣∣|ξ1| +∣∣∣∂α3

∂ξ2ξ2 +

∂α3

∂x1e

∣∣∣(k11|x1e| + |x2e|) +∣∣∣ ∂α3

∂x2e

∣∣∣(|x1e| + k21|x2e| + |θ21||x3e|) +

∣∣∣ ∂α3

∂x3e

∣∣∣(|θ21||x2e| + k31|x3e| + |θ33||x4e|))

≤ imaxH . (38)

Substituting (37) into (38) results in an inequality that all the control and observer gains k11,k12, k21, k22, k31, and k41 must be satisfied to guarantee that no saturation of the current iHoccurs. Since an explicit expression of the range that all the control and observer gains shouldlie in is pretty complicated, we do not write it down here. However, a simple computer programcan be written to give the explicit range of the control and observer gains since it mainly involveswith a recursive substitution.

4.2 Actuator rate saturation case

For this case, we make a practical observation that in an hydraulic system the current inputiH can be changed arbitrarily fast (Yao et al., 2001) and that the rate limitation is placed onthe velocity, x2, of the main piston. As such, we just need to impose a constraint |x2(t)| ≤ RM

x2

where RMx2

is the maximum allowable absolute value of the main piston velocity, x2. Using (37),we can deduce a condition on the control and observer gains to ensure that the rate saturationdoes not occur as follows:

|x2| ≤ |x2e| + |α1| ≤ Ω0 + k11Ω0 +Ω0

δ1+ Cw := xM

2 ≤ RMx2

(39)

which is an additional condition to those conditions derived in Subsection (4.1) to choose thecontrol and observer gains k11, k12, k21, k22, k31, and k41 so that the actuator rate and magnitudesaturations are avoided.

5 Simulations

To illustrate the effectiveness of the controller proposed in the previous section, we carry out somesimulations in this section. The parameters of the hydraulic system are taken based on (Yao et

15

al., 2001) as follows: mH = 1000kg, AH = 0.65m2, bH = 40N/(m/s), 4βHe/VH = 4.53×108N/m5,CHD = 2.21 × 10−14m5/Ns, CHDwH/

√ρ = 3.42 × 10−5m3

√Ns, PHS = 10342500Pa, kHv =

0.0324, τHv = 0.00636. The scale factors are taken as CH3 = 6 × 105, CH4 = 5 × 10−7 to scalePHS down and τHv up as discussed in Subsection 2.1. The riser parameters are taken from (Korde,1998) as follows: length L = 3832m, diameter dr = 0.14m, density ρr = 8200kg/m3, Young’smodulus Er = 2 × 108kg/m2, initial tension Tor = 100KN . Therefore, the force (disturbance),Δ acting from the riser to the piston of the hydraulic system is (Korde, 1998):

Δ = Tor + ErAr

( Nm∑n=1

∓nπCn + δ)

(40)

with Cn being generated by

Cn + cdCn +ErArn

2π2

ρrL2Cn = ∓ 2

nπ

(cdδ + δ

)(41)

where δ = z − xH − L, Ar = πd2r

4, the damping coefficient cd = 0.01m3/s, and the notation ∓

takes the positive sign if n = 2, 4, 6, ... and the negative sign if n = 1, 3, 5, ..., and Nm is numberof modes. The vertical (heave) motion of the vessel z(t) can be represented as a sum of sinusoidsat different frequencies, amplitudes and phases as follows, see (Fossen, 1994) and (Lloyd, 1998):

z(t) = L +Nw∑i=1

(Awikwi sin(wwit − ϕwi)

)(42)

where L is included since z(t) is coordinated in the earth-fixed frame, the amplitude Awi, coef-ficient kwi, frequency wwi, phase ϕwi of the wave ith are given by

wwi = wm +wm − wM

Nwi, Swi =

1.25

4

w4o

w5wi

H2swe

−1.25(

wowwi

)4

Awi =

√2Swi

wmi − wMi

Nw, kwi =

w2wi

9.8, ϕwi = 2πrand(). (43)

In (43), the minimum and maximum wave frequencies are wm = 0.2rand/s, wM = 2.5rand/s; thetwo-parameter Bretschneider spectrum Swi is used with the significant wave height Hsw = 4m;the modal frequency is wo = 2π

Twwith the period Tw = 7.8; Nw = 10; and rand() is a random

number between 0 and 1.The control and disturbance observer gains are chosen as k11 = 2, k12 = 15, k21 = 4, k22 =

25, k31 = 8, k41 = 12. In the simulations, we take the number of modes in (40) as Nm = 5, andinitial values of Cn in (41) are zero. To illustrate the effectiveness of the disturbance observers,we simulate the proposed controller in two cases. In the first case, the disturbance observers areswitched on. The simulation result is presented in Figure 5.

In Figure 5, the top sub-figure plots the error x1e, the bottom sub-figure plots the disturbanceΔ and its estimate Δ. From the bottom sub-figure, we can see that the disturbance observerestimates the disturbance Δ quite well, see also Figure 6 for a short time presentation of Δ andΔ. This results in pretty small error x1e. It is noted that the error x1e does not converge to zeroas said Theorem 1 because Δ and z(t) are time-varying.

In the second case, the disturbance observers are switched-off. The simulation result ispresented in Figure 7. Explanation of the sub-figures is the same as that in the first case. From

16

0 10 20 30 40 50 60−0.5

0

0.5

Time [s]

Erro

r X1e

[m]

0 10 20 30 40 50 60−5

0

5x 10

4

Time [s]

Δ,Δ ha

t

ΔΔ

hat

Figure 5: Simulation result with disturbance observers.

these sub-figures, we can see that when the disturbance observers are switched-off, the errorx1e is pretty large. This is because the controller does not compensate for the disturbances viathe disturbance observers. The above simulation results support the fact in the control designthat the disturbance observers play a vital role in the proposed controller for the active heavecompensation system (i.e. to achieve a small |x1e|) considered in this paper.

6 Conclusions

A nonlinear controller has been designed for an active heave compensation system. One of thekeys to success in the proposed method is the use of the disturbance observers, which are thenproperly embedded in the control design procedure. The heave velocity and the force actingbetween the riser/drill-string and the active heave compensation unit were well estimated by thedisturbances observers. Future work includes extending the proposed control design techniquein this paper to compensate for the local/small motion of the vessel/rig in (x, y) plane as wellto achieve fine tuning of riser/drill-string angles at the bottom and top ends since it is costly touse a dynamic positioning system to adjust small motion of the vessel/rig.

Acknowledgements: We thank the reviewers for their helpful comments, which help usimprove the paper significantly. This work is supported by the ARC Discovery grant No.DP0774645.

Appendix A: Proof of Lemma 1. To prove Lemma 1, we calculate the derivative of

17

0 0.5 1 1.5 2 2.5 3−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5x 10

4

Time [s]

Δ, Δ ha

t

ΔΔ

hat

Figure 6: A short time presentation of Δ and Δ

de(t, x) as follows

de(t, x) = d(t, x) − (ξ + ρ(x))

= K(x)(d(t, x) − ρ(x)) + K(x)(f(x) + u + ρ(x)) − K(x)(f(x) + u + d(t, x)) + d(t, x)

= −K(x)(d(t, x) − d(t, x)) + d(t, x)

= −K(x)de(t, x) + d(t, x). (44)

Consider the Lyapunov function

Ve =1

2dT

e (t, x)de(t, x) (45)

whose first time derivative along the solutions of (44) satisfies

Ve = −1

2dT

e (t, x)(K(x) + KT (x))de(t, x) + dTe (t, x)d(t, x). (46)

Since K(x) is a positive definite matrix, there exists a positive constant � such that

dTe (t, x)

(K(x) + KT (x)

)de(t, x) ≥ �dT

e (t, x)de(t, x), ∀x ∈ R. (47)

Substituting (47) into (46) results in

Ve ≤ −(� − 2ε)Ve +1

4ε‖d(t, x)‖2

≤ −(� − 2ε)Ve +1

4εC2

d (48)

where we have used (45) and |dTe (t, x)d(t, x)| ≤ ε‖de(t, x)‖ + 1

4ε‖d(t, x)‖ with ε a positive con-

stant such that � − 2ε is strictly positive. From the second inequality of (48), it is seen that

Ve exponentially converges to a ball centered the origin with the radius RV e =C2

d

4ε(�−2ε)as long

as the solutions x(t) exist. The existence of the solutions x(t) is to be guaranteed by the design

18

0 10 20 30 40 50 60−2

−1

0

1

2

Time [s]

Erro

r X1e

[m]

0 10 20 30 40 50 60−4

−2

0

2

4x 10

4

Time [s]

Δ,Δ ha

t

ΔΔ

hat

Figure 7: Simulation result without disturbance observers.

of the control input u. This in turn means that the disturbance error de(t, x) exponentially

converges to a ball centered at the origin with the radius Rde =√

12ε(�−2ε)

Cd. Since � can be

chosen arbitrarily large by choosing the function ρ(x), the radius Rde can be made arbitrarilysmall. In the case Cd = 0, the radius RV e = Rde = 0 meaning that the disturbance error de(t, x)exponentially converges to zero.�

Appendix B: Proof of Lemma 2. For the case f(y) = yp, see (Do, 2007) for proof ofProperties 1)-4). We here consider the case where f(y) = yp in (10) is replaced by f(y) = e−1/y.Properties 1)-3) can be proven in the same lines as in (Do, 2007). We focus on proof of Prop-

erty 4’). We first note that f (k)(y) = ∂kf(y)∂yk = Qk(

1y)e−

1y where Qk(

1y) is a polynomial func-

tion of 1y, and k is any positive integer. We will prove Property 4’) by induction. First of

all, we check that f (1)(0) = 0. It is clear that limy→0−f(y)−f(0)

y−0= limy→0−

0y

= 0. On the

other hand, limy→0+f(y)−f(0)

y−0= limy→0+

e− 1

y

y= limξ→∞ 1

eξ = 0 where ξ = 1y

and we have used

l’Hopital’s rule. Since both left- and right-hand limits are equal to 0, we have f (1)(0) = 0.Now assume that f (k)(0) = 0. We now compute f (k+1)(0). The left-hand limit is equal to 0 as

above. The right hand limit is limy→0+f(k)(y)−f(k)(0)

y−0= limy→0+

f(k)(y)−0y−0

= limy→0+1yQk(

1y)e−

1y =

limy→0+ Qk(1y)e−

1y = limξ→∞

Qk(ξ)eξ = 0 (by l’Hopital’s rule), where ξ = 1

yand Qk(ξ) = ξQk(ξ)

is another polynomial of 1y. Since both left- and right-hand limits are equal to 0, we have

f (k+1)(0) = 0, which completes proof of Property 4’).�

Appendix C Proof of Theorem 1. For convenience, we rewrite the closed loop system

19

consisting of (16), (20), (24) and (28) as follows:

x1e = −k11x1e + x2e + we,

x2e = −x1e − k21x2e + θ21x3e − φ1we + Δe,

x3e = −θ21x2e − k31x3e + θ33x4e − φ2we − φ3Δe,

x4e = −θ33x3e − k41x4e − φ4we − φ5Δe,

we = −k12we + w,

Δe = −k22Δe + k22φ1we + Δ (49)

where

φ1 =∂α1

∂x1e, φ2 =

(∂α2

∂x1e− ∂α2

∂x2e

∂α1

∂x1e

), φ3 =

∂α2

∂x2e

φ4 =

(∂α3

∂x1e

− ∂α3

∂x2e

∂α1

∂x1e

− ∂α3

∂x3e

∂α2

∂x1e

+∂α3

∂x3e

∂α2

∂x2e

∂α1

∂x1e

)

φ5 =

(∂α3

∂x2e

− ∂α3

∂x3e

∂α3

∂x2e

). (50)

To investigate stability of the closed loop system (49), we consider the following Lyapunovfunction candidate

V =1

2

(x2

1e + x22e + x2

3e + x24e + δ1w

2e + δ2Δ

2e

)(51)

where δ1 and δ2 are positive constants to be picked later. Differentiating both sides of (51) alongthe solutions of (49) gives

V = −k11x21e − k21x

22e − k31x

23e − k41x

24e − δ1k12w

2e − δ2k22Δ

2e + wex1e − φ1wex2e + Δex2e

−φ2wex3e − φ3Δex3e − φ4wex4e − φ5Δex4e + δ1wwe + δ2k22φ1weΔe + δ2ΔΔe. (52)

From the expressions of α1, α2 and α3, see (15), (19) and (23), there exist nonnegative constantsAi, i = 1, ..., 5 such that

|φi| ≤ Ai, i = 1, ..., 5. (53)

Using (53) and completion of squares, we can write (52) as

V ≤ −k11

2x2

1e −k21

2x2

2e −k31

2x2

3e −k41

2x2

4e

−(δ1k12 − 1

2k11

− A21

k21

− A22

k31

− A24

k41

− δ1ε1 − k22A1δ2

4ε2

)w2

e

−(δ2k22 − 1

k21

− A23

k31

− A25

k41

− ε2k22A1δ2

)Δ2

e +δ1

4ε1

w2 +δ2

4ε3

Δ2 (54)

where εi, i = 1, 2, 3 are positive constants to be picked. Now for chosen control gains k11, k21,k31, k41, k12 and k22, we can always pick the positive constants δ1, δ2 and εi, i = 1, 2, 3 such that

(δ1k12 − 1

2k11

− A21

k21

− A22

k31

− A24

k41

− δ1ε1 − k22A1δ2

4ε2

)≥ γ1

2,

(δ2k22 − 1

k21− A2

3

k31− A2

5

k41− ε2k22A1δ2

)≥ γ2

2(55)

20

where γ1 and γ2 are positive constants. Substituting (55) into (54) results in

V ≤ −cV + λ (56)

where the positive constant c and nonnegative constant λ are given by

c =(min(k11, k21, k31, k41, γ1, γ2)

max(1, δ1, δ2)

), λ = sup

t≥0

( δ1

4ε1w2(t) +

δ2

4ε3Δ2(t)

)(57)

where for simplicity of presentation, we have used only t in the argument of w and Δ. Solvingthe differential inequality (56) shows that V exponentially converges to a ball centered at theorigin with the radius RV = λ/c. This implies that the closed loop system consisting of (16),(20), (24) and (28) is forward complete. Moreover, convergence of V implies that all the errorsxie, i = 1, ..., 4, and the disturbance observer errors we and Δe exponentially converge to a

ball centered at the origin with a radius Re =√

λc min(1,δ1,δ2)

. Moreover, in the case where the

derivatives of the disturbances w and Δ can be ignored, we have λ = 0. This means that allthe state errors xie, i = 1, ..., 4, and the disturbance observer errors we and Δe exponentiallyconverge zero. �

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