MULTI-PURPOSE CONTROL LAW FOR MARINE DYNAMIC POSITIONING SYSTEM UNDER THE INFLUENCE OF SEA WAVES Текст научной статьи по специальности «Физика»

КОГНИТИВНЫЕ ИНФОРМАЦИОННЫЕ ТЕХНОЛОГИИ В СИСТЕМАХ УПРАВЛЕНИЯ / COGNITIVE INFORMATION TECHNOLOGIES IN CONTROL SYSTEMS

УДК 681.5.015

DOI: 10.25559/SITITO.16.202001.72-80

Multi-purpose Control Law for Marine Dynamic Positioning System under the Influence of Sea Waves

A. O. Vedyakova

Saint-Petersburg State University, Saint-Petersburg, Russia 7/9 Universitetskaya Emb., St. Petersburg 199034, Russia vedyakova@gmail.com

Abstract

The paper is devoted to the problem of multi-purpose control law synthesis for marine vessels, which are controlled by a dynamic positioning system under sea wave disturbance. The proposed approach is based on a special control law structure constructed using nonlinear asymptotic observers, that allows decoupling of synthesis into simpler particular optimisation problems. The designed dynamic of a closed-loop system provides an economical mode of vessel motion by reducing general fuel consumption and preventing the wearing down of actuators. The actual value of the external disturbance main frequency is estimated online and used for dynamical corrector tuning. Applicability and efficacy of this approach are illustrated by the practical example of DP system synthesis.

Keywords: Dynamic positioning, control law, stability, external disturbances, frequency estimation.

For citation: Vedyakova A.O. Multi-purpose Control Law for Marine Dynamic Positioning System under the Influence of Sea Waves. Sovremennye informacionnye tehnologii i IT-obrazovanie = Modern Information Technologies and IT-Education. 2020; 16(1):72-80. DOI: https://doi.org/10.25559/ SITITO.16.202001.72-80

Контент доступен под лицензией Creative Commons Attribution 4.0 License. The content is available under Creative Commons Attribution 4.0 License.

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Многоцелевой закон управления морскими системами динамического позиционирования под влиянием морского волнения

А. О. Ведякова

Санкт-Петербургский государственный университет, г. Санкт-Петербург, Россия 199034, Россия, г. Санкт-Петербург, Университетская наб., д. 7/9 vedyakova@gmail.com

Аннотация

Работа посвящена синтезу многоцелевого управления в задаче динамического позиционирования морских судов с учетом морского волнения. Современные системы морского динамического позиционирования, как правило, строятся на основе нелинейных асимптотических наблюдателей, восстанавливающих скорости судна. В статье дополнительно к наблюдателю предлагается использовать динамический корректор, реализующий экономичный режим движения судна с целью снижения общего расхода топлива и предотвращения износа исполнительных механизмов. Для динамической настройки корректора используется оценка основной гармоники возмущающего воздействия. Для этого получена регрессионная модель первого порядка, неизвестный параметр которой зависит от основной частоты морского волнения. На основе метода градиентного спуска строится оценка частоты, обеспечивающая экспоненциальную сходимость ошибки оценивания к нулю. Применимость и эффективность предложенного подхода проиллюстрированы на практическом примере синтеза системы динамического позиционирования.

Ключевые слова: динамическое позиционирование, закон управления, устойчивость, внешние возмущение, оценка частоты.

Для цитирования: Ведякова А. О. Многоцелевой закон управления морскими системами динамического позиционирования под влиянием морского волнения / А. О. Ведякова. - DOI 10.25559/SITITO.16.202001.72-80 // Современные информационные технологии и ИТ-образо-вание. - 2020. - Т. 16, № 1. - С. 72-80.

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Introduction

The problem of dynamic positioning (DP) is one of the most significant problems in marine control. Modern DP systems are widely used in different areas such as hydrography, inspection of marine construction, wreck investigation, underwater cable laying, and so on [1-4].

There are a wide spectrum of publications connected with different questions of DP-control systems design [1-3, 5]. The approaches in [3, 5] propose the structure of DP control law, using nonlinear asymptotic observers, and provide sufficient conditions for global asymptotic stability and validate the possibility of independent tuning for observers and state control laws. In the paper [1], this approach is modified to increase flexibility using the theory of multi-purpose control law synthesis [6-7].

This work is devoted to design an optimal control for providing desired dynamic behaviour of the closed-loop system. The main goal of the dynamical corrector is to suppress the high-frequency signals in the actuators input signals. The slight reaction of the actuators to relatively high frequencies occurring in the sea wave process is achieved. In contrast to [1], where the disturbance main frequency value should be known, in this paper, it is estimated online. The advanced results of frequency estimation provide global exponential convergence of the estimation error to zero. This property is guaranteed for all initial conditions, valid parameters of an algorithm and a measured signal. Such results are described in [8-10]. In this paper a parametrisation proposed in [11] is used to obtain the first-order regression model, where an unknown parameter depends on the external disturbance frequency. The standard gradient approach is used to estimate the regression model parameter value. The frequency estimation error converges to zero exponentially fast. The described algorithm does not require measuring or calculating derivatives of the input signal.

This paper is organised as follows. The problem is formulated in Section 1. In Section 2, the equations of DP vessel motion are presented, the special structure of control law is introduced, and the problem of separate filtering correction is posed. Section 3 presents the computational procedure to implement a filter tuning onboard. In Section 4 we describe the frequency estimation algorithm and prove an exponential convergence of the estimation error to zero. The efficacy of the proposed approach is demonstrated through a set of numerical simulations, which are described in Section 5.

Mathematical model and problem formulation

Consider the 3-DOF horizontal plane nonlinear model [12] of DP-control plant:

Mv(t) = —Dv(t) + τ(ΐ) + d(t),

V(t) =

У(і) = R(t) + ηω (t),

u(t) x(t)

v(f) = v(t) , R(t) = y(t)

r(f) xp(t)

where v(t) Є M3 is the generalised velocity vector defined in a vessel-fixed frame Oxvyvzv that includes linear velocities u(t), v(t) and angular velocity r(t); η(ΐ) Є M3 is the joint vector relative to an earth-fixed frame Oxyz that includes position parameters (x(t), y(t)) and the heading angle ψ(ί); τ(ΐ) Є M3 is a control action generated by the propulsion system; y(t) Є M3

is a measurable output signal; d(t) Є M3 is a disturbance, which describes slowly varying wave, current and wind loads; ηω(t) Є M3 is a measurement error; M Є M3X3, D Є M3X3 are positive definite matrices with constant elements, and M = MT; R(g) is

a orthogonal rotation matrix:

'cos-ψ —ίίηψ 0

ОД = R(ıp) =

sin^ cos^ 00

(2)

The objective is to design a nonlinear dynamic control law ofthe form

Kt) = ffr T УІ (3)

t(t) = g(z, УІ ()

where z(t) Є M! is a state space vector of the controller, ІЄЇ+. The following design requirements must be satisfied for the closed-loop system (1), (3):

1. The system must have the only one equilibrium point, such

that v(t) = 0, η(ΐ) = η*, (4)

2. where η* = [X* у* ψ*]Τ Є M3 is the desired constant position vector.

3. The equilibrium point must be globally asymptotically stable.

4. The controller (3) must provide an integral action with respect to the LF components of the bias vector d(t).

5. The controller (3) must provide a filtering action to the control signal τ(ΐ) for the system (1), (3) with respect to the high frequency components.

Setting the DP-controller with dynamical corrector

Let us construct the nonlinear asymptotic observer to get estimates v(t) and f\(t) of the state vectors v(t) and η(ΐ) of the model (1) correspondingly. The observer should provide a global asymptotic convergence of estimation errors v(t): = v(t) — v(t) and fj(t): = η(ΐ) — f\(t) to zero in the absence of external disturbances and interferences.

A nonlinear asymptotic observer was proposed in the paper [3]: Mv (t) = —Dv(t) + r(t) + RT(y)K1(y(t) — ή(t)), f\(t) = R(y)v(t) + K2(y(t) — η(ί))

where K1 Є M3X3, K2 Є M3X3 are constant matrices, which are chosen to provide global exponential stability (GES) of the zero equilibrium position of the following system in the absence of external disturbances d(t) = 0, ηω(ΐ) = 0:

Mİ)(f) = —Dv(d) — RT(y)Kıή(t), fj(t) = R(y)v(t) — K2fj(t).

A sufficient condition for globally exponentially convergence of the errors v(t) and fj(t) to zero is a diagonal structure and positive definiteness of the matrices K1 and K2 in accordance with [3].

By analogy with the general ideas are proposed in [3, 5, 13], we

construct the feedback control law τ(ΐ) in the following form:

TdpW = —Kdv(t) — RT(y)Kp(fj(t) — ^^,

xf(t) = aXf(t) + βη^Ι

Tf(t) = yxf(t)+ μη(ϊ),

τ(ΐ) = τάρ(ί)+ τf(t),

where rdp(t) is a part of control, which stabilised the desired equilibrium v(t) = 0, η(ΐ) = η* for the closed-loop system (1), (7); Tf (t) is a dynamical corrector output signal; Kp Є M3X3, Kd Є M3X3 , а Є MlXl , β Є MlX3 , γ Є M3Xl , μ Є M3X3 are constant matrices; the matrix a is Hurwitz; Xf(t) Є M! is the state space vector of the corrector.

Note that the dynamical corrector (7) can be rewritten in terms of

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the transfer function by applying the Laplace operator £{·} to

(7):

£{t/(t)} = F(s)L{p(t)}, (8)

P(s) = y(n!x!s - α)-1β + μ,

where Ιίχί is the identity matrix with I x I dimension, s EC1 is a complex variable.

In accordance with the paper [5], the positive definiteness of the symmetric matrices Kp and Kd guarantees that the equilibrium position v(t) = 0, p(t) ξξ η* of the closed loop system (1), (7) with t(t) = rdp(t) is global asymptotically stable (GAS) in the noise-free environment d(t) ξ 0 and ηω(ί) ξ 0.

At this point we have designed a GES observer and a GAS state feedback controller. In [1] it was proved that a separation principle holds for the overall system: the estimation, tracking error, and the correction dynamics can be decoupled yielding a cascaded system. The estimation, position error and the correction dynamics can be analysed separately. Therefore, if the corrector is asymptotically stable, then parts of the controller can be tuned independently.

Let us obtain a requirement for the transfer matrix F(s) of the dynamical corrector (7) that provides astatism property for the closed-loop system with respect to the position error vector fj*(t): = p(t) — η* for any external disturbance with constant or slowly varying components, additionally supposing that ηω(ί) ξ 0. Suppose that the error equations for a constant external disturbance d(t) ξ d0 E M3 Mij(t) = —Dv(F) — RT(y)Kifj(t) + d0, fj(t) = R(y)I>(t) — K2rj(t),

have an equilibrium point with the corresponding heading angle \p(t) ξ ψ0.

Proposition 1 If the transfer matrix F(s) satisfies the equality F(0) = KA = — (D + Kd)RT(f,*)K2 — RT (ψ*)(Κρ + Kf), (10)

and if the following condition holds

det [—Ш — £“4* 0, (И)

where ψ0 is a value of the heading angle in the equilibrium point, then the system (5), (7) is astatic with respect to the position error vector fj*(t) = p(t) — η* for any d0 E M3.

Proof Let us consider the equilibrium point equations from (9):

0 = —Dv(t) — RT (po)Kirj(t) + d0, ( )

0= R(ip0)v(t) — K2fj(t). ( )

If the condition (11) holds, then the linear nonuniform system (12) has a unique solution (νζ rjjff relative to an unknown vector (vT(t) fjT(t))T . Substituting the equilibrium point (Vo RoY to the controller equations (5), (7) yields 0 = —Dv(t) + r(t) + RT(ip0)K1fj0,

0 = R(ipo)Kt) + K2rj0, (13)

YY = —Kdv(t) — RT (тро)Кр(Ш) — R*) +

where in the transfer function of the dynamical corrector F(p) for the equilibrium position we assume p = d =0.

The matrix F(0) can be expressed explicitly from (13):

F(0) = —(D + Kd)RTfpo)K2 — RT(f0)(Kp + Kf). (14)

Substituting ψ0 = ψ* gives (10), which completes the proof.

Αω E M3 is the vector of magnitudes. For this case it is possible to define the intensity functional J(F) = \\Ατ(ω0, ηα, F)||, (15)

where Ατ E M3 is the vector of control actions magnitudes for the closed-loop system (1), (3) with the disturbance ηω(ί). This vector corresponds to the time moment of the DP-process with η (t) = ηα, when the heading angle has the value f>(t) = ψα.

The filter transfer matrix F(s) should satisfy the equality \\Ατ(ω0, ηα, F)\\ = 0. (16)

Rewriting the system (5) with feedback control (7) for the new variables v(t) and rj*(t) = f\(t) — η* gives v(t) = —M-1(D + Kd)v(t) — M-1RT(g)Kpfj*(t) +

+M-1tf(t) + M-1RT(p)K1(p(t) + ηω(ί)),

fj*(t) = R(p)v(t) + K2(fj(t) + ηω (t)j + tf(t), (17)

r(t) = —Kdv(t) — RT (g)Kprj*(t) + F(p)fj(t),

y(t) = η(ί) + ηω(ί).

The system (17) can be rewritten in matrix form

where

AM = \—M-1(D + Kd) —M-1RT(g)Kp] W IRÜ) 03x3 J,

В(П) = \-KM-1«TW1 MQ-1J ,

*-K2 °3X3İ6X6

C(R) = [—Kd —RT(g)Kp]3x6,

(18)

(19)

3x6

where 03x3 is the zero matrix with 3x3 dimension. Let us fix some value of the heading angle ψ = ψα corresponding state space vector ηα. Applying the operator to (18), we get the transfer function model

\Щ(ґ)+ ηω (t)}

ξ [P1(S,Va) P2(S,Vo)] {ц1%+

Pfr Ra) = C(Ra)(hx6S — Λ(ηα))-1Β(ηα) + D,

and the Laplace

(20)

where P(s, ηα) E M3x6 is a block matrix consisting of blocks P^s, ηα) E M3x3 and P2(s, ηα) E M3x3.

Proposition 3 If the block P2(s, ηα) of the matrix P(s, ηα) satisfies the condition

detP2(s, ηα) Φ 0, (21)

then the transfer matrix F* (ω0, ηα) such that condition (16) is satisfied.

Proof. From (8) and (20) we obtain

£{τ(ί)} = ^^,ηα) + P2^, Va)F(s))£{fj(t) + ηω(ί)}. (22)

Choosing F(s) as

P'^ Ra) = —P-1(jωo, Ra)Pı(jш0, ηαI (23)

we get the filter tuned to the frequency ω0 and the angle ψα under condition (21), which is the desired conclusion.

Remark 2 The simplest way to satisfy the requirement (10) is using a corrector with no dynamics, i.e. F(s) ξ Κδ.

The main purpose of the dynamical corrector F(s) is to support an economical regime of motion that provides a filtering effect for some central frequency ω0 of the wave spectrum ηω(ΐ) = Αω$ίηω0ί for the control signal driving a rudder actuator, where

Filter tuning procedure

In this section we construct the transfer matrix F(s) of the corrector, which satisfies the condition (23) and provides the stability and integral action with respect to disturbances, i.e.,

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Р(0) = КА, ҒЦшо) = Ғ\Шо, ηα). (24)

The equations (24) can be rewritten for vector components X{r2(t)} separately

Fi (0) = KAi, Fi(ja>o) = Ғ.(ш о, ηα), і = 13, (25)

where

Pl(s)' \Pai РІ (щ Па )

P2(S) , К a = Pa2 , F*(a>o, na ) = P2 (wo Па )

Fs(s) Раз- Р3 (wo Па )

Let us rewrite transfer function Ft(s) as Pi(s) = Yi(hx2S - αί)-1βι + μι, і = 1,3, (27)

where at Є M2x2 are Hurwitz matrices, βί Є M2x3, yt Є M1x2, μίΕΜ.1χ3 are constant matrices, ί = 1,3.

Taking into account (27) and (25), we obtain

-Yia-^i + Fi = Kao ____ (28)

Yi(jhx2Uo - αί)-1βί + μι = Ғ((Шо, ηα), і = 1,3.

Let us select any Hurwitz's matrices at and matrices yt so that the condition of full observability for (at, yt) is satisfied at ί = 1,3. The system of matrix equations (28) are solving by equating real and imaginary parts:

-Υί»ΐ:1βί + Fi = Pm ,

YiRe{(jhx2^o - Яі)-1}βί + Fi = Re{FÎ(üio, ηα)}, ί = 1,3. (29) YiIm{(Jhx2Uo - αι)-1}βι = 1ш{Ғ;*(шо, ηα)}.

Subtracting the second expression from the first (29), we get YiiReWhx2Vo - аі)-1} + α-1]βί = Re^·^ ηα)} - ΚΑν і = 1,3 YiIm{(j^2x2^o - яд-1Ж = Iш{Fi*(ωо, ηα)}.

(30) _

The matrices βt, ί = 1,3 are found from (30):

0 (,, „ ■) = \Yi [Rei(/12x2"o-“i )-1}+ “Г1]]”1 ГК-е{^г*С^о, ηα)}-Faî1

Pi ( 0 Va) \Yt\m[(jl2x2Mo-ai)-1} ] \1ш{Ғ,'(Шо, ηα)} ] ’

1 = 1,3. (31)

The matrices μü ί = 1,3 are expressed from (29):

Fi(Uo, По) = Раі + Υία-1βί(ωο, Па), ί = 1,3. (32)

Finally, all the matrices a, β, γ and μ are obtained, which allows us to construct the optimal filtering corrector F(s),

adjusted to the frequency ω0, in the following form:

a1 ®2x2 ®2x2 βΐ(Mo, Па)

a = ®2x2 a2 ®2x2 , β(Щ, Па) = β2(Mo, Па)

-®2x2 ®2x2 аз 6x6 Рз^ Па).

Yl ®2x2 ®2x2 Fl^ По)

Y = ®2x2 Y2 ®2x2 , Ғ^ Па) = F2(^o, По)

■®2x2 ®2x2 Y3 3x6 _F3(Vo, Па).

External disturbance frequency estimation

In this section we find the frequency estimate (3o(t) for external harmonic disturbance ηω (t) = Αω$ίηω0ί that provides

exponential convergence of the error S>o(t): = ω0 - (3o(t) to zero and tune-up the dynamical corrector (7).

Let us consider the difference between the output signal y(t) =

η(ί) + ηω(ΐ) of the model (1) and the estimate n(t), obtained

from a non-linear observer (5) with DP controller (7):

ў(Ғ) ■.= y(t) -n(f)= fj(t)+ ηω(t). (34)

Due to globally exponentially convergence of the error fj(t) to

zero, the signal y(t) has the following form:

y(t) = Αω sirniiot + e(t), (35)

where e(t) is the exponentially decaying function.

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Assumption 1. The lower and upper bounds on the signal frequency ω0 are known and equal to ω and ω, where 0 < ω < ω0 < ω. (36)

The assumption is not particularly restrictive. It is required that value of the frequency be distinct from zero and less than infinity. Nevertheless, bounds can be chosen to contain all possible values in each specific case.

Neglecting the exponentially damped term, let us consider a signal Y(t) = Αωsin<D0t and two auxiliary transport delay blocks with the following outputs

where h ЄШ+ is the chosen delay constant.

The signals (37) can be rewritten explicitly as Y1(t) = Aωc1sinω0t - Aωs1cosω0t,

Y2(t) = K^sinMot - A^cosuot, where

c1 = cosωh, c2 = cos2ωh = 2c^ - 1, s1 = sinωh, s2 = sin2ωh = 2c1s1.

Subtracting (38) from Y(t) multiplied by c1 we obtain

c1Y(t) - Y1(t) = s1Aωcosω0t. (40)

Similarly, subtracting Y2(t) from Y(t) multiplied by c2 = 2c1 - 1 gives

(2c1 - 1)Y(t) - Y2(t) = 2ciSiAacosMot. (41)

Subtracting (41) from (40), multiplying by 2c1, we get

Y(t) + Y2(t)= 2ciYi(t). (42)

Equation (42) describes the linear regression model

ip(t) = c^tf (43)

where f>(t) = Y(t) + Y2(t) is the regressand, is the unknown parameter, and φ(ΐ) = 2Y1(t) is the regressor. Parameter c1 can be estimated from equation (43) using standard gradient descent method [14].

Proposition 3 The estimation algorithm

Ci(t) = Κφ(β)(ψ(β) - άι(ί)φ(ί)), (44)

where К Є R+ is the chosen constant, provides exponential convergence of the estimation error to zero \cı-cı(t)\ <bie-ait, (45)

where b1 and a1 are positive constants.

Proof. By [14], if the function φ(ΐ) is bounded and persistently exciting (PE), i.e. there exist positive constants T and γ such that

§*+T φ2(τ)άτ>γ, Vt > 0, (46)

then algorithm (44) provides exponential convergence of the estimation error to zero.

Signal φ(ΐ) is the sum of sine and cosine functions multiplied by constant coefficients, so it is bounded. Let us show that signal φ(ΐ) is also PE. Consider the following integral f^+T φ2(Γ)άτ = 4 f‘+T Yf(r)dr = 4A2ω f^+T sin2(wor - woh)dr =

= 2Α?ω f^+T 1 - cos(2ω0r - 2o>oh)dr = (47)

= 2ЛІC dr - ÎC:^2r2M0h cos(f)dr- =

= 2Α2ωΤ + — sin(2<i)ot - 2ωί2Κ) - — sin(2wo(t + T) - 2ωί2Κ).

If T = n/uo and ω >ω, then

f^+T φ2(τ)άτ > 2π3.ω > 0, Vt > 0. (48)

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Equation (48) shows that PE condition (46) is satisfied with γ = 2π_^ω, τ = —, and the proof is complete.

ω ω0

We can obtain the frequency estimate ω0(ί) from tq(t): ω (t) = 1 arccos(c1 (t)). (49)

Since the domain of the function (49) is the subset of M, it is necessary to put some restrictions on c2(t). Under Assumption 1, the possible values of c1 satisfy the inequality cosωΗ < c1 < cosωΗ. (50)

To provide this property to c1(t), we can use gradient algorithm with projection [14]

1ç(t), if c1(t) > cosωΗ and c1(t) < cosωΗ, or if c1(t) = coscnh and ç(t) > 0, or if c1(t) = cosıüh and ç(t) < 0, (51)

0, otherwise,

ç(t) = Κφ(ί)(ψ(ί) - C1 (ί)φ(ί)).

which retains all properties that are established in the absence of projection.

From Proposition 3 follows that c1 (t) converges exponentially to c1. However, for ω0(ί) this is not obvious.

Proposition 4 If (t) converges to c1 exponentially fast, then S>0(t) converges exponentially to zero and objective |n>0-Öo(t)l <b2e-a2t, Ü2, Ь2ЄШ+ (52)

is fulfilled.

Proof. The arccosine function on [cosıüh, coscnh] is Lipschitz [15] |arccos(x1) — arccos(x2)| < L|x1 — x2l, (53)

where Lipschitz constant L can be calculated as follows

Combining (49), (45) and (53) gives

|So(t)| <L|C1-£1(t)| <b2e-a2t, (55)

where b2 = Lb1, a2 = a1, which is the desired conclusion.

For linear approximation of the system (1), (3), the global asymptotic convergence of the desired equilibrium position η* is preserved using the obtained frequency estimate ω0(ί).

Simulation results

In this section, we present the simulation results that illustrate the efficacy of the proposed DP control low wits the dynamical corrector. All simulations have been performed in MATLAB Simulink.

Consider the DP-control system for the vessel 'Northern Clipper' (the length is L = 76.2 m and mass is m = 4.59 · 106 kg) with the model (1), taken from [3]. The constant matrices in the equation (1) are the following:

5.31 · 106 0 0

M = 0 8.28 · 106 0 ,

-0 0 3.75 · 1095.02 · 104 0 0

D = 0 2.72 · 105 -4.39 · 106 .

0 -4.39 · 106 4.19 · 108 .

The matrices K1 and K2 of the observer (5), and the matrices Kd and Kp of the controller (7) is chosen in accordance with [3]

0.1 0 0 1.1 0 0

к1 = 0 0.1 0 , K2 = 0 1.1 0

О

00 0.01 О 1.1

0.0207 0 0

Kd = 0 0.0155 0.0439 · 108,

0 0.0439 4.05

«г =

0.0213 0 0

0 0.00990 0 · 107.

0 0 4.49

Hurwitz matrices au for ί = 1,3 with the

0.150, s12 = -0.152, s 21 = - °.12°, S22 =

-0.128, s31 = -0.178, s32 = -0.180 and the matrices γϋ і = 1,3 for the parts (27) of dynamical corrector (7) so that the condition of full observability for (at, yt), i = 1,3 is satisfied

a1 L—0.0228 —0.302 J' a2 1-0.0154 -0.0248.

аз = [-0.0320 -0.358],

Yi = [0 1], і = 13.

Desired position and control mode switching

The desired position vector is equal to

(56)

To illustrate that the controller (7) provides the desired features to the closed-loop system, we use a wave disturbance ηω(ί) = [^ω1(ί0 hrn2(t) ηω3(ΐ)]Τ of the ship with harmonic

components

Ίωί(ί) = Αωί sinw0t, і = 1,3, (57)

where ω0 = 0.455, АШі = 3 · 106 , АШг = 3 · 106 , АШз =20 ·

106. Filtering action to the control is shown by comparison with the astatic corrector of the form

t2(f) = + ηω(ϊ)], (58)

which works until 200th second. The controller (7), (58) provides an integral feature, but loses the filtering one. At time t = 200 s, which is marked in figures 1-3 by the black dashed line, we turn on the dynamical corrector (7) instead of (58) and observe a desired effect of filtering. The signal t(t) with components Tj, i = 1,3 are shown in the Figure 1. The control signals is essentially different for control mode (58) and (7).

In Figure 2 the frequency estimate is depicted. The estimate (30(t) is exponentially converges to the true value ω0.

Figure 3 shows the results of the vessel motion simulation for the considered closed-loop DP system.

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ωθί ώο(ί)

\ τ )

|_

к,

1 Î .....

О 200 400 600 800 1000

t [s]

F i g. 2. The frequency ω0 of external disturbance ηω(ί) and estimate ώ0(ί)

-to

(a) X coordinate

(b) Y coordinate

(c) The heading angle F i g. 3. State vector components

External disturbance frequency changing

Let us consider the simulation results for the case with external disturbance frequency changing and presence of additive noise:

„ ,^_{Αωίsinwot + Si (t), if t < 300 s, —

4ωί it) sinWlt + Si (t), if t>300 s, 1,3 (59)

where ω0 = 0.455, ω1 = 0.3, Αω1 = 3 ■ 106, Αω2 = 5 · 106, Αω3 = 20 ■ 106, the additive noise S (t) =

[51(t) S2(t) S3(t)]T, which components are simulated as a uniformly distributed process ranging within [-0.2 ■ Аыі,0.2 ■ Αω1 ], ί = 1,3. The components of the signal ^ω(ί) are shown in Figure 4. Frequency change time t = 300s is marked on Figures 4-7 by the black dashed line.

The output signal y(t) of the model (1), the estimate rç(t), obtained from a non-linear observer (5) which are closed by the DP controller (7) and its difference у(t) (35) are depicted in Figure 5.

0 100 200 300 400 500 600

t [s]

In Figure 6 the frequency estimate is depicted. Figure 7 shows the control actions t1(t), r2(t), and t3(t) for the mentioned process. In this case, we obtain almost the same curves as in Figure 3, which presents the positioning processes.

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ISSN 2411-1473 sitito.cs.msu.ru

F i g. 6. The frequency ω0. and its estimate o>0(t) in case of change the exter-

Conclusions

[12]

The problem of dynamic ship positioning under the action of sea wave disturbances for nonlinear vessel model was considered. The approach proposes the specialised structure of nonlinear DP-control law, and filtering corrector synthesis method, which is oriented [13] to onboard implementation.

To estimate the external disturbance main frequency, we obtain the first-order regression model. The standard gradient approach is used [14] to estimate the regression model parameter value. It is shown that the frequency estimation error converges to zero exponentially fast. The [15] set of simulations illustrates the efficacy of the proposed approach.

References

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Veremey E.I., Korchanov V.M. Multi-objective stabilization of a range of dynamic systems. Automation and Remote Control. 1988; 49(9):1210-1219. (In Eng.)

Veremei E.I. Synthesis of multi-objective control laws for ship motion. Gyroscopy and Navigation. 2010; 1(2):119-125. (In Eng.) DOI: https://doi.org/10.1134/S2075108710020069 Aranovskiy S., Bobtsov A., Kremlev A., Nikolaev N., Slita O. Identification of Frequency of Biased Harmonic Signal. European Journal of Control. 2010; 16(2):129-139. (In Eng.) DOI: https://doi.org/10.3166/ejc.16.129-139 Vedyakov A.A., Vediakova A.O., Bobtsov A.A., Pyrkin A.A. Relaxation for online frequency estimator of bias-affected damped sinusoidal signals based on Dynamic Regressor Extension and Mixing. International Journal of Adaptive Control and Signal Processing. 2019; 33(12):1857-1867. (In Eng.) DOI: https://doi.org/10.1002/acs.3034 Pin G., Chen B., Parisini T. Robust finite-time estimation of biased sinusoidal signals: A volterra operators approach. Automatica. 2017; 77:120-132. (In Eng.) DOI: https://doi. org/10.1016/j.automatica.2016.10.031 Gromov V.S., Vedyakov A.A., Vediakova A.O., Bobtsov A.A., Pyrkin A.A. First-order frequency estimator for a pure sinusoidal signal. In: 2017 25th Mediterranean Conference on Control and Automation (MED), Valletta; 2017. p. 7-11. (In Eng.) DOI: https://doi.org/10.1109/MED.2017.7984087 Hassani V., S0rensen A.J., Pascoal A.M., Aguiar A.P. Multiple model adaptive wave filtering for dynamic positioning of marine vessels. In: 2012 American Control Conference (ACC), Montreal, QC; 2012. p. 6222-6228. (In Eng.) DOI: https://doi.org/10.1109/ACC.2012.6315094 Fossen T.I. Handbook of Marine Craft Hydrodynamics and Motion Control. John Wiley & Sons, Ltd; 2011. (In Eng.) DOI: https://doi.org/10.1002/9781119994138 Ioannou P.A., Sun J. Robust Adaptive Control. Courier Corporation, 2012. (In Eng.)

Vedyakov A.A., Vediakova A.O., Bobtsov A.A., Pyrkin A.A., Kakanov M.A. Frequency estimation of a sinusoidal signal with time-varying amplitude and phase. IFAC-PapersOn-Line. 2018; 51(32):663-668. (In Eng.) DOI: https://doi. org/10.1016/j.ifacol.2018.11.501

[1] Veremey E.I. Separate filtering correction of observer-based marine positioning control laws. International Journal of Control. 2017; 90(8):1561-1575. (In Eng.) DOI: https://doi. org/10.1080/00207179.2016.1214749

[2] S0rensen A.J. Lecture notes on marine control systems. Technical Report UK-12-76. Trondheim: Norwegian University of Science and Technology; 2012. (In Eng.)

[3] Fossen T.I., Strand J.P. Passive nonlinear observer design for ships using Lyapunov methods: full-scale experiments with a supply vessel. Automatica. 1999; 35(1):3-16. (In Eng.) DOI: https://doi.org/10.1016/S0005-1098(98)00121-6

[4] Koschorrek P., Siebert C., Haghani A., Jeinsch T. Dynamic Positioning with Active Roll Reduction using Voith Schneider Propeller. IFAC-PapersOnLine. 2015; 48(16):178-183. (In

Submitted 14.12.2019; revised 10.03.2020; published online 25.05.2020.

Поступила 14.12.2019; принята к публикации 10.03.2020; опубликована онлайн 25.05.2020.

About the author:

Anastasiya O. Vedyakova, Assistant of the Department of Computer Applications and Systems, Faculty of Applied Mathematics and Control Processes, Saint-Petersburg State University (7/9 Universi-tetskaya Emb., St. Petersburg 199034, Russia), ORCID: http://orcid. org/0000-0003-0865-3578, vedyakova@gmail.com

The author has read and approved the final manuscript.

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Ведякова Анастасия Олеговна, ассистент кафедры компьютерных технологий и систем, факультет прикладной математики - процессов управления, Санкт-Петербургский государственный университет (199034, Россия, г. Санкт-Петербург, Университетская наб., д. 7/9), ORCID: http://orcid.org/0000-0003-0865-3578, vedyakova@gmail.com

Автор прочитал и одобрил окончательный вариант рукописи.

Современные информационные технологии и ИТ-образование

Том 16, № 1. 2020 ISSN 2411-1473 sitito.cs.msu.ru