Dynamical Correction of Control Laws for Marine Ships’ Accurate Steering

Evgeny I. Veremey

Dynamical Correction of Control Laws for Marine Ships’ Accurate Steering

Evgeny I. Veremey

Computer Applications and Systems Department, Saint Petersburg University, 198504, Russia

The objective of this work is the analytical synthesis problem for marine vehicles autopilots design. Despite numerous known methods for a solution, the mentioned problem is very complicated due to the presence of an extensive population of certain dynamical conditions, requirements and restrictions, which must be satisfied by the appropriate choice of a steering control law. The aim of this paper is to simplify the procedure of the synthesis, providing accurate steering with desirable dynamics of the control system. The approach proposed here is based on the usage of a special unified multipurpose control law structure that allows decoupling a synthesis into simpler particular optimization problems. In particular, this structure includes a dynamical corrector to support the desirable features for the vehicle’s motion under the action of sea wave disturbances. As a result, a specialized new method for the corrector design is proposed to provide an accurate steering or a trade-off between accurate steering and economical steering of the ship. This method guaranties a certain flexibility of the control law with respect to an actual environment of the sailing; its corresponding turning can be realized in real time onboard.

marine ships; control law; dynamical corrector; accurate steering; autopilot; sea wave

1 Introduction1

Due to the intensive development of the world economy, there constantly appear many problems connected with the safety of a ship’s motion in a range of increasing marine traffic. One of the commonly used approaches to ensuring safety is the application of automatic guidance and control for the ship’s motion (Fossen, 1994; Perez, 2005), including using such popular systems as marine autopilots.

To guarantee safe sailing, the highest quality of automatic ship course keeping is in urgent demand. However, as is common knowledge, a ship’s motion is extensively influenced by different environmental disturbances such as sea waves, winds, sea currents, change of depth under keel, etc. (Fossen, 1994). Therefore, the accurate ship steering under actions of varying external conditions is a complicated control problem.

Let us note that various issues associated with the design of autopilots for marine surface vessels have already been extensively researched (Fossen, 1994; Perez, 2005; Dove and Wright, 1991). However, essentially less attention is paid to the feasibility of the autopilot’s turning during its operation depending on the actual conditions of sailing. For example, one of the ideas is to provide certain flexibility of the autopilot control law with respect to external disturbances rejection as discussed in Hammound and Mohamed (2012), where the multi-controller approach is proposed. This approach is based on the usage of local controllers set which are oriented to the concrete regimes of a ship’s motion. Besides this, an autopilot must include both switching and supervisory systems to change a controller subject to an actual environment of the sailing.

Nevertheless, this type of structure of autopilot has both advantages and disadvantages. In particular, full switching of the controller can lead to the undesirable dynamical splashes, which in general decreases the reliability of the system.

A similar alternative approach is based on the theory of multi-purposes control laws synthesis, which was initially presented in Veremei and Korchanov (1989) and transformed to the modern level in Veremey (2010). In recent years, some new analytical and numerical methods of synthesis have been developed, which are based on the special unified structure of the control laws for marine autopilots.

The mentioned special structure includes some basic parts and several various separate items to be adjusted for an actual environment of the sailing. The basic part is invariant with respect to the environment, but varying elements can be switched on or off as needed to provide the best dynamical behavior of the closed-loop system. As it was shown in Veremei and Korchanov (1989) and Veremey (2010), the central item of the structure is the so called dynamical corrector aimed at the counteraction of sea wave disturbances.

The mentioned dynamical corrector can be designed on the basis of the optimization approach using the specific methods of2H or H∞synthesis, presented in Bokova and Veremei (1996), Veremey (2011), Doyle at al. (1992), Veremey (2012). A goal of this design is a construction ofthe LTI varying item in a framework of the special feedback structure such that the correspondent closed-loop system has optimal features with respect to its behavior for the vehicle motion under action of significant sea wave disturbances.

This paper presents a new mathematical and computational method for a dynamical corrector synthesis based on the2H-optimization theory. Central attention is paid to the situation, when a dynamical corrector provides an accurate regime of the yaw-keeping motion for a ship. An alternative economical regime, when a corrector plays a role of a notch filter, has been considered in Veremey (2012) for linear autopilots and in Veremey (2013) for nonlinear dynamical positioning systems.

The paper is organized as follows. In section 2 the problem of an accurate dynamical correction is posed on the mathematical level. Here a model of the ship and the functionals to be minimized are presented. Section 3 is devoted to specific features of the closed-loop connection with multi-purpose feedback, which are used as a bottom for the synthesis. Section 4 can be treated as the main part of the article presenting the optimal solution to the problem posed above. In section 5, the proposed approach is illustrated by the practical example of the autopilot synthesis for an accurate steering of the transport ship with the displacement of about 4 000 t. Finally, section 6 concludes this paper by discussing the overall results of the investigation.

2 The problem of accurate correction

Standard problems of autopilot design are usually considered with respect to the linear state space model of a ship’s motion. To obtain this model, one can make linearization of the initial nonlinear ship′s equations in the neighborhood of its equilibrium position, which corresponds to the given regime of motion with a constant speed. Let us suppose that the mentioned linearization is done and its result is accepted as the following LTI-model of a ship as a controlled plant:

Here x∈Enis the state space vector, scalar δ∈E1denotes the rudder deflection, d∈E1presents the external disturbance, y∈E1is the yaw angle treated as a measured variable, and e∈E2is the controlled vector, λ≥0 is the given real number. All the matricesA,b,h and c with constant components have correspondent dimensions. Suppose that the system (1) is controllable by δ and observable byy.

In addition to the equations (1) let us also introduce the following linear model of an actuator: whereis the control signal.

Let us accept the following structure of the control law (controller) to stabilize marine ship motion under sea wave disturbance, as it was justified in Veremei and Korchanov (1989) and Veremey (2010):

It is easy to see that the introduced structure consists of three elements: the first equation presents an asymptotic observer, the second one describes the so called "dynamical corrector" and the third equation composes a control signal for the rudder’s actuator. Here z∈Enis the state space vector of the asymptotic observer, g is the given column matrix, such that the matrix A?gc is Hurwitz.

One can easily observe that the Eq. (3) has the equivalent form:

where we introduced the additional notations=μ(A?gc ), k0=μb and=μg+ν. Let us suppose that the coefficients μ and ν in Eq. (3), that is the same,k～, k0and ν～ in Eq. (4), are selected so that the controller u=kx+k0δ stabilizes the plant (1), (2), where k=k～+ν～c=μA+νc . So we have an asymptotic stability of the state-driving closed-loop system, i.e. all the eigenvalues of the matrixare located in the open left half plane.

Let us introduce the following performance indices

to reflect our treatment of the control process quality. Here we suppose that the functional (5) are given on the closed-loop system (1) ÷ (3) movements, which are determined by the sea-wave disturbance action. It is easy to see that the functionalyI presents the accuracy of the stabilization process whereas the functional Iδdetermines the intensity of a control action.

So far as all the controller’s (3) elements are given, excluding a corrector, the informal problem of the accurate control consists of the F～ choice to minimize the functionalyI providing limitation of the control intensity, stability of the closed-loop connection, and integral action of the controller (3) with respect to the output y.

To formalize the problem posed above, let us consider theclosed-loop system (1) ÷ (3) with the block representation shown in Fig. 1.

Fig. 1 Block-scheme of the correction

In accordance with (1) ÷ (3), the local controlled plant in the presented block-scheme is described by the following equations:

The mentioned plant is closed by the corrector, which plays the role of a local controller

Taking into account the control intensity limitation, let us consider a linear convolution of the functional (5)

with the weight multiplier2λ, which is supposed to be chosen in conformity with the limitation.

Therefore, an accurate correction of the control law (3) can be treated as the following optimization problem:

where the admissible set*Ω is a population of the proper rational fractions ()Fs～, providing stability of the system (6), (7) and integral action of the controller (3) with respect to input d and output y.

To complete the correction problem statement we need to specify the features of the external disturbances ()dt, presenting the sea waves actions on the ship. Usually, such disturbances can be treated as ergodic stationary Gaussian processes. In a general case of no regular waves these processes are fully described by the given rational power spectral densities.

However, if sea waves have a regular nature, we can assume that the approximate representation of the actual disturbance is a harmonic oscillation

with the given frequency0ω and magnitudeA. Here we will not go beyond this assumption for the sake of simplicity and the subject of a further discussion is a solution to the problem (9) for the closed-loop system with the disturbance (10).

The statement that the more complicated case of no regular waves is based on the issue mentioned previously can be considered in the analog manner. This statement also relates to the situation when we have no description of d(t). For this case we need to replace functional I for the problem (9) by the norm of the transfer matrix Hde(s) from d to e for the system (6), (7) in the sense of the spaces H2or H∞.

3 Dynamics of the closed-loop connection

Most importantly, let us discuss some dynamical features of the closed-loop system (1) ÷ (3). To begin with, consider its representation in accordance with Fig. 2, which is somewhat different from the system (6), (7).

Fig. 2 Block-scheme of the closed-loop system

Here one can see a feedback connection with the controlled plant (1) and the "global controller"

Its basic part has the following state-space model:

which can be transformed to tf-form:

Taking into account equation (7) of the corrector treated as a local controller for the plant (13), one can easily obtain a transfer function of the global controller as follows:

The closed-loop systems (1), (12) and (7) have specific dynamical properties, which simplify the solution of the problem (9).

Lemma 1: If a transfer function of the corrector satisfiesthe identity

where ()Fs is any strictly proper rational fraction with a Hurwitz denominator, then the closed-loop systems (1), (12) and (7) are stable and possess the astatic property with respect to input d and output y.

Proof: First we present an equation of the corrector (7) in the state-space form:

where p∈En1is the corrector state vector, α,β,γ,ε are the matrices satisfying the identity γ(En1s?α)?1β+ε≡F～(s), moreover the matrix α is Hurwitz. In this case we have the following state equations of the closed-loop system:

The characteristic polynomial of this system is

Because all of the determinants here are Hurwitz polynomials, Δ(s) also is.

As for the integral action of the controller (3), one can easily see that in conformity with Eqs. (2) and (3) we have δ˙=μz˙+νy +ξ, hence if the closed-loop system has the equilibrium position for the disturbance d with constant components, on the base of (15) we obtainy=0, i.e. this system is astatic.

Theorem 1: If the following conditions hold

then there exists the transfer matrix F～∈Ω*of the corrector (7) such that

where r is any given complex number.

Proof: Let us substitute (14) with (20) obtaining the following equation with respect to the unknown complex value0(j)Fω～:

One can easily see that since we have (19), the Eq. (21) has the unique solution

So under conditions (19), for any dynamical corrector satisfying (20), the equality (22) holds as well as backwards.

Now let us show that there exists a function F～∈Ω*such that the equality (22) holds. To this end, let us take any Hurwitz polynomial Φ(s) with degΦ≥2 and construct the transfer function of a form F～(s)=s(f1s+f0)Φ(s), where the evidence belongs to Ω*for any real f1andf0. Then, denoting the right part of Eq. (22) as the complex numberf, obtains jω0(f1jω0+f0)Φ(jω0)=f that allows finding the unknown coefficients uniquely:

Thus, the function F～(s)=s(f1s+f0)Φ(s)∈Ω*satisfies the conditions (22) and (20).

4 Mean-square synthesis based approach

Let us present the equations of the closed-loop system in accordance with Fig. 2 for a global controller with any transfer function ()Ws as follows:

using notations:

If we consider the functional

depending on the choice of the global controller, we can pose a traditional mean-square synthesis problem (Bokova and Veremei, 1996) as follows:

for the closed-loop system (24) on the set Ω of the stabilizing controllers (11). It is evident that such a problem has the same nature as the problem (9) but is more general then this one. Really, for any F～∈Ω*the global controller (12), (7) belongs to the set Ω, but not vice versa.

Nevertheless, solutions of the problems (9) and (26) with the harmonic disturbance (10) are connected by the following statement.

Theorem 2: If the conditions of theorem 1 hold, then there exists the transfer matrix F～∈Ω*of the corrector such that the global controller (12), (7) is the solution of the problem (26).

Proof: First, it has been stated that mean-square problem (26) has a lot of solutions. This statement was discussed by Bokova and Veremei (1996) in detail and the main point is that all these solutions satisfy the following equation

At that time, according to the theorem 1 we could find the transfer matrix F～∈Ω*of the corrector (7) such that the transfer function W of the global controller (11) or (12) satisfies the equation W(jω0,F～)=r*. So, such a controller is one of the problem (26) solutions. Theorem 2 is proven.

Corollary: For any Hurwitz polynomial Φ(s) with degΦ≥2 the transfer function of the corrector

is the solution of the optimization problem (9), if its coefficients are determined by the formulas (23) with the complex multiplier

where the value*r is determined by Eq. (27).

Proof: This statement directly follows from the proofs of the Theorems 1 and 2.

5 Example of the corrector synthesis

It is possible to form a numerical algorithm of the corrector design on the basis of the approach proposed above. Let us illustrate its practical implementation by the example of the course-keeping autopilot synthesis for the transport ship with the displacement of about 4 000 t.

Assume that we have given the mathematical model of the ship motion with a constant speed 10V= m/s under sea wave action:

Here β is a drift angle, ω is an angular velocity, φ is a yaw angle, and δ is a rudder deflection. Let us accept the following values of the coefficients: a11=?0.093 6, a12= 0.634, a21= 0.048 0, a22=?0.717, b1= 0.019 0, b2= 0.016 0, h1= 0.008 30, h2= 0.477.

Let the basic control law be u=kx+k0δ, where,

corresponds to the eigenvalues s1,2=?0.240±0.51j, s3=?0.337, s4=?0.133 of the matrix Ac. Simple computations allow us to obtain the following coefficients μ and ν for the controller (3) and also the correspondent coefficients for the controller (4):

Besides this, let us construct the asymptotic observer in the range of the structure (4), providing its stability with the eigenvaluess1=s2=s3?0.266, that give the vector g =[g1g2g3]T,where g1= 0.040 7, g2= 0.186, g3=?0.011 6.

So, all the basic elements of the controller (4) are computed, i.e. the local controlled plant (6) is fully determined.

Next, let us solve the optimization problem (9) by finding the optimal transfer function

of the corrector.

To this end, firstly accept the value 37.3λ= of the weight multiplier for the mean-square functionals (8) and (25). This allows us to compute the complex value (27)

for the mean frequency ω0=0.31/s of the waves spectrum,

Let us take into account that for the local plant (13) we have:

Selecting the Hurwitz polynomial

as a denominator for the transfer function (30), in accordance with (28) and (31) obtains

Then on the basis of (23) we have f1=?22.4,f0=1.88, so we obtained the transfer function

of the optimal corrector.

In accordance with the results of the computations, let us finally present the controller (3) in the normal form as follows:

where α21=?0.206, α22=?0.908, m2=f0?0.908f1, θ=1, if the corrector is switched on, θ=0 otherwise.

Fig. 3 represents a magnitude partAdf(ω)=Hdf(jω) of the Nyquist diagram with respect to the transfer function Hdf(s) of the closed-loop systems (29), (33). Observe that the curve Adfhas a pronounced minimum for the turning frequency ω0due to the action of the optimal corrector (32).

To illustrate the dynamics of the closed-loop system (29), (33), accept the following representation for sea wave actions with the 5-th number intensity on the Beaufort scale:

Let us consider the stabilization process presented by the graph of the function y(t) in Fig. 4 for this system. Before the 750-th second the controller (33) works with no corrector (θ=0). Then we switch-on this one to provide waves compensation. Comparison of both parts of the process illustrates the significant effectiveness of the proposed control law correction.

Fig. 3 Frequency response of the closed-loop system

Fig. 4 Yaw angle ()yt for the closed-loop system

6 Conclusions

The main goal of this work is to propose a constructive method for marine autopilot synthesis to provide accurate ship steering. In contrast to the well-known approaches, we achieved this goal using a dynamical correction of the basic control law with the special multi-purposes structure. In our opinion, this method provides a certain flexibility of the control law with respect to an actual environment of the sailing. The mentioned dynamical features of the control laws (3), (4) with a special structure grant such flexibility. Here we can select the considered corrective term subject to a current regime of the motion in the following variants:

(1) If the ship moves under the condition of quiet water, we can fully switch off the dynamical corrector in control law (4) ((p)≡0), i.e. there is no correction, and controller (4) works in the spared regime.

(2) If we have motion under significant bias disturbances, but no sea waves, it is quite suitable to accept the control law (3) with no corrector, providing an integral action of the controller does not overload the system by additional useless dynamics.

(3) If sea waves also influence the vessel motion, we can switch on the corrector for the controller (3), using them in an accurate steering regime, and keeping an integral property to react against bias.

(4) Finally, to use an economical regime of motion, we can change a transfer matrix of the corrector for the notch filtering action in accordance with the recommendations of Veremey (2012 and 2013), also keeping integral action of the controller.

We believe that the approach proposed here can be useful not only for the surface ships, but also for various kinds of AUVs and flying offshore structure stabilization systems designs. The results of the investigations presented in this paper can be developed to take into account transport delays and robust features of the control law.

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Author’s biography

Evgeny I. Veremey is a Professor, and Head of the Computer Applications and Systems Department at Saint Petersburg University. He is the author or coauthor of more than 130 scientific publications (mostly in Russian). Research interests include control theory, optimization approaches, computer modeling and theirs applications to marine vessels, robotics, and tokomaks plasma control. Professor Veremey is an honored worker of the Higher Educational System of Russia. Since 2004, he has been a member of the International Public Association "Academy of Navigation and Motion Control".

1671-9433(2014)02-0127-07

date: 2014-03-01.

Accepted date: 2014-03-31.

Partially supported by Russian Foundation for Basic Research (Research project No. 14-07-00083a).

＊Corresponding author Email: e_veremey@mail.ru

? Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2014