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Automatica 40 (2004) 1845 – 1863 www.elsevier.com/locate/automatica

Nonlinear optimal tracking control with application to super-tankers for autopilot design夡 Tayfun Çimen∗ , Stephen P. Banks Department of Automatic Control and Systems Engineering, The University of Shefﬁeld, Mappin Street, Shefﬁeld S1 3JD, UK Received 10 August 2002; received in revised form 2 March 2004; accepted 4 May 2004 Available online 12 August 2004

Abstract A new method is introduced to design optimal tracking controllers for a general class of nonlinear systems. A recently developed recursive approximation theory is applied to solve the nonlinear optimal tracking control problem explicitly by classical means. This reduces the nonlinear problem to a sequence of linear-quadratic and time-varying approximating problems which, under very mild conditions, globally converge in the limit to the nonlinear systems considered. The converged control input from the approximating sequence is then applied to the nonlinear system. The method is used to design an autopilot for the ESSO 190,000-dwt oil tanker. This multi-input–multi-output nonlinear super-tanker model is well established in the literature and represents a challenging problem for control design, where the design requirement is to follow a commanded maneuver at a desired speed. The performance index is selected so as to minimize: (a) the tracking error for a desired course heading, and (b) the rudder deﬂection angle to ensure that actuators operate within their operating limits. This will present a trade-off between accurate tracking and reduced actuator usage (fuel consumption) as they are both mutually dependent on each other. Simulations of the nonlinear super-tanker control model are conducted to illustrate the effectiveness of the nonlinear tracking controller. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Nonlinear systems; Optimal control; Tracking; Stabilization/Regulation; Global convergence; Continuous-time systems; Time-varying systems; Ship control; Autopilots; Maneuvering; Marine systems

1. Introduction Many control systems of practical importance are inherently nonlinear. However, nonlinearities in the model are often overlooked in the design of control systems. Local linearization techniques, which linearize a system around some equilibrium or operating point through small perturbation state approximations, are common practice and are often used in the design of suitable controllers that yield the 夡 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor T. Glad under the direction of Editor Hassan Khalil. ∗ Corresponding author. Tel.: +44-114-2225250; fax: +44-1142225661. E-mail addresses: tayfun.c@shefﬁeld.ac.uk (T. Çimen), s.banks@shefﬁeld.ac.uk (S.P. Banks).

0005-1098/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2004.05.015

desired motion for the linearized model. Indeed, to apply classical linear and other well-known nonlinear control laws (for instance, see Slotine & Li, 1991) to practical models, a linear representation of the nonlinear dynamics to be controlled must be used. The key assumption is that the range of operation is small for the linear model to be valid. As a consequence the controller will only be effective in the neighborhood of the equilibrium point. However, when the required operating range is large, the nonlinearities in the system cannot be properly compensated for by using a linear model. The linearization process therefore results in the loss of vital mathematical information from the dynamics of the physical system. As demands for better performance continue to increase over the years, the need to take into account the nonlinearities becomes increasingly important.

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Multi-input–multi-output systems do not possess the properties usually assumed in classical control approach, therefore, limiting the possibilities of achieving satisfactory design. Modern, as opposed to classical, control presents solutions to a much wider class of control problems than classical control can tackle. Optimal control is one particular branch of modern control that sets out to provide design where the end result is stable, has a certain bandwidth, satisﬁes any of the desirable constraints associated with classical control and also provides the “best” possible system of a particularly appealing type. In particular, linear optimal control theory is a well-documented branch of general optimal control, where the plant that is controlled is assumed linear and the feedback controller is constrained to be linear with respect to its input. Optimal control of general autonomous nonlinear regulator systems of the form x˙ = f(x, u),

x(t0 ) = x0

with the ﬁnite-time, linear-quadratic cost functional J (u)= 21 xT (tf )Fx(tf ) + 21 tf × xT (t)Qx(t) + uT (t)Ru(t) dt t0

can be solved, in principle, by the use of Lie series and inﬁnite-dimensional bilinear systems theory (Banks, 1986, 1992; Banks &Yew, 1985). However, the solution is complex and difﬁcult to implement. As a result, some fundamental contributions have been made to the theory of nonlinear dynamical systems in the past decade, particularly in the area of nonlinear input-afﬁne systems in the form x˙ = A(x)x + B(x)u.

(1)

One of the uses of “pseudo-linear” systems of the form (1) in control theory was in 1992 (Banks & Mhana, 1992), where a “freezing” technique was used to develop near-optimal, (locally) asymptotically stabilizing controllers for a variety of systems. Many authors have considered this approximation to nonlinear optimal control based on solving a “statedependent Riccati equation” or SDRE. The computational simplicity and effectiveness of this control algorithm for autonomous nonlinear systems caught the interest of many control theorists and engineers, so that its use is now common in many practical applications (see, for instance, McCaffrey & Banks, 2001; Mracek & Cloutier, 1998; Sznaier, Cloutier, Hull, Jacques, & Mracek, 2000). However, the SDRE feedback provides a locally optimal control policy, which can only be applied to autonomous regulator problems. This is because the approach requires solving the inﬁnite-time algebraic Riccati equation and, unfortunately, the theory which deals with this for the optimal tracking problem is not available. Recently, Banks and McCaffrey (1998) have introduced a new approach to nonlinear systems theory based on systems

of the form x˙ = A(x)x. Associated with such a system is the Lie algebra generated by the set of all the matrices A(x), for all x. Moreover, one can consider such a system as the limit of the sequence of linear, time-varying (LTV) approximations x˙ [i] (t) = A x[i−1] (t) x[i] (t), which has been shown to converge in the space of continuous functions in very general circumstances; technically all that is required is that the function x → A(x) is locally Lipschitz, the usual condition for uniqueness of solutions of the equation. The great advantage here, of course, is that nonlinear systems are approximated arbitrarily closely by linear ones, which means that all the usual linear machinery can be brought to bear. Of course, LTV systems are much more difﬁcult than time-invariant ones, but recent developments (Banks, 2002) have led to explicit representations of the solutions in terms of the Lie brackets of the matrices A(x[i−1] (t)). The approximation theory has been extensively used in the study of Lie algebras, chaotic motion and in the theory of nonlinear delay systems (see Banks, 2002; Banks & McCaffrey, 1998). Naturally, the same ideas can be applied to control systems (1), by using a sequence of LTV approximations x˙ [i] (t) = A x[i−1] (t) x[i] (t) + B x[i−1] (t) u[i] (t), (2) where controls are determined by any of the classical methods, such as optimal control (see Banks & Dinesh, 2000), robust H ∞ control, etc. In the case of control systems (1), the approximation technique described by (2) has proved very successful in controlling a number of nonlinear systems, including aircraft systems (Banks, Salamci, & McCaffrey, 2000; Salamci, Özgören, & Banks, 2000; Çimen & Banks, in press) and nonlinear solitary wave motion (Banks, 2001). Salamci et al. (2000) have used the approximation technique in designing sliding mode controls for nonlinear systems with optimally selected sliding surfaces and successfully tested the resulting controller performance on an autopilot design for a missile. In a recent paper, Banks and Dinesh (2000) have also applied the approximation theory proposed by Banks and McCaffrey (1998) to ﬁnd approximate optimal nonlinear time-varying feedback controllers for input-afﬁne nonlinear regulator systems (1), by introducing the sequence of LTV approximations (2). The approximations have been shown to converge under very mild conditions of local Lipschitz continuity. In this paper, the results of Banks and Dinesh (2000) on optimal regulation of control-afﬁne nonlinear systems are generalized by extending the control problem to deterministic optimal tracking of continuous-time nonlinear dynamic systems. In order to verify a good control design it is necessary to simulate the control law against a realistic model of a realworld application. The prerequisite is that the motions of the

T. Çimen, S.P. Banks / Automatica 40 (2004) 1845 – 1863

real-world application are accurately described by a mathematical model. This is the principal point and also one of the main difﬁculties. A great deal of effort has been devoted to the construction of mathematical models describing the motions of ships (see Crane, 1973; Fossen, 1994; Norrbin, 1970; Van Berlekom & Goddard, 1972). New super-tankers are built to transport ever-larger quantities of crude oil, containers or bulk cargoes between the continents to meet demands at a minimum of ﬁnancial expense for industries. This article will also investigate a model representing the highly nonlinear and coupled motions of the heading and propulsion dynamics of a 304.8 m-long, 190,000-deadweight ton (dwt) oil tanker; the ESSO 190,000-dwt tanker. By the end of 1969 ﬁve ships of this class were in operation and their maneuvering characteristics were well documented. The overall handling of these ships was also considered to be excellent by their masters. McGookin, Murray-Smith, Li, and Fossen (2000) have recently attempted to design an autopilot by sliding mode control law for the nonlinear ESSO 190,000-dwt tanker dynamics. An automated technique based on genetic algorithms, for optimizing controller parameters, has also been presented in their paper in order to obtain optimal performance from such a nonlinear controller. However, the control law was based on the representation of a single-input–single-output linear dynamic system. So in order to apply this, the yaw rate (r) and the heading angle () dynamics were decoupled from the entire system and the resulting subsystem linearized through small perturbation state approximations about nominal equilibrium points, making it only a valid representation close to these chosen points. In their control conﬁguration, the propeller input was also assumed to obey simple step commands at its maximum speed, and another controller was then used to regulate its rotational speed. As well as a nominal linear control element, the sliding mode controller used in the investigation provided an additional nonlinear controller term. The purpose of this nonlinear term was to provide enough additional control effort to compensate for some of the unmodeled dynamics of the tanker that were neglected during the linearization process. In this paper, the proposed design will not require decoupling or linearization of the nonlinear dynamics, unlike other well-known control methods. In contrast to these works, the dynamics studied will be nonlinear. The nonlinear optimal tracking control law will be applied to the nonlinear tanker dynamics for an autopilot design that provides tracking of a commanded course heading at a desired shaft velocity. The rest of this paper is organized as follows. Section 2 serves as preliminary background theory and recalls the classical linear optimal tracking problem with linear-quadratic performance criteria to set up the basic mathematical results associated with them. The classical linear-quadratic optimal tracking theory is then used in Section 3 as a design tool in an attempt to solve the ﬁnite-time nonlinear deterministic optimal tracking control problem. The method introduces an “Approximating Sequence of Riccati Equations” (ASRE)

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to ﬁnd time-varying feedback controllers for such nonlinear systems, and will be referred to as ASRE control throughout the paper. A brief discussion on how to solve these approximating sequences is given at the end of the section. In Section 4, the resulting approximations are shown to converge to the true systems in an appropriate space under very mild conditions that each operator is locally Lipschitz. However, Banks and Dinesh (2000) have only presented a local (in time) proof of convergence to the approximation theory for control systems. In this paper, the local proof is extended to a global proof for dynamical control systems that do not have ﬁnite escape time. In Section 5, the ESSO 190,000dwt tanker model is derived, the problem formulation stated and an ASRE feedback controller designed for the nonlinear tanker dynamics for an autopilot design that provides tracking of a commanded course heading at a desired shaft velocity. Simulations for illustrating the effectiveness of the proposed controller are presented in Section 6, with the main results stated. This is followed by concluding remarks in Section 7. A complete nomenclature, which contains all the symbology used in the manuscript, is provided in Appendix A. The principal particulars and hydrodynamic coefﬁcients for the ESSO 190,000-dwt tanker are contained in Appendix B.

2. Background Let us now summarize classical linear optimal tracking control theory where a linear observable system is described by the equations x˙ (t) = A(t)x(t) + B(t)u(t),

x(t0 ) = x0 ,

y(t) = C(t)x(t)

(3)

with the ﬁnite-time linear-quadratic cost functional J (u)= 21 eT (tf )Fe(tf ) tf 1 eT (t)Q(t)e(t) + uT (t)R(t)u(t) dt, +2 t0

(4)

which minimizes the error given by e(t) = z(t) − y(t) = z(t) − C(t)x(t). Here F, Q ∈ Rl×l are positive-semideﬁnite, R ∈ Rm×m is positive-deﬁnite, u is unconstrained, A ∈ Rn×n , B ∈ Rn×m , C ∈ Rl×n , and the objective is to control the system (3) so that the output vector y(t) is “near” the desired output vector z(t). This problem of minimizing the cost functional (4) subject to the dynamics (3) to keep the output y(t) near a time-varying desired output z(t) is called the linear-quadratic optimal tracking control problem. In fact, the optimal control is a time-varying linear state-feedback control law given by (see Athans & Falb, 1966) u(t) = −R−1 (t)BT (t){P(t)x(t) − s(t)},

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where the real, symmetric and positive-deﬁnite matrix P ∈ Rn×n is the solution of the Riccati-type matrix differential equation ˙ P(t)= − CT (t)Q(t)C(t) − P(t)A(t) − AT (t)P(t) + P(t)B(t)R−1 (t)BT (t)P(t), P(tf ) = CT (tf )FC(tf ). The additional vector s ∈ Rn is a feed-forward term, which is the solution of the linear vector differential equation T s˙(t)= − A(t) − B(t)R−1 (t)BT (t)P(t) s(t)

s(tf ) = CT (tf )Fz(tf ). The state of the optimal system is then given by the solution of the linear differential equation x˙ (t)= A(t) − B(t)R−1 (t)BT (t)P(t) x(t) + B(t)R−1 (t)BT (t)s(t)

3. The nonlinear optimal tracking problem Consider the class of nonlinear observable systems represented in the general form y = g(x).

Assuming f(0, 0) = 0 and g(0) = 0, that is, the origin is an equilibrium point, suppose that the system can be described in the factored control-afﬁne form by the equations x(t0 ) = x0 ,

y(t) = C(x(t))x(t),

(5)

together with the ﬁnite-time nonlinear (nonquadratic in x) cost functional J (u)= 21 eT (tf )Fe(tf ) tf 1 eT (t)Qe(t) + uT (t)Ru(t) dt, +2

y[i] (t) = C x[i−1] (t) x[i] (t),

(7)

J [i] (u)= 21 e[i]T (tf )Fe[i] (tf ), tf 1 +2 e[i]T (t)Qe[i] (t) + u[i]T (t)Ru[i] (t) dt, t0

e[i] (t)=z(t) − C(x[i−1] (t))x[i] (t)

(8)

for i 0. The ﬁrst approximation in each sequence in (7) and (8) is given by x˙ [0] (t) = A(x0 )x[0] (t) + B(x0 )u[0] (t),

x[0] (t0 ) = x0

y[0] (t) = C(x0 )x[0] (t) J [0] (u)= 21 e[0]T (tf )Fe[0] (tf ) tf 1 + 2 e[0]T (t)Qe[0] (t) + u[0]T (t)Ru[0] (t) dt t0

e[0] (t) = z(t) − C(x0 )x[0] (t),

t0

e(t)=z(t) − y(t) = z(t) − C(x(t))x(t).

x˙ [i] (t) = A x[i−1] (t) x[i] (t) + B x[i−1] (t) u[i] (t),

where the initial state x[i] (t0 ) = x0 , and the corresponding linear-quadratic cost functional is given by

starting at the known initial state x(t0 ) = x0 .

x˙ (t) = A(x(t))x(t) + B(x(t))u(t),

3.1. The ASRE feedback design strategy For the ﬁnite-time nonlinear-nonquadratic optimal tracking problem (5) and (6), the following sequences of LTV approximations are introduced:

− CT (t)Q(t)z(t),

x˙ = f(x, u),

systems, that is systems having nonlinear control inputs (see Çimen, 2003; Çimen & Banks, in press). In fact, even the nonquadratic and time-varying cost functional (6) can be extended to include state- and controldependent timevarying weighting matrices F(x(tf )), Q(x(t), u(t), t) and R(x(t), u(t), t). However, the super-tanker model considered in this article can be described by the factored inputafﬁne Eqs. (5) and (6). Therefore, particular interest will be drawn in the paper to systems whose equations can be represented in this form.

(6)

The objective of the nonlinear-nonquadratic optimal tracking problem is therefore to control system (5) by minimizing the error e(t) in order to keep the output y(t) near a desired output z(t). Note that the factored state-space representation A(x)x in (5) is not unique. In addition, provided the origin is an equilibrium point, the ASRE control design strategy proposed in the subsequent discussion can be generalized to nonlinear nonautonomous and control-nonafﬁne

thus assuming that the initial function x[i−1] (t) when i = 0 is x0 . Since each approximating problem in (7) and (8) is time-varying (with the exception of the ﬁrst approximating sequence) and linear-quadratic, from Section 2, the optimal tracking control law for the nonlinear problem (5), (6) can be given in the form u[i] (t) = −R−1 BT x[i−1] (t) P[i] (t)x[i] (t) − s[i] (t) (9)

T. Çimen, S.P. Banks / Automatica 40 (2004) 1845 – 1863

for i 0, where the n×n real, symmetric and positive-deﬁnite matrix P[i] (t) is the solution of the ASRE P˙ [i] (t)=−CT x[i−1] (t) QC x[i−1] (t) − P[i] (t)A x[i−1] (t) − AT x[i−1] (t) P[i] (t) + P[i] (t)B x[i−1] (t) R−1 BT x[i−1] (t) P[i] (t), P[i] (tf )=CT x[i−1] (tf ) FC x[i−1] (tf ) .

(10)

The vector s[i] (t) (with n components) is the solution of the linear vector differential equation s˙[i] (t)=−CT x[i−1] (t) Qz(t) − A x[i−1] (t) T − B x[i−1] (t) R−1 BT x[i−1] (t) P[i] (t) s[i] (t), s[i] (tf )=CT x[i−1] (tf ) Fz(tf ),

(11)

and the optimal trajectory becomes the limit of the solution of the linear differential equation x˙ [i] (t)=B x[i−1] (t) R−1 BT x[i−1] (t) s[i] (t) + A x[i−1] (t) − B x[i−1] (t) × R−1 BT x[i−1] (t) P[i] (t) x[i] (t)

(12)

starting at the known initial state x[i] (t0 ) = x0 . The feedforward part of the optimal tracking controller provides the necessary input for following the speciﬁed motion trajectory and canceling the effects of known disturbances. The feedback part then stabilizes the tracking error dynamics. Theorem 1. (Nonlinear optimal tracking ASRE control). Given the nonlinear observable optimal tracking problem (5), (6) and the desired output z(t), the sequence of linearquadratic and time-varying approximations (7), (8) can be introduced. Provided that u(t) is unconstrained, the terminal time tf is speciﬁed, F and Q are positive-semideﬁnite, R is positive-deﬁnite, and since each of the approximating systems (5), (6) are (time-varying) linear-quadratic, then the optimal control exists, is unique for each sequence of approximations, and is given by (9). Under bounded and local Lipschitz conditions of A(x), B(x), C(x) and z(t), the sequence of approximations x[i] (t), y[i] (t) and u[i] (t) globally converge in C([t0 , tf ]; Rn ), C([t0 , tf ]; Rl ) and C([t0 , tf ]; Rm ), respectively, to the functions x(t), y(t) and u(t), which minimize (6) over the set of feedback controls of the form −B(x(t))R−1 BT (x(t)){P(t)x(t) − s(t)}.

(13)

Remark 1. The nonlinear optimal stabilization (or regulator) problem is a special case of the nonlinear optimal track-

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ing problem, with the desired trajectory taken to be zero. Therefore, the state- and output-regulator problems can each be solved in the same way as Theorem 1 for the tracking problem. If z(t) = 0, the cost functional (6) reduces to the output-regulator problem, which requires bringing and keeping the output y(t), rather than the state x(t), near zero without using an excessive amount of control energy. In addition, if C(x(t)) = In×n , then y(t) = x(t) = −e(t) and the cost functional (6) reduces to the state-regulator problem (Banks & Dinesh, 2000). The physical interpretation of this is that it is desired to keep the state near zero without excessive control-energy expenditure. This means that the feedback structure of the nonlinear optimal tracking system is the same as the feedback structure of the nonlinear optimal regulator system. The essential difference is provided by the vector s[i] (t), which is the forcing function of the system and tends to counterbalance the regulator features of the system. 3.2. Solving the approximating sequences In solving the set of coupled Eqs. (9)–(12), a sequence of time-varying (with the exception of the ﬁrst approximating sequence) and linear-quadratic equations is obtained where each sequence is solved as a standard numerical problem. For each sequence, optimization has to be carried out on the system trajectory at every numerical integration timestep resulting in time-varying feedback controls. In order to calculate the optimal solution for each sequence of approximations, it is necessary to solve the ASRE (10) and the approximating sequence of linear vector differential Eqs. (11), storing the values of P(t) and s(t) from the last sequence at every discrete time-step. In practice this will be done in a computer and it will be necessary to solve the equations using standard numerical integration procedures, starting at the ﬁnal time t =tf and integrating backwards in time by taking negative time-steps. The control u[i] (t) that minimizes the ﬁnite-time linear-quadratic cost functional (8) at every time-step for each sequence is then given by (9), where the approximate optimal state trajectory x[i] (t) is obtained by solving (12), once again storing the values from the last sequence at every time-step. Note that in solving for the ﬁrst sequence, that is, when i = 0, x[i−1] (t) in Eqs. (7)–(12) is set to x0 for all t ∈ [t0 , tf ]. Under this assumption, the ﬁrst control sequence is given from (9) by u[0] (t) = −R−1 BT (x0 ) P[0] (t)x[0] (t) − s[0] (t) , where P[0] (t), s[0] (t) and x[0] (t) are obtained from (10)–(12) in a similar way by assuming x[i−1] (t) = x0 , ∀t ∈ [t0 , tf ]. The ﬁrst sequence is therefore linear, time-invariant (LTI). However, subsequent approximations for i > 0 become LTV since these are obtained by replacing x[i−1] (t) at every timestep with corresponding values from the previous stored sequence. When both x[i] (t) and u[i] (t) converge, say on the kth approximating sequence, applying P[k] (t) and s[k] (t) from the converged sequence to the original nonlinear sys-

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tem (5) will yield the optimal state trajectory x and control u that minimizes the ﬁnite-time nonlinear cost functional (6). Remark 2. The initial function x[i−1] (t) when i = 0 in the ASRE iteration technique has been ﬁxed at x0 for t ∈ [t0 , tf ], and u[0] (t) is the corresponding control. In many cases, it may be better to use a linear representation of the nonlinear dynamics (5) to obtain a better guess for this initial function (instead of x0 ), for instance, by applying linear-quadratic optimal tracking theory to the linearized dynamics and replacing x[i−1] (t) with the corresponding controlled state of the linear system. This is likely to improve the accuracy of the ﬁrst approximating sequence and hence reduce the number of iterations required to achieve convergence, but this will largely depend on the particular system being considered.

Proof. [i−1] and [i−2] are solutions of the respective LTV equations x˙ [i] (t) = A x[i−1] (t) x[i] (t), x[i] (t0 ) = x0 , x˙ [i−1] (t) = A x[i−2] (t) x[i−1] (t), Hence,

x[i] (t) − x[i−1] (t) = [i−1] (t, t0 ) − [i−2] (t, t0 ) x0 . (15) Now, consider writing x˙ [i] (t) − x˙ [i−1] (t) in the form d [i] x (t) − x[i−1] (t) dt = A x[i−1] (t) x[i] (t) − x[i−1] (t) + A x[i−1] (t) − A x[i−2] (t) x[i−1] (t).

4. Proof of global convergence Let us now brieﬂy outline the proof of Theorem 1, that is, prove that the sequences of coupled Eqs. (9)–(12) converge, under certain conditions on A(x(t)), B(x(t)), C(x(t)) and z(t) (that each operator is bounded and locally Lipschitz) and for small enough horizon time tf or small enough initial condition x0 . For a thorough discussion, the interested reader may refer to Çimen (2003). Suppose that [i] (t) sup x[i] (s) − x[i−1] (s) s∈[t0 ,t]

and let [i−1] (t, t0 ) and [i−2] (t, t0 ) denote the transition matrices generated by A(x[i−1] (t)) and A(x[i−2] (t)), respectively. It is well-known (Brauer, 1966, 1967) that the fundamental matrix [i−1] (t, t0 ) of the linear system

Using the variation of constants formula, t x[i] (t) − x[i−1] (t)= [i−1] (t, s) t0

× A x[i−1] (s) − A x[i−2] (s) ×[i−2] (s, t0 )x0 ds

t t0 ,

where (A) denotes the measure of the matrix A. Next, an estimate for [i−1] (t, t0 ) − [i−2] (t, t0 ) is required. Lemma 1. Suppose the following conditions hold:

∀x1 , x2 ∈ R

t

t0

0 d x[i−1] (s) − x[i−2] (s) ds

This completes the proof.

n

for some ﬁnite constant 0 and > 0. Then [i−1] (t, t0 )−[i−2] (t, t0 ) e0 (t−t0 ) (t − t0 )[i−1] (t).

In order to prove the sequence converges to a solution, it is required to show that lim x[i] (t) − x[i−1] (t) , lim y[i] (t) − y[i−1] (t) =0. i→∞

and

(A2) A(x1 ) − A(x2 )x1 − x2 ,

t0

exp

s∈[t0 ,t]

(14)

∀x ∈ Rn ,

t

e0 (t−t0 ) (t − t0 ) sup x[i−1] (s) − x[i−2] (s) .

t0

(A(x))0 ,

t0

− A x[i−2] (s) [i−2] (s, t0 ) ds

(t0 , t0 ) = I, satisﬁes t

[i−1] [i−1] (t, t0 ) exp A x () d ,

(A1)

t [i−1] (t, s) A x[i−1] (s)

[i−1]

(16)

since x[i−1] (t)= [i−2] (t, t0 )x[i−1] (t0 ). Therefore, from (15) and (16), by using (14) and assuming (A1) and (A2) hold, [i−1] (t, t0 ) − [i−2] (t, t0 )

x˙ [i] (t) = A(x[i−1] (t))x[i] (t), such that

x[i−1] (t0 ) = x0 .

i→∞

Let s¯[i] (t)s[i] (tf − t) satisfy the approximating sequence of linear vector differential equations (11), so that [i] s˙¯ (t)=AT x[i−1] (t) s[i] (t) + ET x[i−1] (t) s[i] (t) + CT x[i−1] (t) Qz(t),

(17)

T. Çimen, S.P. Banks / Automatica 40 (2004) 1845 – 1863

1851

s¯[i] (t0 ) = CT x[i−1] (tf ) Fz(tf ),

and

where s¯[0] (t0 ) = CT (x0 )Fz(tf ) and E x[i−1] (t) − B x[i−1] (t) R−1 BT x[i−1] (t) P[i] (t).

x[i] (t)=[i−1] (t, t0 )x[i] (t0 ) t + [i−1] (t, s)E x[i−1] (s) x[i] (s) ds

(18) In addition, consider the approximating output sequence in (7) together with the controlled state sequence (12), which is perturbed by x[i] (t) and s[i] (t), given here in more compact form by the approximating sequences x˙ [i] (t)=A x[i−1] (t) x[i] (t) + E x[i−1] (t) x[i] (t) +B x[i−1] (t) R−1 BT x[i−1] (t) s[i] (t), x[i] (t0 ) = x0 , y[i] (t)=C x[i−1] (t) x[i] (t)

(19)

corresponding to the perturbed nonlinear system x˙ (t)=A(x(t))x(t) + E(x(t))x(t) +B(x(t))R−1 BT (x(t))s(t),

y(t)=C(x(t))x(t).

(20)

(A5)

B(x)1 ,

∀x1 , x2 ∈ Rn ,

∀x ∈ R , ∀x1 , x2 ∈ Rn ,

∀x ∈ Rn ,

C(x)1 ,

(A8)

C(x1 ) − C(x2 )2 x1 − x2 ,

(A9)

z(t)z,

∀x1 , x2 ∈ Rn ,

Proof. By variation of constants, integrating (17) and (19), [i] for s˙¯ (t) and x˙ [i] (t), from [t0 , tf ] gives s¯[i] (t)=[i−1]T (t, t0 )¯s[i] (t0 ) t + [i−1]T (t, s)ET x[i−1] (s) s¯[i] (s) ds t

t0

(23)

x[i] (t) 2 (t),

(24)

which imply boundedness by a small interval time t ∈ [t0 , tf ] and for small x0 , provided that F, Q and R−1 are bounded. To show s¯[i] (t) is locally Lipschitz, ﬁrst s¯[i] (t) − s¯[i−1] (t) is written in the form

= [i−1]T (t, t0 )¯s[i] (t0 ) − [i−2]T (t, t0 )¯s[i−1] (t0 ) t [i−1]T (t, s)ET x[i−1] (s) s¯[i] (s)−¯s[i−1] (s) ds +

[i−1]T (t, s)CT x[i−1] (s) Qz(s) ds

t t0

[i−1]T (t, s) E x[i−1] (s)

T s¯[i−1] (s) ds − E x[i−2] (s) +

t t0

T

[i−1] (t, s) − [i−2] (t, s)

× ET x[i−2] (s) s¯[i−1] (s) ds +

t t0

[i−1]T (t, s) C(x[i−1] (s))

T Qz(s) ds − C x[i−2] (s) +

t t0

T

[i−1] (t, s) − [i−2] (t, s)

× CT x[i−2] (s) Qz(s) ds. Then from Lemma 1, inequalities (14) and (23), and assumptions (A1)–(A9), extensive algebraic manipulations lead to the expression

t0

+

¯s[i] (t) 1 (t),

for ﬁnite positive numbers 1 , 2 , 1 , 2 , 1 , 2 and z, the limit of the solutions of the approximating sequences (19) converges to the unique solutions of (20) on [t0 , tf ].

(22)

respectively. By using (14), assumptions (A1)–(A9) and Gronwall–Bellman’s Lemma, bounds on s¯[i] (t) (thus s[i] (t)) and x[i] (t) can now be obtained from (21) and (22), respectively, in the form

+

n

(A7)

×R−1 BT x[i−1] (s) s[i] (s) ds,

∀x ∈ Rn ,

(A6) B(x1 ) − B(x2 )2 x1 − x2 ,

t0

[i−1] (t, s)B x[i−1] (s)

t0

Lemma 2. Under conditions (A1), (A2) and

(A4) E(x1 ) − E(x2 )2 x1 − x2 ,

t

s¯[i] (t) − s¯[i−1] (t)

x(t0 ) = x0 ,

(A3) E(x)1 ,

+

t0

(21)

[i] (t) 3 (t)[i−1] (t),

(25)

1852

T. Çimen, S.P. Banks / Automatica 40 (2004) 1845 – 1863

where

gives

[i] (t) sup s¯[i] (s) − s¯[i−1] (s) s∈[t0 ,t]

and 3 (t) represents a complex time-dependent function of the upper bounds and Lipschitz constants in (A1)–(A9), as well as the upper bounds on F, Q and R−1 . Therefore, (25) implies that s¯[i] (t) (and so s[i] (t)) converges on C([t0 , tf ]; Rn ) iff x[i] (t) is a Cauchy sequence. The latter is proved from (22), where a similar representation scheme for x[i] (t) − x[i−1] (t) is used to ﬁnd an expression for [i] (t) in the form

[i] (t) 4 (t)[i−1] (t),

(26)

where 4 (t) represents the upper bound on the rate of convergence. Thus, if | 4 (t)| < 1 for t ∈ [t0 , tf ], then on C([t0 , tf ]; Rn ), x[i] (t) → x(t) and hence s[i] (t) → s(t) since from (26), by induction, [i] (t) satisﬁes

P¯ [i] (t)=[i−1]T (t, t0 )P¯ [i] (t0 )[i−1] (t, t0 ) t + [i−1]T (t, s)CT x[i−1] (s) t0

×QC x[i−1] (s) [i−1] (t, s) ds +

t t0

[i−1]T (t, s)P¯ [i] (s)B x[i−1] (s)

×R−1 BT x[i−1] (s) P¯ [i] (s)[i−1] (t, s) ds. Hence, if (t)P¯ [i] (t), under conditions (A1), (A5), (A7) and by inequality (14),

(t)e20 (t−t0 ) 21 F t + 21 Q + 21 R−1 2 (s) e20 (t−s) ds t0

[1] [i] (t) i−1 4 (t) (t)

(27) x[i] (t)

is a Cauchy sequence in the and this implies that Banach space C([t0 , tf ]; Rn ). The proof of convergence for the approximate output sequence in (19) follows directly from the above result and conditions (A7) and (A8), which gives [i] y (t) 1 x[i] (t) , [i] y (t) − y[i−1] (t) 1 + 2 x[i−1] (t) x[i] (t) − x[i−1] (t) . Hence from (24) and (27), y[i] (t) → y(t) on C([t0 , tf ]; Rl ). Remark 3. The assumptions (A1)–(A9) are global, and can be replaced by local versions. Therefore, the results are not as restrictive as might appear. In the case of the controlled sequence (12), E x[i−1] (t) is deﬁned by (18) and so 2 E x[i−1] (t) B x[i−1] (t) R−1 P[i] (t) . The following lemma provides a boundon P[i] (t) directly from the ASRE (10). The bound on P[i] (t) − P[i−1] (t) will follow in a similar way.

so that, by using a well-known comparison argument, differentiating this inequality with respect to t gives

˙ (t)c1 + c2 (t) + c3 2 (t),

where c1 21 Q, c2 20 and c3 21 R−1 , together with (t0 ) = CT x[i−1] (tf ) FC x[i−1] (tf ) 21 F. This is a nonlinear equation in (t), so for some ∈ [t0 , tf ) if the solution (t) < CT x[i−1] (tf ) FC x[i−1] (tf ) +

21 F + for t ∈ [, tf ] for any > 0, then on [, tf ] ˙ (t) < c1 + c2 + c3 21 F + (t)

(28)

for ﬁnite constants c1 , c2 , c3 > 0. Hence, the solution to the scalar differential inequality (28) is given by

(t) <e

c2 +c3 (21 F+ ) (t−t0 )

+

t t0

(t0 )

e

c2 +c3 (21 F+ ) (t−s)

c1 ds,

that is, c2 +c3 (21 F+ ) (tf −t−t0 ) 2 [i] 1 F P (t) e + c1

t0

c2 +c3 (21 F+ ) (tf −t−s)

tf −t

e

ds

Lemma 3. P[i] (t) is bounded on [t0 , tf ] for small enough tf .

and so for small enough tf (or small c2 , c3 ) it follows that P[i] (t) is bounded on [t0 , tf ].

Proof. Let P¯ [i] (t)P[i] (tf − t) satisfy the ASRE (10) such that, for any sequence P¯ [i] (t), integrating this from [t0 , tf ]

Lemma 3 shows that P[i] (t) is bounded by a small interval time t ∈ [t0 , tf ] (or small c2 , c3 ). This suggests that the

T. Çimen, S.P. Banks / Automatica 40 (2004) 1845 – 1863

Lipschitz condition for P[i] (t) should be satisﬁed within this interval. The following lemma provides the proof for this condition. Lemma 4. P[i] (t) is Lipschitz continuous on [t0 , tf ]. Proof. Let the matrix P[i] (t) satisfy the ASRE (10) such that −1 with X[i] (tf ) = I and Y[i] (tf ) = P[i] (t)Y[i] (t) X[i] (t) [i−1] −1 [i−1] T (tf ) FC x (tf ) , where X[i] (t) represents C x [i] the inverse of X (t). Hence,

−1 −1 [i] ˙P[i] (t) = Y ˙ [i] (t) − Y[i] (t) X[i] (t) ˙ . X (t) X[i] (t) Since this equation satisﬁes the ASRE (10), separating ˙ [i] (t) and Y ˙ [i] (t) terms after substitution gives the coupled X set of equations ˙ [i] (t)=A x[i−1] (t) X[i] (t) X − B x[i−1] (t) R−1 BT x[i−1] (t) Y[i] (t), ˙ [i] (t)=−CT x[i−1] (t) QC x[i−1] (t) X[i] (t) Y − AT x[i−1] (t) Y[i] (t),

(29)

which are linear in X[i] (t) and Y[i] (t). Therefore, provided that A(x), B(x) and C(x) are Lipschitz continuous, the pair of linear Eq. (29) imply that X[i] (t) and Y[i] (t) also satisfy the Lipschitz condition. It is easy to verify that −1 also holds. Therefore, Lipschitz continuity for X[i] (t) [i] −1 [i] both satisfy the Lipschitz consince Y (t) and X (t) −1 [i] is also dition, it follows that P (t) = Y[i] (t) X[i] (t) Lipschitz. Given any initial state x0 , the sequence of linear-quadratic, time-varying approximations obtained by classical linearquadratic methods for the nonlinear optimal tracking control problem have been shown to converge uniformly on some small interval [t0 , tf ], where tf may depend on x0 . This has been shown in the space of continuous functions, under local Lipschitz continuity of the nonlinear dynamical operators A(x), B(x) and C(x), and for small ﬁnite-time tf . As yet there is no clear indication of how small this interval should be taken for the method to work and hence be of any practical use for nonlinear optimal control. A global convergence theory is therefore required, which removes this strong restriction. This is obtained by applying a similar principle to that used by Tomas-Rodriguez & Banks (2003) for nonlinear homogeneous equations. That is to say, if a solution to the nonlinear optimal control problem exists, for which the cost is ﬁnite and the trajectory is bounded in the interval [t0 , ] ⊆ R, then the approximating sequences are shown to converge uniformly on [t0 , ].

1853

Lemma 5. (Global convergence). Suppose that the nonlinear optimal control problem has a continuous feedback control on the interval [t0 , ]. Then the controlled sequence of functions {x[i] (t)}, {y[i] (t)} and feedback controls {u[i] (t)} deﬁned by the linear-quadratic, time-varying approximations converge uniformly on [t0 , ]. Proof. It is clear from the above proofs that tf can be chosen to be locally constant, that is, for any x¯ there exists a neighborhood Bx¯ of x¯ such that the sequences of linear-quadratic, time-varying approximations with initial state x0 ∈ Bx¯ converge uniformly on some interval [t0 , tx¯ ], where tx¯ is independent of x0 . Now suppose the result is false, so that there is a maximal time interval [t0 , t¯) such that for any tf < t¯, the quadratic and LTV sequences converge uniformly on [t0 , tf ]. Let us consider the controlled trajectory x(t; x0 ) of the original nonlinear optimal control problem on the interval [t0 , ]. Deﬁne the set S {x(t; x0 )|t ∈ [t0 , ]}. For each x¯ ∈ S, choose a neighborhood Bx¯ as above, that is, the sequences of LTV and quadratic approximations converge uniformly on the interval [t0 , tx¯ ] for any x0 ∈ Bx¯ and for tx¯ independent of the initial state x0 . Since S is

compact and x¯ ∈S Bx¯ is an open cover of S, there exists a ﬁnite sub-cover {Bx¯ 1 , . . . , Bx¯ p } with corresponding times {tx¯ 1 , . . . , tx¯ p }. Let tmin min{tx¯ 1 , . . . , tx¯ p }. Now since, by assumption, the approximating sequence of Riccati operators and feedback controls converge on [t0 , tf ], the controlled sequence x[i] (t) converges uniformly on [t0 , t¯ − tmin /2]. Let x0,i x[i] (t¯ − tmin /2). Since these converge to x(t¯ − tmin /2), they can be assumed to belong to Bx¯ p , so that another sequence of solutions given by the linear-quadratic, time-varying approximations can be obtained from the initial states x0,i and converging uniformly on the interval [t¯ − tmin /2, t¯ + tmin /2] to the corresponding solutions of the nonlinear optimal control problem (with initial state x0,i ). Let us denote these solutions by x[i,j ] (t), which converge to x[i] (t) on the interval [t¯ − tmin /2, t¯ + tmin /2] as shown in Fig. 1. Now let us use a Cantor-like diagonal argument. Consider the functions [i] x (t), t0 t t¯ − tmin /2, [i] (t) x[i,j ] (t), t¯ − tmin /2t t¯ + tmin /2. Then [i] (t) converges uniformly to x(t) on [t0 , t¯ + tmin /2] and is arbitrarily close to x[i] (t) on [t0 , t¯], which contradicts the assumption that {x[i] (t)}, and therefore {y[i] (t)}, is not uniformly convergent on [t0 , t¯]. Since the controls are expressed in feedback form, it follows that {u[i] (t)} also converges on [t0 , t¯].

1854

T. Çimen, S.P. Banks / Automatica 40 (2004) 1845 – 1863

ψ Body-Fixed Coordinate System

u

v

yp

xp

h

r XEARTH

Earth-Fixed Coordinate System

YEARTH ZEARTH Fig. 1. Extending the approximating control sequence.

Fig. 2. The ESSO 190,000-dwt tanker coordinate systems.

Proof of Theorem 1. The proof of the nonlinear optimal tracking ASRE feedback control algorithm follows directly from Lemmas 1–5. Since the sequence of functions {x[i] (t)} and {y[i] (t)} converge in C([t0 , tf ]; Rn ) and C([t0 , tf ]; Rl ), respectively, and since the sequence of vectors {s[i] (t)} and Riccati operators {P[i] (t)} are bounded and Lipschitz continuous, the feedback controls {u[i] (t)} also converge in C([t0 , tf ]; Rm ) when expressed in the form (13).

Crane (1973). The principal particulars for this model are given in Table 1 in Appendix B. Fig. 2 shows the ship-ﬁxed and earth-ﬁxed coordinate systems of the tanker. The forces and moment for the kinetic states (velocities u, v, r) of the system are given with reference to a ship-ﬁxed coordinate system. The kinematic components (xp , yp , ) for the kinematic states of the system are deﬁned along the earth-ﬁxed inertial reference frame. The hydrodynamic derivatives/coefﬁcients can partly be calculated using results from potential theory and semiempirical formulas (Norrbin, 1970). The data used here is given by Van Berlekom & Goddard (1972), which has been obtained from the Planar Motion Mechanism (PMM) tests at the Hydro-og Aerodynamisk Laboratium (HyA) in Lyngby, Denmark. The original data given by HyA were in cubic ﬁt as coefﬁcients for the X-, Y- and N-equations. Results from these tests have been recalculated at the Swedish State Shipbuilding Experimental Tank (SSPA) in Goteborg, Sweden, according to an essentially quadratic ﬁt. The coefﬁcients for the quadratic ﬁt are given in Tables 2 and 3 in Appendix B in the so-called “Bis” system (see Norrbin, 1970) as different from the SNAME (Society of Naval Architects and Marine Engineers) “prime” system generally used. This means that, for example, forces are non-dimensionalized by dividing by the product ∇g and moments by ∇gL, where the unit for mass is m = ∇ and = 1 in normal (surface) ship dynamics. The use of non-dimensional coefﬁcients is accepted in all branches of ship theory, and when motion studies are considered even the variables of the equations are often normalized. For more details on the Bis-system normalization procedure refer to Fossen (1994) and Norrbin (1970). The basic mathematical model describes the ship’s motion in calm water. The hydrodynamic forces and moment in calm water are the sum of forces and moment in calm deep water and effects due to conﬁnements in the waterway, that is, shallow water and banks. Thus in the evaluation of maneuvering performance of large tankers, the effect of changes in the water depth must also be considered. The representation of conﬁnement effects has been studied ex-

Remark 4. Suppose that a unique Lipschitz continuous optimal feedback control exists on the interval [t0 , tf ], minimized over a restricted set of nonlinear feedbacks of the form (13). Assuming that the costate variable has a factorization of the form Px , and P satisﬁes a standard (quadratic) matrix Riccati differentail equation, then the ASRE feedback algorithm in Theorem 1 for the nonlinear optimal tracking problem will converge to this unique feedback. (General conditions are given in an upcoming paper.) 5. The ESSO 190,000-dwt tanker The area of greatest importance for very large tankers is their maneuverability. Nonlinear mathematical models describing the maneuverability of large tankers in deep and conﬁned waters are given by Crane (1973), Norrbin (1970) and Van Berlekom & Goddard (1972) in the general statespace form x˙ = f(x, u), where x = [u v r xp yp n]T represents the state vector and u = [ c nc ]T represents the (control) input vector to the tanker. The maneuverability performance of one of these models, the ESSO 190,000-dwt tanker, involving the ship control system will be investigated in this study, with the exception of stopping. While stopping is certainly of great importance, it has not been considered here, and the interested reader may refer to

T. Çimen, S.P. Banks / Automatica 40 (2004) 1845 – 1863

tensively by Norrbin (1970). Bank effects have been ignored throughout this paper, however, shallow-water effects have been introduced by a depth parameter

=

Td , h − Td

where Td = 18.46 m is ship’s draft, which is the depth that the vessel occupies in the water, and h is the water depth. From Table 3, note that there is a transition point where the changes value, which is at hydrodynamic coefﬁcient Yuv h = 41.54 m. This transition changes the dynamics of the sway equation by increasing the surge-sway coupling by an amount related to the depth ratio . A graphical representation of against h indicates that this relationship does not vary considerably for depths over 50 m (refer to McGookin et al., 2000). Even though the average depth of the oceans is some 3800 m and, in practical situations, it is recommended that tankers are given an operating restriction for water depths that are less than three times their draft (Crane, 1973; Fossen, 1994, Van Berlekom & Goddard, 1972), which in this case is 55.38 m, the operating restriction will be disregarded in this investigation to test the ASRE controller performance. Shallow-water effects are accounted for by introducing additional terms in the equations. So these effects are all regarded as disturbances superposed on the calm, deep water case. This terminology of introducing shallow-water effects can also be adopted for other conﬁnements in the waterway, and the effects of current, wind, waves, etc., providing a basis for a systematic development of such effects. However, in this application, the modeling and simulation of external disturbances will be very time consuming and are therefore not investigated. 5.1. Tanker dynamics The speed and steering equations of motion describing the maneuverability of the ESSO 190,000-dwt tanker are given by the following three equations (Bis-system):

(30)

(31)

u˙ − vr = gX (u, ˙ u, v, r, T , , c, ), ˙ u, v, r, T , , c, ), v˙ + ur = gY (u,

˙ u, v, r, T , , c, ), (32) (Lk zz )2 r˙ + Lx G ur = gLN (u, √ = L−1 I /m is the non-dimensional radius of where kzz zz gyration, xG = L−1 xG , and X , Y and N are nonlinear non-dimensional functions representing the surge, sway and yaw dynamics, respectively. It can be noted that the roll equation is neglected, as these angles are known to be quite insigniﬁcant in the tanker case (Norrbin, 1970). In addition to these equations, the inﬂuence of engine maneuvers on the ship’s motions is expressed by the propeller thrust T and ﬂow velocity c at the rudder, deﬁned as (see Norrbin, 1970 for details) 2 gT = L−1 Tuu u + Tun un + LT |n|n |n|n

(33)

1855

2 2 2 un + cnn n . c2 = cun

(34)

Expressions for the non-dimensional surge force, sway force and yaw moment can be obtained by modeling deep water and conﬁnement effects (Fossen, 1994; Norrbin, 1970) so that, from (30)–(34), the resulting surge, sway and yaw dynamics are expressed, respectively, as 1 2 2 u= ˙ Xuu u + L(1 + Xvr )vr + Xvv v L(1 − Xu˙ − Xu˙ ) 2 +Xc|c|

|c|c + Xc|c| |c|c + LgT (1 − td ) 2 2 2 u +Xuu + LX vr + X v vr vv

v= ˙

1 L(1 − Yv˙

(35)

− Yv˙ )

uv + Yv|v| |v|v + Yc|c| Yuv

|c|c

+L(Yur − 1)ur + Yc|c| ||| | |c|c||| | + LY T gT +LY ur ur + Yuv uv + Yv|v| |v|v +Yc|c| ||| | |c|c||| |

r˙ =

(36)

1 uv + LN |v|r |v|r Nuv 2 − N − N ) L2 (kzz r˙ r˙ +Nc|c|

|c|c + L(Nur − xG )ur + Nc|c|||| | |c|c||| |

+LN T gT + LN ur ur + Nuv uv + LN |v|r |v|r +Nc|c| |c|c| | |

| . (37) ||| |

The kinematic states of the tanker dynamics can be described by the following set of equations: x˙p = u cos − v sin ,

(38)

y˙p = u sin + v cos ,

(39)

˙ = r.

(40)

Eqs. (35)–(40) represent the kinetic and kinematic states of the system, together making up the rigid-body dynamics of the vessel. The mathematical model of the tanker also includes the input actuator dynamics given by the mechanics of the rudder and the propeller. The rudder model is represented by

˙ = c − ,

(41)

which incorporates saturation for both rudder deﬂection and rudder rate at maximum values of 20◦ and 2.33◦ s−1 , respectively, for this model (see Fossen, 1994; McGookin et al., 2000, Van Berlekom & Goddard, 1972). For the propeller, n˙ =

1 (nc − n), Tm

(42)

1856

T. Çimen, S.P. Banks / Automatica 40 (2004) 1845 – 1863

where the input nc is constrained, in the same way as in the case of the commanded rudder input. The ship is assumed to be propelled by a steam turbine giving the full speed of 16 knots (8.2320 m s−1 ) at 80 rpm. (Note that the values of n and nc in (42) are given in rps, not rpm.) The rate limit for the propeller is not given since n˙ is governed by a time constant Tm = 50 (Fossen, 1994). Therefore, equations (35)–(42) represent the nonlinear mathematical model of the ESSO 190,000-dwt tanker where the motion of the vessel will be constrained as a result of the restricted motion of the actuators due to these dynamic limits.

5.2. The control problem Improving the navigational efﬁciency of vessels is not an easy task due to maneuverability difﬁculties caused by their bulk. This is mainly due to the restricted size of the rudder, which has to be deﬂected by a large amount in order to change the vessel’s course signiﬁcantly (Crane, 1973; Norrbin, 1970; Van Berlekom & Goddard, 1972). For instance, consider the ﬂow velocity at the rudder represented by (34), which in turn depends on the surge velocity (u) and the shaft velocity (n). It is easy to see that these quantities are relatively small compared to the size of the vessel, resulting in a small ﬂow velocity and turning moment. Therefore, in order to turn quickly or make a signiﬁcant change in the course of the tanker, the input for the commanded rudder angle from a control system or the helmsman/pilot will be large and may exceed the limitations of the tanker. Excessive rudder motions will result in speed loss and the saturation may also result in the uncontrollability of the vessel (Van Berlekom & Goddard, 1972). Rudder motions should be kept small by regulating the deﬂection of the rudder to alter the course of the vessel in the desired manner by using an automatic control system. Some decades ago, an analytical study published also indicated that manual control of ships would be impossible beyond a certain size (Norrbin, 1970). In actual operation, autopilot control will be used for most open-sea steaming as it provides more signiﬁcant results (Van Berlekom & Goddard, 1972). The aim is therefore to design an autopilot where the purpose of a controller is to track the desired course heading of the tanker at a commanded speed by manipulating the vessel’s rudder deﬂection angle and propeller speed (Fossen, 1994; McGookin et al., 2000). Accuracy and actuator saturation are both design criteria which the controller must satisfy. The amount of change in the heading angle is determined from the desired heading the controller needs to track, which is commanded either by a helmsman/pilot (usually in the form of step changes in the heading reference) or an autopilot (see McGookin et al., 2000). Having established that the controlled output by the autopilot is the heading angle () and propeller speed (n), a desired heading response must be deﬁned. A tanker of the size considered here will not be able to exactly track instantaneous changes in the heading given

by step commands (by the helmsman/pilot or the autopilot) due to its mass and inertia. This is parallel to practical applications because pure step inputs cannot be obtained due to time delays. The resulting heading change will be more gradual. Therefore the desired response should not be a step command but be similar to the actual response of the vessel to a step command to prevent sudden jumps in the simulations, which may cause saturation in the actuators due to their dynamic limits. In practice, a desired response would be to consider a critically damped second-order response (Fossen, 1994), where any overshoot would be damped out by the mass and inertia of the vessel. In the simulations performed in this paper, the desired heading response has been commanded by the equation d 1 − exp(−1.5 × 10−4 t 2 ) , where the large rise time has been chosen due to the limited maneuverability of the tanker. Note that the heading angle follows the sign convention where positive angles (0 < d rad) are to starboard and negative angles (− rad < d < 0) are to port. 5.3. Nonlinear optimal tracking ASRE control design One feature of the SSPA mathematical model is that principally it uses second-order terms. Most mathematical models used instead include third-order terms for the nonlinearities. The main reason for using third-order terms instead of second-order terms appears to be that mathematics are less complicated, as no absolute value terms are necessary (Van Berlekom & Goddard, 1972), since these are not continuously differentiable at all points. From physical reasoning (see Norrbin, 1970), second-order terms appear to be more appropriate for the ESSO 190,000-dwt tanker. However, since the mathematical model for the tanker includes absolute value terms of the states, the application of the nonlinear ﬁnite-time optimal ASRE control design strategy would require Lipschitz continuity of the state-modulus function. The proof for this is trivial and follows directly from the triangle inequality. Note that this is yet another improvement over common nonlinear control-design algorithms that require continuously differentiable functions f(x, u) and g(x), which, therefore, cannot be applied to nonlinear models incorporating absolute value terms of the states. The ASRE, on the other hand, can cope with such terms, since the algorithm does not require differentiability; it only requires local Lipschitz continuity. The equations of motion of the ESSO 190,000-dwt tanker (35)–(42) can be represented in state-space in the factored form A(x)x as given by (5), where A becomes a nonlinear autonomous matrix function in x, which can be chosen as one of many possible representations. Here the extra design degrees of freedom, arising from the nonuniqueness of the factored representation A(x)x, can be utilized to enhance ASRE controller performance and hence improve

T. Çimen, S.P. Banks / Automatica 40 (2004) 1845 – 1863

maneuvering ability. (A systematic procedure for choosing A(x) will be presented in a future paper.) In this application, B(x) and C(x) take the form of constant LTI matrices B and C, respectively. In order to minimize both the tracking error for a desired course heading and the rudder deﬂection angle to ensure that actuators operate within their operating limits, the control conﬁguration for the tanker is required to minimize the cost functional (6). The recursive LTV approximations of the set of equations (5) can thus be formed as stated in Section 3.1 where B(x[i−1] (t)) and C(x[i−1] (t)) are replaced by constants B and C, respectively. For the controller to provide accurate tracking, F, Q and R are chosen as diag(1, 2.5 × 104 ), diag(1, 103 ) and diag(1, 1), respectively. Note that the state and input weighting matrices Q and R are selected just to present the theory and, provided that the commanded actuator inputs remain within their dynamic limits, they do not correspond to any speciﬁc design criteria. However, the end-point weighting matrix F had to be appropriately designed in order to remove an undesired peak exhibited by the super-tanker control system at the ﬁnal time. Recall that the purpose of the terminal cost 1/2eT (tf )Fe(tf ) is to guarantee that the error e(tf ) at the terminal time tf is small. The advantage of the ﬁnite-time horizon control problem should therefore be evident here.

6. Simulations and results The simulation of the nonlinear optimal ASRE control problem for the tanker model has been performed in MATLAB䉸 by using a simple Euler numerical integration technique. An integration step-length of 0.1 s was selected, which is suitable for the tanker model due to its long response time. However, the choice of step-length should be much smaller if the application being considered has a quick response time, like that of a missile or an aircraft (see Çimen & Banks, in press). The short period motion of the ˙ ) and propeller speed tanker comprises the rudder rate ( (n), with a response time of the order of seconds. Therefore, for a clear observation of the early fast trajectories corresponding to these variables, only the initial 10 s from the simulations have been presented in the ﬁgures. Course angle, yaw rate, rudder angle, propeller rpm, and the vessel’s speed have been recorded as functions of time similar to the original SSPA Steering and Maneuvering Simulator, which consisted of recording and display devices including a multichannel recorder and x − y plotters. The x − y plotter is used to record the ship’s track. Scaling factors of 180/ and 60 have been used in the simulations to convert radians to degrees and rps to rpm, respectively. Consider simulating the sequence of approximations for a commanded vector T z(t) = d 1 − exp(−1.5 × 10−4 t 2 ) nd

(43)

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for the controller to track, where d = 90◦ and nd = 80 rpm. The ﬁrst approximation has been obtained by evaluating the nonlinear system at the initial condition

80 x0 = 8.2320 0 0 0 0 0 0 60

T ,

which results in a LTI system. After the ﬁrst approximation, the recursive approximation procedure results in LTV systems. The number of approximations required will vary for each application depending on its convergence rate, which has an upper bound 4 (t) deﬁned in (26) that depends largely on the choice of A(x), x0 , z(t) and tf . A larger choice of initial state or horizon time, for example, will cause a moderate increase in the number of approximations required to achieve convergence. For the representation of A(x) adopted in these simulation studies, the approximate systems (for tf = 250 s and for almost all d ) converged when i 20, for which x[20] (t) − x[19] (t) < 0.1. To illustrate convergence, the responses of the kinetic states and the actual rudder motion from the ﬁrst, ﬁfth and 20th approximated systems are shown in Fig. 3. Note that the response of the nonlinear tanker dynamics to the 20th approximated feedback, P[20] (t) and s[20] (t), will be exactly the same as the response of the twentieth approximated LTV system. This is because the approximate systems converged to the nonlinear system (5) and thus the converged solution represents the nonlinear system’s responses. The converged solution in Fig. 3 shows that steady-state conditions have not been reached since the surge speed is still increasing. Even though the propeller input is kept constant at the maximum speed, there is an initial decrease in the surge speed. This is due to the initial decrease in the sway velocity, since the equations for surge and sway are coupled. The surge speed eventually starts to increase towards its maximum speed of 16 knots (equivalent to the maximum propeller speed of 80 rpm) as the sway velocity approaches zero. This conﬁrms the ﬁndings of the tanker simulation study by Van Berlekom & Goddard (1972) where they claim large tankers, rarely, if ever, reach steady-state conditions while maneuvering. The nonlinear sliding mode control law proposed by McGookin et al. (2000) was able to execute a desired heading of 45◦ after 700 s by a maximum commanded rudder deﬂection of about 10◦ . In his work, Norrbin (1970) also reported that before entering a 90◦ starboard turn, the speed is required to be brought down to less than 2 knots, and the tanker will then proceed under slow acceleration by its own power. When the desired output z(t) is given by (43), commanded maneuvers at maximum speed can be achieved for any given desired heading d using the ASRE controller, thus demonstrating the signiﬁcant performance improvement that can be accomplished over neglected nonlinear dynamics. For |d |110◦ , however, the dynamic limit of the rudder rate of 2.33◦ s−1 is exceeded (up to 6◦ s−1 when d = 180◦ ) using the ASRE autopilot.

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Fig. 3. Response of the ESSO 190,000-dwt tanker sequence of states x[i] (t) for a commanded course heading of d = 90◦ and a desired propeller speed of nd = 80 rpm.

Consider increasing the rise time of the desired heading response in (43) by about 100 s, which is now commanded by the equation d {1 − exp(−7 × 10−5 t 2 )}, so that z(t) = [d {1 − exp(−7 × 10−5 t 2 )} nd ]T . For tf =350 s, d =180◦ and nd =80 rpm, the ASRE systems converge (that is, x[i] (t) − x[i−1] (t) < 0.1) when i 29. Fig. 4 shows responses of the nonlinear tanker dynamics, obtained in simulating the tanker control system for a constant water-depth of 500 m, by using the thirtieth approximated ASRE feedback, P[30] (t), and s[30] (t) (computed from the thirtieth approximated LTV system), for a commanded heading of 180◦ at the maximum propeller speed. It can clearly be seen that the ASRE automatic control system is able to execute a commanded turn extremely fast (in under 300 s) and accurately at high speeds while keeping the rudder deﬂection and the rudder rate well within their operational limits, with max | | < 8◦ and max | ˙ | < 2.2◦ s−1 , thus satisfying the saturation criteria even when d = 180◦ . This would ensure that there is plenty of additional deﬂection available if more control effort is required, in an emergency for example, to avoid coastal or hazardous waters where shallow-water regions could cause the tanker to run aground. It would also enable the tanker to reject disturbances caused by external factors such as waves, wind, etc., thus meeting the control objective and at the same time minimizing rudder usage. Now consider applying the converged control that was obtained at h = 500 m to the nonlinear tanker model, but when the simulation is conducted in shallow water. Fig. 4 also illustrates the effect of the control solution from h = 500 m for a 180◦ starboard maneuver in shallow-water-depths for

h=50 m and 20 m. It can be seen that in applying the control from deep-water effects, up to depths as low as 20 meters (recall that the ship’s draft is 18.46 m), the heading is still tracked quite accurately, indicating that the control obtained is very robust to such uncertainties in the model. The state that is most affected by the water depth is the sway velocity. However, even on this there is very little effect and it does not cause any signiﬁcant change in the desired heading or the rest of the states. When compared to the deep-water case, Fig. 4 indicates that in shallow water a speed reduction (in sway) is obtained due to the greater resistance. As the water depth decreases, the vessel will become more stable and the steady-state turning ability decreases for small rudder angles, hence the slight increase in the peak of the response of rudder angle caused in shallow-water. With regard to the yaw rate it can be concluded that in shallow water large rudder angles are necessary to obtain the same turning rate as in deep water. All these effects are quite small and insigniﬁcant up to h = 50 m, that is, just before the transi changes tion point where the hydrodynamic coefﬁcient Yuv value. For very shallow water when h = 20 m, slight differences in the responses can be noticed. Therefore, effects of conﬁnements do not in particular inﬂuence the ASRE control solution and so, for maneuvering large tankers in open sea waters, the ASRE control may be simulated for constant depths. To achieve optimal performance, however, this should be chosen as the depth the ship will most frequently maneuver in. Additional simulation studies show that, during straight course keeping, any commanded speed can be executed very efﬁciently by the ASRE autopilot, and within only a

T. Çimen, S.P. Banks / Automatica 40 (2004) 1845 – 1863

few seconds. Furthermore, commanding a decrease in the propeller speed from its initial value of 80 rpm to a desired speed of 55 rpm while maneuvering a 45◦ starboard turn, for instance, was executed successfully by keeping the commanded inputs within their dynamic limits. However, in the simulations conducted, further reductions in propeller speed while maneuvering caused saturation in the rudder rate, which was later avoided by properly weighting Q and R, trading-off with larger heading errors.

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7. Conclusions A traditional approach to control when the nonlinear dynamic equations of motion of a practical application are known is to stabilize the linearized version of the nonlinear system differential equations. The success of this approach depends heavily on the match between the actual system dynamics and the linearized approximation. Indeed, the usefulness of most linearly designed controllers for nonlinear systems would be restricted by the operational limits of the

Fig. 4. Response of the nonlinear tanker model subject to the converged ASRE feedback P[30] (t) and s[30] (t), obtained when h = 500 m, for a desired starboard maneuver of d = 180◦ at the maximum speed nd = 80 rpm.

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Fig. 4. (continued).

linear model that it is based on. Optimal control effort cannot be achieved due to the neglected dynamics of the system (during the linearization process) which has not been optimally compensated for. In this paper, a new method has been proposed to solve the nonlinear ﬁnite-time optimal tracking (and stabilization/regulator) control problem associated with a general class of continuous-time nonlinear deterministic systems and nonquadratic performance criteria. The requirement is that, without loss of generality, there is an equilibrium at the origin, so that factorization is possible, resulting in a (factored) state-space representation which may not be unique. The method introduces a sequence of linearquadratic and time-varying approximations, which globally converge to the nonlinear systems considered. The nonlinear-nonquadratic optimal control problem has, therefore, been transformed into a sequence of linear-quadratic optimal control problems, so that optimization can be carried out for each sequence, which requires solving an “approximating sequence of Riccati equations” (ASRE) by classical methods, leading to an optimal control when one exists. The local Lipschitz continuity of the commanded input vector z(t) and the nonlinear operators A(x), B(x) and C(x) has been used to prove the convergence of the optimal controls and the feedback Riccati operators in the weak form in the interval t ∈ [t0 , tf ]. If the problem is coercive, so that a unique optimum exists, the limit of the ASRE gives this optimal control. (Precise conditions are presented in a forthcoming paper.) Therefore, for Lipschitz continuous systems with a unique optimal feedback solution, the ASRE algorithm for different factorizations of the optimal nonlinear tracking problem will converge to this unique solution. In general, however, these systems may only con-

verge to some local optimum (refer to Çimen & Banks, in press). The ASRE method is very general, promising and simple to implement and applies to a wide range of nonlinear (nonautonomous) systems. The theory for the nonlinear optimal tracking ASRE control design strategy has been illustrated on a nonlinear tanker dynamics for an autopilot design that provides tracking of a time-varying commanded course heading and propeller speed. In utilizing the nonlinear optimal control law for the design and implementation of automatic control systems, control is achieved for the whole state vector by using the entire nonlinear state-space model rather than a subsystem and/or a linearized model. It has been shown that automatic nonlinear ﬁnite-time optimal ASRE control systems can be used effectively to maneuver large tankers for any desired heading and so they are considered as effective alternatives to the manual open-loop operation of a tanker. Because of the powerful effect that the proposed optimal control system can have upon the maneuverability of such super-tankers, its utility in evaluating the controllability of these ships could signiﬁcantly improve the navigation of such tankers, thus beneﬁting from safer and the cheapest transportation of crude oil. The proposed nonlinear optimal control problem is a ﬁxed-interval problem, in that the ﬁnal time for the integral in the cost functional must be speciﬁed. This makes realtime implementation of the ASRE technique difﬁcult, due to the need for powerful computer hardware and software, because of the sequence of recursive approximations required. Indeed, the ASRE procedure can be performed offline, which would require pre-computation and storage of computed data for control implementation, and then implemented by, for example, scheduled-type control. However,

T. Çimen, S.P. Banks / Automatica 40 (2004) 1845 – 1863

for investigations of the maneuverability of large tankers, real-time simulations provide an accurate and versatile tool. Real-time simulation is the method of investigating ships’ maneuverability that has been used at the Swedish State Shipbuilding Experimental Tank in Goteborg, Sweden and elsewhere. Therefore, further research is essential to develop a similar control algorithm, ideal for real-time computer implementation. The extension of the ASRE control-design framework to more general systems, including partially observable stochastic nonafﬁne controlled nonlinear systems, is also under investigation by the authors and will be presented in a future paper. Appendix A. Nomenclature

m L B Td ∇ CB g Izz kzz xG t td T Tm xp yp

u v r

n c h

J X

mass of ship, kg ship length between perpendiculars, m ship beam, m ship draft to design waterline, m ship volume of displacement, m3 block coefﬁcient: CB = ∇/(LBT d ) mass density of water acceleration due to gravity: g = 9.81 m s−2 moment of inertia with respect to z-axis radius of gyration distance of ship center of gravity from origin of coordinate system time, s thrust deduction factor propeller thrust time constant, s earth-ﬁxed x-coordinate of the tanker, m earth-ﬁxed y-coordinate of the tanker, m yaw (heading) angle of ship, rad surge velocity (ship speed in xp direction), m s−1 sway velocity (ship speed in yp direction), m s−1 ˙ , rad s−1 yaw rate: r = drift angle: = v/u actual rudder angle, rad actual propeller angular (shaft) velocity, revolutions per second (rps) ﬂow velocity at rudder, m s−1 water depth, m restricted water depth parameter: = Td /(h − Td ) body mass density ratio: = m/(∇) cost (or performance) functional to be minimized surge hydrodynamic force

Y N x u y z e Subscripts c d f 0 Xu , Yv , Nr , . . .

Superscripts [k] T (˙)

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sway hydrodynamic force yawing hydrodynamic moment n-dimensional state vector of dynamic system m-dimensional system control input vector l-dimensional system output vector l-dimensional desired output vector l-dimensional tracking error vector state transition matrix of linear dynamic system commanded input value desired value ﬁnal value initial value surge, sway and yaw hydrodynamic derivatives/coefﬁcients: *X *Y *N , , , . . . *u *v *r kth approximating sequence vector or matrix transpose time derivative the “Bis” system used to denote the non-dimensionalized variables

Appendix B. ESSO-190,000-dwt tanker data For the ESSO 190,000-dwt tanker the principal particulars are given in Table 1 (Van Berlekom & Goddard, 1972). The hydrodynamic coefﬁcients follow standard hydrodynamic notation and are given here for two auxiliary equations and three principal equations in Tables 2 and 3, respectively. Here YT represents the propeller side force Table 1 Principal particulars of the ESSO 190,000-dwt tanker L B Td ∇ CB Ship’s full speed Nominal propeller Dead weight

304.8 47.17 18.46 220,000 0.83 16 knots 80 rpm 190, 000 tons

Table 2 Hydrodynamic coefﬁcients for auxiliary tanker equations T -Equation = −0.00695 Tuu = −0.000630 Tun = 0.0000354 T|n|n

c-Equation cun = 0.605 cnn = 38.2 c = 0}n < 0

n0

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Table 3 Hydrodynamic coefﬁcients for principal tanker equations X-Equation

Y -Equation

N-Equation

1 − Xu˙ = 1.050 = −0.0377 Xuu = 2.020 1 + Xvr = 0.300 Xvv Xc|c| = −0.093

Xc|c| = 0.152 td = 0.22

1 − Yv˙ = 2.020 − 1 = −0.752 Yur = −1.205 Yuv = −2.400 Yv|v| Yc|c| = 0.208

Yc|c| = −2.16 ||| | YT = 0.04

2 − N = 0.1232 kzz r˙ − x = −0.231 Nur G = −0.451 Nuv = −0.300 N|v|r Nc|c| = −0.098

Nc|c| = 0.688 ||| | NT = −0.02

Yv˙ = −0.387 = 0.182 Yur   −0.85 1 − 0.8 , 0.8 Yuv =  < 0.8 0, Yv|v| = −1.50 Yc|c| = −0.191 ||| |

Nr˙ = −0.0045 = −0.047 Nur

Additional terms in shallow water ( = 0) Xu˙ = −0.05 = −0.0061 Xuu = 0.387 Xvr Xvv = 0.0125

and NT represents the propeller yaw moment. Shallow-water effects are accounted for by the parameter . The hydrodynamic coefﬁcients and their derivation will not be further discussed here, as the subject is not within the scope of this paper. References Athans, M., & Falb, P. L. (1966). Optimal control: An introduction to the theory and its applications. New York, McGraw-Hill. Banks, S. P. (1986). On the optimal control of nonlinear systems. Systems and Control Letters, 6, 337–343. Banks, S. P. (1992). Inﬁnite-dimensional Carleman linearization, the Lie series and optimal control of nonlinear partial differential equations. International Journal of Systems Science, 23, 663–675. Banks, S. P. (2001). Exact boundary controllability and optimal control for a generalized Korteweg de Vries equation. International Journal of Nonlinear Analysis, Methods and Applications, 47, 5537–5546. Banks, S. P. (2002). Nonlinear delay systems, Lie algebras and Lyapunov transformations. IMA Journal of Mathematical Control and Information, 19, 59–72. Banks, S. P., & Dinesh, K. (2000). Approximate optimal control and stability of nonlinear ﬁnite- and inﬁnite-dimensional systems. Annals of Operations Research, 98, 19–44. Banks, S. P., & McCaffrey, D. (1998). Lie algebras, structure of nonlinear systems and chaotic motion. International Journal of Bifurcation and Chaos, 8(7), 1437–1462. Banks, S. P., & Mhana, K. J. (1992). Optimal control and stabilization for nonlinear systems. IMA Journal of Mathematical Control and Information, 9, 179–196. Banks, S. P., Salamci, M. U., & McCaffrey, D. (2000). Nonlocal stabilisation of nonlinear systems using switching manifolds. International Journal of Systems Science, 31(2), 243–254. Banks, S. P., & Yew, M. K. (1985). On a class of suboptimal controls for inﬁnite-dimensional bilinear systems. Systems and Control Letters, 5, 327–333. Brauer, F. (1966). Perturbations of nonlinear systems of differential equations. Journal of Mathematical Analysis and Applications, 14, 198–206.

= −0.241 Nuv N|v|r = −0.120 Nc|c| = 0.344 ||| |

Brauer, F. (1967). Perturbations of nonlinear systems of differential equations, II. Journal of Mathematical Analysis and Applications, 17, 418–434. Çimen, T., 2003. Global optimal feedback control of nonlinear systems and viscosity solutions of Hamilton–Jacobi–Bellman equations. Ph.D. thesis, The University of Shefﬁeld, United Kingdom. Çimen, T., & Banks, S.P. Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria. Systems and Control Letters, in press. Crane, C. L. (1973). Maneuvering safety of large tankers: Stopping turning and speed selection. Transactions of SNAME, 81, 218–242. Fossen, T. I. (1994). Guidance and control of ocean vehicles. Chichester, Wiley. McCaffrey, D., & Banks, S. P. (2001). Lagrangian manifolds and asymptotically optimal stabilizing feedback control. Systems and Control Letters, 43, 219–224. McGookin, E. W., Murray-Smith, D. J., Li, Y., & Fossen, T. I. (2000). Ship steering control system optimisation using genetic algorithms. Control Engineering Practice, 8, 429–443. Mracek, C. P., & Cloutier, J. R. (1998). Control designs for the nonlinear bench-mark problem via the state-dependent Riccati equation method. International Journal of Robust and Nonlinear Control, 8, 401–433. Norrbin, N.H. (1970). Theory and observation on the use of a mathematical model for ship maneuvering in deep and conﬁned waters. Eighth Symposium on naval hydrodynamics, Pasadena, California (pp. 807–904). Also available as Swedish State Shipbuilding Experimental Tank Publication, No. 68, Gothenburg, Sweden. Salamci, M. U., Özgören, M. K., & Banks, S. P. (2000). Sliding mode control with optimal sliding surfaces for missile autopilot design. AIAA Journal of Guidance, Control, and Dynamics, 23(4), 719–727. Slotine, J. -J. E., & Li, W. (1991). Applied nonlinear control. New Jersey, Prentice-Hall. Sznaier, M., Cloutier, J., Hull, R., Jacques, D., & Mracek, C. (2000). Receding horizon control Lyapunov function approach to suboptimal regulation of nonlinear systems. AIAA Journal of Guidance, Control, and Dynamics, 23(3), 399–405. Tomas-Rodriguez, M., & Banks, S. P. (2003). Linear approximations to nonlinear dynamical systems with applications to stability and spectral theory. IMA Journal of Mathematical Control and Information, 20, 89–103. Van Berlekom, W. B., & Goddard, T. A. (1972). Maneuvering of large tankers. Transactions of SNAME, 80, 264–298.

T. Çimen, S.P. Banks / Automatica 40 (2004) 1845 – 1863 Tayfun Çimen was born in Ankara, Turkey on 15 July 1979. He received the B.Eng. degree with First Class Honors in Computer Systems Engineering in July 2000, and the Ph.D. degree in Systems and Control Theory in September 2003, both from the Department of Automatic Control and Systems Engineering at the University of Shefﬁeld, United Kingdom. Since September 2003, he has been involved in research work at the same Department as a Postdoctoral Research Associate in the stability and design of nonlinear marine and aircraft control systems. His current research interests are in the areas of continuous-time nonlinear, optimal, deterministic, and stochastic estimation and control theory, with emphasis in modeling, simulation and control of complex nonlinear dynamical engineering systems, particularly those related to aerospace and maritime applications.

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Stephen P. Banks received the B.Sc., M.Sc. and Ph.D. degrees from the University of Shefﬁeld, United Kingdom in 1970, 1971 and 1973, respectively. He spent some time as a Research Associate in the Control Theory Centre, University of Warwick and is currently Professor of Systems Theory at the Department of Automatic Control and Systems Engineering, University of Shefﬁeld. He is the Editor of the IMA Journal of Mathematical Control and Information and Associate Editor of Computational and Applied Mathematics. His research interests are in nonlinear systems theory (both ﬁnite and inﬁnite dimensional), optimal control, delay systems, Hamilton-Jacobi-Bellman equations and viscosity solutions, and chaos and knots in dynamical systems. He has published over 200 papers and 5 books.