Ship Motion Control

These specific designs provide a good overview of the problems faced by designers of ship motion control systems and thus serve well as an example-driven introduction to the field. This continuing interest is a result of the significant influence that the control strategy has on the performance and the issues related to control system design.

The Fundamental Problem of Ship Motion Control

This chapter provides a brief introduction to the fundamental problem of controlling a ship's motion and issues related to the design of a control system. The solution to the problem of controlling the movement of the ship depends on the requirements of the individual operations carried out by the ship, e.g. transit, dynamic positioning, mooring assistance, diving (for underwater vehicles), and is based on the interconnection of three systems. as shown in Figure 1.1.

Ship Motion Control Problems and Control Designs

Mathematical Models for Control

The purpose of these models is to describe the behavior of the system as accurately as possible. Thus, the HFi models can be used to assess both the quality of the controller and the quality of the CD models.

State-space and Input-output Models Revisited

State-Space Models

If the value of all the components of the state vector are necessary and sufficient to determine the value of any variable in the system at a time (together with the knowledge of the input at that time), the state vector is said to be the minimal state vector [156 ]. The number of components in any minimal state vector is the order of the system, which is a unique characteristic of the system.

Laplace-Transform Models

When the matrices in (1.6) are constant in time, the model or system is said to be a Linear Time Invariant (LTI) model or system.

Computer-Controlled Systems

From a control design perspective, one can perform the continuous-time design and then implement a discrete-time approximation of the resulting controller. However, in many cases, such models may not be possible to obtain due to the high complexity of the systems we want to control.

The Road Ahead

Ship Modelling for Control

Basic Hydrodynamic Assumptions

Fluid Flow and Continuity
Material Derivative
Navier-Stokes Equations
Potential Flows and The Bernoulli Equation

Regular Waves in Deep Water
Encounter Frequency
Ocean Waves and Wave Spectra

Statistics of Wave Period
Statistics of Maxima
A Note on the Units of the Spectral Density

Standard Spectrum Formulae
Linear Representation of Long-crested Irregular Seas
The Encounter Spectrum
Short-crested Irregular Seas
Long-term Statistics of Ocean Waves
Simulation of Wave Elevation
Reference Frames
Vector Notation
Coordinates Used to Describe Ship Motion

Manoeuvring and Seakeeping
Manoeuvring Coordinates and Reference Frames
Seakeeping Coordinates and Reference Frames
Angles About the z-axis

Velocity Transformations

Rotation Matrices
Kinematic Transformation Between the b- and the
Kinematic Transformation Between the b- and the

An Overview of Ship Modeling for Control
Seakeeping Theory Models

Equations of Motion and Hydrodynamic Forces in the
Wave Force Response Amplitude Operator (Force RAO) 66
Ship Motion Spectra and Statistics of Ship Motion
Time-series of Ship Motion using Seakeeping Models

Manoeuvring Theory Models

Rigid Body Dynamics in the b-frame
Manoeuvring Hydrodynamics
Nonlinear Manoeuvring State-space Models
Linear Manoeuvring State-space Models

A Force-superposition Model for Slow Manoeuvring in a Seaway 86

Seakeeping Model in the b-frame
A Uniﬁed Nonlinear State-pace Model

Geometry of Fin and Rudder Hydrofoils
Hydrodynamic Forces Acting on a Foil
Unsteady Hydrodynamics
Forces and Moments Acting on the Hull

Rudder

Rudder-Propeller Interaction

Fins

Hydraulic Machinery
Part I Summary and Discussion

It discusses the main wave characteristics relevant to ship motion control system design and presents various modeling and simulation techniques. This chapter presents a new state space model for maneuvering in a seaway, which is believed to be the basis for a new generation of model-based ship motion control systems.

Introduction to Ship Roll Stabilisation

Eﬀects of Roll Motion on Ship Performance

Damping or Stabilising Systems?

Ship Roll Stabilisation Techniques

Gyroscopes
Bilge Keels
Anti-rolling Tanks
Active Fin Stabilisers
Rudder Roll Stabilisation RRS

A Note on the Early Days of Ship Roll Stabilisation

Reduction of Roll at Resonance—RRR
Reduction of Statistics of Roll—RSR
Reduction of Probability of Roll Peak Occurrence–RRO 128

Seakeeping Indices Aﬀected by Roll

Lateral Force Estimator—LFE
Motion-induced Interruptions–MII
Motion Sickness Incidence—MSI

Implications for Stabiliser Control System Design

Part II Summary and Discussion

Performance Limitations in Feedback Control with

Introduction to Fundamental Limitation in Feedback Control

This became the foundation of the frequency-domain approach to fundamental constraint analysis in feedback control system design. Consider the linear feedback system shown in Figure 8.1, in which G(s) is the transfer function of the device or system to be controlled and C(s) is the transfer function of the controller. The scheme shown in Figure 8.1 can be taken as a simplified version of the block diagram of the stabilizer control system.

One way to interpret the steering effect is to analyze the characteristics of the feedback structure shown in Figure 8.1 in the frequency domain. The motivation example above demonstrates the delicate interplay between the various parts of a control system and the basic constraints and trade-offs that a planner typically faces.

Non-minimum Phase Dynamics in Ship Response

Note that special plant characteristics (eg delays, unstable poles and non-minimum phase zeros) were not mentioned. The torque response to this steering input is quick due to the smaller moment of inertia and damping in rolling relative to cornering. The rest of the oscillations in the roll response shown in Figure 8.4 appear due to a complex conjugate pole pair in the wheel-wheel response.

From Figure 8.6 we see that the location of the NMP zeroq1in (8.7) is affected by the speed. We can also see from this figure that the damping of the complex poles in the roll response increases with speed.

Deterministic SISO Performance Limitations of RRS

Sensitivity Integrals–Frequency Domain Approach
Performance Trade-oﬀs of Non-adaptive Feedback

Furthermore, due to the weighting factor in the above integral, this surface equilibrium must be achieved over a limited frequency band, which depends on the position of the MNP zeros. Expression (8.16) can also be used to analyze the worst location of the interval [ω1, ω2] of sensitivity reduction relative to the NMP zero. The above description of the interference attenuation problem is formulated from a deterministic point of view.

The above discussion demonstrates the limitations of controllers that aim for a large reduction in sensitivity in the frequency range where the NMP is zero and uncertainty about the actual disturbance frequencies. If the controller is designed to reject interference at some fixed range of frequencies (usually around the natural rotation frequency), it is very likely that there will be cases (combinations of wave frequency, ship speed and heading). ), where the interference power will be in the interference amplification range.

Stochastic SISO Performance Limitations of RRS

Limiting Optimal Control Performance Limitations
Stochastic SISO Results and RRS

Cheap control is a case of limiting optimal control where the weight of the control effort in the cost is (or tends to) zero. Using the Fourier transform pair between the autocorrelation and the power spectral density, the variance of the output can be expressed as. The result given in Theorem 8.2 is a new characterization, in stochastic terms, of the trade-offs given in the previous sections by means of integrals on the sensitivity function.

The same effect is now quantified in the result of Theorem 8.2, in which the optimal controller attempts to eliminate the effect of the disturbance on the output. However, the ability to do so is limited when most of the perturbation energy is concentrated near zero NMP (ie, when PSD or |H(s)|has large values ats=q).

Optimal Roll Reduction vs. Yaw Interference Trade-oﬀ

SITO Control Problems in the Frequency Domain
Limiting Stochastic LQR

The PSD Sdφdφ(ω) of the WSS stochastic disturbance dφ allows a rational approximation so that it can be represented by filtered white noise, i.e. By defining the cost in this way, the parameterλ weighs the importance of minimizing the variance of one output relative to that of the other. Unfortunately, the direction of the NMP zero associated with Gφα(s) is canonical, due to the rank of the installation for the SITO case under consideration, meaning that the integral constraint affects only Sφφ(s) – see [173]. .

In other words, the trade-off associated with the roll sensitivity function cannot be alleviated by considering a multivariable design approach. Thus, Poisson's integral constraint applies in the SISO case. Therefore, we can expect other more realistic controllers to provide performance on top of the curves shown in Figures 8.11 and 8.12.

Comments on the Applicability of Rudder Stabilisers

The only feasible modification may then be the position of the rudder relative to the CG. In fact, it follows from all of the above that the greater the vertical distance from the center of rudder pressure to the CG, the better. Likewise, the shorter the longitudinal distance from the center of rudder pressure to the CG, the better.

The rear of the ship must be able to withstand the large forces and moments generated by the rudder. This is to specify the stirring machinery; in practice the maximum angle will be set depending on the speed of the vessel and may be limited to less than the stall angle.

NMP Dynamics in Fin Stabilizers

In the previous chapter we saw that if we consider the problem of disturbance rejection for linear systems at a single frequency, there are no limits on the achievable closed-loop performance. In this context, good performance is associated with a reduced value for the magnitude of the sensitivity transfer function, and no limitations in the achievable performance refer to the fact that the effect of the disturbances on the output can be completely eliminated. This is accomplished by forcing the magnitude of the sensitivity function to be zero at the specific frequency of interest.

In these cases, good performance can be associated with a reduced variance of the output of the closed-loop system, and eliminating the effect of the disturbance on the output is not always possible. These limitations are the result of the feedback structure and the dynamic properties of the plant, especially non-minimum phase dynamics.

Input Constraints and Saturation Eﬀects

Indeed, we have discussed situations where the reduction of output variance is limited, and these limits apply regardless of how much power is available to implement the control action. In this chapter we analyze what additional limitations and design compromises arise due to limited control action (constraints) and actuator saturation effects. If the dynamics of the actuator are fast compared to the dynamics of the system, the effect of a saturating actuator can be modeled using the static function sat∆(·) defined (for the scalar control case) as.

Saturation usually occurs in situations where the control command is large and the control strategy ignores the fact that the actuator has limited power. Control strategies that avoid saturation effects or limit degradation due to saturation are usually called constrained control strategies because the control command is limited by design to avoid saturation.

Input Constraints and Performance at a Single Frequency

Magnitude Limitations
Rate Limitations

We can then use (9.3) to determine, for example, what value of ¯d0 would result in saturation effects, and also what implications this might have on performance if we want to avoid saturation by reducing the control command such that the system remains linear. The amplitude ¯d0∗ of the largest perturbation that we can possibly completely reject without saturating the actuator can be determined from (9.3) and the constraint (9.1) as. If the performance is measured as the percentage reduction in the output amplitude of the closed loop system relative to that of the open loop system, then we can express the performance by. where ¯y is the steady-state amplitude of the output.

9.7) Expression (9.7) shows an explicit relationship between the performance of the closed-loop system and the size limitation. Under similar assumptions as in the case of a magnitude constraint, we have that the amplitude ¯d0∗∗ is of the largest output disturbance that we can possibly completely reject without saturating the actuator in rate.

Application to Rudder-Based Stabilizers

The pitch disturbances correspond to a wave height of 4 m, and the maximum rudder angle limit is 25 degrees and the rudder speed is 12 degrees/s.

Stochastic Approach: Variance Constraints

IVC Optimal Control Problem Formulation
IVC Application to RRS

Given a matrixQ= diag(λ1, ..., λp), we look for the stabilizing state feedback controlleru=Kx, (i.e. A+BKis Hurwitz) that minimizes a weighted output variance, i.e. The IVC problem is always feasible because any specified bound on the input energy or input variance (var[ui] =< γi2) is achievable because no performance limit is required on the cost (9.11) (we simply minimize this cost without any constraint.). The algorithm given above solves an equivalent LQR problem, where Q is given and the weighting matrix Ris varies such that the variance of the input is adjusted to be close to the specified tolerance for the constraint. 9.13) Consequently, the IVC problem defined above can be used to solve a reduced version of the SISO and SITO problems from Chapter 8, provided that the system given by (9.10) is the extended system to control the factory and the disruption model – see section 8.5.2.

In the case of rate constraints, we can use the same algorithm by expanding the condition with integrators.

Part III Summary and Discussion

Control System Design for Autopilot with Rudder Roll

Rudder Roll Stabilisation in the 1970s

Rudder Roll Stabilisation from 2000 to 2004

Work on Fin and Combined Rudder and Fin Stabiliser Control 204

Constraint Classiﬁcation

Diﬀerent Approaches to Constrained Control Problems

Finite-horizon Sequential-decision Problems

Inﬁnite Horizons and Receding-horizon Implementation

Model Predictive Control

Constrained Linear Systems

Explicit and Implicit Implementations of QP-MPC

Stability of Model Predictive Control

Constrained Control of Uncertain Systems

Overview of Autopilot Functions and their Inﬂuence on

RRS: A Challenging Control Problem

Control System Architecture

Control Design Models

Control to Motion Model
Wave-induced Motion Model

Disturbance Parameter Estimation and Forecasting

Observer Design: State Estimation and Wave Filtering

Autopilot Control System Design

Autopilot Control Problem and Assumptions for the Design

A Model Predictive Control Solution

Performance of Model Predictive RRS

Choosing the Prediction Horizon
Penalising Roll Acceleration in the Cost
Case A: Beam Seas at the Top of Sea State 4
Case B: Quartering Seas at the Top of Sea State 5
Case C: Bow Seas at the Top of Sea State 5
The Role of Adaptation
A Comment About the Simulation Results

Performance and Control of Rudder and Fins

A Model for Fin Stabilizer Control Design

Output Constraints to avoid Dynamic Stall

A MPC Fin-Stabiliser Controller

Numerical Simulations

Integrated Control of Rudder and Fins

Summary and Discussion

State Estimation via Observers

Kalman Filtering

Optimality of Kalman Filters

Correlated Disturbances

Practical Kalman Filter: Tuning

Steady State Kalman ﬁlter

Implementation Issues

Hull Shape

Adopted Reference frames

Principal Hull Data and Loading Condition

Rudder, Fins and Bilge Keels

Manoeuvring Coeﬃcients and Motion RAO