SHIP ROLL STABILIZATION VIA SWITCHED CONTROL SYSTEM

Transcription

1 SHIP ROLL STABILIZATION VIA SWITCHED CONTROL SYSTEM Anna-Zaïra Engeln, Ali J. Koshkouei, Geoff Roberts, Keith Burnham Control Theory and Applications Centre, Coventry University, Coventry CV1 5FB, UK {a.engeln, a.koshkouei, Keywords: Ship roll stabilisation, switched systems, PID controllers, hybrid systems, parallel multi-model control system. Abstract Switched control systems consist of a set of subsystems (models) in which for each subsystem (mode), a controller is designed so that stability of the entire system is achieved. In a switched system, termed a parallel multi-model control (PMMC) system, the number of controllers and subsystems is the same. In fact, PMMC is a hybrid system consisting of a family of subsystems with a set of logical rules (decisionmaking process) orchestrating between them. The activation of the switching mechanism between controllers is dependent on the dynamic behaviour of the system. A controller is active for a period of time and then switched to another depending on the operating condition of the system. The period of time for switching, for stabilisation of ship motion may vary. The stabilization of ship roll using a family of switched controllers is considered in this paper. The system includes subsystems (models) resulting from environment and different positions of roll, rudder and fin as well as the sea waves. Controllers are designed to stabilise the entire system for all conditions of a ship s journey. Switching of models of the system and switching of controllers are not necessarily achieved simultaneously and there may be some delays for switching from one controller to another. In this paper, use is made of a family of PID controllers, each being designed for a specific condition including/accommodating measured changes in the ship model and/or changes in the environmental conditions. 1 Introduction A PMMC system consists of a family of subsystems with a class of switched controllers so that each subsystem corresponds to a specific controller. Therefore, each controller is designed based on a certain model corresponding to an operating condition. Well-designed PMMC systems should have superior performance over single-controller systems. The stability of such switched systems and design of the decision-making algorithm has attracted a great deal of interest in the recent years, see for example [4-6,10]. There are various approaches to make decisions for switching controllers including the following approaches: (i) minimising an explicit cost function to initialise switching, where the parameters to be minimised are functions of performance and control effort; and (ii) an intelligent approach where switching is initialised in a heuristic manner, based upon interrogation of a set of logical rules. The switched controllers could be optimised to meet normal design considerations, or may be optimised for particular operating conditions. A combination of the availability of inexpensive very fast microprocessors and the relatively s1ow sampling times for ship motion stabilisation means that it would be possible to use almost any of the control designs currently available. A limiting factor is the time required for the sampling process and evaluation of the control algorithms. The development of new algorithms for the evaluation of potential control procedures and the predictive cost function techniques to improve the PMMC decision making process are identified as new techniques/technologies required for a future PMMC system. Control of a ship system has been widely studied in the recent years [1-3,8,9]. The behaviour of these systems is nonlinear, however to simplify the problem, a linear approximation for each subsystem is applied. There are models such as for roll, yaw, engine (speed), rudder as well as for the sea wave behaviour, involved in the control of a ship. These models/behaviours will all interact in some interval of time. These individual sub-models need to be controlled using different controllers at each interval of time. Therefore, for each sub-model, a control system is designed. The number of controllers is based upon the number of sub-models of the system. Switching from one controller to another needs logical rules regarding the environment and different conditions of the ship. Selecting different controllers with respect to the different sub-models of the system requires intelligent/logical rules. Applying these rules may not guarantee that the controllers are simultaneously switched when the associated sub-models of the systems become appropriate. The time interval that a controller is active may differ from the time that an associated sub-model of the system is appropriate. Therefore in a finite time, there are sequences of the switching of system sub-models and a sequence of switching controllers, and it is this overall switching sequence, which is required to be optimised. Switched systems are a class of hybrid systems consisting of a set of continuous- or discrete-time subsystems. If all the models of the system are stable, the sub-systems may not necessarily be stable. Additional conditions are required to ensure the stability of the system [10]. For example, if a controller is switched (or remains active) with a nonassociated sub-model, the behaviour of the overall system may be unstable. Furthermore, since the overall system is

2 nonlinear, an inappropriate switching mechanism may cause the overall system to become chaotic. Therefore, the switching problem of the controllers regarding the various sub-models of the system and the time that they need to remain active before switching to an alternative controller are important factors. The task of a control system supervisor (or decision maker) is to determine which controller should be selected at a certain time depending on environmental operating conditions. The control system supervisor will need to allow a sufficient time for a switched controller deactivation/activation to mitigate the transition effects. A necessary condition for asymptotic stability of the system under arbitrary switching is that all the individual subsystems are asymptotically stable [6]. However, this condition is not sufficient for stability of the switched system. The stability, analysis and control synthesis for switched model controllers have been considered in [4], in which output feedback controllers have been designed for subsystems so that asymptotic stability of the closed-loop system is achieved. Some work has also been done on sea spectra and wave prediction [1,9]. This study is very useful because the sea wave is a very important component in designing controllers and the stabilisation of ship roll. In this paper, the stabilisation of ship roll as a PMMC system is considered by designing a switching mechanism for a family of PID controllers. To stabilise the system, different controllers are activated to accommodate measured changes in the ship model and/or changes in the environmental conditions. In this work, controller design methods for ship roll stabilisation is also considered utilising integrated control of rudder and stabilising fins. The decisionmaking is based upon the output error of the system and the outputs of the given models (subsystems). The first controller is selected initially and it is switched to an alternative controller if the output error becomes significantly large. The decision-making process checks the output error to ensure that the active controller is the best controller for the current model. The paper is organised as follows: Section 2 describes the architecture of a general PMMC system and ship motion stabilisation is considered as an application of PMMC. The structure of models and the decision maker component are considered in Sections 2.1 and 2.2, respectively. Other components that affect the stabilisation a ship motion are studied in Section 2.3. The control design is addressed in Section 2.4. Section 3 deals with simulation results and conclusions are presented in Section 4. 2 PMMC for ship motion stabilisation An indication of possible benefits from adopting a PMMC approach for warship motion stabilisation was given in [3] where the potential improvements in the ability of a ship to perform an anti submarine warfare mission in the North Atlantic has been illustrated. In general, ship motion stabilisation primarily concerns the minimisation of roll motion (roll stabilisation), although in some instances minimisation of lateral forces (LFE stabilisation) [8] may be the goal. Such control is normally achieved through the use of actively controlled stabilising fins. However, this motion control can also be achieved through controlled rudder movements (rudder roll stabilisation (RRS)) or by a combination of rudder and stabilising, fins (integrated control). In addition, because ship motion stabilisation may be considered as a slow system, slow switching between controllers in a PMMC system would be appropriate. 2.1 The models (subsystems) The PMMC structure depicted in Fig. 1, such that models 1 to N are connected to controller pairs 1 to N when the ship is subject to environmental conditions, i.e. wind and waves, provides the basis for initial simulation. Each controller pair consists of a controller for the fins and one for the rudder. Fig. 1: A block diagram of a PMMC (switched) system using the error outputs. In Fig. 1 the term Actual System represents the real ship. Each model consists of various including fin-roll, rudder-roll, rudder-yaw and disturbance including see wave and environment (see Fig. 2). These are represented by second order linear systems T ( s) = s The desired roll angle is zero. 2 as b 2ξω ns ω As far as the application of PMMC ship motion stabilisation (including fin, rudder or rudder and fin stabilisation), is concerned, any designs for ship motion stabilisation should consider an integrated system. Before considering the benefits to be accrued from PMMC it is useful to explore the reasons for considering PMMC. For simplicity, only roll stabilisation will be considered. The arguments presented are equally applicable to LFE stabilisation. The basis for using PMMC for roll stabilisation is because of the difficulty in designing a controller (the control algorithm) that achieves satisfactory roll stabilisation for the complete range of ship operating conditions. 2.2 Decision Maker The decision maker block involves two sub-blocks; the first part makes decisions to determine which model yields the minimal output error and which controller is the most appropriate to be activated; the task of the second sub-block is switching smoothly to the appropriate controller. One can consider the following facts for making decisions for switching smoothly from one controller to another: 2 n

3 Desired roll - Fin controller Fin servo Fin/roll Wave/roll TF roll RRS controller Rudder/roll Disturbance Desired heading - Autopilot Rudder servo Rudder/yaw Wave/yaw TF yaw Fig. 2: The block diagram of the ship system. Sea conditions vary relatively slowly in a particular sea area. Therefore, wave disturbances can be considered as being reasonably constant for a period of time. For a period of time and for many ship operations, a course is steady with constant speed. So for this period of time, the spectrum of the wave disturbances can be assumed to be reasonably constant. Thus initialisation of the PMMC decision maker would occur at regular time intervals, say once per hour or half an hour. Control surfaces at low speeds are ineffective. Therefore, for ship speeds less than 6 knots the PMMC system would be operating but the outputs to the actuators (stabilisers and rudder) would be inhibited. The decision making process would only be taken on current performance rather than current and past performances. The main drawback from predictive simulation models is the need to predict realistic input commands for the simulation model. However, for motion stabilisation, this problem disappears as the desired set point (input command), is always zero. The decision making process again could be accomplished either by the explicit (cost function minimisation) or implicit (soft computing) approach, suitably modified to consider predicted rather that historical data. A control strategy, which is designed to respond to future rather than past conditions, is clearly attractive, but technically more challenging. The key to its success will be the prediction integrity and although a Kalman-type predictor could be used to estimate future ship roll motions, a fast simulation would require predicted sea state conditions for the environmental disturbance inputs to the model. Initialisation of the decision maker would be necessary if there were significant changes in ship speed or course, as these changes would affect ship roll motion and the effective spectrum of the waves. In this case, initialisation of the decision maker would occur if, for example, there was a speed change of more than 3 knots or a course change of greater than Environmental conditions and waves Ship roll motion is affected by the environmental conditions of wind and waves, and this is considered to be very much the heart of the problem. Sea disturbances are stochastic and their underlying energy spectrum is a function of wave height and frequency components. These are factors that vary quite considerably on a local basis but also have different characteristics depending where the ship is operating in mid-ocean or coastal, north or south hemisphere etc. The situation is exacerbated by the fact that even when a ship is operating in a relatively constant sea state (which in any case is rare) the encounter angle, which the ship makes with the direction of the sea will change the encounter frequency. This is particularly significant for ships, which regularly and routinely undertake manoeuvres as part of their normal operations. A suitable way to demonstrate this is to consider a fin control algorithm designed to minimise ship roll motion for a beam sea having a typical North Atlantic wave energy [2]. Two possible problems with controller performance are how well the control algorithm will work with quartering and following seas and how well the control algorithm will work when the ship is operating elsewhere where wave energy spectra is very different to that of the North Atlantic. An interesting example of this problem was presented in [2], where the authors discovered that a ruderto-roll stabilisation (RRS) controller designed and evaluated using the universally accepted Bretschneider wave energy spectrum (based on observations of North Atlantic wave spectra) gave unsatisfactory results when implemented in a ship operating off the Danish coast. Redesign and evaluation of the RRS controller using sea spectrum obtained from local measurements resulted in successful trials. Figure 3 illustrates a block diagram of a ship control system including rudder, roll, yaw and controllers.

4 2.4 Control design Ship roll stabilisation, or ship motion stabilisation, may therefore be described as designing a control algorithm for a nonlinear dynamic system operating in a changing and unpredictable environment. Ship motion stabilisation with PMMC structure, is shown in Fig. 1, in which different controllers are switched into operation to accommodate measured changes in the ship model (which would include changes environmental conditions outlined above). The switched controllers may either be optimised to meet normal design considerations but may have specific qualities at different operating conditions or may be optimised for particular operating conditions. Each controller block involves a pair of controllers: fin-toroll and rudder-to-roll controllers. Whilst a comprehensive and meaningful design study for ship motion stabilisation using PMMC would not only need to address the issues raised in the preceding sections but would also need to investigate which of the many control paradigms available are best suited for the motion stabilisation problem, it should be noted that consideration is restricted here to switched PID controllers. For each subsystem, a PID controller with the transfer function K I G ( s) = K K s C P D s is designed. For each model, a pair of controllers, fin-toroll and rudder to-roll are designed control to stabilise the system. changes over time, e.g. due to course changes. The speed is constant at 13 knots. The middle plot shows the controller indices (1 for 0, 2 for 30, 3 for 60 etc) wave encounter angle at constant speed). The dotted line represents the index of the actual wave encounter angle that changes during course change (from wave encounter angle 90 with controller No 4 to 30 wave encounter angle with controller No 2). It changes again when the wave encounter angle increases to 60 (controller No 3). The solid line represents the controller number estimated by the system: It does not change during course change, but allows the system time to settle down afterwards and then switches correctly from controller No 4 (90 ) to controller No 2 (30 ). For the second course change (starting at 400 seconds) again the estimated controller number follows the index of the wave encounter angle correctly (middle plot of Fig. 4) after the system is settled and re-estimation of the wave encounter angle is completed. The lower plot shows the roll angle in degrees of the ship during this simulation. The reason for the roll angle being higher during the time period the actual system is encountered by waves from an angle of 30 is due to the fact that the disturbance of following seas has higher impact on the ship than beam and/or bow seas. 3 Simulation results Fig. 3 shows the output errors between six different subsystems and the actual system. In this example, the controller corresponding to signal e 4 would be switched into the system, because the mean value of the output error for this model is the smallest. Fig. 4: Ship behaviour on course change. Fig. 3: The error out put of the six different models. Fig. 4 shows the system's behaviour during a simulation study including two course changes at constant speed. The upper plot illustrates the wave encounter angle, which Fig. 5 presents the error outputs of four models (e.g. e 13,0, e 13,30, e 13,60 and e 13,90 ) resulting from the comparison between the roll angle outputs of the models at 13 knots and 0, 30, 60 and 90 wave encounter angles, with the roll angle output of the actual system at the same speed and roll angles, respectively. For the first 150 seconds of her journey, the output error is zero for e 13,90, which means that the ship is experiencing a situation extremely close to

5 Fig. 5: Output errors this sub-model. A zero value can obviously only be obtained during computed simulation. In a real ship trial, this will be the lowest mean value over a certain period of time. After 150 seconds, the difference becomes larger, but this is not yet the time to switch because the course change is still on-going and the system is unsettled. Now the error signal of e 13,60 decreases to zero, because the model for the 60 wave encounter angle at 13 knots behaves very much like the actual system at this time. At 250 seconds e 13,60 becomes larger and e 13,30 reduces to zero. In the middle plot of Fig. 4 the solid line representing the controller indices, switches from 4 to 2 after the system is settled and the error value averages are compared over a certain amount of time. Fig. 6 illustrates the roll output of the corresponding models for 0, 30, and 60 degree (all at 13 knots), in the upper, middle and lower plots, respectively. 4 Conclusions Stabilisation of a ship motion using a parallel multi model control (PMMC) has been studied in this paper. For improving the roll effectiveness and stabilising ship motion, a family of PID controllers has been designed. Ship roll has been stabilised using a switching strategy based upon the error outputs resulting form the output of the actual system (ship) and the output of sub-models. The most appropriate controller is the controller corresponding to the model, which yields the minimum mean error. Each controller can be switched to another depending the environment and other factors such as sea waves. This has been achieved by considering a set of rules, which act as a decision maker process (supervising control system). The switching mechanism has been demonstrated when the model of a ship is subjected to environmental changes. References [1] Belmont, M. R. and J. M. K Horwood. The effect of frequency distribution in sea model spectra on Fig. 6: The output behaviour of three models. simulations of deterministic sea wave prediction, International Ship Building Progress, 46, pp , (1999). [2] Blanke, M., J. Adrian, K. Larsen and J. Bentsen, Rudder roll damping in coastal region sea conditions, Proceedings of 5th IFAC Conference on Manoeuvering and Control of Marine Craft, MCMC 2000, (2000). [3] Crossland, P., The Effect of Roll Stabilisation Controllers on Warship Operational Performance, Control Engineering Practice, , (2003). [4] Daafouz, J., P. Riedinger and C. Lung, Stability analysis and control synthesis for switched systems: A switched Lyapunov function Approach, IEEE Transactions on Automatic Control, 47, pp , (2002). [5] Hespanha, J. P. and A. S. Morse, Stabilisation of nonholonomic integrators via logic-based switching, Automatica, 35, , (1999). [6] Liberzon, D. and A. S. Morse, Basic problems in stability and design of switched systems, IEEE Control Systems Magazine, 19, pp , (1999). [7] Roberts, G. N., M. T. Sharif, R. Sutton, and A. Agarwal, Robust control methodology applied to design of a combined steering/stabilisable system for warships, IEE Proceedings, Control Theory and Applications, 144, pp , (1997). [8] Sharif, M.T., G. N. Roberts, S. A. French, and R. Sutton, Lateral force stabilisation: a comparison of controller designs, Eleventh Ship Control System Symposium, Canada, 5, pp , (1993). [9] Tedeschi, R., Sea state measurements in the Ross Sea based on ship motions, Proceedings of the 9th International Offshore and Polar Engineering Conference, Brest, France, Vol III, pp , (1999). [10] Xie, W., C. Wen, and Z. Li, Input-to-output stabilisation of switched nonlinear systems, IEEE Transaction on Automatic Control, 46, , (2001).