International Journal Volume 5 on Marine Navigation Number 3 and Safety of Sea Transportation September 2011 Kalman-Bucy Filter Design for Multivariable Ship Motion Control M. Tomera Gdynia Maritime University, Faculty of Marine Electrical Engineering, Department of Ship Automation, Poland ABSTRACT: The paper presents a concept of Kalman-Bucy filter which can be used in the multivariable ship motion control system. The navigational system usually measures ship position coordinates and the ship head- ing, while the velocities are to be estimated using an available mathematical model of the ship. The designed Kalman-Bucy filter has been simulated on a computer model and implemented on the training ship to demon- strate the filtering properties. 1 INTRODUCTION of the signal, passes it to the observer system in which the disturbances are filtered out and the ship Modern ships are equipped with complicated ship state variables are calculated. Stochastic nature of motion control systems, the goals of which depend the forces generated by the environment requires the on tasks realised by an individual ship. The tasks ex- use of observers for estimating variables related with ecuted by the control system include, among other the moving ship and for filtering the disturbances in actions, controlling the ship motion along the course order to use the signals in the ship motion control or a given trajectory (path following and trajectory systems. tracking), dynamical positioning and reduction of Filtering and estimating are extremely important ship rolls caused by waves. Figure 1 presents basic properties in the multivariable control systems. In components of the ship motion control system. many cases the ship velocity measurements are not The guidance system generates a required smooth directly available, and the velocity estimates are to reference trajectory, described using given positions, be calculated from the position and heading values velocities, and accelerations. The trajectory is gener- measured by the observer. Unfortunately, these ated by algorithms which make use of the required measurements are burdened with errors generated by and current ship positions, and the mathematical environmental disturbances like wind, sea currents model with complementary information on the exe- and waves, as well as by sensor noise. cuted task and, possibly, the weather. One year after publishing his work on a discrete The control system processes the motion related filter (Kalman, 1960), Rudolph Kalman, this time signals and generates the set values for actuators to together with Richard Bucy, published the second reduce the difference between the desired trajectory work in which they discussed the problem of contin- and the current trajectory. The controller can have a uous filtering (Kalman & Bucy, 1961). This work number of operating modes depending on the exe- has also become the milestone in the field of optimal cuted tasks. On some ships and in some operations filtering. In the present article the continuous Kal- the required control action can be executed in sever- man filter is derived based on the discrete Kalman al ways due to the presence of a number of propel- filter, assuming that the sampling time tends to zero. lers. Different combinations of actuators can gener- A usual tendency in numerical calculations is rather ate the same control action. In those cases the reverse: starting from continuous dynamic equa- control system has also to solve the control alloca- tions, which are digitised to arrive at the discrete dif- tion problem, based on the optimisation criteria ference equations being the approximates of the ini- (Fossen, 2002). tial continuous dynamics. In the Kalman filter idea The navigation system measures the ship position the discrete equations are accurate as they base on and the heading angle, collects data from various accurate difference equations of the model of the sensors, such as GPS, log, compass, gyro-compass, process. radar. The navigation system also checks the quality 345 Way-points Waves, wind, currents Trajectory DGPS Controller Allocation Ship generator Gyro-compass Observer Estimated positions and Guidance Motion-Control velocities Navigation System System System Figure 1. Basic components of modern ship motion control system (Fossen, 2002). The dynamic positioning systems have been de- x(t) = A(t)x(t) + G(t)u(t) (1) veloped since the early sixties of the last century. The first dynamic control systems were designed us- where u is the input vector having the form of white ing conventional PID controllers working in cascade noise. The state transition matrix for equation (1) with low-pass filters or cut-off filters to separate the takes the form: motion components connected with the sea waves. t A(t )(t−t ) τ ( −τ ) However, those systems introduce phase delays = 0 0 + A( ) t τ τ τ x(t) e x(t0 ) ∫ e G( )u( )d (2) which worsen the quality of the control (Fossen, t0 2002). For the discrete model, the objects of analysis are From the middle of 1970s more advanced control process samples recorded at times t0, t1, ..., tk, .... techniques started to be used, which were based on Equation (2) written for a single sampling interval optimal control and the Kalman filter theory. The can be presented as first solution of this type was presented by (Balchen t et al., 1976). It was then modified and extended by = + k +1 τ τ τ τ x(tk+1) F(tk+1,tk )x(tk ) ∫ F(tk+1, )G( )u( )d (3) Balchen himself and other researchers: (Balchen et tk al., 1980a; Balchen et al., 1980b; Fung and Grimble, which can be briefly written as 1983; Saelid et al. 1983; Sorensen et al., 1996; Strand et al. 1997). The new solutions made use of xk+1 = Fk xk + w k (4) the linear theory, according to which the kinematic characteristics of the ship were to be linearized in where Fk is the state transition matrix for the step be- the form of sets of predefined ship heading angles, tween times tk and tk+1 at the absence of the excita- with an usual resolution of 10 degrees. After the lin- tion function ( )⋅ earization of the nonlinear model, the observer based F = F(t ,t ) = e A tk T = I + A(t )⋅T (5) on such a model is only locally correct. This is the k k+1 k k disadvantage of the Kalman filter. The Kalman filter and wk is the excited response at time tk+1 due to the can make use of measurements done by different presence of the white noise at the input in the time sensors at different accuracy levels, and calculate interval (tk, tk+1), i.e. accidental disturbances affect- ship velocity estimates which are not measured in ing the process the majority of ship positioning applications. tk +1 w k = F(tk+1,τ )G(τ )u(τ )dτ (6) The main goal of the article is designing and test- ∫tk ing the observer for the ship motion velocity estima- tion. The white noise is a stochastic signal having the mean value equal to zero and finite variance. The matrix elements wk can reveal non-zero cross corre- 2 DISCRETE MODEL OF THE PROCESS lation at some times tk. The covariance matrix con- nected with wk is denoted as Discussed are time-dependent discrete processes, T E{w w }= Q (7) which are recorded by sampling continuous process- k i k es at discrete times. Let us assume that the continu- The covariance matrix Qk can be determined us- ous process is described by the following equation: ing the formula written in the following integral form 346 T Qk = E{w k wi } T tk+1 tk+1 = ξ ξ ξ ξ η η η η E∫ F(tk+1, )G( )u( )d ∫ F(tk+1, )G( )u( )d tk tk tk +1 tk +1 T T T = F(tk+1,ξ )G(ξ )E[u(ξ )u (η)]G (η)F (tk+1,η)dξdη (8) ∫tk ∫tk T The matrix E[u(ξ)u (η)] is the matrix of the Di- The task is to find the vector amplifications Lk rac delta function, well known from continuous which update the estimate in the optimal way. For models. this purpose the minimisation of the mean square er- ror is done. Then, the covariance matrix is deter- mined for the error relating to the estimate updated 3 DISCRETE KALMAN FILTER by the performed measurement. = T = − ˆ − ˆ T Briefly, the Kalman filter tries to estimate, in an op- Pk E{e k e k } E{(x k x k )(x k x k ) } (15) timal way, the state vector of the controlled process In time intervals between the sampling times, the modelled by the linear and stochastic difference estimates are calculated using the following formula equation having the form given by formula (4). Ob- servations (measurements) of the process are done at x k +1 = Fk xˆ k (16) discrete times and meet the following linear relation Firstly, the covariance matrix P k+1 is calculated y k = H k xk + v k (9) using formula (13) after correcting it by one sample ahead where xk is the state vector of the process at time tk, yk is the vector of the values measured at time tk, Hk T P k+1 = E(x − x k+1 )(x − xk+1 ) (17) is the matrix representing the relation between the k+1 k+1 measurements and the state vector at time tk, and vk represents the measurement errors. It is assumed that After placing relations (4) and (16) into formula (17) we get (Brown & Hwang, 1997) the signals vk and wk have the mean value equal to zero and there is no correlation between them.