Kalman-Bucy Filter Design for Multivariable Ship Motion Control

International Journal Volume 5 on Marine Navigation Number 3

and Safety of Sea Transportation September 2011

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 heading, 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 estimation. 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 connected 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

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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 error is done. Then, the covariance matrix is determined 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. T P k+1 = E{[Fk (x k − xˆ k ) + w k ][Fk (x k − xˆ k ) + w k ] } The covariance matrix for the vector wk is given = F E{(x − xˆ )(x − xˆ )T }FT + F E{(x − xˆ )wT } by formula (7), while that for vk is defined in the fol- k k k k k k k k k k lowing way T T T + E{w k (x k − xˆ k ) }Fk + E{w k w k } (18) T = E{vk vk } R k (10) No correlation is assumed between the estimation error signals e and the disturbances w . After plac- E w vT = 0 , for all k and i (11) k k { k k } ing the covariances defined by formulas (7) and (15) It is assumed that the initial values of the process into relation (18) we get the required error covari- estimates are known at the beginning time tk and that ance matrix between the sampling times these estimates until time tk base on the knowledge T P k+1 = Fk Pk Fk + Q k (19) about the process. Such an estimate is denoted as xk where the bar means that this is the best estimate at The estimation error covariance matrix Pk is cal- time tk before the measurement. The estimation error culated for sampling times in the similar way. After is defined as: placing relations (9) and (14) into formula (15) we get ek = xk − xk (12) = − − − − and the related error covariance matrix is Pk E{[(xk xk ) Lk H k (xk xk ) Lk v k ] T  T   T  ⋅ − − − −  P k = Eek ek  = E(x − x k )(x − x k )  (13) [(xk xk ) Lk H k (xk xk ) Lk v k ]  (20)    k k  

At sampling times tk at which the measurement yk And after similar algebra operations as in formula is done, the possessed estimate xk is corrected using (21) we get the following solution the following relation (Brown & Hwang, 1997; T T P = P k − L H P k − P k H L Franklin et al. 1998) k k k k k T T + L (H P k H + R )L (21) xˆ k = x k + L k (y k − H k x k ) (14) k k k k k where xˆ k is the estimate updated by the performed The final task is to calculate the optimal values of measurement, and Lk is the scaling amplification. the amplification matrix Lk. This task is realised by

347 finding such values of the vector Lk. which minimise Rk (10) in the discrete filter, but have different nu- the trace of the matrix Pk being the sum of the mean merical values. square errors of the estimates of all state vector ele- To do the conversion from discrete to continuous ments. The trace of the matrix Pk is differentiated form, let us first define the relations between Qk and with respect to Lk and made equal to zero. What can Rk, on the one hand, and the corresponding values of be easily noticed, the second and third term of equa- Q and R for small sampling intervals ∆t. Based on tion (21) are linear with respect to L , while the k formula (5) we can notice that for small ∆t values fourth term is quadratic and the trace of LkHkPk is the discrete transition matrix Fk = I. After applying equal to the trace of its transposition LkHk P k P T T this conclusion to the matrix Qk given by formula kHk Lk . We get (Brown & Hwang, 1997) (8) we get (Brown & Hwang, 1997). d(traceP ) k = T T Qk = G(ξ )E[u(ξ )u (η)]G (η)dξdη (30) dLk ∫∫ ∆t→0 T  T  T − 2H P k  + 2L (H P k H + R )= 0 (22) Then placing the equation (27) into the equation  k  k k k k (30) and integrating for a small time interval ∆t we and after some transformations we arrive at the re- get quired form of the matrix Lk, bearing the name of = T ∆ Kalman amplification: Qk GQG t (31) −1 Deriving the equation relating R to R is not = T T + k Lk P k H k (H k P k H k R k ) (23) straightforward. In the continuous model the signal v(t) is the white noise and direct sampling returns Now we remove the matrix Lk from equation (21) by placing the relation (23) to get the measuring noise with infinite variance. In order to get equivalent values at the discrete and continu- T T −1 ous times it is assumed that the continuous meas- Pk = P k − P k H k (H k P k H k + R k ) H k P k (24) urements are averaged over the time interval ∆t in The obtained recurrent calculation algorithm, the sampling process. The state variables x are not based on equations (14), (16), (19) (23) and (24), is “white” and can be approximated as constant along widely known as the Kalman filter. Equation (24) this interval. Hence can be presented in another form taking into account 1 tk 1 tk the Lk relation given by the formula (23) which = = + y k ∫ y(t)dt ∫ [Cx(t) v(t)]dt gives ∆t tk −1 ∆t tk −1

tk P = P k − L H P k = (I − L H )P k (25) 1 k k k k k ≈ Hk xk + v(t)dt (32) ∆t ∫tk −1

where Hk = C(tk). This way a new equivalent is ob- 4 CONVERSION FROM DISCRETE TO tained which defines the relation between the con- CONTINUOUS EQUATIONS DESCRIBING tinuous time and discrete time domains THE KALMAN FILTER ALGORITHM 1 v = v(t)dt (33) k ∆t ∫ The general form of a continuous process is already ∆t→0 given by equation (1), while the equation describing the measuring model for a continuous system is From formula (10) we get y = Cx + v (26) T 1 T E[vk vk ]= R k = E[v(u)v (v)]dudv (34) ∆t 2 ∫∫ Consequently, by analogy with the discrete model ∆t→0 it is assumed that the vectors u(t) and v(t) are the Placing equation (28) into equation (34) and inte- vectors of the stochastic process bearing the name of grating we get the required relation the white noise with the zero cross correlation value. R T τ = δ ( −τ ) R k = (35) E{u(t)u ( )} Q t (27) ∆t E{v(t)vT (τ )}= Rδ (t −τ ) (28) The amplification of the discrete Kalman filter is

T given by formula (23). After placing relation (35) in- E{u(t)v (τ )}= 0 (29) to this formula we get The covariance parameters Q and R play a similar role to that played by the parameters Qk (7) and

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∧ x0 ∧ y x a) L ∫

. T −1 P = AP + PAT − PCT R−1CP + GQG T P L = PC R b) CT R−1 P(0) = P0

Figure 2. Block diagram of the continuous Kalman filter.

T T −1 L = P k H H P k H + R ∆t T 2 k k ( k k ) − ALk H k P k A ∆t + Qk (41) T −1 ≈ Pk H k R ∆t (36) It can be seen from equation (38), than in equation (41) the matrix Lk is of an order of ∆t. And, af- as the second term inside the parentheses is dominat- ter removing all terms containing ∆t of and order ing in formula (36) higher than one from equation (41) we get the sim- T plified form (Brown & Hwang, 1997): R ∆t >> H k P k H k (37) = + ∆ + T ∆ − + When passing from the discrete form to the con- P k+1 P k AP k t P k A t Lk H k P k Qk (42) tinuous form, the error covariance matrix between After substituting formula (38) for Lk and formu- sampling times, given by formula (19), nears to P la (31) for Qk in equation (42) we arrive at k+1 ≈ Pk when ∆t→0. Therefore only one error matrix T P is in force in the continuous filter. Equation (36) P k+1 = P k + AP k ∆t + P k A ∆t takes the form T −1 T − P k H R H P k ∆t + GQG ∆t (43) T −1 k k Lk = PC R ∆t (38) Based on equation (43) we can get the following And, finally, when ∆t→0, we arrive at the formu- difference la for determining the amplification L in the continuous Kalman filter (Brown & Hwang, 1997) P k+1 − P k T = AP k + P k A ∆t L = k = T −1 L PC R (39) T −1 T ∆t − P k H k R H k P k + GQG (44) As the next step, the equation describing the es- Then, after limiting ∆t→0 and removing all sub- timation error covariance matrix is to be derived. scripts and bars over matrices we get the matrix dif- Placing relation (25) into equation (19) we get ferential equation T P k+1 = F (I − L H )P k F + Q . k k k k k P = AP + PAT − PCT R −1CP + GQGT (45) T T = Fk P k Fk − Fk Lk H k P k Fk + Qk (40) with the initial condition

Then the discrete transition matrix Fk in equation P(0) = P0 (46) (40) is substituted by its approximate given by formula (5) (Brown & Hwang, 1997) The last remaining step is to derive the state estimation equation given by formula (14). Placing the T P k+1 = (I + A∆t)P k (I + A∆t) below given relation (47),

T x k = F xˆ (47) − (I + A∆t)Lk H k P k (I + A∆t) + Qk k −1 k −1

T T 2 = P k + AP k ∆t + P k A ∆t + AP k A ∆t

T − L k H k P k − AL k H k P k ∆t − Lk H k P k A ∆t

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Figure 3. Variables used for describing ship motion. derived from equation (16), into formula (14) makes estimation of ship motion parameters. The tests were it possible to arrive at the following form of equa- performed on the training ship Blue Lady owned by tion (17) the Foundation of Sailing Safety and Environment Protection in Ilawa. Blue Lady is the physical model, xˆ = F xˆ + L (y − H F xˆ ) (48) k k −1 k −1 k k k k −1 k −1 in scale 1:24, of a tanker designed for transporting Here again, the matrix Fk−1 is approximated by crude oil. The overall length of Blue Lady is formula (4) L = 13.75 m, the width is B = 2.38 m, and the mass is m = 22.934x103 [kg]. xˆ k = xˆ k −1 + Axˆ k −1∆t + L k (y k − H k xˆ k −1 − H k Axˆ k −1∆t)(49) The ship sailing on the surface of the water region After removing all terms containing ∆t of an or- is considered a rigid body moving in three degrees der higher than one from equation (50) and observ- of freedom. The ship position (x, y) and the ship ing that Lk = L∆t, the equation takes the form course ψ in the horizontal plane with respect to the xˆ − xˆ = Axˆ ∆t + L∆t(y − H xˆ ) (50) stationary, inertial coordinate system {xe, ye} are k k −1 k −1 k k k −1 represented by the vector η=[x,y,ψ]T. The second Finally, after dividing by ∆t, coordinate system {xb, yb} is connected with the moving ship and fixed to its centre of gravity. Ve- xˆ k − xˆ k −1 = Axˆ k −1 + L(y k − H k xˆ k −1 ) (51) locities of the moving ship are represented by the ∆t vector ν=[u,v,r]T, where u is the longitudinal ship reducing by ∆t→0, and removing all subscripts and velocity (surge), v is the lateral velocity (sway), and bars over variable matrices we get the matrix differ- r is the angular speed (yaw). These variables are ential equation of continuous state estimation shown in Fig. 3. xˆ = Axˆ + L(y − Cxˆ ) (52) The position coordinates (x, y) are measured by DGPS (Differential Global Positioning System), Equations (39), (45), (46) and (52) compose the while the ship courseψ is measured by the gyro- continuous Kalman filter. They are shown in Fig. 1. compass. These three measured state variables are Figure 1(a) shows a block diagram illustrating the collected in the vector η=[x,y,ψ]T. The three remain- principle of operation of the continuous Kalman fil- ing state variables, composing the vector ν=[u,v,r]T, ter. The input signal for the filter is the measured are to be estimated. noised output signal of the object. Figure 1(b) pre- The ship motion equations simply express the sents the method of determining the optimal amplifi- Newton’s second law of motion in three degrees of cation L. freedom. These equations, formulated in the stationary coordinate system connected with the map of the water region, have the following form (Clarke, 5 APPLYING THE KALMAN-BUCY FILTER 2003). FOR THE ESTIMATION OF SHIP MOTION VELOCITIES mx = X (53) my = Y (54) The algorithm of the continuous Kalman-Bucy filter described in the previous section was applied for the I zψ = N (55)

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Figure 4. Simulation study: actual position with estimate and actual (black) and estimated (blue) velocity u in surge. Left-hand column – discrete Kalman filter, right-hand column – continuous Kalman-Bucy filter.

where X and Y are forces acting along the xb and yb u  cosψ sinψ 0 u x        axes, respectively, N is the torque, m is the mass of v = − sinψ cosψ 0 ⋅ v (60) the ship, and I is the moment of inertia along the      y  z       lateral axis directed downwards. r  0 0 1  r  The above differential equations can be presented The continuous Kalman-Bucy filter implemented as three sets of dynamic equations having the fol- for estimating the motion parameters on the training lowing general form ship Blue Lady worked based on the system and equations shown in Fig. 2. Moreover, for the pur-  x = Ax + Bu (56) poses of the algorithm of the Kalman-Bucy filter, y = Cx (57) relevant values of the coefficients in the matrices G, Q and R were selected. For the first and second For each degree of freedom the matrices A, B, C degree of freedom the following values were adopt- are identical and take the form ed: 0 0 1     A = , B = , C = [0 1] (58) 1 0 0.1 0     G x = G y =   , Q x = Q y =   , 1 0 0 0 0.01  0 0.2 while the state vectors for consecutive states of free- Rx = R y = 0.01 (61) dom are the following while for the third degree of freedom: u x  u y   r  x1 =   , x2 =   , x3 =   (59)  x   y  ψ  0.2 0  1 0 Gψ =   , Qψ =   , Rψ = 0.1 (62)  0 0.01 0 1 where ux = dx/dt, vy = dy/dt, r = dψ/dt are velocities in the stationary coordinate system. The velocity The covariances of the position coordinates T vector ν=[u,v,r] expressed in the moving coordinate measured by GPS were equal to Rx = Ry = 0.01while system {xb, yb}, can be calculated based on the ve- the covariances of the ship course measurement locities determined in the stationary coordinate sys- were equal to Rψ = 0.1 and were determined based tem {xe, ye} and making use of the relation on the experimental tests done on the training ship Blue Lady.

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Figure 5. Simulation study: actual position with estimate and actual (black) and estimated (blue) velocity v in sway. Left-hand column – discrete Kalman filter, right-hand column – continuous Kalman-Bucy filter.

Figure 6. Simulation study: actual heading angle ψ with estimate and actual (black) and estimated (blue) angular rate r in yaw. Left-hand column – discrete Kalman filter, right-hand column – continuous Kalman-Bucy filter.

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Figure 7. Experimental data: measured position with estimate and estimated velocity u in surge. Left-hand column – discrete Kalman filter, right-hand column – continuous Kalman-Bucy filter.

Figure 8. Experimental data: measured position with estimate and estimated velocity v in sway. Left-hand column – discrete Kalman filter, right-hand column – continuous Kalman-Bucy filter.

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Figure 9. Experimental data: measured heading angle ψ with estimate and estimated angular rate r in yaw. Left-hand column – discrete Kalman filter, right-hand column – continuous Kalman-Bucy filter.

Initially, the simulation investigations were per- hand column refer to the investigations performed formed in the calculating environment using the continuous Kalman-Bucy filter. Matlab/Simulink based on the mathematical model The presented diagrams reveal that the estimates of the training ship Blue Lady, described in detail by of the position coordinates and the course are identi- (Gierusz, 2001; Gierusz. 2005). These investigations cal as the measured values. On the other hand, the were performed at the presence of measurement correspondence between the time-histories of the es- noises, which were added to the positions and head- timated velocities is much worse, as their curves are ing measurements in simulations and at the presence not smooth and are burdened with relatively large of the external disturbances. In simulation study as- errors. All inaccuracies in the measured values are sumed that on ship was acting wind with average reflected in the estimated velocity values. speed equal 2 m/s and in direction 0 degrees. The simulation results are shown in Figs. 4-6. The actual and estimated velocities in surge, sway and yaw are 6 REMARKS AND CONCLUSIONS shown in the bottom of plots. After simulation tests, the algorithm of the dis- The article describes the method of deriving the al- crete Kalman-Bucy filter was implemented on the gorithm of the continuous Kalman-Bucy filter based training ship Blue Lady sailing on the lake Silm near on the discrete Kalman filter. This algorithm does Ilawa. The performed experimental tests aimed at not take into account the model of ship dynamics, testing the quality of filter operation and its re- but only bases on the position coordinates x, y meas- sistance to disturbances. In order to provide good ured by GPS and the ship course angle ψ measured opportunities for comparison, the tests of the train- by the gyro-compass. The estimates of ship position ing ship Blue Lady were also performed using the coordinates x, y and ship course angle ψ shown in discrete Kalman filter described by Tomera Figs. 7, 8 and 9 well correspond to the measured (Tomera, 2010). values. The correspondence is worse for velocity estimates determined for the moving coordinate sys- The results recorded in the experimental tests are tem fixed to the ship and collected in the vector shown in Figs. 7 − 9. The diagrams in the left-hand ν=[u,v,r]T. The determined time-histories of these column present the results obtained for the discrete velocities are burdened with a noise of relatively Kalman filter while the time histories in the right- high level, lower for the Kalman-Bucy controller than for the discrete Kalman filter. The shapes of the

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