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Design of Dynamic Ship Positioning Using Extended H∞ Filtering
Copyright @ IFAC Adaptive Systems in Control and Signal Processing, Glasgow, Scotland. UK. 1998
DESIGN OF DYNAMIC SHIP POSITIONING USING EXTENDED H", FILTERING
M R Katebi and M J Grimble
Industrial Control Cenrre Strathclyde University Glasgow, Scotland.
Abstract This paper is concerned with the extension of linear H", filter to non-linear systems with application to marine control systems. The structure of the Extended Hoo Filter (EHF) is similar to the Extended Kalrnan Filter (EKF) but it can account for uncertainties by exploiting the H", robustness properties. Since combined state and parameter estimation problems can be set up as a non-linear filtering problem, the EHF can provide both parameter and state estimation. EHF gain must be computed on-line using a local linearisation. The filter equations include the non-linear system description but the extended H", filter has the same type of approximations found in the EKF and hence there are similar difficulties in trying to produce stability and convergence results. The EKF has been successful in a wide range of applications. Using a marine control problem, it is shown that the EHF also has similar applications potential. Copyright © 1998 lFAC Key Words: Non-linear filtering, H", control, and robustness, marine motion control
1.
An HOC) control law for NL systems can also be defmed using a separation principle type of result, which applies for linear systems but invoking the EHF rather than the HOC) filter. The resulting adaptive system will be in state-space model form . It can be made robust by introducing uncertainty in the models and by minimising the resulting HOC) criterion. It can
INTRODUCTION
The Extended Kalman Filter (EKF) has been established since the 1960's. It is an extension of the KF for linear systems. The Kalman filter comprises a linear signal model with an outer feedback loop. The EKF uses the same structure with a non-linear signal model and a gain matrix, which is obtained from a local linearisation. Since combined state and parameter estimation problems can be set up as a non-linear filtering problem, the EHF can provide both parameter and state estimation. The problem with the KF and EKF is that they are not robust to modelling uncertainties. Grimble introduced the first Hr filter in 1986 for linear systems. which provides a framework for studying filter robustness properties. Glover, et al (Zhou, et aI, 1996) provided a state space H filter solution, which has a similar structure to the KF . It is a simple extension to introduce an EHF which can be made robust and handle non-linearities and parameter estimation. The definitions of the EHF and its algorithms and its application to the ship Dynamic Positioning (DP) ship control problem (Katebi. et al. 1997) are the main objectives of this paper.
deal with non-linearities through the Extended H'X) filter feature . Parametric uncertainty can also be accommodated through parameter estimation using the extended filter. A locally linearised Hoo controller can be constructed using feedback from the state estimates. Because of the parameter estimation and subsequent updating of the controller the total system will of course be adaptive. The H x control design technique IS particularly suitable for dynamic ship positioning control system design since the variation in the ship dynamics can be significant for different operating conditions. The modelling information for the sea disturbances is also often unreliable. Moreover, a practical DP control system should meet conflicting control design requirement such as low thruster modulation and good current disturbance rejection. The robust control design via the H x technique
X>
171
enables the modelling errors and disturbances to be taken into account. 2.
linear ship model may then be calculated. The complete linear model may then be written as: .\-,(t) = A,x(t) + B,u(t), y, U) = C,x(t) + D,u(t) (5)
SHIP AND SEA STATE MODELS
where x=[u, v, r, x, y,
Consider the following definitions (see Fig. 1 for details): Input disturbance: An input disturbance affects the output to be controlled directly and the controller is tuned to attenuate such disturbances. For the OP problem, these are Wind, second order waves forces, and current forces. Output disturbance and noise: An output disturbance does not directly affect the output to be controlled and the controller does not attempt to attenuate this disturbance. For the OP problem these are first order Low Freq.
High Freq.
Wind, Wave Current
2.2. Wave Disturbances There are three main sources of disturbances on the ship's position, namely, wind, current, wave motion and the measurement noise. The current, constant wind and second-order wave forces act on the ship at low frequencies « 0.5 rad / sec). The first-order wave forces and possibly the turbulent wind force (depending on the infrastructure of the vessel) and the measurement noise also acts at high frequencies (> 0.7 rad / sec). The control design problem is to compensate the low frequency motion whilst suppressing unnecessary high frequency thruster movements (thruster modulation). Since the wind speed is usually measured, it is advantageous to compensate the effect of the wind using feed-forward control action. The wind feedforward controller is usually employed in conventional DP systems. In the following section, the models of the Low Frequency (LF) and the High Frequency (HF) disturbances are formulated. The high frequency disturbances for the three surge, sway and yaw channels are assumed to be decoupled. The disturbances in each channel are modelled by second-order transfer-functions of the form:
I
Wave"
u
---+ Thrusters
Ship Model
Figure 1 The OP Model wave forces, and measurement noise. From Newton's equations of motion all forces on the ship enter at the same point including the thruster forces. However, the wave model forces will not be added at this point and instead the wave motion effects will be moved to the output. The reason lies in the difference in the type of disturbance. The wave motion must not be offset by the control action and it is therefore an output disturbance in the above terminology. In fact it is necessary to separate the low and high frequency motion signals since the cost function depends upon the output y I due to the low frequency motions which are caused by the input disturbances.
G; ( S )
= b;s2 +2t;I,cU,.,S+W;',
(6) 2 2 a;s +2t;2;W m S+W,., where i =X,Y and Z Converting the three transfer functions into state-space form gives the high frequency disturbance model:
2.1. Ship Model
2.3. The Combined Model
Using Newton's equations of motion, the dynamics of a surface ship can be described by:
The state equations for the low frequency vessel dynamics and the wave disturbance can be combined to obtain the overall model of the vessel:
(m - XIi)u
= (m -
Y,,)rv + (u~ + v;c )CHr S Hr + Tx (I)
(m-Y.;. )v=(m-XJrv+(u~ +V~)CHrSHr +Ty (2)
x(t) = Ax(t) + B:u(t) + DI wet) y(t) = C:x(t) + Dzvet)
(3)
x=u,y=v,~=r (4) where u, v and r are the surge, sway and yaw speeds and x, y and
u
(7)
r
3.
172
LINEAR Ha) FILTER
Consider the dynamic system described by: :((t) = A(t)x(t) + B(t)u(t) + D1(t)w(t)
(8)
y(t)=C,(t)x(t)+D,v(t) (Measurements)
(9)
=(1) = (>(t) (Estimated signal)
Consider the class of non-linear systems driven by white noise with white noise-corrupted observations defined by:
(10)
where x E Rn , )' E Rm. =E RP and wet) and v(t) are vectors of independent white noise sequences of zero means and covariances Q and R, representing the measurement and process noises. Assume
:r(/) = f[X(I) , /]+ K[x(t),/]u(t)+ G[X(t), / ]W(/)
and D:,D~, = D~RD~ The filtering problem is to find an estimate =(t) ofz(t) using the measurements ofy up to time t. The filter z(t)=F(s) yet) is required to minimise the cost function: B,B:
J
=
= D,QD:,
sup 1/=1/; < ,,~[o·"')IIWII;
r~ ,x(O) = 0
where wet) and vet) are zero-mean. white, and Gaussian and uncorrelated with the following properties: E{w(t)} = 0, E{v(t)} = O. Cov{w(t)} = Q, Cov{v(t)} = R, Cov{w(t),v(t)} = 0 Assume that the conditional-mean estimate x(t) is known and used to expand the system and
(11 )
=
fory > 0 and (t) = z( I) - i( t) . The filter is required to be casual and the estimate should be unbiased. The above filtering problem can be written in standard LFT fonnulation as shown in Fig. 2: pes)
= [
B,(t)
C,
0
-I
C2 (t)
D~,(t)
0
Assuming the system (C 2 and [A(t) - j wl C2 (t)
w
-'1
0]
A(I)
z
Figure 2 The Standard system representation ,
A) is detectable
B, (t)] has full rank for all D 2 ,(t)
0),
measurement models in a Taylor series about x(t) = x(t) to obtain the following set ofiinear equations in x{t):
the
x{t):::: f[x(t),t] +
Initial Conditions: x(to) = x o, Y(to)
z(t) = h[x{t),t] +
=0
;(1) = A(t)x(t) + B(t)u(t) + K f [y(t) - C 2 (t)x(t)]
= Y(t)C~ (t)R-'(t)
ah~~:i,t] [x(t) -
x(t)] + vet)
= C2 (t)x{t) + v(t) + n(t) z(t) = C\x(t) y(t)
= C~(t)x(t)
HOD Riccati
(14)
where known input and disturbances are described by:
= Y(t)A' (t) + A(t)Y(t) + Y(t)(r -2C\C: -
met) = f[x(t),/]- x(l)
C(t)R-\C~(t»Y(t) + D, (t)QD,' (I)
Note that by making r ~ 00 , the Hac filter converges to a continuous time Kalman filter.
4.
x(t)] + K[x(t), t]
Rewriting this model in the basic state-space fonn gives: x(t) = A{t)x(t) + B(t)u(t) + G(t)W(I) + met)
Filter Algorithm :
Y(t)
Of~~:i' t] [x(t) -
+ G[x(t), t]w(t)
Table I The linear Hac filter Known Inputs: u{t)
Estimated Output: 2(t)
I Z. q-F-(S)---I~ Y pes)
(12)
linear Hac filter can be summarised as shown in Table I (Zbou et ai, 1996 and Grimble, 1995).
Filter gain: K f
(13)
y(t) = h[x(t), I] + v(t)
net) = h[x(/), /j-[ah[x(I),/] 1 ox(t)]X(/)
(15)
The linearised system matrices are defmed by : A(t) = of[x{t),t)] 1 ax(t), B(t) = K[x(t),t)]
( 16) G(t) = G[x(t), t)] , C1 (t) = ah[x(t), I)] 1 ax(t) The detennination of the resulting extended H", filter is now straightforward and is given in Table 2.
EXTENDED Hac FILTER
The H", filter can be used to estimate the state of a non-linear system by linearising the system equations around a nominal operating point. In this paper, attention is concentrated on systems where the system parameters are augmented as additional states to the system's state vector. This type of system is useful for wave filtering in Ship Dynamic Positioning (DP) systems. where the wave centre frequency changes with wave direction and sea state. Similar procedures to extended Kalman filters are adopted here to fonnulate the extended Hac filter.
4.1. Parameter Estimator
The ability to estimate unknown plant parameters is useful Dynamic Ship Positioning, where unknown wave model parameters are required to be identified. These parameters can be modelled as integrators that are driven by white noise and augmented to the system states.
173
reduced in the usual manner until one of the eigenvaiues of Y becomes imaginary or negative. For time-varying systems, the value of y should be found at each time instant. This will impose a high level of computational load on the filter and make it impractical in real applications. An alternative scheme is to use a time decreasing exponential function for y starting from a large value, i.e. y(t)=Ymue -"'+Ymin (17) The rate of decay of the exponential can be set to 3 to 4 times the plant dominant time constants. The scalar Ymin depends on the weighting functions used to design the filter (Fig. 3). The filter gain is similar to KF gain when y is large. As y decreases, the filter converges to the optimal Ha> filter. The Ymin should be usually set to a value to ensure a viable solution to the Riccati equations. This procedure will
Table 2 The Extended H", Filter 1 II
Inputs m() ] r7l"["() ~'( I )]"( x I) x I , I ] / ox XI,I-,e I =In)
I
n(t) = h[ x(t), I] - [ ch[x( I), t] / m( I) ]x( t)
I
Initial Conditions: x(to) = x o, Y(to) = 0 System Matrices: A(t) = Of[.i(t),t)] / m(t), B(t) = K[.i(t),t)]
G(t) = G[.i(t),t)], C2 (t) = Oh[.i(t),I)]/ met) Filter Algorithm: i(t) = A(t).i(t) + B(t)u(t) + K ,[y(I)- C 2 (I)X(t)- n(t)]+ met) Filter Gain: K, = Y(t)C~ R-'
C~ofG.nma
Estimate: £(t) = C~x(t) H""Error Variance: Y(t) = Y(t)A' (t) + A(t)Y(t) + Y(t)(y- 2 C,C; -
. . G
m
C 2 (t)R-'C~ (t»Y(t) + G(t)QG' (t)
m
The state vector may be partitioned so that: x(t)
= [ x(t)
B(t)] also f(x,t)
=[
1;(B,t)
0
______ .::.....-------'-Hx>=-_ _ -!
o][x] 0 B and
60
20
D,
=[~r ~J, C, =[C,
100
0] Figure 3 The Parameter y
The proposed extended H.o filter for the system can now be defmed in Table 3.
significantly reduce the computational burden and ensures that the filter is stable over the estimation horizon.
Table 3 The EHF for parameter estimation Initial Condition: x(to) = x o' Y(to) = 0
6.
System Malrices:
[1; 0(B) ~=q:/(x,t)]/Oxir;i=
~~2
Filter: i(t) = I(x, t) + K, [y(t) - C2x(t)] Filter gain: K, = Y(t)C~R-' Estimate: =(t) = C~xCt) H""Error Variance:
P ARTITIONED FILTER
The structure of the state-space model of the DP system given in equation (7) can be exploited to simplify the filter and reduce the computation requirement. The state matrices can be partltionea into subsystems representing the input disturbance and output disturbance models. The former are normally the disturbances affecting the output to be controlled and are dominantly of low-frequency behaviour, and the latter are the disturbances which affect the measurements but do not affect the outputs to be controlled. The output disturbance model can also represent coloured measurement noise and is usually dominantly of high frequency behaviour. The partitioned model may therefore be represented in the form:
]
where ~12 = 0[/1 (B)x] / ox 2ir;i
2
YCt) = Y(t)~' + ~Y(t)+ Y(tXy- C,C; - C 2 R-'C~ )YCt) + D, (t)QD; (t)
5.
eo
Time
DETERMINA TION OF Y
The filtering problem can be stated as minimising cost function J in eq. (6) for a known value ofy > O. For the time invariant systems, the optimal value of y can be found by selecting a large value of y and solving the algebraic Riccati equation of Table I to calculate the filter gain. The value of y can then be
rX,(t+I)] [Af(t) x 2 (t + I) = 0
l
[
174
0
Ah(t,B)
] [-,",(t)] x 2 Ct) +
Ef(t)] U(I)+ [Df(t)
o
0
0]
Dh(t)
[w,(t)] w~(t)
=(t)
= [H,(t)
y (t)
= [C / (t)
R=R (18)
6.2. Extended H", Filter Equations Note the that output distance model is assumed to be parameterised in terms of the vector () which includes the wave frequency which are often changing and unknown.
The HF model equations involve the parameter estimates, which are treated as state variables, and hence these equations involve the products of states. The model is therefore non-linear and the estimator must have the form :
6.1. Parameter Estimation Models
i"(t) = !(x(t), u(/» + K f (t)(y(t) - C(t)x(t))
The calculation of the extended Kalman filter gains may be considerably simplified by taking advantage of the partitioning of the gain matrix (Grimble, 1982). The HF model is dependent upon the parameter vector () and the assumption is made that these parameters remain approximately constant. Thus, the vector () may be modelled by an integrator with given initial condition vector (}a. That is:
= De(t)wJ(t)
xJ(t)
Using the partitioning given above: x(/) = [XI T(t)
[
=
0 0]0
[AI(t)
0 0
(}(t)
+
Ah(t,(})
0
and HF state/parameter conditional means, respectively, and note that the non-linear function may be partitioned as: AI(/)XI(t)+ B I (t)U(t)]
[D~t)
DhO(t)
o
0
y(t) = [cl(t)
Estimator equations: The Extended Hoc filter (EHF) equations can be partitioned into the sub-equations: i"1 (I)
(20)
where the partitioned equations become:
DII(/) [
o
0]0
0 _0 ] De(t)
y (t)=[CI(/)
[XI (t)] +
(}(/)
[8 0(1)] 1
u(t)+
[~I(t)]
y=
we(t)
0] [~~:J+v(t)
(23)
x evaluated at
0
[
M 2(t,xp
(})
Since the LF dynamics are assumed constant and independent of the parameter vector the term Mdt) = o and since the high frequency model is independent of the low frequency states M 21 (t) = O. Hoo filter gain: The filter gain matrix may be computed from the !inearised equations as follows : The Riccati equation becomes:
i"(t) = A(t)x(t) + B(t)u(t) + D(t)w(t)
[AII(I)
(t)]
AII(t) M(t,x l ) =
Write these equations, including the parameter estimation model, in the form :
= [ ~I(t)] {}(/)
Cl (t)XI
0 0] 0 000
+v(t) (19)
= C(t)x(t) + vet)
+ K\ (/)[Y(/) -
respect to the estimated state vector time t. That is:
xJ(t)
y (t)
(t)
Linearisation: Let M(t) denote the partial derivative matrix (the Jacobian) of A(t,x2 )x(t) with
WJ{t)
O)[;:~:~]
= All (t)XI
(}(t)= K2(t)[y(t)-C I (t)x.{t)]
~ 1[::~:~l De{t)
Ch(t)
[
0
xJ(t)
Ah(t ' ~)X l (t)
!(x(t),u(t» =
[XI(t)] [BI] x 2 (t) + 0 u(t)
0
xi (t)f denote the LF
Let x(t) = [XIT (I)
The total system model and parameter vector model may be combined to obtain:
XI(t)] x?(t)
(}T(t)f
xJ(t) ~ (}(t)
where
(22)
r[ A~~
~] + [ ~ ~ I
]
p+
~]]y + [~I
(21)
The noise sources are assumed to be zero mean, white and mutually independent with the known second moments:
The individual equations may therefore be written as:
r; I
= r; I All
E{ w
r;2
= A11 r; 2 + r; , VI l'; 2 + r; 2V: Y::
l
(r)w; (I)}
E{w 2(r)w ; (t)}
.:..
--
-T
+
-
A ll
--
-
--
r; I + r; YI r; I + r;: V2Y2 1 + QI
- -- - - ...... - - Y: = Y: VIr;: + Ye: V Y:: + Qe
= Q,o" , E{ v( r)v T (t)} = Rt5 :r = Qh O" , E{we(r)w~(t)} = Quo"
2
where
In terms of the partitioned equations (22), the process and noise covariances become:
-..
I
(25)
l
V,
= r-:r~ -C!R-1CIT and V: = y-2 We .
WI
and We are the weightings on the state and parameter estimates. respectively.
175
References
When y is large {EKF), V: becomes very small and the panitioned equations in (25) can be decoupled. Once, the parameters are converged, the coupled Riccati equation can be solved to find the EHF gains. This procedure will simplify the filter structure and reduces the computational requirements.
Fung, P and M Grimble, 1983. Dynamic Ship Positioning Using Self-tuning Kalman Filter, IEEE Trans. AC, 28 (3), 339-349. Nagpal, K and P Khargonekar. 1991, Filtering and smoothing in an H.., sening, IEEE trans. AC 36 (2), 152-166. Katebi. M R, M j Grimble, and Y Zhang,J997. H robust control design for dynamic ship positioning, lEE proc- Control theory Appl, Vol. 144 No 2 March 1997, 110-120. Zhou K, j Doyle and K Glover, 1996, Robust Optimal Control, Prentice Hall, Inc. Grimble, M J, 1994, Robust Industrial Control, Prentice Hall.
aJ
7.
SIMULATION STUDIES
To demonstrate the perfonnance of the extended H", filter, the simple example of Fig. 4 is considered. The filter is used to estimate the wave frequency, O)h . The numerical values used are 0, = 0.1, 0), = 0.05, Oh = l.e-5, O)h = 0.5, cov(w) = cov(~) = 0.004, cov(v) = 0.1. The exponential function for decreasing y is tuned to y = 500 exp(-0.00015t) + 0.01. The sine wave, u(t) = 0.001 sin(0.05t), is used to drive the low frequency system. The estimation results are shown in Fig. 5 and 6. The wave frequency is estimated in less than a minute. The low-frequency output estimate (y,) and the high frequency output estimate (yJ show good approximation to the true outputs. Further simulation results are needed to compare the robustness of the proposed filter with the extended Kalman filter. 8.
Low Fl'llqUllncy Motion
.2 0
SO
100
150
I
sec
HgII Frequency Motion
:~ K!lA AAT\ fAnV\]1~T\ ::YllIDJIVVlV
{I
CONCLUSION
The Extended H", Filter was developed using similar procedures to those for the Extended Kalman Filter. A scheme was proposed to calculate the optimal value for y to reduce the computational requirements. The filter was used in a case study for Dynamic Ship positioning which requires estimating the sea wave frequency on-line. To simplify the filter structure, a panitioned filter was presented, which is specifically suitable for marine control design applications.
-40
so
100
lSO
sec:
Figure 5 The LF and HF Estimation
~ ~(t) 11
High Frequency Gh(s)
+W(t)
.~ u(t)
Low
F~uency
G,(s)
y(t) .J . \"'\." iF
~~E
'\...
~-p"
v() t
Figure 4 The simulation example
00
G(s) = loaf /(i +2&01s+af) 2 G.(s) =/(;,OlrS / (s 2 +20hOlrs+at' )
50
100
~
sec
Figure 6 The wave frequency Estimation
176
150
DESIGN OF DYNAMIC SHIP POSITIONING USING EXTENDED H", FILTERING
M R Katebi and M J Grimble
Industrial Control Cenrre Strathclyde University Glasgow, Scotland.
Abstract This paper is concerned with the extension of linear H", filter to non-linear systems with application to marine control systems. The structure of the Extended Hoo Filter (EHF) is similar to the Extended Kalrnan Filter (EKF) but it can account for uncertainties by exploiting the H", robustness properties. Since combined state and parameter estimation problems can be set up as a non-linear filtering problem, the EHF can provide both parameter and state estimation. EHF gain must be computed on-line using a local linearisation. The filter equations include the non-linear system description but the extended H", filter has the same type of approximations found in the EKF and hence there are similar difficulties in trying to produce stability and convergence results. The EKF has been successful in a wide range of applications. Using a marine control problem, it is shown that the EHF also has similar applications potential. Copyright © 1998 lFAC Key Words: Non-linear filtering, H", control, and robustness, marine motion control
1.
An HOC) control law for NL systems can also be defmed using a separation principle type of result, which applies for linear systems but invoking the EHF rather than the HOC) filter. The resulting adaptive system will be in state-space model form . It can be made robust by introducing uncertainty in the models and by minimising the resulting HOC) criterion. It can
INTRODUCTION
The Extended Kalman Filter (EKF) has been established since the 1960's. It is an extension of the KF for linear systems. The Kalman filter comprises a linear signal model with an outer feedback loop. The EKF uses the same structure with a non-linear signal model and a gain matrix, which is obtained from a local linearisation. Since combined state and parameter estimation problems can be set up as a non-linear filtering problem, the EHF can provide both parameter and state estimation. The problem with the KF and EKF is that they are not robust to modelling uncertainties. Grimble introduced the first Hr filter in 1986 for linear systems. which provides a framework for studying filter robustness properties. Glover, et al (Zhou, et aI, 1996) provided a state space H filter solution, which has a similar structure to the KF . It is a simple extension to introduce an EHF which can be made robust and handle non-linearities and parameter estimation. The definitions of the EHF and its algorithms and its application to the ship Dynamic Positioning (DP) ship control problem (Katebi. et al. 1997) are the main objectives of this paper.
deal with non-linearities through the Extended H'X) filter feature . Parametric uncertainty can also be accommodated through parameter estimation using the extended filter. A locally linearised Hoo controller can be constructed using feedback from the state estimates. Because of the parameter estimation and subsequent updating of the controller the total system will of course be adaptive. The H x control design technique IS particularly suitable for dynamic ship positioning control system design since the variation in the ship dynamics can be significant for different operating conditions. The modelling information for the sea disturbances is also often unreliable. Moreover, a practical DP control system should meet conflicting control design requirement such as low thruster modulation and good current disturbance rejection. The robust control design via the H x technique
X>
171
enables the modelling errors and disturbances to be taken into account. 2.
linear ship model may then be calculated. The complete linear model may then be written as: .\-,(t) = A,x(t) + B,u(t), y, U) = C,x(t) + D,u(t) (5)
SHIP AND SEA STATE MODELS
where x=[u, v, r, x, y,
Consider the following definitions (see Fig. 1 for details): Input disturbance: An input disturbance affects the output to be controlled directly and the controller is tuned to attenuate such disturbances. For the OP problem, these are Wind, second order waves forces, and current forces. Output disturbance and noise: An output disturbance does not directly affect the output to be controlled and the controller does not attempt to attenuate this disturbance. For the OP problem these are first order Low Freq.
High Freq.
Wind, Wave Current
2.2. Wave Disturbances There are three main sources of disturbances on the ship's position, namely, wind, current, wave motion and the measurement noise. The current, constant wind and second-order wave forces act on the ship at low frequencies « 0.5 rad / sec). The first-order wave forces and possibly the turbulent wind force (depending on the infrastructure of the vessel) and the measurement noise also acts at high frequencies (> 0.7 rad / sec). The control design problem is to compensate the low frequency motion whilst suppressing unnecessary high frequency thruster movements (thruster modulation). Since the wind speed is usually measured, it is advantageous to compensate the effect of the wind using feed-forward control action. The wind feedforward controller is usually employed in conventional DP systems. In the following section, the models of the Low Frequency (LF) and the High Frequency (HF) disturbances are formulated. The high frequency disturbances for the three surge, sway and yaw channels are assumed to be decoupled. The disturbances in each channel are modelled by second-order transfer-functions of the form:
I
Wave"
u
---+ Thrusters
Ship Model
Figure 1 The OP Model wave forces, and measurement noise. From Newton's equations of motion all forces on the ship enter at the same point including the thruster forces. However, the wave model forces will not be added at this point and instead the wave motion effects will be moved to the output. The reason lies in the difference in the type of disturbance. The wave motion must not be offset by the control action and it is therefore an output disturbance in the above terminology. In fact it is necessary to separate the low and high frequency motion signals since the cost function depends upon the output y I due to the low frequency motions which are caused by the input disturbances.
G; ( S )
= b;s2 +2t;I,cU,.,S+W;',
(6) 2 2 a;s +2t;2;W m S+W,., where i =X,Y and Z Converting the three transfer functions into state-space form gives the high frequency disturbance model:
2.1. Ship Model
2.3. The Combined Model
Using Newton's equations of motion, the dynamics of a surface ship can be described by:
The state equations for the low frequency vessel dynamics and the wave disturbance can be combined to obtain the overall model of the vessel:
(m - XIi)u
= (m -
Y,,)rv + (u~ + v;c )CHr S Hr + Tx (I)
(m-Y.;. )v=(m-XJrv+(u~ +V~)CHrSHr +Ty (2)
x(t) = Ax(t) + B:u(t) + DI wet) y(t) = C:x(t) + Dzvet)
(3)
x=u,y=v,~=r (4) where u, v and r are the surge, sway and yaw speeds and x, y and
u
(7)
r
3.
172
LINEAR Ha) FILTER
Consider the dynamic system described by: :((t) = A(t)x(t) + B(t)u(t) + D1(t)w(t)
(8)
y(t)=C,(t)x(t)+D,v(t) (Measurements)
(9)
=(1) = (>(t) (Estimated signal)
Consider the class of non-linear systems driven by white noise with white noise-corrupted observations defined by:
(10)
where x E Rn , )' E Rm. =E RP and wet) and v(t) are vectors of independent white noise sequences of zero means and covariances Q and R, representing the measurement and process noises. Assume
:r(/) = f[X(I) , /]+ K[x(t),/]u(t)+ G[X(t), / ]W(/)
and D:,D~, = D~RD~ The filtering problem is to find an estimate =(t) ofz(t) using the measurements ofy up to time t. The filter z(t)=F(s) yet) is required to minimise the cost function: B,B:
J
=
= D,QD:,
sup 1/=1/; < ,,~[o·"')IIWII;
r~ ,x(O) = 0
where wet) and vet) are zero-mean. white, and Gaussian and uncorrelated with the following properties: E{w(t)} = 0, E{v(t)} = O. Cov{w(t)} = Q, Cov{v(t)} = R, Cov{w(t),v(t)} = 0 Assume that the conditional-mean estimate x(t) is known and used to expand the system and
(11 )
=
fory > 0 and (t) = z( I) - i( t) . The filter is required to be casual and the estimate should be unbiased. The above filtering problem can be written in standard LFT fonnulation as shown in Fig. 2: pes)
= [
B,(t)
C,
0
-I
C2 (t)
D~,(t)
0
Assuming the system (C 2 and [A(t) - j wl C2 (t)
w
-'1
0]
A(I)
z
Figure 2 The Standard system representation ,
A) is detectable
B, (t)] has full rank for all D 2 ,(t)
0),
measurement models in a Taylor series about x(t) = x(t) to obtain the following set ofiinear equations in x{t):
the
x{t):::: f[x(t),t] +
Initial Conditions: x(to) = x o, Y(to)
z(t) = h[x{t),t] +
=0
;(1) = A(t)x(t) + B(t)u(t) + K f [y(t) - C 2 (t)x(t)]
= Y(t)C~ (t)R-'(t)
ah~~:i,t] [x(t) -
x(t)] + vet)
= C2 (t)x{t) + v(t) + n(t) z(t) = C\x(t) y(t)
= C~(t)x(t)
HOD Riccati
(14)
where known input and disturbances are described by:
= Y(t)A' (t) + A(t)Y(t) + Y(t)(r -2C\C: -
met) = f[x(t),/]- x(l)
C(t)R-\C~(t»Y(t) + D, (t)QD,' (I)
Note that by making r ~ 00 , the Hac filter converges to a continuous time Kalman filter.
4.
x(t)] + K[x(t), t]
Rewriting this model in the basic state-space fonn gives: x(t) = A{t)x(t) + B(t)u(t) + G(t)W(I) + met)
Filter Algorithm :
Y(t)
Of~~:i' t] [x(t) -
+ G[x(t), t]w(t)
Table I The linear Hac filter Known Inputs: u{t)
Estimated Output: 2(t)
I Z. q-F-(S)---I~ Y pes)
(12)
linear Hac filter can be summarised as shown in Table I (Zbou et ai, 1996 and Grimble, 1995).
Filter gain: K f
(13)
y(t) = h[x(t), I] + v(t)
net) = h[x(/), /j-[ah[x(I),/] 1 ox(t)]X(/)
(15)
The linearised system matrices are defmed by : A(t) = of[x{t),t)] 1 ax(t), B(t) = K[x(t),t)]
( 16) G(t) = G[x(t), t)] , C1 (t) = ah[x(t), I)] 1 ax(t) The detennination of the resulting extended H", filter is now straightforward and is given in Table 2.
EXTENDED Hac FILTER
The H", filter can be used to estimate the state of a non-linear system by linearising the system equations around a nominal operating point. In this paper, attention is concentrated on systems where the system parameters are augmented as additional states to the system's state vector. This type of system is useful for wave filtering in Ship Dynamic Positioning (DP) systems. where the wave centre frequency changes with wave direction and sea state. Similar procedures to extended Kalman filters are adopted here to fonnulate the extended Hac filter.
4.1. Parameter Estimator
The ability to estimate unknown plant parameters is useful Dynamic Ship Positioning, where unknown wave model parameters are required to be identified. These parameters can be modelled as integrators that are driven by white noise and augmented to the system states.
173
reduced in the usual manner until one of the eigenvaiues of Y becomes imaginary or negative. For time-varying systems, the value of y should be found at each time instant. This will impose a high level of computational load on the filter and make it impractical in real applications. An alternative scheme is to use a time decreasing exponential function for y starting from a large value, i.e. y(t)=Ymue -"'+Ymin (17) The rate of decay of the exponential can be set to 3 to 4 times the plant dominant time constants. The scalar Ymin depends on the weighting functions used to design the filter (Fig. 3). The filter gain is similar to KF gain when y is large. As y decreases, the filter converges to the optimal Ha> filter. The Ymin should be usually set to a value to ensure a viable solution to the Riccati equations. This procedure will
Table 2 The Extended H", Filter 1 II
Inputs m() ] r7l"["() ~'( I )]"( x I) x I , I ] / ox XI,I-,e I =In)
I
n(t) = h[ x(t), I] - [ ch[x( I), t] / m( I) ]x( t)
I
Initial Conditions: x(to) = x o, Y(to) = 0 System Matrices: A(t) = Of[.i(t),t)] / m(t), B(t) = K[.i(t),t)]
G(t) = G[.i(t),t)], C2 (t) = Oh[.i(t),I)]/ met) Filter Algorithm: i(t) = A(t).i(t) + B(t)u(t) + K ,[y(I)- C 2 (I)X(t)- n(t)]+ met) Filter Gain: K, = Y(t)C~ R-'
C~ofG.nma
Estimate: £(t) = C~x(t) H""Error Variance: Y(t) = Y(t)A' (t) + A(t)Y(t) + Y(t)(y- 2 C,C; -
. . G
m
C 2 (t)R-'C~ (t»Y(t) + G(t)QG' (t)
m
The state vector may be partitioned so that: x(t)
= [ x(t)
B(t)] also f(x,t)
=[
1;(B,t)
0
______ .::.....-------'-Hx>=-_ _ -!
o][x] 0 B and
60
20
D,
=[~r ~J, C, =[C,
100
0] Figure 3 The Parameter y
The proposed extended H.o filter for the system can now be defmed in Table 3.
significantly reduce the computational burden and ensures that the filter is stable over the estimation horizon.
Table 3 The EHF for parameter estimation Initial Condition: x(to) = x o' Y(to) = 0
6.
System Malrices:
[1; 0(B) ~=q:/(x,t)]/Oxir;i=
~~2
Filter: i(t) = I(x, t) + K, [y(t) - C2x(t)] Filter gain: K, = Y(t)C~R-' Estimate: =(t) = C~xCt) H""Error Variance:
P ARTITIONED FILTER
The structure of the state-space model of the DP system given in equation (7) can be exploited to simplify the filter and reduce the computation requirement. The state matrices can be partltionea into subsystems representing the input disturbance and output disturbance models. The former are normally the disturbances affecting the output to be controlled and are dominantly of low-frequency behaviour, and the latter are the disturbances which affect the measurements but do not affect the outputs to be controlled. The output disturbance model can also represent coloured measurement noise and is usually dominantly of high frequency behaviour. The partitioned model may therefore be represented in the form:
]
where ~12 = 0[/1 (B)x] / ox 2ir;i
2
YCt) = Y(t)~' + ~Y(t)+ Y(tXy- C,C; - C 2 R-'C~ )YCt) + D, (t)QD; (t)
5.
eo
Time
DETERMINA TION OF Y
The filtering problem can be stated as minimising cost function J in eq. (6) for a known value ofy > O. For the time invariant systems, the optimal value of y can be found by selecting a large value of y and solving the algebraic Riccati equation of Table I to calculate the filter gain. The value of y can then be
rX,(t+I)] [Af(t) x 2 (t + I) = 0
l
[
174
0
Ah(t,B)
] [-,",(t)] x 2 Ct) +
Ef(t)] U(I)+ [Df(t)
o
0
0]
Dh(t)
[w,(t)] w~(t)
=(t)
= [H,(t)
y (t)
= [C / (t)
R=R (18)
6.2. Extended H", Filter Equations Note the that output distance model is assumed to be parameterised in terms of the vector () which includes the wave frequency which are often changing and unknown.
The HF model equations involve the parameter estimates, which are treated as state variables, and hence these equations involve the products of states. The model is therefore non-linear and the estimator must have the form :
6.1. Parameter Estimation Models
i"(t) = !(x(t), u(/» + K f (t)(y(t) - C(t)x(t))
The calculation of the extended Kalman filter gains may be considerably simplified by taking advantage of the partitioning of the gain matrix (Grimble, 1982). The HF model is dependent upon the parameter vector () and the assumption is made that these parameters remain approximately constant. Thus, the vector () may be modelled by an integrator with given initial condition vector (}a. That is:
= De(t)wJ(t)
xJ(t)
Using the partitioning given above: x(/) = [XI T(t)
[
=
0 0]0
[AI(t)
0 0
(}(t)
+
Ah(t,(})
0
and HF state/parameter conditional means, respectively, and note that the non-linear function may be partitioned as: AI(/)XI(t)+ B I (t)U(t)]
[D~t)
DhO(t)
o
0
y(t) = [cl(t)
Estimator equations: The Extended Hoc filter (EHF) equations can be partitioned into the sub-equations: i"1 (I)
(20)
where the partitioned equations become:
DII(/) [
o
0]0
0 _0 ] De(t)
y (t)=[CI(/)
[XI (t)] +
(}(/)
[8 0(1)] 1
u(t)+
[~I(t)]
y=
we(t)
0] [~~:J+v(t)
(23)
x evaluated at
0
[
M 2(t,xp
(})
Since the LF dynamics are assumed constant and independent of the parameter vector the term Mdt) = o and since the high frequency model is independent of the low frequency states M 21 (t) = O. Hoo filter gain: The filter gain matrix may be computed from the !inearised equations as follows : The Riccati equation becomes:
i"(t) = A(t)x(t) + B(t)u(t) + D(t)w(t)
[AII(I)
(t)]
AII(t) M(t,x l ) =
Write these equations, including the parameter estimation model, in the form :
= [ ~I(t)] {}(/)
Cl (t)XI
0 0] 0 000
+v(t) (19)
= C(t)x(t) + vet)
+ K\ (/)[Y(/) -
respect to the estimated state vector time t. That is:
xJ(t)
y (t)
(t)
Linearisation: Let M(t) denote the partial derivative matrix (the Jacobian) of A(t,x2 )x(t) with
WJ{t)
O)[;:~:~]
= All (t)XI
(}(t)= K2(t)[y(t)-C I (t)x.{t)]
~ 1[::~:~l De{t)
Ch(t)
[
0
xJ(t)
Ah(t ' ~)X l (t)
!(x(t),u(t» =
[XI(t)] [BI] x 2 (t) + 0 u(t)
0
xi (t)f denote the LF
Let x(t) = [XIT (I)
The total system model and parameter vector model may be combined to obtain:
XI(t)] x?(t)
(}T(t)f
xJ(t) ~ (}(t)
where
(22)
r[ A~~
~] + [ ~ ~ I
]
p+
~]]y + [~I
(21)
The noise sources are assumed to be zero mean, white and mutually independent with the known second moments:
The individual equations may therefore be written as:
r; I
= r; I All
E{ w
r;2
= A11 r; 2 + r; , VI l'; 2 + r; 2V: Y::
l
(r)w; (I)}
E{w 2(r)w ; (t)}
.:..
--
-T
+
-
A ll
--
-
--
r; I + r; YI r; I + r;: V2Y2 1 + QI
- -- - - ...... - - Y: = Y: VIr;: + Ye: V Y:: + Qe
= Q,o" , E{ v( r)v T (t)} = Rt5 :r = Qh O" , E{we(r)w~(t)} = Quo"
2
where
In terms of the partitioned equations (22), the process and noise covariances become:
-..
I
(25)
l
V,
= r-:r~ -C!R-1CIT and V: = y-2 We .
WI
and We are the weightings on the state and parameter estimates. respectively.
175
References
When y is large {EKF), V: becomes very small and the panitioned equations in (25) can be decoupled. Once, the parameters are converged, the coupled Riccati equation can be solved to find the EHF gains. This procedure will simplify the filter structure and reduces the computational requirements.
Fung, P and M Grimble, 1983. Dynamic Ship Positioning Using Self-tuning Kalman Filter, IEEE Trans. AC, 28 (3), 339-349. Nagpal, K and P Khargonekar. 1991, Filtering and smoothing in an H.., sening, IEEE trans. AC 36 (2), 152-166. Katebi. M R, M j Grimble, and Y Zhang,J997. H robust control design for dynamic ship positioning, lEE proc- Control theory Appl, Vol. 144 No 2 March 1997, 110-120. Zhou K, j Doyle and K Glover, 1996, Robust Optimal Control, Prentice Hall, Inc. Grimble, M J, 1994, Robust Industrial Control, Prentice Hall.
aJ
7.
SIMULATION STUDIES
To demonstrate the perfonnance of the extended H", filter, the simple example of Fig. 4 is considered. The filter is used to estimate the wave frequency, O)h . The numerical values used are 0, = 0.1, 0), = 0.05, Oh = l.e-5, O)h = 0.5, cov(w) = cov(~) = 0.004, cov(v) = 0.1. The exponential function for decreasing y is tuned to y = 500 exp(-0.00015t) + 0.01. The sine wave, u(t) = 0.001 sin(0.05t), is used to drive the low frequency system. The estimation results are shown in Fig. 5 and 6. The wave frequency is estimated in less than a minute. The low-frequency output estimate (y,) and the high frequency output estimate (yJ show good approximation to the true outputs. Further simulation results are needed to compare the robustness of the proposed filter with the extended Kalman filter. 8.
Low Fl'llqUllncy Motion
.2 0
SO
100
150
I
sec
HgII Frequency Motion
:~ K!lA AAT\ fAnV\]1~T\ ::YllIDJIVVlV
{I
CONCLUSION
The Extended H", Filter was developed using similar procedures to those for the Extended Kalman Filter. A scheme was proposed to calculate the optimal value for y to reduce the computational requirements. The filter was used in a case study for Dynamic Ship positioning which requires estimating the sea wave frequency on-line. To simplify the filter structure, a panitioned filter was presented, which is specifically suitable for marine control design applications.
-40
so
100
lSO
sec:
Figure 5 The LF and HF Estimation
~ ~(t) 11
High Frequency Gh(s)
+W(t)
.~ u(t)
Low
F~uency
G,(s)
y(t) .J . \"'\." iF
~~E
'\...
~-p"
v() t
Figure 4 The simulation example
00
G(s) = loaf /(i +2&01s+af) 2 G.(s) =/(;,OlrS / (s 2 +20hOlrs+at' )
50
100
~
sec
Figure 6 The wave frequency Estimation
176
150