Dynamical Correction of Positioning Control Laws

9th IFAC Conference on Control Applications in Marine Systems The International Federation of Automatic Control September 17-20, 2013. Osaka, Japan

Dynamical Correction of Positioning Control Laws Evgeny I. Veremey Faculty of Applied Mathematics and Control Processes, Saint-Petersburg State University, Saint-Petersburg, Russia. (e-mail: [email protected]) Abstract: Modern dynamic positioning systems are widely used now to support marine operations of various types. Correspondent control laws include nonlinear asymptotic observers to restore deficient information about the unmeasured velocities of the vessel. Using obtained estimations, it is possible to stabilize closed loop system and to provide its desirable dynamical features. The complexity of this problem is determined by the presence of an extensive population of certain dynamical conditions, requirements and restrictions that essentially hampers the using of traditional synthesis methods. In this connection the mentioned problem can be decoupled into particular separated problems on the base of a special unified multipurpose structure of control law, which includes dynamical corrector to provide desirable features for the DP vessel motion under influence of external disturbances. Keywords: dynamical positioning, control law, observer, stability, corrector, external disturbances. provide automatic control of the vessel under external disturbances actions, taking into account a lot of dynamical requirements, restrictions and conditions. One of the most suitable methods to solve this multiple objective problem is using the resources of the optimization theory. So, the synthesis of modern control systems for marine vessels is substantially based on the various optimization methods with the general goal to raise the effectiveness and quality level of the systems to be designed. The papers of Veremei and Korchanov, 1989, Veremey, 2010, present theoretical basis for multi-purposes approach to control laws synthesis taking into account peculiarities of the mathematical problems connected with practical requirements to marine vehicles motion control. Last years there were developed some new analytical and numerical methods of control theory and computer technologies, which are based on the special unified structure of control laws for marine autopilots.

1. INTRODUCTION The problem of dynamical positioning (DP) vessel control is one of the most practically significant and theoretically interesting problems in the area of marine control systems analysis and design. Now DP systems are widely used for the various types of marine vessels in different marine connected branches such as hydrography, inspection of marine construction, wreck investigation, underwater cable laying, and so on. The exhaustive survey of DP control systems is done in Sørensen, 2011. Some central theoretical and practical issues providing basic background of DP control are presented in Fossen, 1994 and 2011, Sørensen, 2012. Now there exists a wide range of publications devoted to different questions of DP control systems design. Among them the papers of Fossen and Strand, 1999, Loria, Fossen, and Panteley, 2000 hold a significant position. These works give a mathematical validation for the special structure of DP nonlinear control laws on the base of nonlinear asymptotic observers. Sufficient conditions of the global asymptotic stability are derived and the possibility of independent tuning for observers and state space control laws is proven by analogy with a separation principle for LTI systems. Moreover, a construction of the observer here has an expanded treatment, that allows to provide desirable features of the closed loop system, including integral action for low frequency bias and notching effect for wave disturbances filtering.

The special structure includes some basic part and several separate elements to be adjusted for the specified conditions of a vessel motion. These elements can be switched on or off as needed to provide the best dynamical behaviour. The main adjustable element of the structure is so called dynamical corrector. This term of control law provides an integral action of the controller and a notch filtering effect or compensative counteraction effect for closed loop connection. This work can be treated as the first step on the path to join central ideas of nonlinear DP control, proposed in Fossen, 1994 and 2011, Fossen and Strand, 1999, Loria et al., 2000, with various optimization approaches, based on the separated dynamical correction of nonlinear state observer output. In our opinion, such joining can be useful for some practical situations to develop new mathematical and computational methods of dynamical corrector synthesis, for example, based on the ideas of H -optimization theory. Correspondent

One of the alternative approaches to DP control systems design is the theory of multi-purposes control laws synthesis, which was presented in Veremei and Korchanov, 1989. Central ideas of this approach are reflected on the modern level in Veremey, 2010. The essence of the problem consists of the modern marine control systems should first of all 978-3-902823-52-6/2013 © IFAC

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10.3182/20130918-4-JP-3022.00019

IFAC CAMS 2013 September 17-20, 2013. Osaka, Japan

mathematical background for LTI marine autopilot design was discussed in Veremey, 2012.

Mz& v = −Dz v + τ + R T ( η)K 1 ( η − z η ), z& η = R ( η)z v + K 2 ( η − z η ),

2. GENERAL PROBLEM OF CONTROL CORRECTION

τ = −K d z v − R T ( η)K p (z η − ηd ) + F ( p)(η − z η ),

Let consider a nonlinear 3DOF model of a DP vessel in the following form

Mν& = −Dν + τ + d(t ), η& = R ( η) ν, where the vector ν = ( u v p )

(1)

(5)

where p = d / dt , F( p) is a transfer matrix of the LTI dynamical corrector. This term of the control law plays a crucial role in the discussion to be provided below. The equations (4) form a nonlinear asymptotic observer,

T

where z ν ∈ E

represents velocities in a

vessel-fixed frame, η = ( x y ψ ) is a joint position (x, y ) and a heading angle ψ vector relative to an Earth-fixed T

3

3

and z η ∈ E are the estimations of the

vectors ν and η correspondently. As usual, the main reason to use an observer within a control law model is to obtain the estimations of unmeasured vessel velocities. Additional destination of this term is to reject the influence of measurement noise to control processes.

frame. Vector τ ∈ E 3 implies a control action generated by the propulsion system, vector d ∈ R 3 is provided by the external disturbances of any nature. Matrices M and D with the constant elements are positive definite, M = M T .

For the full construction of the control law with the given structure (4), (5), it is necessary to determine four numeric matrices K 1 , K 2 , K d , K p and also a transfer matrix F( p)

The only nonlinearity of the system (1) is determined by the yaw angle orthogonal rotation matrix

with the aim to achieve the mentioned design objective.

 cos ψ − sin ψ 0    R ( η) = R (ψ ) =  sin ψ cos ψ 0  . (2)  0  0 1   Observe that the equations (4) do not contain the models of additional terms, which are taken into account in the basic works of Fossen and Strand, 1999, Loria et al., 2000. These terms represent bias disturbances and additional waveinduced vessel motion. We believe that theirs absence has no vital importance within the framework of dynamical correction to be proposed here. Let us take into account that usually measurements of the vessel velocities are not available for DP automatic system, so any control law must be designed only on the base of position and heading measurements. The goal is the construction of a nonlinear dynamic control law of the form

z& = f (z, τ, η), τ = g(z, η),

(4)

To this end, first of all let us consider an asymptotic observer and form differential equations for error dynamics with respect to the errors of observation ε ν (t ) = ν (t ) − z v (t ) and

ε η (t ) = η(t ) − z η (t ) . On the base of (1) and (4) we have Mε& v = −Dε v − R T ( η)K 1ε η + d(t ), ε& η = R ( η)ε v − K 2ε η .

(6)

Observe that with the assumption d(t ) ≡ 0 system (6) has desirable zero equilibrium position in which connection the choice of the matrices K 1 and K 2 must provide it GAS and GES (globally exponentially stable). As it was proven in Fossen and Strand, 1999, on the base of passivity property and in accordance with Kalman-Yakubovich-Popov lemma, a sufficient condition of desirable stability is a diagonal structure and positive definiteness of the matrices K 1 and

K 2 . Let suppose, that the matrices with mentioned properties are selected by the some way.

(3) The next question is a construction of the basic state feedback control low of the PD-type form

where z ∈ E is a state space vector of the controller, such that the given requirements must be met for the closed-loop system (1) – (3). k

τ * = −K d ν − R T ( η)K p ( η − ηd ) ,

(7)

which stabilizes desirable equilibrium ν = 0 , η = η d for the

Let ηd ∈ E 3 be the desired constant position vector of the vessel. The design objective is to render the only equilibrium point ν = 0 , η = η d of this system globally asymptotically stable (GAS) and to ensure certain desirable features of a closed-loop dynamics. Here we will not go beyond an integral action of control with respect to slow varied components of the external disturbances d(t ) , determined by the drift, current and wind loads.

closed loop system (1), (7) with the assumption that d(t ) ≡ 0 . It was proven in Loria et al., 2000, that positive definiteness of the symmetrical matrices K d and K p guaranties GAS of the mentioned equilibrium position. Let us also suppose that such the matrices are selected somehow or other. So, the general problem of the control law (4), (5) construction is generated by the presence of the dynamical correction term F( p)( η − z η ) in the equation (5). First of all,

On the analogy of Loria et al., 2000, Fossen, 2011, let realize control law (3) as

it is not fully clear whether the separation principle is valid for this case or not. By other words, it is necessary to prove, 32

IFAC CAMS 2013 September 17-20, 2013. Osaka, Japan

Σ1 : x& 1 = A c (x1 )x1 + g (x1 )x 2 , Σ 2 : x& 2 = A o (x1 )x 2 ,

that the replacement of the state feedback control low (7) by the equation (5) on the base of state estimations does not lead to the loss of stability. And the second part of the problem is to ensure certain desirable features of the closed-loop system (1), (4), (5) with the help of dynamical correction. These two issues are the subject of a further discussion.

that is we obtain the same form of the equations, as in Loria et al., 2000. Nevertheless, the matrices of the system Σ1 have some other representation:

3. STABILITY OF THE CLOSED LOOP CONNECTION Uppermost, let consider the problem of stability posed above. To begin with, we introduce the matrices α, β, γ, µ of a corrector such that F( s ) ≡ γ (E l s − α ) −1 β + µ , which allows to present a corrector model in normal form

p& = αp + βε η ,

(8)

ξ = γp + µε η .

(12)

Here p ∈ E l is a state space vector of the corrector, ξ ∈ E 3 is its output vector, l is a corrector order, E l is correspondent identity matrix. Assume that all the mentioned matrices have constant components and that the matrix α is Hurwitz.

0 0  α   −1  −1 T A c (x1 ) =  M γ − M (D + K d ) R ( η)K p  ,  0  R ( η) 0  

(13)

 0  −1 g(x1 ) =  M K d  0 

(14)

β

  R ( η)K p + µ  ,  0  T

 − M −1D − M −1R T ( η)K 1  . A o (x1 ) =  T (15)  − K2  R ( η)  Next we can directly apply the scheme of stability proof proposed in the mentioned paper. First of all, let us consider the system x& 1 = A c (x1 )x1 with the detailed equations

p& = αp,

Now remark that the variables z v , z η can be excluded from

−1

−1

−1

T

ν& = −M ( D + K d ) ν − M R ( η)K p ( η − ηd ) + M γp,

the equation (5) of the control signal τ , using observation errors as follows

η& = R ( η) ν,

τ = −K d ( ν − ε v ) − R T ( η)K p ( η − ε η − ηd ) + F( p)ε η ≡

which can be treated as a cascaded structure

≡ −K d ν − R T ( η)K p ( η − ηd ) + K d ε v + R T ( η)K p ε η +

(9)

−1 S1 : ξ& 1 = A ξ (ξ1 )ξ1 + M γξ 2 ,

(17)

+ F( p)ε η ≡ τ * + K d ε v + R T ( η)K p ε η + γp + µε η .

S 2 : ξ& 2 = αξ 2

Taking into account the observation error dynamics (6), representation (8) and identity (9), the equations of the controller (4), (5) can be transformed to the form

with zero equilibrium position, where ξ1 = ν

(

(10)

ε& η = R ( η)ε v − K 2ε η , τ = τ * + K d ε v + R T ( η)K p ε η + γp + µε η .

Now let consider the full system of differential equations for closed loop connection (1), (10), assuming d(t ) ≡ 0 :

ε& η = R ( η)ε v − K 2ε η , p& = αp + βε η ,

(11)

ν& = −M −1Dν + M −1τ * +

(

)

+ M −1 K d ε v + R T ( η)K p ε η + γp + µε η , It is convenient to introduce the following new notations:

ν

), T

( η − ηd )

T

)

T

, x2 =

(

εTν

all components of the matrix M γ are constant, it is easy to verify, that the system (16) fully satisfies all the conditions of the Theorem 1 from the Appendix to Loria et al., 2000. In accordance with the statement of this theorem, mentioned equilibrium is uniformly globally asymptotically stable (UGAS).

Σ1 : x& 1 = A c (x1 )x1 + g(x1 )x 2 , *

Σ 2 : x& 2 = A o (t )x 2 .

(18)

Then we can directly implement the proof taken from Loria et al., 2000. Since the system Σ 2 has GAS and GES zero equilibrium and all the components of the matrices β and µ

η& = R ( η) ν.

(

T

Taking into account the result obtained above, we can prove that the systems Σ1 and Σ 2 are complete by the same way as in a mentioned paper, that allows to turn from (12) to the cascaded system

ε& v = −M −1Dε v − M −1R T ( η)K 1ε η ,

x1 = p

( η − ηd )

−1

Mε& v = − Dε v − R T ( η)K 1ε η + d(t ),

T

T

ξ 2 = p . Since the matrix α of the corrector is Hurwitz and

p& = αp + βε η ,

T

(16)

in g(x1 ) (14) are constant, the Theorem 1 also is valid. This means that the desirable equilibrium position ν = 0 , η = η d of the closed loop connection (11) is GAS.

)

T εTη

and to rewrite equations (11), (7) as follows:

Obtained result guaranties that not only the estimation and position error dynamics can be investigated separately, but 33

IFAC CAMS 2013 September 17-20, 2013. Osaka, Japan

also that the same separation relates to the correction dynamics under the condition of the corrector asymptotic stability. Therefore, both the observer (4) and the base controller (7) and the corrector (8) can be tuned independently of one another.

T

z v 0 = − R ( ψ 0 )K 2 ε η 0 .

After substitution (21) to the first relation of the system (20) obtain the linear system T

R (ψ 0 )K p ( η − η d ) = Tε η0 ,

4. INTEGRAL ACTION OF CORRECTOR

T

with respect to position error e η = η − ηd . Observe, that the T

matrix R (ψ 0 )K p is not singular, so if T = 0 , then for any vector ε η0 (i.e. for any disturbance d(t ) ≡ d 0 ) the linear system (22) has unique trivial solution e η = η − ηd ≡ 0 , that corresponds to the astatic property. Assuming ψ 0 = ψ d on the base of obtained identity, we can write a requirement to the matrix F (s) , which directly follows from the equality T = 0 , guarantying zero position error of the system:

external disturbance d(t ) ≡ d 0 with constant components. Certainly, astatic property can be realized practically only in the range of the actuators possibilities.

T

It is easy to see, that the simplest way to realize a requirement (23) is to use corrector with no dynamics, so F ( p) ≡ K ∆ , where the matrix K ∆ with constant components is determined by the requirement (23).

To this end, let suppose that there exists equilibrium position of the closed loop system (1), (10) for certain external disturbance d(t ) ≡ d 0 . First of all, consider the observation error equations (6) for equilibrium position: T

(19)

Let illustrate practical implementation of proposed idea on the base of the DP control system for the vessel "Northern Clipper" with the model (1), taken from Fossen and Strand, 1999. The length of this vessel is L = 76.2 m and mass is

 −D − R T (ψ 0 )K 1  ≠0 det  − K2  R (ψ 0 )  the system (19) has a unique solution

(

)

6

m = 4.59 ⋅ 10 kg, correspondent matrices of the equation (1) are presented as follows:

.

 5.31 ⋅ 10  M= 0   0 

Now let us revert to the controller (4), (5) and also consider its equations for equilibrium position: T

0 = −Dz v + τ + R (ψ 0 )K 1ε η0 , 0 = R (ψ 0 ) z v + K 2 ε η 0 ,

 5.02 ⋅ 10  D= 0   0 

T

τ = −K d z v − R (ψ 0 )K p (z η − η d ) + F(0)ε η0 . An evident transformation of these equations implies

[

T

T

]

6

4

0 6 8.28 ⋅ 10 0 0 5 2.72 ⋅ 10 6 − 4.39 ⋅10

 0  , 0 9 3.75 ⋅ 10    0  6 − 4.39 ⋅ 10  . 8  4.19 ⋅ 10  

The matrices K 1 and K 2 of the observer (4) we also accept from the mentioned paper:

T

0 = −( D + K d ) z v − R (ψ 0 )K p ( η − ηd ) + + F ( 0 ) + R ( ψ 0 ) K p + R ( ψ 0 )K 1 ε η 0 ,

Let us note, that, strictly saying, integral action of the controller (4), (5) is achieved by indirect way and actually has compensative nature similar to the approach presented in Loria et al., 2000. 5. DYNAMICAL CORRECTION EXAMPLE

where ψ 0 is equilibrium value of a heading angle. Under the condition that

T εTη0

T

F(0) = K ∆ = −(D + K d )R (ψ d )K 2 − R (ψ d )(K p + K 1 ) . (23)

Let us obtain a requirement to the transfer matrix F(s ) of dynamical corrector (8), which guaranties the mentioned property of a controller.

εTν 0

T

T = (D + K d )R (ψ 0 )K 2 + F(0) + R (ψ 0 )(K p + K 1 ) ,

Hear we will restrain our discussion only by the integral feature of the closed loop system. Firstly recall, that this system is said to be astatic with respect to the position error vector e η = η − ηd , if it tends to zero as t → ∞ for any

0 = R (ψ 0 )ε v − K 2 ε η ,

(22)

where

As it was mentioned above, the main function of the corrector is to provide desirable features of the feedback DP control system. This role of the corrector is substantially supported by the possibility of its relatively independent tuning with respect to the main part of the controller.

0 = −Dε v − R (ψ 0 )K 1ε η + d 0 ,

(21)

(20)

0 = R (ψ 0 ) z v + K 2 ε η 0 , and from the second relation we have equilibrium vector z v 0 of the speed estimation:

0  0  0.1 0 1 . 1 0     K 1 =  0 0.1 0  , K 2 =  0 1.1 0  .     0 0.01 0 1.1  0 0 Let the basic control law (7) be determined by the matrices

34

IFAC CAMS 2013 September 17-20, 2013. Osaka, Japan

p& = αp + βε η ,

0 0   0.0207   8 Kd =  0 0.0155 0.0439  ⋅10 ,   0.0439 4.05   0 0 0   0.0213   7 Kp = 0 0.00990 0  ⋅ 10 ,   0 4.49   0 which provides the following eigenvalues for the closed loop system if heading angle is small, i.e. R ( η) ≈ E3×3 :

with the following matrices: 0 1 0 0 0 0     − 0 . 0228 − 0 . 302 0 0 0 0    0 0 0 1 0 0  , α=  0 0 − 0.0154 − 0.0248 0 0    0 0 0 0 0 1    0 0 0 0 − 0.032 − 0.358  

s1 = s2 = −0.1 , s3 = s4 = −0.12 , s5 = s6 = −0.2 . Calculations accordingly (20) give us the matrix

0 0   0.0193   0 . 00400 0 0    0 0.0182 0  8  ⋅ 10 , β=  0 0.00170 0    0 7.19   0  0 0 1.25  

0   −0.0255 −0.0013   8 K ∆ =  0.0012 − 0.0210 0  ⋅ 10 ,   0 0 − 9.51  which guaranties astatic property with respect to a position error, if the equality F(0) = K ∆ for the corrector holds. Firstly, let accept the simplest variant of the compensator with transfer matrix F ( p) ≡ K ∆ , providing an astatic property for the closed loop system (1), (4), (5). Correspondent transient processes for given T ° ηd = (xd y d ψ d ) , xd = 30 m , y d = 30 m , ψ d = 45 are presented by figure 1. 30

0 1 0 0 0 0   γ = 0 0 0 1 0 0 , 0 0 0 0 0 1   0 0   − 0.0062   µ= 0 − 0.0029 0  ⋅ 108 .  0 0 2.33   This LTI system is asymptotically stable with the eigenvalues

30

10 0

ψ (deg)

y (m)

x (m)

40 20

20 10

0

100 200 t (sec)

0

300

30

0

100 200 t (sec)

s1 = −0.150 , s1 = −0.152 , s3 = −0.120 , s4 = −0.128 ,

20

0

300

s5 = −0.178 , s6 = −0.180 . 0

100 200 t (sec)

300

Besides the corrector provides an astatic property of the closed loop connection, here we also achieve additional notching effect with respect to the suppression of the wave influence to the control variable τ .

30

10 0

ψ (deg)

y (m)

x (m)

40 20

20 10

0

100 200 t (sec)

0

300

30

0

100 200 t (sec)

20

0

300

0

100 200 t (sec)

Though this property of the system is not the matter of the discussion in this work, let consider the stabilization process presented by the graph of the function τ1 (t ) on the figure 2. This process corresponds to the closed loop system (1), (4), (5). Before the 550-th second the controller (5) works with the simplest variant F ( p) ≡ K ∆ of the corrector, but then we use the corrector (24) with additional filtering feature.

300

30

ψ (deg)

y (m)

x (m)

40 20

20

10

10

0

0

(24)

ξ = γp + µε η

20

5

x 10

0

100 200 t (sec)

300

0

100 200 t (sec)

300

0

0

100 200 t (sec)

300

5

τ1 (N)

Fig.1. Transient processes for the closed loop system. Hear the first row presents processes with no external disturbances, the second one goes with d(t ) ≡ d 0 and with no corrector, and at last, the third row shows the processes when corrector is switched on and provides integral action.

0

-5

100

200

300

400

500

600

700

800

900

1000

t (sec)

Fig. 2. Control action τ1 (t ) for the closed loop system.

The more complicated variant of the corrector satisfying the condition F(0) = K ∆ can be presented by the equations

Comparison of the both parts of the process, presented by figure 2, illustrates the significant effectiveness of the filtering with the help of the proposed control law. 35

IFAC CAMS 2013 September 17-20, 2013. Osaka, Japan

Automation and Remote Control, 49 (9 pt 2), pp. 1210 – 1219. Veremey E.I. (2010) Synthesis of multiobjective control laws for ship motion. Gyroscopy and Navigation, 1 (2), pp. 119 – 125. Veremey E.I. (2012) H ∞ -Approach to Wave Disturbance Filtering for Marine Autopilots. Proceedings of 9th IFAC Conference on Maneuvering and Control of Marine Craft. Arenzano, Italy, September 19–21.

6. CONCLUSIONS The main goal of this work is to expand the idea of a separate tuning for all units of nonlinear DP control law. In our opinion, this idea is not fully realized in existing control laws, where the integral actors and the dynamical filters are incorporated into the asymptotic observer. Such a combination can be treated as a comprehensive whole and a separation principle here is applied only to the basic state control law and to the generalized observer turning. The mentioned combination is not always convenient for some situations, where we need to have a certain flexibility of the control law. Such flexibility is granted by the structure (4), (5), where the proposed corrective term can be selected subject to a current regime of the vessel motion in the following variants: 1. If the vessel moves under the condition of quiet water, dynamical corrector can be fully switched off, i.e. we have no any correction, so controller works in the spared regime. 2. If we have a motion under significant bias disturbances, but no sea wave, it is quite suitable to assign constant transfer function of the corrector, providing an integral action, that does not overload the system by additional useless dynamics. 3. If also sea wave influences to the vessel motion, dynamical corrector can be commuted to the filtering regime, keeping an integral property to react against bias. 4. At last, if there is any possibility to compensate external sea wave disturbances, the transfer matrix of the corrector can be correspondently changed also keeping integral action of the controller. The results of investigations presented above can be developed to take into account transport delays and robust features of the DP control law. REFERENCES Sørensen, A. J. (2011) A survey of dynamic positioning control systems. Annual Reviews in Control, 35, pp. 23– 136. Fossen T. I. (1994) Guidance and Control of Ocean Vehicles. John Wiley & Sons. New York. Fossen T. I. (2011) Handbook of Marine Craft Hydrodynamics and Motion Control. John Wiley & Sons, Ltd. Sørensen, A. J. (2012) Lecture notes on marine control systems. Technical Report UK-12-76, Norwegian University of Science and Technology, 2012. Fossen, T. I. and J. P. Strand. (1999) Passive Nonlinear Оbserver Design for Ships Using Lyapunov Methods: Experimental Results with a Supply Vessel. Automatica, Vol. (35), No. (1), pp. 3-16. Loria A., T. I. Fossen, and E. Panteley. (2000) A Separation Principle for Dynamic Positioning of Ships: Theoretical and Experimental Results. IEEE Transactions of Control Systems Technology, Vol. 8, No. 2, pp. 332-343. Veremei E.I., Korchanov V.M. (1989) Multiobjective stabilization of a certain class of dynamic systems. 36