Int. J. Appl. Math. Comput. Sci., 2013, Vol. 23, No. 1, 4763DOI: 10.2478/amcs-2013-0005

LINEAR ADAPTIVE STRUCTURE FOR CONTROL OF A NONLINEAR MIMODYNAMIC PLANT

STANISAW BANKA, PAWE DWORAK, KRZYSZTOF JAROSZEWSKI

Faculty of Electrical EngineeringWest Pomeranian University of Technology, Szczecin, 26 Kwietnia 10, 71-126 Szczecin, Poland

e-mail: {stanislaw.banka,pawel.dworak,krzysztof.jaroszewski}@zut.edu.pl

In the paper an adaptive linear control system structure with modal controllers for a MIMO nonlinear dynamic process ispresented and various methods for synthesis of those controllers are analyzed. The problems under study are exemplified bythe synthesis of a position and yaw angle control system for a drillship described by a 3DOF nonlinear mathematical modelof low-frequency motions made by the drillship over the drilling point. In the proposed control system, use is made of aset of (stable) linear modal controllers that create a linear adaptive controller with variable parameters tuned appropriatelyto operation conditions chosen on the basis of two measured auxiliary signals. These are the ships current forward speedmeasured in reference to the water and the systematically calculated difference between the course angle and the sea current(yaw angle). The system synthesis is carried out by means of four different methods for system pole placement after havinglinearized the model of low-frequency motions made by the vessel at its nominal operating points in steady states that aredependent on the specified yaw angle and the sea current velocity. The final part of the paper includes simulation results ofsystem operation with an adaptive controller of (stepwise) varying parameters along with conclusions and final remarks.

Keywords: MIMO multivariable control systems, nonlinear systems, modal control.

1. Introduction

Control of multivariable dynamic plants is still the subjectof studies and is the source of many unresolved issues,especially those concerning nonlinear systems. Nonlinearcontrol systems are commonly encountered in manydifferent areas of science and technology. In particular,problems difficult to solve arise in motion and/or positioncontrol of various vessels, like drilling platforms andships, sea ferries, special purpose ships as well assubmarines. Complex motions and/or complex-shapedbodies moving in the water, and in the case of ships also atthe boundary between water and air, give rise to resistanceforces dependent in a nonlinear way on velocities andpositions, thus causing the floating bodies to becomestrongly nonlinear dynamic plants.

In general, there are two basic approaches to solvethe control problem for nonlinear plants. The first one,called nonlinear, consists in synthesizing a nonlinearcontroller that would meet certain requirements overthe entire range of control signals variability. Thesecond approach, called linear, consists in designingan adaptive linear controller with varying parameters to

be systematically tuned up in keeping with changingplant operating conditions determined by system nominaloperating points. Nominal operating points areusually defined in steady states of the plant; however,these also can be determined in its transient regimes.

The nonlinear approach may include techniquesbased on the second Lyapunov method, for example,by employing the sequential backstepping procedure(Fossen and Strand, 1999; Witkowska et al., 2007)or methods that consist in system linearizingthrough a plant output (or state) related nonlinearfeedback, supported by feedforward compensatorswith characteristics being inverse to nonlinearfunctions contained in the plant description (Fabriand Kadrikamanathan, 2001; Zwierzewicz, 2008). In thecase when nonlinear descriptions of the plant are notknown accurately, advantage can be taken of methodsemploying artificial intelligence techniques, for example,those using neural approximators (Tzirkel-Hancock andFallside, 1992; Fabri and Kadrikamanathan, 2001; Pedroand Dahunsi, 2011). Substantial difficulties encounteredin employing this nonlinear approach are due to the factthat control plants are multivariable.

{stanislaw.banka, pawel.dworak, krzysztof.jaroszewski}@zut.edu.pl

48 S. Banka et al.

However, in practice, the second approach, calledlinear, is more convenient to use, since advantage can betaken of already proven procedures and commonly knownmathematical methods employed in the design (synthesis)of linear controllers. Here, linearization of nonlinearMIMO plants is a prerequisite for the methods to beemployed. The most frequently used way of linearizationconsists in taking a Taylor series expansion of a nonlinearfunction and then taking only the first order term ofthe expansion. After linearization, local linear modelsare obtained, valid for small deviations from operatingpoints of the plant.

The obtained linear models with known parametersor those to be identified are the starting point for applyingmany known methods for linear control system design.These can be both traditional ones to design classic PIDcontrol systems, although being difficult to implement inthe case of MIMO plants, and relatively simple ones tosynthesize systems with multivariable modal controllers(or possibly LQR/LQG) based on the Luenberger observeror the Kalman filter (Antoniou and Vardulakis, 2005;Banka, 2007; Kaczorek, 1992; Wolovich, 1974). Sinceproperties exhibited by linear models at different (distant)operating points of the plant may substantially vary, thecontrollers used should be either robust (usually of a veryhigh order, as has been observed by Gierusz (2005)) oradaptive, switched (Zhai and Xu, 2010; Tomera, 2010;Banka et al. 2010b; 2011a) or with parameters being tunedin the process of operation (Astrm and Wittenmark,1995).

If the description of the nonlinear plant is known,then it is possible to make use of systems with linearcontrollers prepared earlier for possibly all operatingpoints of the plant. Such controllers can create either a setof controllers with switchable outputs from among whichone controller designed for the given system operatingpoint (Banka et al., 2010a) is chosen, or multi-controllerstructures whose control signal components are formedas weighted means of outputs of a selected controllergroup (fuzzy cluster) according to TakagiSugenoKangrules. The weights could be proportional to the degree oftheir membership of appropriately fuzzified areas of plantoutputs or other auxiliary measured signals (Tatjewski,2007).

What all the above-mentioned multi-controllerstructures, where not all controllers at the moment areutilized in a closed-loop system, have in common isthat all controllers employed in these structures must bestable by themselves, as opposed to a single adaptivecontroller with varying (tuned) parameters. This meansthat system strong stability conditions should be fulfilled(Vidyasagar, 1985).

In the paper an adaptive modal MIMO controllerwith (stepwise) varying parameters in the process ofoperation is studied. As already mentioned, the controller

can also be physically realized as a multi-controllerstructure of (stable) modal controllers with switchableoutputs. In such a case, the number of controllers shouldbe limited to a cluster of controllers with fewer numberof controllers. This cluster should be designed for thenear surroundings of the current operating point ofthe system. The remaining controllers, in such a case,could be stored on the disk or redesigned in on-linemode adequately to the needs. The modal controllersmaking up the adaptive (multi-controller) control systemconsidered will be designed for all possible operatingpoints of the nonlinear MIMO plant. The appropriatecontroller (appropriate set of parameter values of the tunedcontroller) will be selected during system operation onthe basis of two auxiliary measured signals, on which theoperating points of the nonlinear plant are dependent.

The organization of this paper is as follows. InSection 2 a mathematical description of the adoptednonlinear control plant is presented. In Section 3 wediscuss the structure of the proposed control system basedon a set of linear modal controllers that may create anadaptive controller with (stepwise) varying parametersconditioned by two additional auxiliary signals, namely,the ship transitional velocity measured with respectto water and the calculated difference between thesea current angle and the actual ships yaw angle. InSection 4 we carry out a survey of synthesis methodsfor multivariable modal controllers in both time andfrequency domains using the polynomial approach withand without solving Diophantine polynomial matrixequations. Section 5 contains results of controllersynthesis obtained by means of the methods presented inSection 4. The operation of the found controller sets istested in Section 6 by simulation of the designed tunedcontroller system with the nonlinear plant model. We endthe paper in Section 7 with conclusions.

2. Description of the control plant

The MIMO nonlinear dynamic control plant isexemplified here by the drillship Wimpey Sealab havingLpp = 94.49 [m] in length, B = 15.24 [m] in beam,with an average draught of H = 5.49 [m ] and with adisplacement of m = 5670 DWT . When operated, theship was equipped with a simple (clinometric) DynamicPositioning System (DPS) with classical autonomous PIDcontrollers. The system enabled the vessel to keep oncourse and position over the sea bed drilling point withthe help of a 2013 [kW ] main engine and four azimuthSchottel propellers of 746 [kW] each.

The adaptive (multi-controller) control systemstructure considered is studied by means of a 3DOFnonlinear mathematical model of the ships low-frequencymotions, which has been developed on the basis of testscarried out on a physical model on a scale of 1:20 in

Linear adaptive structure for control of a nonlinear MIMO dynamic plant 49

an American ship model basin (Wise and English, 1975).The yaw angle and the ships position in DSP are definedin an Earth-based fixed reference system whose axes aredirected northwards (N) and eastwards (E), and whoseorigin is located over the drilling point on the seabed.By contrast, force and speed components with respect towater are determined in a moving system related with theships body and the axes directed to the front and thestarboard of the ship with the origin placed at its gravitycenter. These are shown in Fig. 1.

x y1 1=

vc

c

x y3 3=

x ,u6 3

x ,u5 2

x ,u4 1

N

S

W E

x y2 2=

Fig. 1. Ships co-ordinate systems.

The mathematical description of the plant is givenin the form of nonlinear state space and linear outputequations:

x1 = x4 cosx3 x5 sinx3 + Vc cosc,x2 = x4 sin x3 + x5 cosx3 + Vc sinc,x3 = x6,

x4 = 0.088x25 0.132x4Vs + 0.958x5x6 + 0.958u1,(1)

x5 = 1.4x5Vs 0.978x35/Vs 0.543x4x6+ 0.037x6 |x6| + 0.544u2,

x6 = (0.258x5Vs 0.764x4x5 0.162x6 |x6| + u3),y1 = x1, y2 = x2, y3 = x3,

where Vs =

x24(t) + x25(t) is the translational velocity

of the ship measured with respect to water. The coefficienta = k2zz + 0.0431 describes the ships inertia momenttogether with water associated with the angle motion ofthe ship around its vertical axis, where kzz is the relativeinertia radius referenced to the ships length Lpp. Vc andc are, respectively, the velocity and direction of the seacurrent as indicated in Fig. 1.

All the signals appearing in (1) are dimensionless,i.e., referenced to the ships dimensions and displacementas follows:

u1(t) =Fx(t)mg

, u2(t) =Fy(t)mg

,

u3(t) =Mz(t)mgLpp

,

x1(t) =y1(t)Lpp

, x2(t) =y2(t)Lpp

,

x3(t) [rad] , (2)

x4(t) =vx(t)gLpp

, x5(t) =vy(t)gLpp

,

x6 =z(t)gLpp

,

together with the dimensionless time t = tr/

Lpp/g 0.32 tr.

It should be noted that dividing by a signalrepresenting the ships translational velocity Vs(t) withrespect to water takes place in the above nonlinear shipmotion model. This accounts for undefined behavior ofthe nonlinear model at zero-valued ship velocity, i.e.,when dividing by Vs(t) = 0 occurs. This has someconsequences not only during system simulation, but alsofor control system synthesis, since linear models becomeundefined at Vs = 0. Hence, controllers with a structurelike that determined for normal operation conditions atVs(t) = 0 cannot be found in this case. This is attributableto the fact that hydrodynamic resistance disappears atVs(t) = 0, which substantially affects the character of thedescribed phenomena and brings about, among others, thezeroing of respective terms in Eqn. (1). Such a situationtakes place when the ship is carried along by currents orwhen the ship stands still over the drilling point in calmwater at Vc = 0.

According to the linear approach adopted in thepaper, the linearization of the model (1) is performedfor ship typical locations within the area of admissiblepositions over the drilling point in steady state whenVs(t) = Vc. The nominal values of the state vectorxo and forces, as well as the moment uo enabling toovercome hydrodynamic resistances of the ships hull,given the known values of Vc = 0 and c, canbe calculated from the system of nonlinear algebraicequations

0 = f(xo,uo, Vc, c), (3)yo = Cxo.

As a result of the linearization performed in thewhole range of the yaw angle x30 [, ] [rad], undervarious sea current velocities Vc [0.053.5] [knot] and

50 S. Banka et al.

at c = [rad], the linear state-space models are obtained

x(t) = A[x(t) xo] + B[u(t) uo], (4)y(t) yo = C[x(t) xo],

where

A =[

xfT(x,u, )

]T

x=xou=uo

=

0 0 0 a14 a15 00 0 a23 a24 a25 00 0 0 0 0 10 0 0 a44 a45 a460 0 0 a54 a55 a560 0 0 a64 a65 0

,

B =

0 0 00 0 00 0 0

0.958 0 00 0.544 00 0 1/a

,

C =

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 0

,

with the entries aij depending on the difference betweenthe sea current angle c and values of the ships yaw angley30 = x30 adopted for the purpose of linearization, andon the current velocity Vc. All the obtained models of theship are unstable with three invariable poles s1 = s2 =s3 = 0.0 and three variable poles s4, s5, s6. Their matrixtransfer functions in the complex domain can be presentedin the form of a relatively right prime (r.r.p.) polynomialMatrix Fraction Description (MFD),

T(s) = B1(s)A11 (s), (5)

where

B1(s) =

b1 b2 0b3 b4 00 0 1/a

,

A1(s) =

s2 + a1s a2s a3s + a4

a5s s2 + a6s a7s + a8

a9s a10s s2 + a11

,

with variable parameters bi, i = 1, 2, 3, 4 and aj , j =1, 2, . . . , 11. The gain matrix of the plant is defined as

Kp = [B1(0)][A11 (0)

]

=

b1 b2 0b3 b4 00 0 1/a

0 0 a40 0 a80 0 a11

1

, (6)

which demonstrates in general the integration propertyof the control plant. However, this observation is not

u( )t-

Modal

controllers

c

Plant

u( )tSaturation

block+

+

uo

e( )t

+

yref

y( )t

-

+

Vc

V (t)s x (t)3

Fig. 2. Block diagram of the proposed control system structure.

applicable to all control paths, among others, to thoseacting on the ships course angle. The coefficient aappearing in Eqns. (1) and (6) depends on the extent towhich the ship is loaded and on the mass distributionon board the ship. Since the numerator matrix B1(s) inthe transfer function (5) is a real matrix, all ship linearmodels are minimum phase, i.e., non-minimum phasetransmission zeroes do not occur there.

3. Description of the proposed controlsystem structure

The block diagram of the control system for ship courseand position over the drilling point is depicted in Fig. 2.The above control system for the nonlinear MIMOplant with specified set points yref consists of a set ofmultivariable modal controllers realized either as a singleadaptive controller with stepwise switchable parametervalues or as a set of controllers with a common inpute(t) and switchable outputs u(t). All modal controllersmaking up the above structure are designed for differentship linear models obtained for adopted operating pointsof the plant at different sea current velocities Vc and yawangles y3o = x3o of the ship standing still over the drillingpoint. The points are determined by nominal values of theplant state vector xo and nominal values of the controlsignals uo in steady states. These are found from thesystem of algebraic equations (3). For such a kind ofplants, xo and uo depend exclusively on the yaw angle setpoint y30, as well as on the velocity Vc and the sea currentangle c.

In the proposed multi-controller structure, thecontroller parameter values are changed (or controlleroutputs are changed over, respectively) on the basisof auxiliary variables measured. These are in the caseunder study: the ships current transitional velocity Vs(t)measured with respect to water (it is negative if the shipsails astern, i.e., at x4(t) < 0) and the systematicallycalculated difference between the sea current angle andthe ships yaw angle c x3(t). During the systemoperation the incremental values u(t) generated by theadaptive controller are added to the nominal values uo.

Modal controllers used in the proposed control

Linear adaptive structure for control of a nonlinear MIMO dynamic plant 51

system structure are multivariable dynamic systems withparameters defined in time domain by

xr(t) = Arxr(t) + Bre(t), (7)u(t) = Crxr(t) + Dre(t).

These can be presented in their natural form, whichis called standard, with the following matrices:

Ar = A BF LC, Br = L, (8)Cr = F, Dr = 0,

where F is the matrix of proportional feedbacks thatare related to state vector components (reconstructedby the observer) of the plant linear models, and L isthe gain matrix of full order Luenberger observers thatreconstruct the state vector of the plant linear models(4). Another possibility (although this is a necessity ifthe polynomial approach with solving polynomial matrixequations is employed) is to present Eqn. (7) in anappropriate canonical form (most common an observableone) with the matrices

Aro, Bro, Cro and Dr = 0. (9)

Unlike the matrices in the standard form, theseare characterized by a minimal number of parametersdifferent from 0 or 1. The above controllers representstrictly causal dynamic systems with Dr = 0. In thes-domain they are described by strictly proper matricesof rational transfer functions in the form of relatively leftprime (l.r.p.) polynomial matrix fractions

Tc(s) = Cr(sIn Ar)1Br (10)= Cro(sIn Aro)1Bro = M12 (s)N2(s),

with the polynomial matrices: M2(s) R[s]mmis a nonsingular row-reduced denominator matrix andN2(s) R[s]ml is a numerator matrix that fulfills thestrict inequalities

degrjN2(s) < degrjM2(s), j = 1, 2, . . . , m, (11)

where degrj [] denote row degrees of [].Static properties of MIMO modal controllers under

discussion depend directly on their gain matrices

Kc = Cr(Ar)1Br = M12 (0)N2(0), (12)and the dynamic properties are determined by poles po-le_reg, defined by the eigenvalues of the matrix Ar ofeach of the controllers, which represent zeroes of thedeterminants

detM2(s) = det [sIn Ar] = 0. (13)In general, the controllers considered can be stable

or unstable. By definition, they cannot exhibit integration

properties and should be stable in the proposed structure.In the case under discussion these will be multivariable(MIMO) controllers whose behavior is close to that PDones with time lag.

In order to limit the effect of excessive forcesand moments produced by the adaptive set of modalcontrollers, we introduce constraints imposed on themaximal values of control signals u(t) = u(t) + uo. In areal ship control system, a block of propulsion distributionamong individual propellers and the main engine has beenused instead of a block for constraining control signalsu(t).

If the values of uo are known and the modalcontrollers are properly designed (for the given operatingpoints), there exists a theoretical possibility that theresidual steady-state error will tend to zero est(t) 0 as u(t) 0. In real situations, the values uo canbe corrected manually in the block for compensation ofsteady-state errors in such a way as to eliminate (orreduce) possible deviations of the ships course and/orposition in steady state for the reason that the effectof some environmental disturbances (wind, motion ofthe sea) has been neglected here and/or the actual shipoperation conditions differ from the nominal adoptedfor linearization. Another reason is the lack of knowledgeabout real nominal values of control signals required tomaintain the ships position in steady state.

It should be noted that steady state errors maybe brought about not only by the effect of additionallong-lasting forces and moments turning the shipproduced by, among others, the (averaged) action ofwind and sea waves, but also for the reason that not allpaths of the ships multivariable model exhibit integrationproperties. This is the case with modal PD controllers withtime lag (Banka and Latawiec, 2009).

4. Methods of modal controller synthesis

The synthesis of modal controllers is based on usingthe technique of pole placement in stable regions of thes-plane. In the case of SISO, pole placement determinesthe system dynamics in one control path only, so thetask is easy to accomplish and results of calculations areunambiguous, i.e., they are independent of the structure ofsource data.

The synthesis of modal controllers with MIMOplants is much more complex, since the dynamics of manycontrol paths are to be shaped. The system poles in eachpath may take different values in accordance with to thedynamics required for each of the paths. This raises thequestion of how to provide the location of a specificpole for an appropriate path of the control system to bedesigned. The task is not easy to perform and, as it turnsout, the final result depends not only on selecting anappropriate design method, but also on setting a concrete

52 S. Banka et al.

data structure used for the design. Additionally, the resultsmay depend on whether the poles are real or complexand on the order in which the poles occur in the setof data taken for design. In a polynomial approach it isrequired to divide the set of pre-determined poles intoappropriate pole blocks with a specific number of poles ineach block. If the poles are conjugate complex, each pairof them must be an element of the same block. This isparticularly essential for plants of odd order n, and also ifan odd number of poles is required for individual blocksof adopted pole values. As a result, completely differentmodal controllers may be obtained for the same input datadepending on the adopted design method and the adopteddata structure used for design.

The synthesis of modal controllers can be performeddirectly in time domain with the plant linear models (4)as a starting point and in s-domain using the polynomialapproach with the transfer function matrices (5) asa starting point. Using the polynomial approach withsolving polynomial matrix equations usually yields causalcontrollers described in s-domain by matrices of properrational transfer functions obtained directly from solutionsof Diophantine polynomial matrix equations. If we decideon (strictly causal) modal controllers based on full-orderLuenberger observers, the design performed directly intime domain (and also in s-domain without solvingpolynomial matrix equations) boils down to separatedetermination of the feedback matrix F, which forces theclosed-loop eigenvalues to the pole locations specifiedby the adopted (stable) pole values pole_sys, and theweight matrix L of the full-order Luenberger observer forappropriately chosen observer poles pole_obs. The realparts of the latter should be more negative than thoseselected for the pole_sys set.

Assuming the modal control plant is given by thelinear MIMO system described by differential state-spaceequations (4), the first step on the road to synthesizing amodal control system in time domain is to determine thestate feedback gain matrix F Rmn in

u(t) = Fx(t), (14)

which shifts the poles of a linear plant model to desiredlocations specified by the preassigned a priori values ofpole_sys. These correspond to respective eigenvalues i,i = 1, 2, . . . , n, of the matrix A and si, i = 1, 2, . . . , n,for the matrix A BF. The latter are the roots of thecharacteristic equation

det [sIn A + BF]= sn + an1sn1 + + a1s + a0 = 0 (15)

with coefficients a0, a1, . . . , an1 calculated frompreassigned eigenvalues si (poles pole_sys) of the matrixA BF.

The eigenvalues i of the plant matrix A correspondto the eigenvectors mi, i = 1, 2, . . . , n, which representthe solution of the equation system

[A iIn]mi = 0 for i = 1, 2, . . . , n. (16)Usually they are found by taking mi as a nonzero(arbitrary) column of the adjugate matrix [A Ini]ad.From them a matrix of eigenvectors

M =[m1 m2 . . . mn

](17)

can be created, which will be nonsingular, providedmi are chosen as linear independent columns fromconsecutive matrices [A Ini]ad for i = 1, 2, . . . , n.

Hence, the sought-for matrix F can be determinedin time domain through the eigenvalues of the matricesA and A BF or on the basis of eigenvectorscorresponding to eigenvalues of the matrices (Kaczorek,1992). Determining the weight matrices L for fullorder observers is carried out in a dual way by usingeigenvalues or eigenvectors of the matrices A and A LC, respectively.

4.1. Eigenvalues method. Making an exclusive useof eigenvalues requires that the plant description (4)be converted into the controllable second canonicalform with matrices A = PAP1 and B = PB,where characteristic nonzero rows occur having numbersni =

di, i = 1, 2, . . . , m, and di are Kronecker

controllability indices of the plant. The form may beobtained by a homothetic transformation with matrix Pcreated appropriately from the controllability matrix forthe pair (A,B) of the plant model (4).

Taking into consideration the nonzero rows of thematrices A and B, denoted respectively by Ani and Bni ,ni =

di, i = 1, 2, . . . , m, the following matrix is

created:

F =

An1 eTn1+1An2 eTn2+1

Anm1 eTnm1+1Anm aT

, (18)

where eTi is the i-th row of the identity matrix In,and aT := [a0, a1, . . . , an1] is the row made up ofcoefficients of the characteristic polynomial (15), and thematrix

Bm =

Bn1Bn2

Bnm

=

1 0 1 . . . 0 0 1

(19)

is formed from nonzero rows of the matrix B. Then thesought-for feedback matrix F, which shifts the poles of

Linear adaptive structure for control of a nonlinear MIMO dynamic plant 53

the closed-loop system to desired locations on the left halfplane s C, can be determined from

F = B1m FP. (20)

This can be done by calling the function [F] =modal(A,B,C,D, pole_sys), which represents animplementation of the above described procedure in thePolynomial Toolbox for MATLAB. Determining weightmatrices L Rnl for the full-order Luenberger observerin time domain can be carried out by utilizing the alreadymentioned function modal.m in a dual way, namely, bycalling [L] = modal(A,C,B,D, pole_obs).

4.2. Eigenvectors method. In the event that matrix Ahas different eigenvalues i, i = 1, 2, . . . , n, the eige-nvectors method comes down to determining the matrixof eigenvectors (17) and creating a diagonal matrix

=

1 s1 0 00 2 s2 0 . . . 0 0 n sn

, (21)

whose elements are differences between eigenvalues i ofthe matrix A and roots si of the characteristic equation(15). Then the feedback matrix F can be calculateddirectly from

F = MM1. (22)

However, this way of calculating the feedback matrixbecomes complicated if eigenvalues of the plant matrixA are complex or multiple real, and if, for some reasonor other, such eigenvalues are preassigned for the controlsystem to be designed. The standard function place.m ofthe Control Toolbox for MATLAB/Simulink representsan implementation of the above-mentioned procedurewith restrictions imposed on the maximal multiplicity ofpreassigned poles pole_sys, which may not excess thenumber of plant inputs m.

Determining weight matrices L Rnl for thefull-order Luenberger observer in time domain can becarried out utilizing the already mentioned function pla-ce.m in a dual way, namely, by respective calling [L] =place(A,C, pole_obs), where pole_obs is the set ofeigenvalues (poles) si, i = 1, 2, . . . , n, specified for thematrix A LC.

It is easy to note that calculations performed with theuse of the above functions do not ensure that eigenvaluesof A BF and A LC, i.e., the poles pole_sys andpole_obs, will be located in a priori specified controlsystem paths since in MIMO systems many differentmatrices A BF and A LC of identical determinantsdet(sIn A + BF) and det(sIn A + LC) may exist.The actual pole location can be verified only throughsimulations of the designed control system, preferably

with a modal controller in its standard form (8). Then thepole location can be assessed whether or not it is properfrom observations of time responses of state variablesfor the plant model and the Luenberger observer, wherexr(t) = x(t) x(t) for t .

If we use the eigenvector method, the final result mayadditionally depend on the order the elements in sets po-le_sys and pole_obs are listed due to the freedom of choiceof the sequence of eigenvectors mi, i = 1, 2, . . . , n, in(16). This means that, depending on the method used,many different matrices F and L may exist for the samepoles pole_sys and pole_obs yielding, as a result, entirelydifferent modal controllers (7).

Furthermore, when employing synthesis methods intime domain, there is no impediment to make use ofdifferent functions, e.g., place.m of the eigenvector me-thod while determining the matrix F and modal.m ofthe eigenvalue method when finding the matrix L orvice versa, which further extends the range of solutionspossible to obtain. However, this makes the results ofdesign ambiguous in the sense that many different modalcontrollers are obtained for the same input data.

4.3. Polynomial method. The matrices F and L canbe found using the polynomial approach without solvingpolynomial matrix equations utilizing the well-knownWolovich structure theorem (Wolovich, 1974). Accordingto this theorem and the procedure described by Banka(2007) as well as Banka and Dworak (2011a), thesematrices can be determined directly from the followingrelationships:

C1(s) A1(s) = F(s) = FPS(s) (23)and

C2(s) A2(s) = S(s)PL, (24)where C1(s) R[s]mm and C2(s) R[s]ll aregenerated on the basis of specified (stable) pole values ofpole_sys and pole_obs, respectively.

The structure of the polynomial matrices C1(s) andC2(s) should comply with those of the denominatormatrices A1(s) and A2(s) of the plant transfer functionmatrix (5), that is, the matrix C1(s) generated on the basisof poles pole_sys should have a column matrix of leadingcoefficients

c(C1(s)) = c(A1(s)), (25)

with degciC1(s) = degciA1(s) = di, i = 1, 2, . . . , m,and the row structure of the matrix C2(s) generatedfrom the poles pole_obs should comply with that of thepolynomial matrix A2(s), namely,

r(C2(s)) = r(A2(s)) (26)

at degrjC2(s) = degrjA2(s) = dj , j = 1, 2, . . . , l.

54 S. Banka et al.

The matrix A2(s) is the denominator matrix of theplant transfer function matrix (5) converted to the dual(r.l.p.) form of transfer matrix T(s) = A12 (s)B2(s).

The polynomial matrices S(s) and S(s) occurring in(23) and (24) have the following form:

ST (s)

=

1 s sd11 0 00 0 0 0 0...

... ... ... ...0 0 0 1 sdm1

,

(27)

S(s)

=

1 s sd11 0 00 0 0 0 0...

... ... ... ...0 0 0 1 sdl1

.

(28)

The structure of S(s) depends on controllabilityindices di, i = 1, 2, . . . , m,

di = n, and that of

S(s) on plant observability indices dj , j = 1, 2, . . . , l,dj = n. The matrices P and P are transformation

matrices obtained in the process of transforming theoriginal plant state-space equations (4) into the secondLuenberger-Brunovsky canonical forms, controllable andobservable, respectively.

Unlike the eigenvalue and the eigenvector methods,where no possibility exists to locate intentionally the polesin specified paths of the MIMO system to be designed, themethod considered here permits the poles pole_sys to beassigned to plant inputs, and the poles pole_obs to plantoutputs. This can be done in the process of generating thematrices C1(s) and C2(s) first in diagonal structures withpolynomials of orders equal to controllability indices di,i = 1, 2, . . . , m, for the first matrix and to observabilityindices dj , j = 1, 2, . . . , l, for the second matrix, andthen bringing these diagonal matrices to forms that satisfythe conditions (25) and (26), respectively. To this end theset of preassigned poles pole_sys should be divided intosubsets with di, i = 1, 2, . . . , m, elements, and the set ofpole_obs into l subsets with dj j = 1, 2, . . . , l, elements.The sequence in which the individual pole subsets areused in the process of system synthesis does matter andhas a pronounced effect on static and dynamic propertiesof the obtained controllers. Although the above proceduredoes not provide full possibility to locate the system andobserver poles in specified paths of the control system,the design of modal control systems with MIMO plants ismade thereby easier.

4.4. Polynomial matrix equations method. In theabove presented methods the synthesis of MIMO modalcontrollers has been based on separately finding thematrices F and L for which, according to (8), theirstandard state-space equations have been formulated.These equations can be converted to appropriatestate-space canonical forms with the matrices (9), andthen, if desired, the matrices of controller transferfunctions (10) can be determined on their basis.

However, instead of separately calculating thematrices F and L when designing modal controllers ins-domain, a more typical polynomial procedure may beemployed, where the controller transfer function matrixTc(s) = M12 (s)N2(s) is directly obtained at one go bysolving the Diophantine left polynomial matrix equation

M2(s)A1(s) + N2(s)B1(s) = (s) = Q(s)C1(s),(29)

where A1(s) and B1(s) are known polynomial matricesdescribing the control plant (5), and M2(s) and N2(s) area pair of unknown polynomial matrices that Eqn. (29) isto be solved for. In the case of MIMO systems obtainingminimal solutions of Eqn. (29) (of minimal degree withrespect to the matrix N2(s)), which should satisfy theconditions degrjN2(s) degrjM2(s), j = 1, 2, . . . , m,is much more complex than in the case of SISO systems.In SISO systems, Q(s) and C1(s) are generated in asimple way as stable Hurwitz polynomials on the basis ofpreassigned respective pole values pole_obs and pole_sys.For solutions of the polynomial equation (29) to exist,only the necessary condition is to be met that roots ofthese polynomials be separable with those of polynomialsA1(s) and possibly B1(s).

However, in MIMO systems the polynomial matrix(s) = Q(s)C1(s) in addition to that it should be relativeright prime (r.r.p.) with matrices A1(s) and B1(s), itshould also have a row-column-reduced structure with anonsingular matrix of the highest (diagonal) coefficients(Callier and Kraffer, 2005; Banka, 2007)

h((s)) = r(Q(s))c(C1(s)) (30)= r(M2(s))c(A1(s)).

The matrices Q(s) and C1(s) should havedeterminants detQ(s) and detC1(s) with zeroesequal to the preassigned poles, i.e., pole_obs for theobserver and pole_sys for the system.

In selecting the matrices Q(s) and C1(s) we havea great freedom of choice of their structure, since, aspreviously, many different polynomial matrices may existof the given dimensions with identical determinants.In the method proposed here the matrix C1(s) canbe generated as in the polynomial method, i.e., inaccordance with the column structure of the denominatormatrix A1(s). A circumstance that may present someproblems is the choice of an appropriate structure of the

Linear adaptive structure for control of a nonlinear MIMO dynamic plant 55

matrix Q(s) so that the structure of the matrix (s) isrow-column-reduced, which guarantees that the obtainedsolutions will have the form of proper transfer functionmatrices (10) for each sought modal controller.

This is not an easy task and requires great skillsor additional a priori information acquired, for example,in the process of system synthesis in time domain.Fortunately, the matrix Q(s) in systems with controllersof full order n may frequently have a diagonal structurewith polynomials of orders rj = degrjQ(s), j =1, 2, . . . , m, selected so that

rj = n. As was reported

by Banka (2007), it is also possible to obtain solutionsof Eqn. (29) in the form of strict proper transferfunction matrices for full-order controllers. However, thisis feasible if the polynomial matrix is selected in a specialway, and in general, only for plant models described bythe strict proper transfer function matrices (5) (Callier andKraffer, 2005; Banka, 2007).

Furthermore, Eqn. (29) may also deliver modalcontrollers of reduced order built on the basis of functionalLuenberger observers of reduced order n1 = m( 1),where = max{dj, j = 1, 2, . . . , l}. Then the matrixQ(s) assumes a regular structure, i.e., with identical rowdegrees rj = degrjQ(s) = 1;

rj = n1 (Banka,

2007; Wolovich, 1974). In this case the controllers ofreduced order n1 = m( 1) will always be obtainedin the form of matrices of proper transfer functions. Theycan be realized in time domain exclusively in canonicalobservable forms of state-space equations with Dro = 0.

Additionally, it might be good to mention that thereexists a possibility to design modal control systems bysolving the Diophantine (dual) right polynomial matrixequation

A2(s)M1(s) + B2(s)N1(s) = (s) = C2(s)Z(s)(31)

with the use of equivalent polynomial descriptionsconcerning the plant and the controller in the formsT(s) = A12 (s)B2(s) and Tc(s) = N1(s)M

11 (s),

and with an appropriately chosen row-column-reducedmatrix (s) R[s]ll, where zeroes of the matrix C2(s)correspond to the preassigned values of pole_obs, andzeroes of the matrix Z(s) correspond to the values ofpole_sys (Banka, 2007). These will not be consideredin this paper, as well as structures with reduced-ordercontrollers, mainly because matrices Dro = 0 occur intime domain realizations of the matrix transfer functionof such controllers, thus increasing quite significantly thenumber of parameters to be tuned.

5. Synthesis of ship modal controllers

In the case of linear models obtained in the form ofstate-space equations (4) or transfer function matrices(5) for the drillship Wimpey Sealab given by nonlinear

state-space equations (1) with the effects of wind gustsand wave action having been neglected for clarityssake, each of the above discussed synthesis methodsleads to yielding strict causal modal full-order controllersdescribed by the space-state equations (7) with matricesDro = 0, which are defined by the strict proper transferfunction matrices (10) in s-domain. In order to obtainsolutions with the minimal number of parameters whosevalues are different from 0 or 1, the state-spaceequations for all controllers to be yielded will be presentedexclusively in canonical forms with matrices (9). Thefollowing sets of stable pole values have been adopted forthe system and the full-order Luenberger observer:

pole_sys

= {0.40,0.45,0.14,0.15,0.15,0.16}and

pole_obs

= {0.80,0.90,0.28,0.30,0.30,0.32}.Such a choice of the poles pole_sys was performed

experimentally to obtain control processes withoutexcessive overshoots on the course and the shipscoordinate position with reasonable times needed toachieve steady-state control conditions and possiblewithout crossing the limits on the control signals. Onthe other hand, the values of pole_obs were chosen withnegative values of its real parts, twice larger than thenegative values of the corresponding values of pole_systhat ensure the vanishing of transitional processes inthe Luenberger observer two times faster than processesoccurring in particular paths of the closed-loop controlsystem.

Employing the above synthesis methods yielded fourdifferent sets of 3650 modal controllers describedby the state-space equations (7) in the secondLuenbergerBrunovsky canonical observable formwith the matrices

Aro =

0 a12 0 a14 0 a161 a22 0 a24 0 a260 a32 0 a34 0 a360 a42 1 a44 0 a460 a52 0 a54 0 a560 a62 0 a64 1 a66

, (32)

Bro =

b11 b12 b13b21 b22 b23b31 b32 b33b41 b42 b43b51 b52 b53b61 b62 b63

,

Cro =

0 1 0 0 0 00 0 0 1 0 00 0 0 0 0 1

, Dro =

0 0 00 0 00 0 0

.

56 S. Banka et al.

They have 36 variable entries: aij , i = 1, 2, . . . , 6,j = 2, 4, 6, and bij , i = 1, 2, . . . , 6, j = 1, 2, 3, dependenton the ships velocity Vs = sign(x4)

x24 + x

25 and on

deviations of the ships yaw angle y30 = x30 from the seacurrent angle c. The controllers have been synthesizedfor velocities lying in the range Vs [4.94.9] [knots]with the resolution of 0.2 [knot] over the entire range ofround angle, that is, over the range c x30 [0 360] with the resolution of 0.0873[rd], for the adoptedship relative radius of gyration kzz = 1/4.

As might be expected, the use of different synthesismethods for modal controllers yielded different resultsfor the same data taken for calculations. The differencesin the obtained results are fundamental both in termsof constructing from them an adaptive controllerwith stepwise varying parameters (or a switchablemulti-controller structure) and also in terms of operationquality provided by these controllers in the designedcontrol system. Nonetheless, all the obtained modalcontrollers are stable exhibiting a time lag affected PDbehavior. Their dynamic and static properties for differentship velocities Vs and yaw angles c x3 within thegroup of controllers obtained by one of the discusseddesign methods experience some fluctuations, since theyhave variable poles pole_reg defined as eigenvalues of thematrices Aro and variable gain matrices

Kc =

k11 k12 k13k21 k22 k23k31 k32 k33

= Cro(Aro)1Bro. (33)

It is worth noting that, despite employing alwaysthe same pole values pole_sys and pole_obs withoutintroducing any changes in their sequence, there havebeen obtained entirely different sets of controllers withvarying entries of Aro and Bro, which yield different(variable) gain matrices Kc for these controllers at varying(over different ranges) values of always stable poles po-le_reg. The variable entries of matrices Aro, Bro and Kcmay be depicted in the form of 3-D surfaces as functionsof ship velocity Vs and yaw angles c x3.

The synthesis results both the most unreliable inoperation and the most difficult to realize an adaptivecontroller (or a switchable multi-controller structure) havebeen obtained by means of the eigenvalues method. Theywill not be shown in the paper. On the other hand, the bestcontrol responses, i.e., smooth and overshoot-free ones,were provided by controllers obtained by the eigenvectormethod. Unfortunately, the parameters of such obtainedcontrollers depend in a complex way on the ships velocityVs and yaw angles cx3, which makes realization of theproposed control system structure difficult. The characterof parameter variability for controllers obtained by thismethod is illustrated by 3-D surfaces published earlier byBanka et al. (2011a).

The most promising results both in terms of easeof realization of the proposed control system structureand also in terms of the quality of controller operationin the multi-controller structure are delivered by the poly-nomial matrix equation method. The controllers obtainedby this method are characterized by moderately smoothsurfaces of parameters variation and, at the same time,meet sufficiently the quality requirements placed oncontrol of the ships nonlinear model. The variability ofentries of matrices Aro, Bro versus the ships velocityVs and yaw angle c x3 obtained by this methodis illustrated by 3-D surfaces shown in Figs. 3 and 4,respectively.

For comparison, 3-D surfaces for all entries of gainmatrices Kc of modal controllers obtained by means ofall the above-mentioned synthesis methods (except forresults yielded by the eigenvalue method, which wereunacceptable from every point of view) are depicted inFigs. 57.

From these plots it can be seen that parameters ofall obtained modal controllers change their values (boththe absolute value and the sign) at different values of yawangle y3 = x3 and the ships velocity Vs. Particularlyviolent changes, especially for controllers obtained by theeigenvector method, take place in the vicinity of valuesthat correspond to yaw angles 0, 90, 180 and 270 andat the ships velocities close to Vs = 0. This particularlyconcerns parameters which constitute the last columnsof matrices Aro and Bro and the last column of gainmatrix Kc (not presented here), i.e., the entries havinga direct influence on signals associated with the shipscourse control.

For parameters of controllers obtained by theremaining methods, i.e., the polynomial and the polyno-mial matrix equation methods, the yielded surfaces arealready more smooth except for controllers obtained bythe eigenvalue method. The latter exhibit sharp spikes (notpresented here) in canonical forms for yaw angles equal to90 and 270 occurring at high ship velocities Vs.

All of this makes a quite complex picture ofproblems connected with implementation of the proposedmulti-controller structures of linear modal controllersdesigned for steady states, but actually utilized for controltransients. This is possible as evidenced below by resultsof simulations carried out with the ships nonlinear model(1) for all obtained sets of modal controllers realized hereas a single adaptive controller with tuned parameters.

6. Results of simulation tests

All simulation tests have been carried out without regardfor the effect produced by the wind and wave actionin the presence of sea current of Vc = 2 [knots] atc = 180 with the use of the ships nonlinear model(1) that describes low frequency varying ship motions in

Linear adaptive structure for control of a nonlinear MIMO dynamic plant 57

090

180270

360

52.5

02.5

5

2.34

2.32

2.3

2.28

2.26

PSIcy3

Aro12diophan

Vs [knot] 090

180270

360

52.5

02.5

50.04

0.02

0

0.02

0.04

PSIcy3

Aro14diophan

Vs [knot] 090

180270

360

52.5

02.5

50.1

0.05

0

0.05

0.1

PSIcy3

Aro16diophan

Vs [knot]

090

180270

360

52.5

02.5

52.55

2.545

2.54

2.535

2.53

2.525

PSIcy3

Aro22diophan

Vs [knot] 090

180270

360

52.5

02.5

50.01

0.005

0

0.005

0.01

PSIcy3

Aro24diophan

Vs [knot] 090

180270

360

52.5

02.5

50.04

0.02

0

0.02

0.04

PSIcy3

Aro26diophan

Vs [knot]

090

180270

360

52.5

02.5

50.1

0.05

0

0.05

0.1

PSIcy3

Aro32diophan

Vs [knot] 090

180270

360

52.5

02.5

50.3

0.25

0.2

0.15

0.1

0.05

PSIcy3

Aro34diophan

Vs [knot] 090

180270

360

52.5

02.5

51

0.5

0

0.5

1

PSIcy3

Aro36diophan

Vs [knot]

090

180270

360

52.5

02.5

50.1

0.05

0

0.05

0.1

PSIcy3

Aro42diophan

Vs [knot] 090

180270

360

52.5

02.5

50.9

0.8

0.7

0.6

0.5

PSIcy3

Aro44diophan

Vs [knot] 090

180270

360

52.5

02.5

51

0.5

0

0.5

1

PSIcy3

Aro46diophan

Vs [knot]

090

180270

360

52.5

02.5

50.1

0.05

0

0.05

0.1

PSIcy3

Aro52diophan

Vs [knot] 090

180270

360

52.5

02.5

50.06

0.04

0.02

0

0.02

PSIcy3

Aro54diophan

Vs [knot] 090

180270

360

52.5

02.5

50.4

0.38

0.36

0.34

0.32

0.3

PSIcy3

Aro56diophan

Vs [knot]

090

180270

360

52.5

02.5

50.1

0.05

0

0.05

0.1

PSIcy3

Aro62diophan

Vs [knot] 090

180270

360

52.5

02.5

50.06

0.04

0.02

0

0.02

0.04

PSIcy3

Aro64diophan

Vs [knot] 090

180270

360

52.5

02.5

52

1.5

1

0.5

0

0.5

Aro66diophan

PSIcy3Vs [knot]

Fig. 3. Entries of the matrix Aro vs. ship velocity and yaw angle obtained by the polynomial matrix equation method.

58 S. Banka et al.

090

180270

360

52.5

02.5

50.2

0.1

0

0.1

0.2

PSIcy3

Bro11diophan

Vs [knot] 090

180270

360

52.5

02.5

50.2

0.1

0

0.1

0.2

PSIcy3

Bro12diophan

Vs [knot] 090

180270

360

52.5

02.5

54

2

0

2

4

x 103

PSIcy3

Bro13diophan

Vs [knot]

090

180270

360

52.5

02.5

51

0.5

0

0.5

1

PSIcy3

Bro21diophan

Vs [knot] 090

180270

360

52.5

02.5

51

0.5

0

0.5

1

PSIcy3

Bro22diophan

Vs [knot] 090

180270

360

52.5

02.5

50.02

0.01

0

0.01

0.02

PSIcy3

Bro23diophan

Vs [knot]

090

180270

360

52.5

02.5

54

2

0

2

4

x 103

PSIcy3

Bro31diophan

Vs [knot] 090

180270

360

52.5

02.5

54

2

0

2

4

x 103

PSIcy3

Bro32diophan

Vs [knot] 090

180270

360

52.5

02.5

50.01

0.005

0

0.005

0.01

PSIcy3

Bro33diophan

Vs [knot]

090

180270

360

52.5

02.5

50.1

0.05

0

0.05

0.1

PSIcy3

Bro41diophan

Vs [knot] 090

180270

360

52.5

02.5

50.1

0.05

0

0.05

0.1

PSIcy3

Bro42diophan

Vs [knot] 090

180270

360

52.5

02.5

50.04

0.02

0

0.02

0.04

PSIcy3

Bro43diophan

Vs [knot]

090

180270

360

52.5

02.5

51

0.5

0

0.5

1

Bro51diophan

PSIcy3Vs [knot] 090

180270

360

52.5

02.5

51

0.5

0

0.5

1

Bro52diophan

PSIcy3Vs [knot] 090

180270

360

52.5

02.5

52

1

0

1

2

3

x 103

PSIcy3

Bro53diophan

Vs [knot]

090

180270

360

52.5

02.5

50.02

0.01

0

0.01

0.02

PSIcy3

Bro61diophan

Vs [knot] 090

180270

360

52.5

02.5

50.04

0.02

0

0.02

0.04

PSIcy3

Bro62diophan

Vs [knot] 090

180270

360

52.5

02.5

50

0.002

0.004

0.006

0.008

0.01

PSIcy3

Bro63diophan

Vs [knot]

Fig. 4. Entries of the matrix Bro vs. ship velocity and yaw angle obtained by the polynomial matrix equation method.

Linear adaptive structure for control of a nonlinear MIMO dynamic plant 59

090

180270

360

52.5

02.5

50.1

0.05

0

0.05

0.1

PSIcy3

KR11diophan

Vs [knot] 090

180270

360

52.5

02.5

50.1

0.05

0

0.05

0.1

PSIcy3

KR12diophan

Vs [knot] 090

180270

360

52.5

02.5

52

1

0

1

2

x 103

PSIcy3

KR13diophan

Vs [knot]

090

180270

360

52.5

02.5

50.04

0.02

0

0.02

0.04

PSIcy3

KR21diophan

Vs [knot] 090

180270

360

52.5

02.5

50.04

0.02

0

0.02

0.04

PSIcy3

KR22diophan

Vs [knot] 090

180270

360

52.5

02.5

50.04

0.02

0

0.02

0.04

PSIcy3

KR23diophan

Vs [knot]

090

180270

360

52.5

02.5

50.01

0.005

0

0.005

0.01

PSIcy3

KR31diophan

Vs [knot] 090

180270

360

52.5

02.5

50.02

0.01

0

0.01

0.02

PSIcy3

KR32diophan

Vs [knot] 090

180270

360

52.5

02.5

55

0

5

10

x 103

PSIcy3

KR33diophan

Vs [knot]

Fig. 5. Entries of the matrix Kc vs. ship velocity and yaw angle obtained by the polynomial matrix equation method.

3DOF. The tests have been conducted for many differentinitial states defined by appropriate ship positions, yawangle and ship initial velocities. In typical situations, thatis, when the ship sailed bow on against the current atVs = 0, all controllers behaved quite correctly yieldingtime responses without excessive oscillations experiencedby the control signals u(t). In that case the ship couldalways be brought to the drilling point and assumed thepreset yaw angle, and then she could be moved to anyspecified position. This is demonstrated by time responsesdepicted in Fig. 8 obtained with controllers found by theeigenvector and the polynomial matrix equation methods.

During these simulations the ship was brought to thedrilling point from a position at a distance of about 100[m] (r = 1) distant situated on the left below the drillingpoint with the adopted initial yaw angle x3(0) = 35 andvelocity components x4(0) and x5(0), which correspondsto the ships sailing bow on against the current Vs(0) =Vc = 2 [knots]. After reaching the drilling point withthe specified yaw angle equaling y3ref = x30 = 0 after150 dimensionless time units (corresponding to about 7.5min. of real time), the yaw angle was changed to y3ref =x30 = 60. Then at t = 200 of time units the referencevalues have been changed stepwise for both ship positioncoordinates so that the ship moved through a distance ofabout 100 [m] from the right over the drilling point and

come to a standstill at a distance of 100 [m] with a velocityof Vs = Vc = 2 [knots] relative to water and bringingthe ships yaw angle y3(t) = x3(t) to the preset valuey3ref = x30 = 60.

However, more interesting and instructive areresponses obtained for a typical situations when the shipsails stern-first against the current, especially at changesin the sign of the linear ship velocity Vs in the vicinityof Vs = 0. This may happen when the ship is broughtto the drilling point with the current (conditioned, forexample, by an unfavorable direction of the wind or seawaves being in opposition to the sea current direction c)or when changing the ships yaw angle over the drillingpoint caused by a change in the wind or waves direction.

To investigate the matter, the remaining simulationshave been performed for the ship situated initially about100 [m] on the left over the drilling point with initial yawangle x3(0) = 125 and velocity components x4(0) andx5(0), which corresponds at the beginning of simulationsto moving astern at Vs(0) = Vc = 2 [knots]. In thesesimulations the preset ship yaw angles y3ref = x30 overthe drilling point, as well as making later changes in theyaw angle and ship final positions, have been performedjust in the same way as earlier, keeping as far as possiblethe same time conditions for manoeuvring.

Simulation results obtained for sets of controllers

60 S. Banka et al.

090

180270

360

52.5

02.5

50.01

0.005

0

0.005

0.01

PSIcy3

KR11place

Vs [knot] 090

180270

360

52.5

02.5

50.02

0.01

0

0.01

0.02

PSIcy3

KR12place

Vs [knot] 090

180270

360

52.5

02.5

52

1

0

1

2

x 103

PSIcy3

KR13place

Vs [knot]

090

180270

360

52.5

02.5

50.1

0.05

0

0.05

0.1

PSIcy3

KR21place

Vs [knot] 090

180270

360

52.5

02.5

50.04

0.02

0

0.02

0.04

PSIcy3

KR22place

Vs [knot] 090

180270

360

52.5

02.5

50.04

0.02

0

0.02

0.04

PSIcy3

KR23place

Vs [knot]

090

180270

360

52.5

02.5

55

0

5

x 103

PSIcy3

KR31place

Vs [knot] 090

180270

360

52.5

02.5

54

2

0

2

4

x 103

PSIcy3

KR32place

Vs [knot] 090

180270

360

52.5

02.5

55

0

5

10

x 103

PSIcy3

KR33place

Vs [knot]

Fig. 6. Entries of the matrix Kc vs. ship velocity and yaw angle obtained by the eigenvector method.

090

180270

360

52.5

02.5

50.1

0.05

0

0.05

0.1

PSIcy3

KR11poly

Vs [knot] 090

180270

360

52.5

02.5

50.04

0.02

0

0.02

0.04

PSIcy3

KR12poly

Vs [knot] 090

180270

360

52.5

02.5

54

2

0

2

4

x 103

PSIcy3

KR13poly

Vs [knot]

090

180270

360

52.5

02.5

50.05

0

0.05

PSIcy3

KR21poly

Vs [knot] 090

180270

360

52.5

02.5

50.04

0.02

0

0.02

0.04

PSIcy3

KR22poly

Vs [knot] 090

180270

360

52.5

02.5

50.04

0.02

0

0.02

0.04

PSIcy3

KR23poly

Vs [knot]

090

180270

360

52.5

02.5

54

2

0

2

4

x 103

PSIcy3

KR31poly

Vs [knot] 090

180270

360

52.5

02.5

55

0

5

x 103

PSIcy3

KR32poly

Vs [knot] 090

180270

360

52.5

02.5

55

0

5

10

x 103

PSIcy3

KR33poly

Vs [knot]

Fig. 7. Entries of the matrix Kc vs. ship velocity and yaw angle obtained by the polynomial method.

Linear adaptive structure for control of a nonlinear MIMO dynamic plant 61

Eigenvectors method

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Fig. 8. Plots of controlled variables and control signals u(t) in the process of bringing the ship to the drilling point bow on against thecurrent.

designed by the eigenvector and the polynomial matrixequation methods are shown in Fig. 9.

Steady-state errors in each of the tested systems havebeen eliminated by applying nominal values uo calculatedfrom the system of equations (3) for preset final shippositions at the yaw angle y3ref = x30 = 60.

The time responses presented above feature smallerovershoots and shorter settling times than those deliveredby other (switchable) structures described in earlier papers(e.g., Banka et al., 2010a; 2010b). A drawback to theseresponses is that the responses of control signals u(t)are (oscillating) nonsmooth, caused mainly by stepwisechanges in parameter values of the controller in thevicinity of operating points that cause trouble.

7. Concluding remarks

It follows from the simulation tests carried out that theproposed concept of control of a nonlinear model ofa MIMO drillship by the use of an adaptive structureof a linear MIMO controller with tuned parameters onthe basis of two auxiliary measured signals is feasible.It can also be realized by a set of linear MIMOmodal controllers in a multi-controller structure with

switchable outputs. Modal controllers, which must bestable in these systems, though predesigned for steadystates, operate properly despite the fact that they actuallyoperate in transient states (in a quasi-steady-state mode).Nevertheless, stepwise changes of parameter values inan adaptive single controller, as well as the switchingover of controller outputs in a multi-controller structure,are accompanied by nonsmooth responses of controlsignals u(t). This especially concerns control transientstaking place at the ships velocities close to Vs = 0 and/orat yaw angles corresponding to values close to 0, 90,180 and 270. The propellers and main engine of a realship will not be able to realize such control signals. Hence,such a problem is a main aim of researching the solutionsthat could give a chance to obtain smoother surfaces ofparameters change shown in Figs. 3 and 4.

Apart from using different design method as wellas employing different description forms of designedmodal controllers, the possibility of replacing the obtained(original) surfaces of parameter change with the surfacegenerated by means of artificial neural networks exists.As training data for neural networks the obtained surfacesof parameter changes can be used. Such a controller withneural parameters should provide a smoother system

62 S. Banka et al.

Eigenvectors method

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Fig. 9. Plots of controlled variables and control signals u(t) obtained for the ships approaching the drilling point with the current.

operation (without any switching over) and also

generalize the parameter values of matrices Aroand Bro to untrained (unknown) values followingfrom quantization of signals Vs and c x30 withacceptable resolution,

smooth out ridges and precipices seen on thesurfaces that describe variable controller parameters,

eliminate the problem of ambiguity in operationof controllers, which assume different parametervalues in the process of switching over depending onwhether the auxiliary signals Vs(t) and c x3(t)increase or decrease.

A first attempt of replacing a discussed linearstructure of controller with tuned parameters (or aswitchable structure of modal controllers) with anadaptive neural controller trained on the basis of knownparameter values contained in matrices Aro and Bro,designed by the eigenvector method, has been presentedby Banka et al. (2011b). Such an approach will be takeninto consideration during further research.

Acknowledgment

The study has been partly supported by Grant No. N N514679240 on Advanced control systems and algorithms fornonlinear dynamic MIMO plants financed by the PolishMinistry of Science and Higher Education.

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Stanisaw Banka received the M.Sc., Ph.D,and D.Sc. degrees in control engineering in1969, 1976 and 1992, respectively. In the years19962002 he was the dean of the Electrical En-gineering Faculty at the Technical University ofSzczecin and in 20032005 the director of theInstitute of Control Engineering of that univer-sity. Since 2009 a professor, currently the head ofthe Chair of Control Engineering and Robotics atthe West Pomeranian University of Technology

in Szczecin. Member of the Committee on Automatic Control and Robo-tics of the Polish Academy of Sciences (since 2003). Author of over 80journal papers and three books. Main scientific interests: multivariablecontrol systems, adaptive and optimal control, estimation and parameteridentification, and applications of informatics in automation.

Pawe Dworak received the M.Sc. and Ph.D de-grees in control engineering in 1999 and 2005,respectively. Currently an assistant professor inthe Department of Control Engineering and Ro-botics, West Pomeranian University of Technolo-gy in Szczecin. Main scientific interests: multiva-riable control systems, adaptive control, processdata acquisition and visualization and industrialapplications of modern control algorithms.

Krzysztof Jaroszewski received the M.Sc.(2001) and Ph.D. (2007) degrees, both in controlengineering. Currently an assistant professor inthe Department of Control Engineering and Ro-botics, West Pomeranian University of Technolo-gy in Szczecin. Main scientific interests: artificialintelligence, especially neural networks, controlsystems and visualization and industrial diagno-stic.

Received: 2 March 2012Revised: 6 July 2012Re-revised: 16 October 2012

IntroductionDescription of the control plantDescription of the proposed control system structureMethods of modal controller synthesisEigenvalues methodEigenvectors methodPolynomial methodPolynomial matrix equations method

Synthesis of ship modal controllersResults of simulation testsConcluding remarks

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