## Abstract

A reasonable knowledge about the response of nonlinear offshore structural systems under environmental loads is necessary but challenging. This is due to the coupling of internal forces along with external excitations. In this paper, a mathematical model of nonlinear offshore systems is studied with the intention of keeping the response close to the desired one. This is achieved using a novel sub-optimal control mechanism derived from nonlinear quadratic regulator (NQR) theory. Herein, two linearized functions of nonlinear motions—displacement and velocity—are introduced such that the parametrization of the state-dependent system matrices is obtained. By doing so, the system becomes conditioned only on the present state and therefore one needs to solve only an algebraic state-dependent Riccati problem. This results in a control law which may either be partial or full rank for the dynamical system depending on measurable states. The performance of the controller is compared with conventional NQR. The performance of the proposed control strategy is illustrated through a range of models of nonlinear offshore problems. The motions (generalized displacements and velocities) show that the proposed controller was not only able to restrict the undesirable behaviour but also provide means of shaping the transient performance.

## 1. Introduction and background

In ocean engineering, the concept of control refers to the devices that can automatically manoeuvre or control vehicles moving under the ocean, or over the surface. The first application of feedback-based controlled motion of ships was as early as 1908 [1]. These vehicles or structures [2] require active control so as to adapt themselves to the direction or frequency of the excitation forces (wind, wave, current); making control engineering an attractive field for offshore structural systems. Hence, active control is obligatory for optimized response of offshore/marine structures. Surveys on techniques and developments of structural control have been reported in articles [3–5]. A major thrust in developing algorithms is related to developing an optimal control algorithm which works for linear structures. An offshore structure is usually afflicted with nonlinearity and therefore, in order for these algorithms to show desired performance, the control algorithms should either be able to handle structural nonlinearity [6] or the existing algorithms can be used by linearization [7]. Recently, many advanced nonlinear regulation schemes such as recursive backstepping, sliding mode control, dynamic inversion, etc., are used to control of nonlinear systems [8,9] though their widespread usage is still limited as they are attempting to exploit the linear system theory.

State-dependent Riccati equation (SDRE) [10,11] is a nonlinear variant of the linear quadratic regulator that has shown considerable promise. In this method, the differential equations are written as state-dependent coefficient (SDC) matrices such that at each time instant they are treated as linear time-invariant (LTI) systems conditionally dependent on the current state of the system. Thereafter, using the control mechanism, one can solve these LTI systems over each time step as an algebraic SDRE problem. In fact, most of the offshore dynamical systems allow non-unique factorization (or parametrization) of the mathematical model into product of SDC (state-dependent matrix-valued functions) and a state vector [12]. Therefore, they can become an ideal candidate for offshore structural control as most of the engineering systems are solved online in the same fashion. By this factorization, the SDRE algorithm not only makes an attempt to account for the nonlinearities in the system but also recasts the nonlinear system into a linear one. Note that usually the control law is sub-optimal [13], while the optimal ones are also available [14]. In optimal control for nonlinear problems, one solves the steady-state partial differential equations of Riccati-type which would not be easy to implement in real-time applications [15]. With the transformation to SDC matrices, one requires solving the SDRE at each time step along with feed-back controller rather than the Hamilton–Jacobi–Bellman control equation [8,9,16]. Moreover, the computational complexity increases polynomially with increase in state dimension which again makes this formulation attractive for application to high dimensional nonlinear systems. Though SDRE has been applied in other fields, their potential applications in offshore engineering are yet to be exploited.

To successfully implement a control algorithm in offshore engineering applications, one has to consider the issues related to the size and complexity, mathematical modelling, uncertainty and so on. There exists a considerable number of studies related to sub-optimal control techniques for nonlinear structural systems [17–21]. However, very few studies exist for SDRE [22,23]. In many analyses such as reliability and noise filtering, sub-optimal controls are preferred as one would be interested in solutions lying in a domain rather than a precise one. Additionally, obtaining optimal controls is not an easy task. However, studies do exist for optimal control of nonlinear structural systems [24,25]. In recent years with the need to harness deeper water, compliant and floating structures have become popular. These structures being parametrically excited usually behave as a mechanism under environmental loading [26]. Owing to the effects of nonlinearity, they exhibit resonant and chaotic behaviour at a range of excitations [27,28]. Therefore, one needs to explore advanced control algorithms that can be easily implementable in the ocean environment.

The present idea of the active offshore structural control hinges on the fact that if one can construct conditionally linear SDC type vector fields which are integrable over each time step, then the advantages of algebraic SDRE can be exploited. The trajectories would be controlled at each point of time and therefore, one can consider the system conditioned on the present state as LTI system. So, the stabilization of trajectories would eventually evolve into a SDRE nonlinear control law. One can therefore, apply the algorithm online, provided an algebraic solution is available. To emphasize, this is in contrast with the nonlinear regulator theory wherein one solves Riccati-type PDEs. Hence, the solution would depend on how one converts the nonlinear vector field to a conditionally linear one. This is achieved by accounting for the nonlinear vector field into the fundamental solution matrix (FSM) of the linearized system by replacing the nonlinear vector fields using explicit local linearization. By doing so, one does not cancel out the beneficial nonlinearities already present in the dynamical system. This method also provides a simplistic framework of deriving more accurate schemes by incorporating more terms (hierarchically) through higher order expansions. In the process, the nonlinear terms of higher orders are replaced in the linearized system in the conventional nonlinear quadratic regulator (NQR) methods. On comparison of the results with the conventional NQR technique, the proposed method with local linearization demonstrates superiority in the sense that the conventional SDRE solutions are unable to appropriately tackle chaotic behaviour in nonlinear systems. The potential of the present method is demonstrated through nonlinear problems of offshore engineering interest under deterministic excitations. The expositions points towards stability and consistency of the proposed technique through various offshore engineering problems.

The present work focuses on developing a control strategy for a given mathematical model related to dynamics of offshore structures such as compliant platforms, geometrically nonlinear mooring lines and platform motion of floating offshore wind turbines [29,30]. A numerical algorithm is proposed to obtain the response under deterministic disturbances through local linearization, which is in turn merged with the controller algorithm to limit the unwanted response (e.g. chaotic behaviour, period doubling, etc.). The present linearization is unique in the sense that one can bypass the calculation of Magnus expansions in time-dependent state transition matrices and also make the parametrization of state-dependent matrices. This is particularly useful in real-time applications. The controller algorithm is based on NQR method [13]. The paper is organized in the following way: A basic mathematical model of offshore platforms is constructed in §2 with relevant assumptions. The controller formulation runs through the §2a to §2b. The illustrative expositions of the present method are given in §3. The paper ends with the main conclusion summarized in §4.

## 2. Numerical modelling of the offshore structure

Any offshore engineering system can be represented as a coupled *m*-dimensional mass–spring–damper model. Such mathematical models are usually derived using the concept of Lagrange and the equations of motion are written as
*m* states of the system; [*M*], [*K*] and [*C*] are *m*×*m* constant mass, stiffness and damping matrices with real values and *F*(*t*) is the external excitation force vector of deterministic nature. Let us now divide the total time [0,*T*] into smaller intervals 0=*t*_{0}<*t*_{1}⋯<*t*_{i}⋯<*t*_{f}=*T*. Let one small interval be defined as *T*_{i}=[*t*_{i},*t*_{i+1}) for *h*_{i}=*h* ∀ *i* is constant. Now one can rewrite equation (2.1) in incremental state–space form (2*m*-dimensional) as follows:
*X*_{2}) has arguments of both displacement (*X*_{1}:=*X*) and velocity (*A*_{nl} as {*A*^{(p)}_{nl}| *p*=1,2,…,*m*}. Without any loss of generality, one can assume the mass matrix to be the identity matrix. In order that the solution does not change rapidly, any function of response vector *M* and

### (a) Nonlinear regulator problem

Consider the dynamical system governed by the state–space equation in the form
*t*_{0} is the starting time and *A*(•) is a nonlinear function of arguments *t*_{i},*t*_{f}). Now one needs to define a performance index (energy function) such that the cost function is minimized with respect to *t*_{f} from an initial state *t* for notational simplicity) as:
*t*_{i} and *t*_{f}. To achieve the optimality, the necessary condition is that the total variation of

### (b) Controller design through state-dependent Riccati equation

As mentioned in the introduction, the SDRE is a nonlinear variant of the linear quadratic method. The details of the controller design using SDRE can be obtained in works [10–12], and recently there are a few applications to ocean engineering problems [31]. The method is briefly mentioned here. One may write the nonlinear equation (2.4) into an SDC matrix form (linear structure) as

The next step would be to solve the SDRE

### (c) Parametrization of nonlinear dynamical equations

The key idea behind recasting the nonlinear equations so as to implement the control strategy would be to replace the nonlinear equations by an equivalent linearized set of equations so that the nonlinear and the linearized response vectors remain transversal everywhere in the space of state vectors. The linearized response (*i*th discretized time-interval *T*_{i}:=(*t*_{i},*t*_{i+1}]. In this interval, the nonlinear solution would be ‘close’ to that of the solution of the linearized system. The solution therefore obtained *t*=*t*_{i+1} would therefore satisfy the initial conditions *t*=*t*_{i} as both the nonlinear solutions as well as linearized solutions transversally intersect each other [34].

Within the interval *T*_{i}, let the initial conditions be *k*. This is particularly important for controlled responses wherein one is interested in solutions in an open ball *B*_{r} with radius *r*>0 measured in Euclidean norm rather than the pointwise solution.

In order to obtain the linearized vector field for *X*_{1},*X*_{2}}^{T}. The first assumption for the parametrization is that function *A*_{nl}(*X*_{1}=0,*X*_{2}=0,*t*)=0) is trivial in the sense that if this assumption does not hold then one would not treat *A*_{nl} as a nonlinear vector but rather as a forcing function. Such a parametrization is non-unique and there would be infinite ways of writing the equation as

The vector field can be linearized, based on the current state of vector, such that the nonlinear vector

In the first form, local linearization equation (2.14),

Upon such replacement of *T*_{i} is given by
*m*-dimensional (padded with zero vector {0}_{m×1}) force vector and nonlinear vector. Moreover, the FSM *Φ*(*t*,*t*_{i}):=*Φ*_{t,ti} is given by _{m×m} and [*I*]_{m×m} are, respectively, zero and identity matrices of size *m*×*m*. One can write the inverse

Towards accounting for the effects of nonlinear vector field in STM, the integration *s*’.

As an example, the parametrization of the nonlinear term

Presently, the displacement and velocity vectors are scalar and therefore the superscripts within brackets are not necessary. To achieve the same, one may also replace *t*∈*T*_{i} as

Following linearization using equation (2.15) is rewritten as

We now recast the term

Observe that this transformation is non-unique (as given in equation (2.25)) and the matrix

After decomposition of

## 3. Numerical illustrations

The performance of the proposed NQR using full and output state feedback concepts is demonstrated through the applications to different offshore engineering problems under externally applied harmonic forces. In particular, the efficacy of the control algorithm is demonstrated via mathematical models of (i) a jacket which can house offshore structures; (ii) the platform motions supporting offshore wind turbines; (iii) control of Mathieu's instability; and (iv) a moored floating structure under Morison force. By implementing the algorithm to these examples, the emphasis is on the numerical and computational advantages of the proposed NQR and also to provide a framework such that one does not rule out its applications in real time. Here, the output feedback concept is employed only in moored structure with nonlinear geometric stiffness and other examples use a full state feedback concept.

### (a) A jacket offshore structure

As the first example, a two-storey jacket offshore structure is considered with two layers of horizontal braces. The jacket is excited by lateral forces such as wave and current. At the point of action of lateral forces, a horizontal bracing is provided. The discretized mathematical model for the jacket structure is shown in figure 1. The jacket is assumed to have nonlinear stiffness only. This generic model may also be used in the context of other offshore structures as articulated towers, tension leg platforms, etc. The masses are taken to be lumped masses and displacement variables *x*_{1} and *x*_{2} correspond to the translation at these mass points. The equations of motion governing the nonlinear spring–mass model (cf. figure 1) are given by
*m*_{i} and *C*_{i} are mass and damping matrices, respectively, *K*_{i} is position-dependent hydrostatic stiffness, *α*_{i} is a positive nonlinear constant, and *F*_{i} and *Ω*_{i} are amplitude and frequency of the external excitation force. The equation of motion is derived based on equilibrium of restoring forces and induced by body motion under wave and current excitation. Equation (??) would be written in state–space form as follows:
*m*_{1}=*m*_{2}=1.1 kg, *C*_{1}=*C*_{2}=0.1 N-s m^{−1}, *K*_{1}=*K*_{2}=10.0 N m^{−1}. The nonlinear constant values *α*_{1}=*α*_{2}=10.0 are chosen such that system can be termed as ‘strongly’ nonlinear. The forcing amplitude is taken as *F*_{1}=10.0 N, *F*_{2}=10.0 N with frequency values as *Ω*_{1}=*Ω*_{2}=1.0. *x*_{1},*x*_{2}}. One can again see, the ‘Proposed NQR’ effectively limits the chaotic motion of the system with respect to the ‘Conventional NQR’.

In order to further elucidate the superior performance of the proposed method, the controller forces are tabulated in table 1. From the table, taking the account the total motion reduction, one can say that ‘Proposed NQR’ method is effective in controlling motions [39].

### (b) Floating platform supporting offshore wind turbine

Another example chosen is to control the platform response of a floating 5 MW NREL offshore wind turbines [40] to obtain the desired power output. Note that the power control, can also be done in other ways, i.e. controlling the feathering of the blades, generator rotor control of nacelle or yaw control of the nacelle. A simplified mathematical model is taken which represents motion of the platform supporting the floating offshore wind turbines [21]. A schematic picture, along with coordinate directions, is shown in figure 5. The mathematical model can be represented as a cart–pendulum system [40–42]. The cart represents the offshore floating platform, whereas the inverted pendulum represent the turbine tower installed on the floating platform. The nacelle, along with blades represent the point mass at the top of the pendulum (tower).

The system consists of a tower (pendulum) of length *l* supporting the nacelle and blade mass *m* on the top. The platform (cart) with mass *M* moves slowly such that the tower makes an angle of *ϕ* with the vertical and thereby maintaining the overall stability. Wave excitation force *F*(*t*) is applied on the platform in the *x*-direction along with the gravity force. The rotational link connection of the tower with the platform has a friction co-efficient *μ*. *x*(*t*) represents the cart position and *ϕ*(*t*) is the tilt angle reference to the vertical. Herein, the goal is to keep the inverted pendulum in an upright position in the neighbourhood of *ϕ*=0°, using the control technique.

The dynamical model can be derived from the Lagrange equation
*K* and *V* being the kinetic and potential energy, respectively. For the inverted pendulum, the kinetic energy is given by

For numerical simulation of the nonlinear model for the offshore wind turbine system aka the cart–pendulum system, it is required to represent the equation (3.5) into the standard state–space form,

### (c) Mathieu's instability problem

The next problem deals with the response control of parametrically excited systems like lateral motions of tension leg tethers excited by the buoyancy fluctuations; heave and pitch coupling of spars; and rolling motion of ships [43,44]. One of the important features of such systems is the occurrence of Mathieu's instability at various excitation frequencies [45]. Additionally, the damping is assumed to be of non-viscous type, i.e. Coulomb's friction type damping, for this problem. A mathematical model representing the lateral motion of tethers of the TLP [46] is thereby written as
*χ* is the coefficient of frictional damping, *δ* and *ϑ* are stiffness and stiffness-dependent force. *Ω* is frequency of the external wave excitation. *y*_{1}(*t*=0)=0,*y*_{2}(*t*=0)=1.0} and the parameters chosen are *χ*=10; *δ*=10 N m^{−1}; *ϑ*=20 N; and *Ω*=*π* rad s^{−1}. The results shown in figure 8 and the response spectrum with and without control are shown in figure 9. The figures clearly demonstrates the superiority of the ‘Proposed NQR’. The control forces are reported in table 4 which also show that the proposed algorithm does work for non-viscous damping cases.

### (d) A Moored floating structure with geometric nonlinearity

The last problem addressed is a floating structure that is moored in open seas using symmetric multi-point catenary mooring using output state feedback concept. Herein, it is assumed that all the states are not available to measure. For the present problem, therefore only velocity is given as output feedback. This nonlinearity is of geometric type due to the mooring lines under the hydrodynamic excitations. Such structures are restricted to move in translational directions. The mathematical model can be represented by a nonlinear model [47] as shown in figure 10. From a practical point, these nonlinearities though undesirable are unavoidable and sometimes they have beneficial effects in dampening out the motion. The model is assumed to be linearly damped and the restoring force consists of linear and cubic components. With restraints for vertical and rotational motion, this system is modelled as a single-degree-of-freedom system for the surge *x*. The governing equation of motion is given by

where the dot denotes differentiation with respect to time, *M* is the total mass including added mass effects, *C* is the hydrodynamic viscous damping, *K* is the geometric mooring linear stiffness and *α* is a positive constant contributing towards the nonlinear stiffness. *F* and *Ω* are amplitude and frequency of the external wave excitation and *C*_{M} and *C*_{D} are inertia and drag coefficient, respectively, *ρ* is the density of water, *V* is the immersed volume, *A* is the projected area normal to the direction of flow and *φ* and *y*_{1}(*t*=0)=0,*y*_{2}(*t*=0)=1.0} and the parameters [47] are *M*=855 kg; *K*=3000 N m^{−1}; *C*=150 N-s m^{−1}; *α*=2 00 000; *C*_{M}=1.5; *C*_{D}=1.2434; *ρ*=1024 kg m^{−3}; *V* =4.1888 m^{−3}; *A*=π m^{2} and *Ω*=*π* rad s^{−1}. The results are shown for ‘Conventional NQR’ also, which clearly illustrates the better performance of the ‘Proposed NQR’ method. The controller input is tabulated in table 5.

## 4. Conclusion

In this paper, a control strategy based on NQR theory is proposed for offshore structural systems. To apply the control law, one requires parametrization of nonlinear function of—displacement and velocity—vector fields as SDC matrix. By doing so, the system becomes conditioned only on the present state and one can obtain the solution of the differential system over each time step as an algebraic state-dependent Riccati problem. This parametrization is achieved through the linearized replacement of nonlinear functions so that the nonlinear and the linearized response vectors remain locally transversal (at each time step) in the space of state vectors. These linearized replacements are obtained through Euler expansions. For the controller gain to be updated at every time instant, the state transition matrices is plugged in with the linearized replacements of the nonlinear functions without taking recourse to Magnus expansions. Four different mathematical models of offshore phenomena are studied: the jacket structure under periodic wave excitations, the motions of a floating platform supporting offshore 5 MW wind turbine, Mathieu's instability phenomena and geometric nonlinearity of moored offshore platform under Morison's forces. These mathematical models are representative of offshore systems and exhibit different types of nonlinearity and instability. The proposed controller was able to effectively restrict the motions in comparison to the conventional quadratic regulator. Taking into account the controlled motions, the controller force required for proposed controller is small compared to conventional quadratic regulator. Simultaneously, through the expositions it was clear that the algorithm was also able to tackle the chaos in strongly nonlinear systems. Note that, the present algorithm assumes nonlinear forms of mathematical models are available; which otherwise would require extensive model updating. Also since linearized replacements are done, the applications of the strategy to rigid body rotation may require further studies. Moreover, taking into account that the relevance of the proposed method would be to ocean engineering problems in an irregular oceanic environment (arrived through finite difference/element method), it is justifiable to apply these algorithms to such problems. The extension of this method is underway by the authors and being reported elsewhere.

## Author' contributions

R.M. led the research, implemented the numerical methods, carried out numerical studies and drafted the manuscript; N.S. participated in the mathematical formulation and manuscript preparation.

## Competing interests

We have no competing interests.

## Funding

The research was supported by Government of India fellowship for R.M. and a project grant from Earth Systems Science Organization, Ministry of Earth Sciences to N.S.

## Acknowledgements

The authors also gratefully acknowledge partial financial support given by the Earth Systems Science Organization, Ministry of Earth Sciences, Government of India through National Institute of Ocean Technology to conduct part of the research.

- Received August 27, 2015.
- Accepted October 28, 2015.

- © 2015 The Author(s)