An analytical solution is derived to describe the wave-induced flow field and surge motion of a deformable platform structure controlled with fuzzy controllers in an oceanic environment. In the controller design procedure, a parallel distributed compensation (PDC) scheme is utilized to construct a global fuzzy logic controller by blending all local state feedback controllers. The Lyapunov method is used to carry out stability analysis of a real system structure. The corresponding boundary value problems are then incorporated into scattering and radiation problems. These are analytically solved, based on the separation of variables, to obtain a series of solutions showing the harmonic incident wave motion and surge motion. The dependence of the wave-induced flow field and its resonant frequency on wave characteristics and structural properties including platform width, thickness and mass can thus be drawn with a parametric approach. The wave-induced displacement of the surge motion is determined from these mathematical models. The vibration of the floating structure and mechanical motion caused by the wave force are also discussed analytically based on fuzzy logic theory and the mathematical framework to find the decay in amplitude of the surge motion in the tension leg platform (TLP) system. The expected effects of the damping in amplitude of the surge motion due to the control force on the structural response are obvious.
The supply and demand for the products of natural energy resources have been rapidly increasing in recent years. This has motivated oil/gas producers to explore even the ocean depths to obtain these resources. This has necessitated the in-depth study and analysis of deep water structures, such as the tension leg platform (TLP), which are particularly suited for water depths greater than 300 meters [
The reduction of the vibration of structures by passive methods such as mass, stiffness, and damping has long been a subject of study in the fields of oceanic, mechanical, and civil engineering. Interest in the usage of tuned mass dampers and tuned liquid column dampers for reducing vibration in oceanic structures has also increased (see [
Much work has been carried out in both industry and academic fields since Zadeh [
In recent years, many control methods have been proposed and there are increasing research activities in the field of structural systems. These methods include optimal control, fuzzy control, pole placement, and sliding mode control (see [
The flow field is first divided into three columns with two artificial boundaries at
Sketch of a deformable tension leg platform subjected to the wave force.
Boundary value problems for scattering and radiation problems.
For a discussion of the kinematic boundary condition, dynamic boundary condition and radiation condition please refer to the stability analysis in [
Analytical solutions of the scattering problem, motion radiation problem, and vibration radiation problem can also be derived using a similar approach [
Recently, fuzzy-rule-based modeling has become an active research field because of its unique merits for solving complex nonlinear system identification and control problems. Unlike traditional modeling, fuzzy rule-based modeling is essentially a multimodel approach in which individual rules are combined to describe the global behavior of the system [
To ensure the stability of the structural system, Takagi-Sugeno (T-S) fuzzy models and the stability analysis are recalled. There are two approaches for constructing fuzzy models mentioned in literature [
identification (fuzzy modeling) using input-output data,
derivation from given nonlinear interconnected system equations.
There has been extensive work done on fuzzy modeling using input-output data following Takagi’s, Sugeno’s, and Kang’s outstanding work (see [
The momentum equation can be obtained from the motion of the floating structure, extensively derived from Newton's second law. Assume that the momentum equation of a TLP system controlled by actuators can be characterized by the following differential equation:
For the controller design, the standard first-order state equation corresponding to (
To ensure the stability of the TLP system, T-S fuzzy models and some stability analysis are utilized. To design fuzzy controllers, the structural systems are represented by Takagi-Sugeno fuzzy models. The concept of PDC is employed to determine the structures of the fuzzy controllers from the T-S fuzzy models in this section. Some detailed steps for designing PDC fuzzy controllers are described in literature [
Rule
The PDC is adopted to design a global controller for the T-S fuzzy model (
Controller Rule
A typical stability condition for a fuzzy system (
The equilibrium point of the fuzzy control system (
The proof is lengthy and can be derived by the similar approach by Hsiao et al
In the physical world, engineers measure and observe the effects induced by embedded structures. They might conduct an experiment or carry out real scale numerical simulations. We do not especially consider engineering oriented issues. Instead, we utilize sensitive concepts to explain the effects of physical parameters on a natural transformation. The dynamic response of a platform in a TLP system is dependent on a large set of parameters that include the characteristics of the waves and the material properties of the structure. Several parameters are particularly important for systematic consideration in TLP stability and stabilization problems. These include the amplitude and frequency of the waves (wave characteristics) as well as the structural properties of the platform (including draft, width, thickness mass, etc.). Only platform mass is considered in the following discussion. The parameters are given in Table
Input data for case study of surge motion calculation.
Initial wave conditions | ||||
Wave frequency | Variable | |||
Wave amplitude | 0.5 m | |||
Environmental conditions | ||||
Water depth | 30 m | |||
Gravitational force | 9.8 m/ | |||
Structural parameters | ||||
Mass | 10, 100, and 1000 kg | |||
Draft | 2 m | |||
Width | 10 m | |||
Young's modulus | ||||
Mass moment of inertia |
Input data for three parametric cases.
Subsystem 1 | ||||
Wave period | 10 s | |||
Structure mass | 100 kg | |||
Structure width | 4 m | |||
Subsystem 2 | ||||
Wave period | 12 s | |||
Structure mass | 200 kg | |||
Structure width | 6 m | |||
Subsystem 3 | ||||
Wave period | 8 s | |||
Structure mass | 150 kg | |||
Structure width | 2 m |
Others are as follows.
Wave amplitude
Comparison of the coefficients of reflection and dimensionless frequency for various structure masses.
Comparison of the coefficients of transmission and dimensionless frequency for various structure masses.
Comparison of the dimensionless amplitude of the surge motion between the platforms with and without controller interactions with respect to the variation of the structure mass.
State response of the states and harmonic waves for three conditions.
Control forces for three conditions.
In order to exploit rich new sources of oil and gas lying beneath the ocean, we must be ready to delve into very deep waters. Many promising fields are situated under water that is in the 4000–7000 ft depth range. A favored platform design in that environment is the TLP system, where the hull is connected to the seabed by strings of tendon pipes. Since tendon length as well as water pressure increases in proportion to water depth, it is essential to produce a stable structure that the tendon pipes be highly resistant to collapse, especially in deep water environments. We present here a new control force concept for the stabilization of TLP systems, an alternative to the tether drag effects used previously. The results of the present study show that this new controller improves the limitations of steel performance for the maximum water depth attainable with the old TLP system. The dependence of wave reflection, transmission, as well as structure surge motion on incident wave conditions and structure properties has been demonstrated. The results show that the response of the floating structure is influenced by the its mass (as demonstrated by the different surge motion profiles). In particular, the response of a floating structure reaches the largest level of displacement during resonance. The resonant phenomenon takes place when the dimensionless wave frequency moves to lower values, approximately in the 0.5–1.0 range. The response to the wave-structure interaction results in platform vibration, but this can be slowed down and stabilized by means of the control force.
The author would like to thank the National Science Council of the Republic of China, Taiwan, for their financial support of this research under Contract no. NSC 98-2221-E-366-006-MY2. The author is also most grateful for the kind assistance of Professor Balthazar, Editor of special issue, and the constructive suggestions from anonymous reviewers which led to the making of several corrections and suggestions that have greatly aided in the presentation of this paper.