Abstract
This paper poses and solves a stabilization problem of a rigid body governed by nonlinear differential equations in two viscous incompressible fluids governed by Navier–Stokes equations (NSEs), where surface tension of the interface between the two fluids is considered, in a bounded domain in three dimensional space. Since only weak solutions of the NSEs exist globally while global existence of their strong/smooth solutions is a millennium problem, point-wise fluid forces and moments acting on the rigid body are not able to bound. This difficulty is overcome by designing an appropriate control law and performing stability analysis of the closed-loop system including the NSEs and surface tension, where “work and power of the two fluids” instead of forces and moments on the rigid body are used. A simulation is included to illustrate the results.
1 Introduction
The most common rigid bodies, which are operating between two fluids, are floating ones, where one fluid is water and the other fluid is air. Such rigid bodies include surface ships [1,2], buoys of wave energy converters [3,4], floating wind turbines [5], floating breakwater and jacket platform [6]. Stabilization of rigid bodies operating between two fluids is an important and challenging area due to their applications and complex loads of fluids on them. An appropriate determination of the fluid loads on the rigid bodies is essential for control design. There are two main approaches to determine loads of fluids on the rigid bodies: (1) approximation approach, where the forces and moments of fluids on the rigid bodies are approximated by using linear (Stokes) and/or potential flow theory (e.g., Refs. [1,2,7,8]); and (2) fundamental approach based on Navier–Stokes equations (NSEs) by using the original NSEs (e.g., Refs. [9–23]).
In the approximation approach, the loads of fluids are referred to as fluid forces and fluid moments induced by air (wind) and water (waves, currents), see Refs. [1,2,7,8] and references therein. For example, the fluid forces are approximated by the sum of Froude-Krylov force, diffraction force, radiation force, hydrostatic force, and viscous drag while wind force is approximately calculated from wind spectra. The fluid forces and fluid moments are assumed to be point-wise bounded. This requires a strong solution of the NSEs for a viscous incompressible fluid. Currently, existence of a strong/smooth solution of NSEs is local in either time or small initial fluid velocity [24,25] under sufficient regularity of initial fluid velocity. Global existence of a strong/smooth solution of NSEs is a millennium problem [26].
In the fundamental NSEs approach, fully coupled dynamics of both rigid bodies and fluids, where motion of rigid bodies are governed by nonlinear differential equations (ODEs) and motion of the fluids are governed by the NSEs in three dimensional space. This approach is much more complex than the approximation approach but gives insides of fluid–structure interactions. The complexities include: (i) existence of an appropriate solution of the fluid and rigid body system; (ii) time-varying domain of the fluids due to motion of the rigid bodies; (iii) interaction between fluids with tension on their interface; (iv) interaction between the fluids and rigid bodies; (v) bound of the forces and moments induced by the fluids on the rigid body. There are several works related to the fundamental NSEs approach. Existence of a weak solution for a system of a fluid and multiple rigid bodies was proved in Refs. [13–21], where dynamics of the fluid and rigid bodies is written as a global fluid and the initial value of the fluid velocity is assumed in , see “Notation of functional spaces” below for a definition of space , and a coordinate transformation is used to handle difficulty caused by the fluid time-varying domain. Several works on stabilization of a rigid body in a fluid include: [27] on a linear control for stabilization of a rigid ball and [28–30] on stabilization of a rigid ball in one-dimensional, two-dimensional, and three-dimensional with the initial value of the fluid velocity in space .
The most relevant references to the present paper are [22] and [23]. In Ref. [22], a control law is designed to stabilize a rigid body completely submerged in a single incompressible fluid based on Lyapunov's direct method, where the initial fluid velocity is assumed in space H. In Ref. [23], a control law is designed to stabilize a rigid body on the interface between two incompressible fluids. The initial fluid velocity of the fluids is also assumed in the space H. The surface tension between two fluid is set to zero. The control design is based on the forwarding method [31] with some modification because the dynamics of a rigid body is not of a strict feed-forward form.
Existence of a weak solution of the closed-loop system in both [22] and [23] is carried out based on a penalized method [32] while stability is analyzed by using fluid work and fluid power to overcome difficulties caused by less regularity of the initial fluid velocity. However, proof of existence of a weak solution and stability analysis of the closed-loop system in Ref. [23] is much more involved than in Ref. [22] due to composition of the two fluids.
The present paper proposes new contributions in design of a control law to almost globally practically exponentially stabilize a rigid body on the interface between two fluids with surface tension in a domain Ω, where “almost global” means that there is no collision between the rigid body and the boundary of the domain Ω, existence of a weak solution and analysis of stability and convergence of the closed-loop system. The initial velocity of the fluids is only assumed in space H, which is less regularity than space , see Eq. (12) for a definition of space . Since the initial velocity of the fluids is only assumed in space H, there is no information on (point-wise) bound of the forces and moments induced by the fluids on the rigid body. Thus, we consider the effect of the work and power of the fluids (instead of the fluid forces and moments) on the rigid body, see the paragraph just below Eq. (74) for the concept of the work and power of the fluids, which was introduced in Ref. [22]. The control design is based on the backstepping method in Ref. [33] and the forwarding method [31].
In comparison with the works in Refs. [22] and [23], the present paper is significant from both practical and theoretical points of view. Practically, the work in Ref. [22] is only applicable to a rigid body in a single incompressible fluid (such as in water or in air) while the work in Ref. [23] allows a rigid body on the interface between two incompressible fluids such as surface ships and floating off-shore structures. The work in Ref. [23] covers the work in Ref. [22] but not vice versa but the surface tension is set to zero in Ref. [23]. In practice, surface tension of a fluid is never zero. Hence, setting surface tension of a fluid to zero significantly simplifies the real scenario. This deteriorates modeling accuracy and performance of a control system. The work in the present paper considers a rigid body on the interface between two incompressible fluid, where nonzero surface tension is allowed. Hence, the present paper covers both [22] and [23] but not vice versa. In addition, since surface tension is allowed to be nonzero, it reflects reality in modeling and hence improves performance of a control system.
Theoretically, nonzero surface tension (see the second equation in Eq. (4)) makes the normal component of the stress tensor discontinuous. This is the main source that makes formulation and existence proof of a weak solution and analysis stability of the closed-loop system difficult, see also Remark 2.1. In fact, formulation of a weak solution of the closed-loop system in Sec. 4.1 is much harder than those in Refs. [22] and [23], see the Eq. (42) with the extra term defined in Eq. (43) due to nonzero surface tension. Nonzero surface tension also makes stability analysis of the closed-loop system in Sec. 5 very involved, see the term in Eq. (90).
The rest of the paper is organized as follows: In Sec. 2, a problem of stabilizing a rigid body in two fluids in a domain Ω is posed. In Sec. 3, a control law is designed to achieve almost global practical exponential stabilization of the rigid body, and is of an inverse pre-optimal form, which can be amended to be inverse optimal [34] without having to solve a Hamilton-Jacobi-Bellman or a Hamilton-Jacobi-Isaacs equation. In Sec. 4, we formulate the closed-loop system as a global fluid, of which existence of weak solution is shown via a penalizing method. In Sec. 5, almost global practical exponential stability of the closed-loop system is proved. A simulation is given in Sec. 6 to illustrate the results.
1.1 Notation of Functional Spaces.
Let Ω be an open bounded set in , and a time constant T > 0. , where , denotes the standard Lebesgue space of measurable p-integrable functions; denotes the space of essentially bounded functions; is the usual Sobolev space of order 1, see Ref. [35]; denotes with compact support; with and ; , where and X is a Banach space with the norm denoted by , denotes a Bochner space with the norm . Let be the dual space of a Banach space X. The space is defined as all weak- measurable functions , i.e., , where denotes a family of probability measures, is measurable for each , such that . The space of k-times differentiable functions on a set Ω is denoted by . We also use to denote the Euclidean (point-wise) norm, i.e., . For a scalar, we use to denote the absolute value.
2 Problem Statement
Let be a domain occupied by fluid 1 (which is usually the air), fluid 2 (which is usually the water), and a rigid body. The fluid 1 and fluid 2 occupy bounded open domains and at time , see Fig. 1. These fluid are separated by the interface and surround a rigid body represented by , which is a bounded open connected subdomain of Ω. We have . The domain will be defined and used in Sec. 5. The time-dependence of the domains and Ωs is due to motion of the rigid body. We assume that . We will drop the argument t of , Ωs, and Γ when there is no confusion.
where is the orthogonal projection onto the tangential space, which is defined point-wisely on Γ, the symbol ⊗ denotes the outer product operator, i.e., with being the transpose of b for , and for 3 × 3 symmetric matrices A, B with being the trace of matrices. The mean curvature is defined as .
Remark 2.1. Due to the surface tension term in Eq. (4), the normal component of the stress tensor is discontinuous. This introduces difficulties in proof of existence of a weak solution and stability analysis of the closed-loop system in Secs. 4 and 5 as discussed in Sec. 1. The surface tension term also makes the present paper more general from both practical and theoretical points of view than the works in Refs. [22] and [23].
where with being the position vector of actuator forces such that and (i.e., the rigid body is fully actuated).
In derivation of Eq. (9), we have used a transmission condition that the stress is continuous in normal direction on the interface, i.e., on with being the Cauchy stress tensor, i.e., is the force applied by the rigid body on the fluid i.
where . Note that the elements of K(t) are given by the rigid body velocity in Ωs. One can prove the following lemma on the space K(t), see Refs. [32,37].
In this paper, we address the following control objective.
whereand, design the control force vectorand control moment vectorto almost globally practically exponentially stabilize the rigid body at the origin. Moreover, the designed control forces and control moments must ensure existence of a weak solution, which is defined in Definition 4.1, of the closed-loop system consisting of both the rigid body and the fluids.
Note that the initial values was imposed to be compatible with , and was made due to the domain of the modified Rodrigues parameter vector .
where . Here, is a finite Radon measure and , see Ref. [39]. There are no boundary conditions for χf, ρf, and μf. As shown in Refs. [13,38,39], the nonmiscibility condition in Eq. (4) is equivalent to a transport equation for , where the surface tension is taken into account due to the term in Eq. (17), see also [40], and the transmission conditions in Eqs. (4) and (11) will hold as soon as belong to .
3 Control Design
We extend the backstepping method in Ref. [33] and the forwarding method [31] to design the controls and with necessary modifications so that these controls are of a form that can be amended to be inverse optimal. Existence of a weak solution of the closed-loop system will be shown in Sec. 4. Analysis of stability and convergence of the closed-loop system will be given in 5. Since the rigid body dynamics Eq. (18) is of a second-order system, the control design consists of two steps.
3.1 Step 1.
3.2 Step 2.
ε01 and ε02 being positive constants to be chosen.
where and are positive definite matrices to be chosen. Note that the nonlinear term is due to the nonlinear terms and in the rigid body dynamics Eq. (18). From Eqs. (36) and (10), we can solve for the actuator forces . The controls Eq. (36) do not cancel any system functions. This is important for robustness.
It can be easily shown that there always exist sufficiently large control gain matrices , and such that all the inequalities in Eq. (38) hold for given positive constants .
where W is a positive definite matrix. This form means that the controls and are inverse pre-optimal and can be easily extended to be inverse optimal by using the Legendre-Fenchel transform, see Ref. [34], which has many desired properties such as an infinite gain margin and minimization of a cost function without having to solve a Hamilton-Jacobi-Bellman equation or a Hamilton-Jacobi-Isaacs equation. However, we do not detail this issue here as we focus on the interaction of the controlled rigid body and two phase flows of two viscous incompressible fluids in this paper.
Remark 3.2. We cannot obtain a bound for the fluid force and the fluid moment , and hence the term ϖ defined in Eq. (29), i.e., we cannot conclude any stability of the controlled rigid body dynamics Eq. (18) with actuator forces obtained from Eqs. (10) and (36) based on Eqs. (26) and (39) at this point. This is a major difficulty compared with the approximation approach, see Refs. [1,2,7,8] and references therein. Therefore, we will analyze stability of the rigid body dynamics and convergence of its states in Sec. 5 after we show existence of the weak solution of the dynamics of the fluids and the controlled rigid body in Sec. 4.
4 Existence of a Weak Solution of the Closed-Loop System
4.1 Formulation of a Weak Solution.
with being the two-dimensional Hausdorff measure.
due to .
where means that and with being the space of functions with bounded variation [41]; means that and ; means that and ; and means and .
The above derivations motivate the following weak solution definition.
for all.
The above definition of a weak solution is similar to a definition of a weak solution to a two phase flow of two viscous incompressible fluids, [39] except for the fact that we take the constraint of rigidity into account.
4.2 Penalized System.
In order to find the tuple , χf, ρf, μf, such that they satisfy Definition 4.1, we use a penalization approach. Due to Lemma 2.1, there are three methods to penalize the rigidity constraint: (i) penalizing the difference between the fluid velocity and rigid body velocity [32]; (ii) penalizing the spatial derivative of the rigid body velocity [37]; (iii) combination of methods i and ii [42]. We use the third method.
and define . Several observations on the penalized system Eq. (63) are given in the following remark.
Remark 4.1. The penalized system Eq. (63) is based on [32], where action of the controls and given by Eq. (36) and nonhomogeneous viscosity and surface tension are taken into account.
Since is positive, and the matrix is symmetric and positive definite, see Eq. (7), and hence is invertible.
where is a rigid velocity field, i.e., there exist such that , and is given by Eq. (64). Hence, the penalized term in the first equation of Eq. (63) is the difference between and its projection onto velocity fields rigid in the rigid body domain, i.e., . Moreover, the viscosity term μn vanishes asymptotically in the solid part. Hence, there will be no uniform bound in V for . This does not conflict with the penalized term because of the space K.
The density, viscosity, and fluid character function χfn are transported with the velocity field . This eases calculations in estimating bounds for the penalized system.
Existence of a weak solution to the penalized system Eq. (63) is stated in the following lemma.
Lemma 4.1. There is at least one weak solution , χfn, ρfn, μfn, to the penalized system Eq. (63) that satisfies Eq. (62) for all , where T is such that and for all .
Proof . Proof of this lemma principally follows the part of a priori estimates and convergence arguments in proof of Theorem 2.1 in Ref. [32] for a rigid body and fluid system, proof of Theorem 1.1 in Ref. [38] for a multifluid system (particularly, the renormalized solution of the transport equations is used), and proof of Theorem 1.6 in Ref. [39]. The only main difference is that a priori estimates should use the penalized energy as in Eq. (52) with being substituted by . This is due to inclusion of and controls in this paper.▪
4.3 Existence of a Weak Solution.
Having obtained a weak solution of the penalized system Eq. (63) in Lemma 4.1, existence of a weak solution stated in Definition 4.1 is given in the following theorem.
in, where.
▪
5 Stability and Convergence of the Closed-Loop System
5.1 The Term ϖ.
Since we already showed existence of a weak solution of the closed-loop system (including both the fluids and rigid body) in Theorem 4.1, the idea to handle the term ϖ is as follows. For the term A1, we will multiply the fourth equation in Eq. (17) by to detail A1, where we will use the transmission condition (19). For the terms A2 and A3, we proceed in two steps. In the first step, we use regularity of the Stokes problem, see Ref. [43], to obtain vector fields over Ωs for and . In the second step, we use the extension theorem, see Ref. [44, Theorem 1, Sec. 5.4] to extend the vector fields to a virtual rigid body , see Fig. 1, to detail the terms A2 and A3.
5.1.1 Detail of A1.
where ε20 is a positive constant to be chosen.
5.1.2 Detail of A2 and A3.
We define the domain , which is an extension of the domain Ωs in the normal direction, such that , see Fig. 1. We cite the following results for use in detailing A2 and A3.
whereis a constant depending only on Ωs, k, r.
Proof . See Ref. [43].▪
such that for each: (i)a.e. in Ωs, (ii)has support in, and iii), whereis a constant depending on only r, Ωs, and.
Proof . See Ref. [44, Theorem 1, Sec. 5.4].▪
due to the embedding because is a bounded domain, where c is a positive embedding constant.
where and denotes the size of .
5.1.3 Detail of ϖ.
5.2 Convergence of the Closed-Loop System.
where and are positive constants by choosing a sufficiently large ε0.
Next, we need the following lemma.
where.
where, c1 is a positive constant, and c0 is a non-negative constant (depending on the initial data).
6 Simulations
A simulation is carried out to illustrate the effectiveness of the control law given by Eq. (36). For the rigid body, we take the physical shape of a neutrally buoyant cube with dimensions: and the mass: ms = 1 kg, which give . For the air with and and water with and , the domains and are also taken as cubes subtracting the rigid body. The dimensions of and are respectively taken as , where is the part of the rigid body submerged in the air, and , where is the part of the rigid body submerged in the water. The surface tension constant is . Thus, the domain Ω is a cube with dimensions . We approximate all the sharp corners of Ω and Ωs by rounding them off to make and Lipschitz, where recalling that . We assume that the rigid body is neutrally buoyant otherwise we can pre-eliminate the difference between buoyancy and gravity forces before applying Eq. (36). The body forces are taken as and , which satisfy the condition (2).
where are scalar functions of time, are eigenfunctions of the Stokes operator. We substitute Eq. (109) into the first equation of Eq. (63) and multiply it by to obtain a system of ODEs for , which is numerically solvable. The transport equations (the third, fourth, and fifth equations of Eq. (63)) are solved by using the characteristic method. We now need to derive eigenfunctions for our domain Ω. To do so, we need the following lemma [45, Theorem III.2.3].
Lemma 6.1. If Ω is a bounded open set inwith Lipschitz boundary, then H coincides with the space of divergence free functions insuch thaton, wherenis the normal unit vector to.
where such that , which are taken into account to have summing combination in calculating Eq. (109), and with . It is easy to show that A > 0 for all , and that satisfies all the properties listed in Eq. (110) with .
For the initial values of the fluid velocity, we take to be random number in if and in if . The initial value of the interface between the air and water is neutral, i.e., and . This results in if if if ; and if if if . The initial values of the rigid body are taken as m, , which yields a principal axis/angle pair and rad. The initial values of the velocities and of the rigid body are determined via Eqs. (7) and (61), where is substituted by . We perform open-loop and closed-loop simulations for comparison.
We first simulate the open-loop system, i.e., we set the controls and . The position vector , orientation vector , linear velocity vector , and angular velocity vector are plotted in Fig. 2. The H-norm of the global velocity , the control force vector , and control moment vector are plotted in Fig. 3. It is seen from these figures that tend to drift away from the origin due to forces and moments from the fluids; tend to the origin due to (restoring) forces and moments from the fluids; , and decay slowly to zero (decay is due to resistance of the fluids); and indeed the controls and are zero.
In the closed-loop simulations, we choose the control gains as follows: . It can be checked that the conditions in Eq. (38) hold. The position vector , orientation vector , linear velocity vector , and angular velocity vector are plotted in Fig. 4. The H-norm of the global velocity , the control force vector , and control moment vector are plotted in Fig. 5. It is seen from these figures that all the states , and the velocity norm ; and the controls and converge to the origin (note that the body forces and decay to zero). In comparison with the open-loop simulation results, see Figs. 2 and 3, the effectiveness of the controls is clearly illustrated. Note that the overshoot in is higher for the closed-loop due to the control action.
7 Conclusions
Almost global practical exponential stabilization problem of a rigid body between two fluids with surface tension was posed and solved. The main ingredients include (i) formulation of two fluids as a single fluid; (ii) design of an appropriate feedback stabilization control law; (iii) formulation of the two fluids and rigid body as a global fluid; (iv) derivation of work and power of the two fluids, (v) derivation of a Lyapunov function, including stabilization errors, energy of the two fluids and surface tension. Future work is to consider an unbounded domain.
Funding Data
Australian Research Council (Grant No. DP230100390; Funder ID: 10.13039/501100000923).
Data Availability Statement
The authors attest that all data for this study are included in the paper.