## Abstract

The use of solar chimneys for energy production was suggested more than 100 years ago. Unfortunately, this technology has not been realized on a commercial scale, in large part due to the high cost of erecting tall towers using traditional methods of construction. Recent works have suggested a radical decrease in tower cost by using an inflatable self-supported tower consisting of stacked toroidal bladders. While the statics deflections of such towers under constant wind have been investigated before, the key for further development of this technology lies in the analysis of dynamics, which is the main point of this paper. Using Lagrangian reduction by symmetry, we develop a fully three-dimensional theory of motion for such towers and study the tower's stability and dynamics. Next, we derive a geometric theory of optimal control for the tower dynamics using variable pressure inside the bladders and perform detailed analytical and numerical studies of the control in two dimensions. Finally, we report on the results of experiments demonstrating the remarkable stability of the tower in real-life conditions, showing good agreement with theoretical results.

## 1. Introduction

Solar updraft towers, also known as solar chimneys, provide a simple and reliable way of generating electricity from solar energy [1–3]. With the main principle behind these devices dating back more than 100 years, solar chimneys are perhaps one of the simplest and most robust ways to generate power from solar radiation, even compared with the current photovoltaic (PV) technology. The principle of energy production by solar towers is very simple: solar radiation heats up air in a collector (a greenhouse) occupying a large area. The air escapes the collector through a tall (hundreds of metres) pipe, which connects the hot volume of the collector with the cooler air above the ground. The temperature difference induces convection, and a turbine within the pipe harvests the energy of the updraft. This method of energy generation has certain advantages over the traditional ways of generating solar energy. Unlike the PV producing maximum electricity during the peak of the day and no electricity at night, the solar tower output is smoothed by a huge thermal mass of the ground underneath the greenhouse, making it possible to produce electricity continuously. In addition, the only components of the design requiring regular maintenance are the generating turbines located at ground level, provided that the tower is stable and not needing additional support work (see below).

One of the major flaws of this design is the extremely low efficiency, especially compared with PV or wind turbine systems, which has severely limited the application of solar towers. The efficiency for a typical installation considered in the literature, defined as the ratio of power output to total power of the solar radiation impacting the collector, is estimated to be in the range of 0.5–5% [4]. One of the ways to improve this efficiency would be by increasing the temperature differential between the greenhouse on the bottom and the exhaust on top of the updraft tower, which would mean making the tower considerably taller. Current state-of-the-art design for a ‘traditional’ updraft tower is the Jinshawan Tower [5] currently under construction in north China's Inner Mongolia Autonomous Region. The operation of the first stage of the facility was supposed to start in December 2010, and the full height of the updraft tower is expected to reach about 900 m, which is over two times the height of the Empire State Building in New York, USA. The first operational updraft tower prototype [6] built in Manzanares, Spain, had a tower height of 195 m, with thin iron sheet used for construction. The main reason for decommissioning the facility in 1989 [7] was the failure of the tower's guy wires in a storm, which led to the tower's collapse. Towers for proposed full-scale facilities, such as the Ciudad Real Torre Solar in Spain [2,8], are much taller (750–1500 m). Construction challenges faced by this design are so severe that the costs of erecting the tower can dominate the overall cost of the facility and increase the levellized electricity cost from the tower. Detailed computations show that for a 100 MW plant the cost of a chimney capable of withstanding high winds already represents about half of the overall cost [6]. However, this estimate does not take into account construction difficulties such as soil composition, motion, local seismic activity and so forth. By the time these effects are taken into account, the cost of the tower may increase even further and reach a prohibitively high value. Furthermore, it is not clear how to protect the tower from, say, a hurricane-class wind. Finally, once the tower is erected successfully, any repairs to the structure, especially near the top, become complicated and dangerous.

Previous authors have tried to circumvent the challenge of building a rigid solar tower by making it free-floating and supported by a lighter-than-air gas [9] while using an accordion-like structure near the bottom to accommodate for wind-driven movement of the tower. While this design is appealing on some levels, the supporting element needs to be properly supplied with lift gas, which may be expensive in the case of helium and dangerous in the case of hydrogen. If the lift by toroidal elements supporting the tower is decreased, the tower becomes unstable and will collapse. Moreover, the accordion structure at the bottom will be subject to constant deformation and thus mechanical wear. Other designs suggested in the literature that are based on buoyant support of the tower face similar problems in their eventual practical realization.

By contrast, a recent paper [10] suggests making a flexible tower out of stacked inflatable toroidal elements pressurized with air. The shape of the tower was computed to spread the deformation due to wind loading uniformly along the tower. The wind was assumed to have a constant magnitude and not to vary with elevation. This idea is an extension of the ideas used by Eiffel in the design of his famous tower in Paris, France. A summary of Eiffel's considerations [11] was recently extended further for realistic wind profiles [12]. This stacked-torus approach has a potential for affording a much simpler way of erecting tall towers (http://www.bbc.co.uk/news/technology-25015030) which are an order of magnitude cheaper than traditional rigid construction. In addition, the tower can be deflated and re-inflated for repairs and in the event of approaching dangerously high winds. Finally, such towers, filled with air, may be controlled dynamically by locally inflating and deflating each individual element. The dynamics and control theory for such towers has a particular mathematical interest and is the main point of this paper. As it turns out, this theory is connected with the theory of optimal control on manifolds, where the control variables are the parameters of the linked elements, i.e. the pressure of the tori controlling stiffness and mass. These variables need to be changed in a specific way to control the position of the element on a group; in our particular case, the special Euclidean group *SE*(3) or the rotation group *SO*(3). Note that this paper only deals with the dynamics and control of the tower itself. For other aspects, such as thermodynamic considerations and energy production in static towers, we refer the reader to the studies in [10]. Energy production in moving, flexible towers is a more challenging topic and will be considered in future works.

The plan of the paper is as follows. In §2, we set up the basic framework for further considerations and outline basic physical and geometric assumptions behind our theory. In §3, we derive the equations of motion in the fully three-dimensional case using the method of Lagrangian reduction by symmetry. Further, §4 describes the reduction of motion to two dimensions and numerical simulations of tower dynamics. Section 5 is dedicated to deriving geometric optimal control methods for stabilizing the dynamics using variable pressure in the tori. Finally, in §6, we outline the results of experiments demonstrating some important aspects of the stability and dynamics underpinning our theoretical considerations.

## 2. Set-up of dynamical equations

Let us consider a flexible tower consisting of *N* toroidal elements with inner radius *d*_{n} and outer radius *q*_{n}, *n*=1,…,*N*, as shown in figure 1. In practice, these toroidal elements have different sizes so that the tower profile is optimized for both energy production and resistance to wind. The latter requirement leads to the tower profile becoming narrower at the top following a certain law, as described in [10]. In our theoretical considerations, we will use general *d*_{n} and *q*_{n}; however, all numerical simulations performed in this paper will use identical tori *q*_{n}=*q* and *d*_{n}=*d*, so that the tower is cylindrical in its equilibrium form. We describe the dynamics of each torus by giving the position **r**_{n} of its centre, and the orientation of a moving basis **E**_{1},**E**_{2},**E**_{3}). The moving basis is described by means of a rotation *Λ*_{n} such that
*Λ*,**r**) of the special Euclidean group *SE*(3) with semidirect product multiplication
*U*_{d}.

### Remark 2.1 (on the symmetry invariance of tori interaction)

Given the orientation and position of two adjacent tori (*Λ*_{n},**r**_{n}) and (*Λ*_{n+1},**r**_{n+1}), the potential energy can only depend on the relative position and orientation, i.e. on the combination
*left* rotation
*SE*(3) invariance of dynamics in the context of non-locally interacting rods and its physical consequences.

This principle of invariance, stating that the potential energy *U*_{d} of the elastic deformation is invariant with respect to rotations and translations of the origin, must be verified for any physical model used in the quantitative description of the elastic energy. For example, one can obtain this energy through a computation of the volume overlap of two neighbouring undeformed tori. While this computation is straightforward, it is rather cumbersome and implicit for two tori in a generic position **r**_{n} and orientation *Λ*_{n}. Let us now provide one example of such a computation in the particular situation where the tori are in persistent contact. Let us choose the system of coordinates for the *n*th toroidal element so that *θ* is the angle sweeping along the large radius *q*_{n}, and *α* is the angle sweeping the small radius *d*_{n}. The deflection of the torus perpendicular to the plane swept by *θ* will be called *h*_{n}(*μ*_{n},*θ*) defining the shaded annulus-like area shown in figure 1.

### Remark 2.2 (on contact between tori)

If the position of the tori is unrestricted, for some (large) deformations *h*_{n}(*μ*_{n},*θ*) may degenerate to 0 for |*θ*|>*θ*_{m} simply because one part of the torus loses contact with the neighbouring torus. We shall consider the case where such loss of contact cannot occur. For example, the experimental realization of the tower, as explained in §6, consists of rubber inner tubes glued together along the large circle's perimeter. In that case, the tube is assumed to be in contact at all *θ* values, which leads to *h*_{n}(*μ*_{n},*θ*)≥0 for −*π*<*θ*<*π*. Technically, the persistence of contact means that the variables **r**_{n}, *μ*_{n} satisfy a certain relationship and are no longer independent variables. We shall emphasize, however, that the persistence of contact is determined by the details of tower construction; other situations such as tori being attached with soft springs will allow the loss of contact.

The potential energy of deformation of an individual torus is given by the work required to change the volume of a given element, i.e. *U*_{n}≃*p*_{n}Δ*V* _{n}. If the normal extent of the torus overlap is *z*, and the local curvature is *d*_{n}, then the width of the shaded area in figure 1 can be estimated as *h*_{n}(*μ*_{n},*θ*)^{3/2} by its typical value *C*, which will be dropped from further considerations. Thus, the total potential energy is

### Remark 2.3 (alternative energy expressions: symmetry invariance)

The reader may note that in our original setting *U*_{d} is a function of both *μ*_{n} and *ρ*_{n},*ρ*_{n+1}, whereas in (2.5) the energy *U*_{d} depends only on *μ*_{n}. This happens because we assumed a perfect contact of tori which fixes the values of *ρ*_{n} and *ρ*_{n+1} given *μ*_{n}. In the description of other mechanical realizations of the tower, for example tori connected by springs or elastic material, *U*_{d} can depend on both *μ*_{n} and *ρ*_{n}, *ρ*_{n+1}. Thus, our expression for the potential energy (2.5), while being rooted in physics, is necessarily simplistic. However, no matter how complex the expression for the inter-tori potential energy *U*_{n} is considered, whether it is with or without persistent contact, it *must* satisfy the symmetry invariance, i.e. *U*_{n} reduce to be a function of *μ*_{n}, *ρ*_{n} and *ρ*_{n+1} only. Thus, we shall derive all the formulae in the most general case of a symmetry-invariant potential, and only use expressions (2.5) for computations and comparison with experiments. Gravity terms in the potential energy breaking this rotational and translational invariance are considered separately in (2.8).

For future computations, it is advantageous to simplify (2.5) for the case of two-dimensional dynamics that we will use later. If **E**_{3} is the axis of symmetry of the torus, then the relative tilt, denoted as *ϕ*_{n}=Δ*φ*_{n} to match the notation later in this paper, is
*φ*_{n} yields the same deformed volume and thus the same *U*_{d}, which leads to the absolute value sign in (2.7). We shall remark that a two-dimensional analogue of (2.7) has been derived in [10] without the use of symmetry-reduced variables. While this approach was reasonable for static considerations in [10], the use of the powerful methods of Euler–Poincaré theory is advantageous for the derivation of geometrically exact dynamical equations. We shall now proceed to recasting the rest of the Lagrangian in the coordinate invariant form.

To consider the effect of gravity for three-dimensional dynamics, the potential energy (2.7) is augmented by the gravity potential
*M*_{n} is the mass of the *n*th torus, *κ*_{n} verifies the additional dynamics *ω*_{n} via the map *κ*_{n} is an example of an *advected* variable in the system. The theory of dynamics incorporating such additional variables in the Euler–Poincaré framework has been considered before (e.g. [14]) in the context of heavy tops and compressible fluids. For simplicity, this potential energy contribution will be neglected in our numerical simulations. The kinetic energy is
*U*_{d} is proportional to the excessive pressure in each torus *p*_{n}. Similarly, if we denote the ambient atmospheric pressure as *p*_{0}, the density of air at each torus is proportional to the total pressure *p*_{0}+*p*_{n} assuming temperature is constant. Thus, for a given volume, we obtain that both the mass and moment of inertia of the *n*th torus are proportional to *p*_{n}+*p*_{0}, i.e. *M*_{n}=(*p*_{n}+*p*_{0})*M*_{0,n} and **I**_{n}=(*p*_{n}+*p*_{0})**I**_{0}. We can thus rewrite the complete symmetry-reduced Lagrangian ℓ:=ℓ(** ω**,

**,**

*γ***,**

*ρ***,**

*μ***) as follows:**

*κ**U*

_{0,n}(

*μ*

_{n},

*ρ*_{n},

*ρ*_{n+1}) as the term multiplying the pressure in the relative deformation energy

*U*

_{n}, i.e.

*U*

_{n}=

*p*

_{n}

*U*

_{0,n}(

*μ*

_{n},

*ρ*_{n},

*ρ*_{n+1}). In what follows, we shall assume for simplicity that the tori are identical,

*q*

_{n}=

*q*and

*d*

_{n}=

*d*, leading to the universal energy function

*U*

_{0}(

*μ*

_{n},

*ρ*_{n},

*ρ*_{n+1}); the formulae generalize in a rather straightforward way to more complex geometries. The quantities with the subscript 0 denote the variables independent of the pressure and depending only on the spatial dimension of each torus.

An interesting feature of the potential energy is the singularity at equilibrium. This is typical for the interaction of deformable spherical objects in contact, such as Hertzian chains [15], and presents interesting challenges for the simulations and opportunities for new solutions; in particular, since the restoring force is, loosely speaking, infinitely stronger than any linear force at equilibrium, growing as the square root of the deformation. Thus, the straight equilibrium state is stable with respect to small disturbances. This can be proved by taking a regularization of the potential at equilibrium and tending the corresponding regularization to zero (see (4.9)).

### Remark 2.4 (notation)

In (2.10), we used the notation

## 3. Derivation of the equations of motion

In this derivation, we follow the Euler–Poincaré theory for the interacting elastic components. The Hamiltonian form of this framework was pioneered by Simó *et al.* [16], while the Lagrangian formulation of this theory including reduction by symmetries and discrete interactions was derived subsequently in [13,17,18]. The equations of motion are derived using a symmetry reduced version of Hamilton's principle
*TQ* of the configuration space *Q*. The critical point condition in (3.1) is computed with respect to arbitrary variations *δq*(*t*) of a curve *q*(*t*) defined on the time interval [0,*T*], such that *δq*(0)=*δq*(*T*)=0, and *T*>0 is a given point in time that can be chosen arbitrarily.

In our case, we have *Λ*_{k}∈*SO*(3) and *Λ*_{k},**r**_{k})∈*SE*(3), for *k*=1,…,*N*. Therefore, the configuration space is

In the present case, it turns out that the Lagrangian *reduced Lagrangian*. This property of *L* is due to its *SO*(3) symmetry, on which we will comment further in remark 3.4.

The equations of motion in terms of the variables (3.4) are derived by expressing Hamilton's principle (3.2) in terms of ℓ. This is done by computing the variations of the variables (3.4) induced from the free variations *δΛ*_{k}, *δ***r**_{k} of the variables *Λ*_{k}, **r**_{k}. We get the variations
*Σ*_{k}(0)=*Σ*_{k}(*T*)=0, *Ψ*_{k}(0)=*Ψ*_{k}(*T*)=0. We refer to [19] for more details on these variations. We thus get the reduced critical action principle

The equations of motion are obtained by applying the variational principle (3.7). This is done by taking the variations
*Σ*_{k} (giving the equation for angular momentum) and *Ψ*_{k} (defining the linear momentum equation). We get
*k*=1,2,…,*N*, where we defined the vector
*k*=0 or *k*=*N*, then *μ*_{k}=0. In that formula, *p*_{n} are considered as fixed parameters. We will use them as time-varying control variables in §5.

### Remark 3.1

The net contribution of the gravity term to the angular momentum part of equations (3.9) vanishes, since for

### Remark 3.2 (functional derivatives)

The functional derivatives *δ*ℓ/*δ**ω*_{k}, *δ*ℓ/*δ**γ*_{k},

### (a) Including wind torque

The torque due to the wind on the element *n* is given from the torques due to wind acting on all the elements above *n*. For simplicity, we can assume that the velocity of the wind is the same for all elevations, and the module of the force of the wind on the *n*th element is *C*_{D}*ρS*_{n}*U*^{2}_{w}, with *ρ* being the density of air and *C*_{D} being the drag coefficient. This formula is adequate as a first-order approximation of the force and will be used in our simulations. The drag force due to the wind resistance is acting in the direction of the wind, i.e. the **E**_{2} direction. The other components of wind resistance, acting perpendicular to the wind when the *n*th torus is inclined, cause tower compression but do not contribute to the total torque on the tower and thus will be ignored. The torque due to the wind is computed as follows. In the spatial (steady) frame, the wind force on the *k*th torus is *U*_{w} is the wind speed, and the wind is assumed to be always in the **e**_{2} direction. The torque applied by this force, with respect to the centre of the *n*th torus (*n*<*k*), is (**r**_{k}−**r**_{n})×**F**_{k}. So the total torque in the spatial frame due to the tori *n*+1,*n*+2,…,*N*, with respect to the centre of the *n*th torus, is

The equations in the presence of the wind torque

### Remark 3.3 (conservation laws)

From the symmetry reduction approach, in the absence of wind but in the presence of gravity, one can derive the conservation of angular momentum about the **e**_{3}-axis. This can be achieved by considering the cotangent lift momentum maps of the system corresponding to the rotation about the **e**_{3}-axis [20], giving the conservation law for the quantity
*z*-axis.

### Remark 3.4 (Lagrangian reduction by symmetry)

The passage from the variational principle (3.2)–(3.7) is rigorously justified by applying the process of Lagrangian reduction by symmetry. In general, this process is described as follows.

Consider a Lie group action *g*,*q*)↦*Φ*_{g}(*q*) of the Lie group *G* on the configuration manifold *Q*. The action *Φ* induces an action *G* on the tangent bundle *TQ* of *Q*, defined by *g*∈*G*, *v*_{q}∈*TQ*, where *TΦ*_{g} denotes the tangent map (derivative) of *Φ*_{g}.

A Lagrangian *G*-invariant if *g*∈*G* and all *v*_{q}∈*TQ*. In this case, *L* induces a function *reduced Lagrangian*. When the action is free and proper, the process of Lagrangian reduction allows one to rewrite the Euler–Lagrange equation of *L*, uniquely in terms of *TQ*)/*G*. Roughly speaking, this is obtained by computing the variations of the reduced curve in (*TQ*)/*G*, induced by the variations *δq*(*t*) of the curve *q*(*t*)∈*Q*.

In our case, we have the Lie group *G*=*SO*(3)∋*A* acting on the configuration manifold **g**. Thus, in our case the reduced Euler–Lagrange equations are obtained by computing the variations of

### Remark 3.5 (incorporating air motion)

Let us also note that, in what follows, the dynamical effects of the moving air inside the tube will be neglected. This is for two reasons. First, for the smaller (several metres) size towers and correspondingly sized greenhouses realized in the experiments in §6, the motion of air is slow, and the forces produced by the combination of air motion and the tube's dynamics are small compared with the elastic, gravity and wind resistance forces. Second, for larger towers suitable for energy production, the air motion may become important, but it is impossible to predict without some knowledge of the greenhouse heating up the air; moreover, care must be taken in trusting numerical simulations because of the sensitivity of the problem to boundary conditions [10]. A detailed geometric theory of flexible tubes carrying fluid has been developed by two of the co-authors of this paper only recently [21] and includes interesting mathematical concepts by mixing the left-invariant tube motion, right-invariant fluid motion and volume constraint. Extension of this paper, especially the control theory part, to include the air motion inside is interesting but is also quite mathematically challenging and technical, and thus will be pursued in future studies.

## 4. Two-dimensional dynamics

### (a) Reduction of equations of motion in two dimensions

To make further progress, let us apply the theory derived in the previous section, which is valid for arbitrary Lagrangians, to the particular case of the tower consisting of inflatable tori. In that case, we shall assume that the main deformation of the tori comes from bending the tower and not mutual compression. The bending is assumed to be always in the plane which is collinear with the wind direction. We shall also make the non-essential assumption that the mass of the material of the tori is much less than the excessive mass coming from overpressurizing the torus itself. This leads to some simplifications of the equations, although more general cases can be considered easily. A more general case of heavy wall material and gravity is considered later in the discussion of pressure control.

All dynamics is assumed to occur in the (**E**_{1},**E**_{2}) plane with **E**_{1} being the undisturbed (spatial) vertical axis of the tower. In this framework, denoting the tilt of the axis for the *k*th torus from the vertical axis as *φ*_{k}, we have
**E**_{3} is orthogonal to the plane of motion, unlike the three-dimensional discussion above, where **E**_{3} denoted the vertical axis. The position of the centre of the *n*th torus is given by
*φ*_{k}|≪1 gives
** φ**=(

*φ*

_{1},…,

*φ*

_{N})

^{T}, the Lagrangian is given by

*I*

_{n}=(

*p*

_{0}+

*p*

_{n})

*I*

_{0},

*M*

_{n}=(

*p*

_{n}+

*p*

_{0})

*M*

_{0,n}and we have assumed that, due to the identical nature of the tori, the dependence of the potential energy on relative deformation is exactly the same for each torus and is given by the two-dimensional analogue of (2.7), i.e.

*N*×

*N*matrix

Applying Hamilton's principle yields the equations of motion
*n*=1 (bottom torus) and *n*=*N* (top torus).

In the presence of wind, the external torques added to the right-hand side of (4.6) are computed by simplifying expression (3.10) as follows. Assume that the deformations from equilibrium are small and so **r**_{k}≃**E**_{3}*z*_{k}, with *z*_{k} being a scalar height variable. In that approximation, using (3.10) we also get **E**_{3}×**E**_{2}=−**E**_{1}, we get the approximate, but easy to use, expression for the wind torque

### (b) Numerical simulations

We shall confine all numerical simulations to two-dimensional dynamics, where the motion is limited to the (**E**_{1},**E**_{2}) plane and all the rotations are about the **E**_{3} axis, as in §4*a*. In this case *d*_{n}=1 m, *q*_{n}=3 m, temperature *T*=300 K and overpressure *p*_{n}=10^{5} Pa=1 atm.

For further consideration, it is convenient to explicitly separate the dependence on the pressure *p*_{n} in each torus, which will be later treated as the control parameter. We define *U*_{0}:=*U*/*p*_{n}, the scaled energy having dimensions of volume. Then the scaled energy for the *n*th torus as well as corresponding derivatives are given by

We compute the dynamics given by (4.6) for a tower consisting of *N*=20 identical tori. For the purpose of regularization and error control, we use the following smoothing of the *h*_{n}∼|Δ*ϕ*|:
*a*, we present snapshots of the dynamics of the 20 tori. A video in the electronic supplementary material shows that the system undergoes chaotic, irregular motions.

## 5. Optimal control

### (a) General considerations from optimal control theory

Before considering the optimal control of flexible towers, we first recall some general facts about the Pontryagin maximum principle and explain the application to our problem.

#### (i) Pontryagin maximum principle

Given a state manifold *M* and a control set *x*(*t*)∈*M*, *u*(*t*)∈*U*, and *S*_{0}, *S*_{T}⊂*M* are given submanifolds. Here, *u*-dependent vector field on *M*. It is well known that a necessary condition for an extremal of this optimal control problem is provided by the Pontryagin maximum principle. Consider a real number *v*(*t*)∈*T***M*, having the physical meaning of the Lagrange multiplier enforcing the equations of motion. Then, consider the Pontryagin Hamiltonian *H*_{p0}(*x*,*v*,*u*):=〈*v*,*X*(*x*,*u*)〉−*p*_{0}*C*(*x*,*u*). If (*x*(*t*),*u*(*t*)) is an extremal, then by the Pontryagin maximum principle there exist *p*_{0}≥0 and a curve *v*(*t*)∈*T**_{x(t)}*M* such that
*t*. Here (*T*_{x(0)}*S*_{0})°⊂*T**_{x(0)}*M* denotes the annihilator of the tangent space *T*_{x(0)}*S*_{0} to *S*_{0} at *x*(0), similarly at *t*=*T*. It is clear that a necessary condition for *H*_{p0} to reach the maximum (if the maximum of *H*_{p0} is not on the boundary of the control set) is
*p*_{0}=0, then the solution is called abnormal. In this paper, we shall only consider normal solutions and take *p*_{0}=1. We refer to [22], theorem 12.4 for a proof of the Pontryagin maximum principle, together with an account of the appropriate regularity assumption assumed on the various curves. We refer for example to [23,24] for extended reviews on optimal control theory. For application to our problem, we shall make crucial use of the fact that conditions (5.4), (5.5) and (5.7) can be obtained as the critical points of the variational principle
*δx*, *δp*, *δu*, with *δx*(0)∈*T*_{x(0)}*S*_{0} and *δx*(*T*)∈*T*_{x(T)}*S*_{T}, and where we assumed *p*_{0}=1.

The optimal control problem (5.1) and (5.2) can be extended to the case where the terminal time *T* is not given but has to be determined. Both the Pontryagin maximum principle and the variational formulation extend to this case. We shall also make use of this fact later in the paper.

#### (ii) Symmetries in optimal control problems

Lie group symmetries, which will play an important role in our approach, have been highly used in control and optimal control problems, starting for example with [25–28].

As we shall see below, the optimal control problem for the flexible tower can be seen as a symmetry-reduced version of an optimal control problem of the above form. We shall thus derive a necessary condition on the extremal solution by using a symmetry-reduced version of the variational principle (5.8). Note that the differential equation constraint (5.2) will be of second order since it is given by controlled Euler–Lagrange equations. Various intrinsic geometric descriptions of symmetry and reduction in optimal control problems have been proposed [29–32] by using, for example, pre-symplectic formulation. In our approach, we shall perform the reduction by using exclusively variational principles. It would be interesting to study how our approach relates to the geometric setting for symmetries mentioned above as in [29–32]. Appendix A in the electronic supplementary material describes the symmetry-reduced exposition of the optimal control problem for a case when the centres of the tori are fixed in space, using for example a rigid rod and a system of wires. The general case can be considered as well, but is quite cumbersome and we do not treat it here.

One of the advantages of the flexible tower lies in the possibility to control the parameters for each individual torus as a function of time, in order to achieve the motion within a particular range by minimizing a certain cost function. In particular, the advantage of air-filled tori lies in the fact that one can inflate and deflate these elements over time, thus controlling the pressure. Note that this control mechanism is different from applying external control forces to each element. Changing of pressures affects the cost of relative deformation of the tori, but does not introduce extra forces *per se*. This parametric control of geometric objects is interesting from the mathematical point of view, and we will describe it here in some detail. For simplicity, we shall consider **r**_{n}−**r**_{n−1}=*const*. so there is no relative vertical deformation of the tori, and all deformations are strictly in the angular direction. A more general theory can be derived as well in a straightforward manner; however, the equations tend to be quite cumbersome. In reality, due to the wind, most of the potential energy will be caused by the relative tilt deformations. Also, for simplicity, this particular section of control neglects the effects of gravity.

### Remark 5.1 (on wind prediction)

In what follows, we shall explicitly assume that the wind direction and amplitude are known for a short period of time (minutes). This can be achieved by positioning anemometers at a certain distance from the tower. For example, an anemometer positioned at a distance of 1 km from the tower can predict the variations of wind with a typical speed of 10 m s^{−1} at the tower for a time period of about 1 min. This time interval will get shorter if the wind is stronger; however, we assume that a certain minimum prediction time is guaranteed.

### Remark 5.2 (on energy efficiency and pressure controls)

The usual formulae of work being the area under the graph in the (*p*,*V*) plane are not applicable here, since each torus by itself, or the whole assembly of tori, is not a thermodynamically closed system owing to the presence of actively working pumps. Moreover, it is likely that the relevant thermodynamics of the system will be irreversible. The work necessary for control will be determined primarily by the power and efficiency of the pumps, the pressure losses in the supply lines, etc., and thus will depend on a specific realization of the control mechanism. In any case, it is clear that the control will cost some energy and thus should be used sparingly for energy-producing applications. In reality, owing to the high stability of the tower to ambient winds, as described in §6, the control will probably be needed for only a relatively small fraction of time and should not be energetically taxing. One of the methods for a realization of the control mechanism could be having a high-pressure reservoir continuously charged by a relatively low-power pump. Air from this reservoir can be used to inflate the tori quickly. Another worthwhile consideration is whether some periodic control (at a relatively low power) can be used to suppress flow-induced vibrations of the chimney due to vortex shedding, but this matter is well beyond the scope of this study. It is also worth noting that the precise form of the cost function will again depend on the particular realization of the control mechanism. Thus, we have kept our discussion of the control procedure as general as possible.

### (b) Optimal control of arbitrary motions in two dimensions

While the results above can also be derived in the case when the motion of the centre of mass is present, we shall not pursue this here. The main difficulty lies in the corresponding kinetic energy expressions for the motion of the centre of mass, leading to the kinetic energy being a highly non-trivial function of *ω*_{k} and *μ*_{k}. Detailed expressions for the optimal control equations are thus quite cumbersome to present and, in our opinion, do not lead to better understanding of the problem.

As in §4, we write *p*_{k} now allowed to depend on time. We see that the matrix **p**=(*p*_{1},…,*p*_{n}). Thus, we write

The cost function is
*α*_{k}, *β*_{k}>0. The first term penalizes the deviation from the straight configuration, and the second term optimizes the pressure used in each torus close to the desirable value, so *F* is chosen as a convex function on *v*_{k} enforce the equations *E*_{n}=0, i.e. (5.9). The variations with respect to *p*_{n} give
*v*_{n}(0)=*v*_{n}(*T*)=0 as we will obtain below. Variations with respect to *φ*_{n} yield the evolution equation for *v*_{k}
**v**(0)=**v**(*T*)=0 from the arbitrariness of

Equations (5.9), (5.11) and (5.12) form a closed system of equations for the variables (**p**,**v**,** φ**). This, in principle, completes the optimal control for the cost function (5.10). However, such a cost function, while being simple, can be criticized on physical grounds. Indeed, as equation (5.11) shows, the pressure is assumed to change instantaneously according to other variables. In reality, however, the pressure change is provided by remotely controlled pumps which only specify the

*rate*of pressure change. Thus, an alternative approach is to consider the cost function

*α*

_{k},

*β*

_{k}>0, which optimizes the work done by pumps rather than pressure itself. In principle, a cost function depending on the combination of pressure and its rate of change can also be considered without difficulty. The system of motion to solve is then given by equations (5.9), (5.12) and the second-order equation for the pressure given by

**v**(0)=

**v**(

*T*)=0 and

### (c) Control of dynamics in a simplified case

In this section, we shall note that, for physical reasons, the inter-tori motion tends to generate fast motions and dissipate quickly. These dissipation effects are not included in our theory. From physical considerations, it is thus useful to consider the motions where the dissipation is minimal, i.e. the large-scale motions of the system where the tower bends ‘as a whole’. More precisely, we consider the analogue of self-similar solutions for the discrete height, assuming that *φ*_{n}(*t*)=*K*_{n}*ϕ*(*t*). It is interesting to note that if the potential energy is the power law, *U*_{0}(*ϕ*)∼|*ϕ*|^{q}, and *P*_{n}(*t*)=*P*_{n}=const., these solutions represent an exact reduction of the equations of motion, with a recursive relation for *P*_{n} that can be solved in some cases. Moreover, we shall also assume that the pressures *p*_{n}(*t*)=*G*_{n}*p*(*t*) are proportional to a given pressure; for example, the whole tower is inflated by a single pump through a sequence of connected chambers, as is indeed the case with real-life inflatable structures.

The Lagrangian of the whole system can be computed as follows. For simplicity, we shall approximate the axis of the bent tower by a circular arc, which leads to *φ*_{n}≃*nϕ*, i.e. Δ*φ*_{n}=*ϕ*, where *ϕ*(*t*) is the unknown variable. It is interesting to note that the experiments provide almost a linear growth of *φ*_{n} except for the top few tori, as will be shown in §6. In principle, we can substitute more complex shapes as *φ*_{n}=*K*_{n}*ϕ*, where *K*_{n} are given constants increasing with *n*. In this case, all the computations below will be equivalent except for dimensionless constants of order unity. For simplicity, we shall also assume that all tori have equal sizes *q*_{n}=*q*, *d*_{n}=*d*. The coordinates of the *n*th torus are
*M*_{s} and inertia *I*_{s}, respectively, and for a shell of given density *ρ*_{s} and thickness *h* they are computed as
*p*(*t*), atmospheric pressure as *p*_{0} and air density at the given temperature as *ρ*_{0}, the components of mass *M*_{a} and inertia *I*_{a} due to air weight are given by

The potential energy, as before, depends on the energy of relative deformations given by the relative angles, here assumed to be equal exactly to *ϕ* for all tori. We thus take, as before, *U*_{n}(*ϕ*)=*pU*(*ϕ*), where *U*(*ϕ*) is independent of the pressure. The total potential energy due to relative deformations is thus *NpU*(*ϕ*). It is also possible to incorporate the potential energy due to gravity as
*p*, namely *U*(*ϕ*,*p*)=*pU*_{1}(*ϕ*)+*U*_{0}(*ϕ*). The variational principle then reads
*C*(*ϕ*,*p*)=*αF*(*p*)+*βG*(*ϕ*) and for variations *δp*, *δv*, *δϕ* with *δϕ*(0)=*δϕ*(*T*)=0. Thus, the variables (*p*,*v*,*ϕ*) satisfy the following system of equations:
*ϕ*(0) and *ϕ*(*T*) are known and the final time *T* is given. In this case, the variational principle yields the boundary conditions *v*(0)=*v*(*T*)=0. As it turns out, other boundary conditions are possible, especially when the interval time *T* is not known and has to be determined. A discussion of these boundary conditions is given in the electronic supplementary material, appendix B. Before we proceed with numerical solutions of equations (5.23), we shall prove that in the absence of wind, *τ*^{w}=0, these equations possess a first integral for general functions *F*(*p*), *G*(*ϕ*) and *U*_{0,1}(*ϕ*). The procedure of finding this integral is quite non-trivial so we shall outline it in some detail. Multiplying the first equation by *p*, *v* and *ϕ* only
*p*,*v*,*ϕ*) space. The constant of integration *D* can be obtained using the initial conditions *ϕ*(0)=*ϕ*_{0}, *p*(0)=*p*_{0}, and *v*(0)=0 as
*p*-independent control function to have a physical meaning.

One could in principle solve problem (5.23) as a differential algebraic equation (DAE). However, one should keep in mind that the solution of (5.23) is behaving in a rather stiff manner, so special care must be taken when using standard DAE packages. The first integral (5.26) obtained above is useful for checking the solution, as the lack of conservation of this integral indicates corresponding inaccuracies in the numerics in the absence of wind. As it turns out, commercial computational packages with DAE capabilities yield uncontrolled loss of accuracy even for moderate times. This loss of accuracy is not apparent in the solution, and without the existence of integral (5.26), verifying the accuracy would have been difficult. In future work, we will try to construct a scheme that can provide a more accurate answer, as this will be important for the control procedure in the presence of wind. Here, we present an alternative scheme using the integral (5.26) directly. If we express *v*=*v*(*p*,*ϕ*) from (5.26) and substitute into the third equation of (5.23), we obtain an expression *p*(*t*) (first order) and angle *ϕ*(*t*) (second order). This equation does have singularities whenever

We then proceed as follows. We choose *T*_{0}>*T* sufficiently large, and some small values of *ϕ*(*T*_{0}), *ϕ*(*t*) and *p*(*T*_{0}) close to the desired equilibrium pressure, and integrate backwards to *t*=0. Select points *T*_{1} and *T*_{2}, 0<*T*_{1}<*T*_{2}<*T*, where *v*(*T*_{i})=0. We can choose *p*(*T*_{0}) and *ϕ*(*T*_{0}) so *v*(*T*_{0})=0 according to (5.26); in that case, we will have exactly *v*(*T*_{0})=0, and *T*_{2}=*T*_{0}, although it is not the only option. Shift the time variable so *T*=*T*_{2}−*T*_{1}. If *ϕ*(0)=*ϕ*_{0} is given, an iteration can be set up to find the initial conditions *ϕ*(*T*), *p*(*T*) so that the corresponding *ϕ*(0) has the desired value. Here, we simply consider the values of *ϕ*(0) and *ϕ*_{f}(*t*) of the initial value problem with no control by solving the second equation of (5.23) with *p*(*t*)=*const*. We had to use a combination of forward and backward integrations because of the presence of instabilities and singularities in the control system. In practice, it is probably more advantageous to provide a table of short-term control functions *p*_{ϕ}(*t*) based on the values of *ϕ*(0) and

All simulations are performed for *N*=20 rubber tori with dimensions *q*=1 m, *d*=0.2 m, wall thickness of 3.3 mm and excessive pressure of *p*_{e}=0.25 *atm*. The potential energy for this case is shown in figure 3: it has two minima corresponding to two stable equilibria, one being straight and the other one twisted to the side due to gravity. For small excessive pressures, the experimental apparatus also shows this instability to large perturbation due to the presence of gravity; see §6. The function *F*(*p*) was taken to be *G*(*ϕ*)=*ϕ*^{2}/2 and *α*=1 and *β*=1. As it turns out, it is possible to control the system even in this case, as the results in figure 3 illustrate.

## 6. Experiments

An experimental prototype of a solar chimney power plant was assembled and tested at the University of New Mexico, USA, on the roof of the mechanical engineering building to maximize exposure of the apparatus to wind. The chimney proper (3 m tall) was constructed from a stack of 40 motorcycle and scooter tyre inner tubes (figure 4) glued together following the proportional design scheme [10]. The apparatus was designed to serve as a proof of principle for the inflatable tower concept, and to provide performance benchmarks for numerical modelling under realistic insolation and wind conditions; thus, all the testing of wind resistance and dynamics was done outside. During the experiments, the chimney was exposed to a variety of weather conditions, including sustained winds exceeding 20 m s^{−1}. With higher values of overpressure, the tower demonstrated remarkable stability; unfortunately, deflections are very hard to measure in that case. Thus, in what follows, we concentrate on data collected at a modest overpressure of 0.17 atm, which is far below the maximum rated pressure for the inner tubes in excess of 30 psi—or about 2 atm. This lower overpressure leads to a variety of interesting behaviours. Below, we focus on the static deflections only.

The overall viability of the inflatable-chimney design has been demonstrated, with even the small prototype producing measurable power and continuing operation after sunset because thermal storage elements under the greenhouse (water bladders) continued to release heat accumulated during the day. Moreover, the chimney survived some very strong winds, with gusts up to 27 m s^{−1}.

While the focus of this preliminary study was on the proof of concept, some measurements of the tower behaviour under wind loading were also acquired. First, deflection of the tower under wind loading was found to be consistent with the theoretical assumptions, with the tower deformation distributed smoothly along the stack of inflatable tori. In table 1, we present the horizontal deflection of the top of the tower with several measured wind speeds.

We have also performed a detailed comparison of the tower deflection with wind speed in the static configuration corresponding to the medium wind speed (figure 4*b*). To do so, we assumed a constant wind speed throughout the vertical direction and introduced a friction term proportional to −*ϕ*′(*t*) in the right-hand side of (4.6), which made the system converge to a steady shape, independent of the friction coefficient. The result is shown in figure 5. All the sizes of the tori and overpressures were taken to be exactly equal to their experimental values. To match modelling with the experimental images, the horizontal and vertical scales on the right-hand side of the figure were chosen to fit the immobilized bottom torus. Thus, the only fitting parameter was the regularization constant *ε* in (4.9), which was chosen to be *ε*=0.06. The reason for the deviation of the scale at the top part of the experimental comparison is the perspective effect due to the fact that the deflection of the real tower is not fully two dimensional, and a small deflection of the tower in the direction normal to the plane of the photograph exists.

We shall also note that the steady states we have observed were not truly steady for the small overpressures in the experiment. First, along with the overall deflection, the tower did manifest modest-amplitude periodic motions, most probably due to vortex shedding and the inherent unsteadiness of atmospheric wind flow. Second, after deflection due to high winds for small overpressures, the prototype tower did not return to its original position, instead remaining bent (as in figure 4*c*), as predicted by the bi-stable case in figure 3. According to the theoretical calculations, the bi-stability disappears for high overpressures; on the other hand, if the towers get very tall, the overpressure required to remove the bistability is also large, thus requiring stronger, and more expensive materials and more powerful pumps. Finding the right balance between the requirements of stability and control is thus highly desirable and warrants further studies.

## 7. Conclusion

We have developed a mathematical theory describing the dynamics and optimal control procedure for an inflatable solar chimney design. The viability of this design has also been demonstrated on a small scale in experiment. Future work should include, on the theoretical side, consideration of fully three-dimensional dynamics and control. On the other hand, further studies of control should follow experimental designs of the control mechanism as they will influence the cost function used. The correct choice of cost function is thus influenced by both theory and experiment.

There are two factors to be taken into account. First, the cost of the control system must not exceed a certain percentage of the overall cost of the solar chimney facility. Second, the power consumption of the control system under normal operation must likewise be commensurately small in comparison with the output power of the facility. These constraints will impose limits on, for example, how fast the air can be pumped in and out of the tori. Some of these limitations can be overcome by careful design, e.g. by incorporating a reservoir of compressed air which can be used for occasional fast supply and slowly replenished during regular operation.

## Funding statement

V.P. and M.C. gratefully acknowledge the support provided by the University of Alberta and an NSERC Discovery grant. F.G.-B. was partially supported by a ‘Projet Incitatif de Recherche’ contract from the Ecole Normale Supérieure de Paris, and by the ANR project GEOMFLUID. The solar chimney prototype was constructed using funding from a University of New Mexico Research Allocation Council (RAC) grant under the leadership of Prof. A. A. Mammoli by a senior student team (D. Ziraksari, C. Delatorre, C. Been, A. Al Ansari, B. Mestas, T. Humphrey) as a part of a student senior design competition supervised by Prof. J. Wood.

## Acknowledgements

We acknowledge fruitful discussions with Professors D. D. Holm, P. Kevrikidis, A. Mammoli, T. S. Ratiu and E. Stechel.

- Received July 16, 2014.
- Accepted October 17, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.