Invariant manifolds in control problems

Invariant manifolds are useful tools for the investigation of nearly all nonlinear systems. Especially for the determination of stabilizing controls the center‐stable manifold characterizes the proper feedback controls.


Introduction
In this article it will be explained, how invariant manifolds can be used to determine feedback laws for control problems, such that the state variables approach a steady solution. The method is well established for equilibria, at which the linearization has stable and unstable eigenvalues. In this case it is usually sufficient to calculate the eigenvectors of the stable eigenvalues, but for higher precision also nonlinear expansions of the stable manifold might be required. We will focus on an application, where the eigenvalues of the linearized system lie on the imaginary axis and higher expansions are necessary to enforce the desired behaviour.

Stabilization of a tethered satellite
As a demonstration example we consider a tethered satellite, which is connected by a massless tether to a space station rotating around the earth along a Keplerian circle with constant angular speed (Fig. 1). By applying a tension force on the tether we try to steer the satellite from a nearby configuration to the steady local vertical position. The scaled nonlinear dynamics of the tether is given by ( [1]) [(θ + 3/2 sin 2ϑ) cos ψ − 2(θ − 1)ψ sin ψ] + 2˙ (θ − 1) cos ψ = 0, (1b) The variables ϑ and ψ denote the in-plane and out-of-plane angles and is the scaled tether length; the control variable u denotes the tension force with u = 3 in the steady vertical configuration. In order to find a smooth control, which minimizes the cost we apply Optimal Control theory ( [2]): We state the Hamiltonian where q = (ϑ, ψ, ,θ,ψ,˙ ) T , f (q, u) denotes the right hand side of the first order system corresponding to (1) and p are the adjoint variables. The optimal control u is obtained from the Maximum principle The adjoint variables satisfy the differential equationṡ At the start all values of the state variables q i are given: q i (0) = q 0 i . The controlled system should converge to the steady configuration The eigenvalues of the Jacobian of the Hamiltonian system at the equilibrium point z e (q e , p e ) with p e = 0 are displayed in Fig. 2. There are two pairs of real eigenvalues (black diamonds), a quadruple of complex eigenvalues (blue circles) and a double pair of imaginary eigenvalues ±2i, displayed by red squares. This purely imaginary pair occurs, because the out-ofplane oscillation ψ(t) is influenced by the parametric excitation term 2˙ ψ .

Pure out-of-plane oscillation
To leading order the dynamics of the out-of-plane oscillation is given by the parametrically excited oscillation equation Regarding v =˙ / as control variable, we state the optimal control problem subject to the first order systeṁ Introducing the Hamiltonian we find the optimal control and the corresponding Hamiltonian The Jacobian of the canonical equations It has a non-semisimple pair of purely imaginary eigenvalues ±2i and is already in real Jordan Normal Form. In [3] the Hamiltonian Hopf bifurcation, which occurs precisely with this critical matrix, was investigated and a new set of variables which are the invariant functions for the flow generated by the purely rotatory semi-simple part of A, was introduced. These functions satisfy the relation Using Normal Form theory ( [3,4]) it is possible to simplify the Hamiltonian (12) tõ Remark 2.1 In [3] it is shown, that by Normal Form simplification also the term Z 2 /8 could be eliminated. In this case one would need higher order terms to obtain the proper behaviour of the system, so we keep this term in the reduced Hamiltonian.
Since S is a first integral and we are looking for solutions converging to ψ = 0, we restrict our attention to the leaf S = 0. By (14) the reduced dynamics therefore takes place on the cone Z 2 = 4XY , and the trajectories of the system are the level curves ofH on this manifold, as displayed in  state variables ψ i oscillate with slowly decaying amplitude. The control function v oscillates with frequency 2ω; from the conditions S = 0 and Z < 0 it follows, that ψ and p oscillate in paraphase. In this example the dichotomy of the Mathieu equation in primary resonance is used to extinguish the out of plane oscillations and the stable manifold in the reduced system helped us to find the proper initial values for p i .

Treatment of the 3-dimensional system
Since the control v =˙ / of the simple problem is always positive, the tether length would increase indefinitely; furthermore the single control simultaneously influences the in-plane and out-of-plane oscillations. Since we found a two-dimensional center-stable manifold for the pure out-of-plane system, we expect to find a 3-dimensional stable manifold for the full system (1) to (5). As the formulas quickly become quite involved, the calculations are carried out using a symbolic algebra system ( [5]). www.gamm-proceedings.com In order to determine the dynamics of the out-of-plane subsystem and the center manifold of the system, we apply Normal Form calculations for the Hamilton function using the recursive algorithm explained in detail in [4]: Let be the initial power series expansion of the Hamiltonian, where ε is a scaling parameter; H 0 0 contains the quadratic parts of H, H 0 1 and H 0 2 the cubic and quartic ones, respectively. Let be the transformed Hamiltonian and an auxiliary function for the coordinate transform, respectively. The functions H i j satisfy the recursive relations where {H, W } denotes the Poisson bracket. At lowest order we obtain the relations By a proper choice of W 1 all cubic terms can be eliminated; the coefficient of Z 2 in H 2 0 is given by a = 1248/18725; the amplitude √ 2Y of the periodic orbit for the costate variables is therefore given by 4/a. The lowest order approximation of the center manifold is given by in Figs. 5 and 6. The variable˙ oscillates with twice the frequency ofψ. Fig. 6 demonstrates, that the numerical solution (solid curve) converges quickly to the approximate center manifold, depicted by the dashed line.