Sum-of-Squares Flight Control Synthesis for Deep-Stall Recovery

Cunis, Torbjørn; Condomines, Jean-Philippe; Burlion, Laurent

doi:10.2514/1.g004753

Cited by 12 publications

(12 citation statements)

References 27 publications

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“…The problem of computing attractor sets is related to the problem of certifying the asymptotic stability of equilibrium points of an ODE (1); since certifying A * = {0} is an attractor set of an ODE (1) is equivalent to showing the asymptotic stability of the ODE (1) about 0 ∈ R n . The use of SOS Lyapunov functions to certify the asymptotic stability of equilibrium points of an ODE (1) has been well treated in the literature [13], [14], [15], [16], [17], [18], [19].…”

Section: Introductionmentioning

confidence: 99%

A Converse Sum of Squares Lyapunov Function for Outer Approximation of Minimal Attractor Sets of Nonlinear Systems

Jones¹,

Peet²

2021

Preprint

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Many dynamical systems described by nonlinear ODEs are unstable. Their associated solutions do not converge towards an equilibrium point, but rather converge towards some invariant subset of the state space called an attractor set. For a given ODE, in general, the existence, shape and structure of the attractor sets of the ODE are unknown. Fortunately, the sublevel sets of Lyapunov functions can provide bounds on the attractor sets of ODEs. In this paper we propose a new Lyapunov characterization of attractor sets that is well suited to the problem of finding the minimal attractor set. We show our Lyapunov characterization is non-conservative even when restricted to Sum-of-Squares (SOS) Lyapunov functions. Given these results, we propose a SOS programming problem based on determinant maximization that yields an SOS Lyapunov function whose 1-sublevel set has minimal volume, is an attractor set itself, and provides an optimal outer approximation of the minimal attractor set of the ODE. Several numerical examples are presented including the Lorenz attractor and Van-der-Pol oscillator.

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Section: Introductionmentioning

confidence: 99%

A Converse Sum of Squares Lyapunov Function for Outer Approximation of Minimal Attractor Sets of Nonlinear Systems

Jones¹,

Peet²

2021

Preprint

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“…The first converse Lyapunov function (given in Eqn. (10)) is a special case of those first found in [22] that have the form…”

Section: A Globally Lipschitz Continuous Converse Lyapunov Function T...mentioning

confidence: 94%

“…Over the years, many SOS optimization problems have been proposed for ROA estimation [6], [7], [8]. Recently in [10], SOS was used to estimate the region of attraction of an uncrewed aircraft; in [11] an SOS based algorithm was proposed to construct a rational Lyapunov function that yields an estimate of the ROA; in [12] a recursive procedure for constructing the polynomial Lyapunov functions was proposed.…”

Section: Introductionmentioning

confidence: 99%

Converse Lyapunov Functions and Converging Inner Approximations to Maximal Regions of Attraction of Nonlinear Systems

Jones¹,

Peet²

2021

Preprint

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This paper considers the problem of approximating the "maximal" region of attraction (the set that contains all asymptotically stable sets) of any given set of locally exponentially stable nonlinear Ordinary Differential Equations (ODEs) with a sufficiently smooth vector field. Given a locally exponential stable ODE with a differentiable vector field, we show that there exists a globally Lipschitz continuous converse Lyapunov function whose 1-sublevel set is equal to the maximal region of attraction of the ODE. We then propose a sequence of d-degree Sum-of-Squares (SOS) programming problems that yields a sequence of polynomials that converges to our proposed converse Lyapunov function uniformly from above in the L 1 norm. We show that each member of the sequence of 1-sublevel sets of the polynomial solutions to our proposed sequence of SOS programming problems are certifiably contained inside the maximal region of attraction of the ODE, and moreover, we show that this sequence of sublevel sets converges to the maximal region of attraction of the ODE with respect to the volume metric. We provide numerical examples of estimations of the maximal region of attraction for the Van der Pol oscillator and a three dimensional servomechanism.

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“…These studies either involve simplified flight dynamics models [11][12][13][14][15] or empirical methods [10,16], making it hard to determine a safe and consistent procedure to guarantee recovery. Moreover, recent research into advanced landing techniques for small unmanned aerial vehicles involves deliberately bringing the aircraft into a deep stall to minimise the landing distance [17][18][19]. These developments emphasise the need to improve our understanding of the flight characteristics in the deep stall regime.…”

Section: Introductionmentioning

confidence: 99%

Analysing dynamic deep stall recovery using a nonlinear frequency approach

2022

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Based on bifurcation theory, nonlinear frequency response analysis is a recent development in the field of flight dynamics studies. Here, we consider how this method can be used to inform us on how to devise the control input such that the system transitions from an undesirable equilibrium solution—an aircraft deep stall solution in our case—to a desirable solution. We show that it is still possible to induce a large-amplitude oscillation via harmonic forcing of the pitch control device and escape the otherwise unrecoverable deep stall, despite very little control power available in such a high angle-of-attack flight condition. The forcing frequencies that excite these resonances are reflected as asymptotically unstable solutions using bifurcation analysis and Floquet theory. Due to the softening behaviour observed in the frequency response, these unstable (divergent) solutions have slightly lower frequencies than the value predicted using linear analysis. Subharmonic resonances are also detected, which are reflected in the time-domain unforced responses. These nonlinear phenomena show strong dependency on the forcing/perturbation amplitude and result in complex dynamics that can impede recovery if the existing procedures are followed. The proposed method is shown to be a useful tool for nonlinear flight dynamics analysis as well as to complement the rather thin literature on deep stall analysis—a topic of relevance for recent research on unconventional landing techniques in unmanned aerial vehicles. A full description of the aircraft model used, the unstable F-16 fighter jet, is provided in the appendix.

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Sum-of-Squares Flight Control Synthesis for Deep-Stall Recovery

Cited by 12 publications

References 27 publications

A Converse Sum of Squares Lyapunov Function for Outer Approximation of Minimal Attractor Sets of Nonlinear Systems

A Converse Sum of Squares Lyapunov Function for Outer Approximation of Minimal Attractor Sets of Nonlinear Systems

Converse Lyapunov Functions and Converging Inner Approximations to Maximal Regions of Attraction of Nonlinear Systems

Analysing dynamic deep stall recovery using a nonlinear frequency approach

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