A class of bounded continuous time-invariant finite-time stabilizing feedback laws is given for the double integrator. Lyapunov theory is used to prove finite-time convergence. For the rotational double integrator, these controllers are modified to obtain finite-time-stabilizing feedback that avoid "unwinding".

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Continuous finite-time stabilization of the translational and rotational double integrators

Ambient energy harvesting has been in recent years the recurring object of a number of research efforts aimed at providing an autonomous solution to the powering of small-scale electronic mobile devices. Among the different solutions, vibration energy harvesting has played a major role due to the almost universal presence of mechanical vibrations. Here we propose a new method based on the exploitation of the dynamical features of stochastic nonlinear oscillators. Such a method is shown to outperform standard linear oscillators and to overcome some of the most severe limitations of present approaches. We demonstrate the superior performances of this method by applying it to piezoelectric energy harvesting from ambient vibration.

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Nonlinear energy harvesting.

We present two constructions of controllers that globally stabilize linear systems subject to control saturation. We allow essentially arbitrary saturation functions. The only conditions imposed on the system are the obvious necessary ones, namely that no eigenvalues of the uncontrolled system have positive real part and that the standard stabilizability rank condition hold. One of the constructions is in terms of a "neural-network type" one-hidden layer architecture, while the other one is in terms of cascades of linear maps and saturations. >

A general result on the stabilization of linear systems using bounded controls

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Statistical Theory of Communication

A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: THE GENERAL THEORY OF ORBITS

It is shown that when the plant of a relay control system consists of a pure integrators, with n ≥ 3, and the control signal proceding the relay is a linear combination of state coordinates, the system is unstable in-the-large for all values of the controller coefficients. The same result holds if the relay is replaced by a. saturator. The method used is to prove that (except for a degenerate case) state trajectories exist which go to infinity. This method enables more general systems to be treated than those discussed in a previous paper, which demonstrated the existence of a periodic solution. Plants not consisting of pure integrators are also considered. The question of when it is worthwhile to incorporate non-linearities in the controller is discussed.

In-the-large stability of relay and saturating control systems with linear controllers

Nonlinear systems disturbed by Gaussian white noises (or by signals obtained from Gaussian white noises) can sometimes be analysed by setting up and solving the Fokker–Planck equation for the probability density in state space. In the present paper a simplified derivation of the Fokker–Planck equation is given. The uniqueness of the steady-state solution is discussed. Steady-state solutions are obtained for certain classes of system. These solutions correspond to or slightly generalize the Maxwell–Boltzmann distribution which is well known in classical statistical mechanics

Analysis of nonlinear stochastic systems by means of the Fokker–Planck equation†

When the plant contains a pure delay and the control signal is subject to a saturation constraint, the optimal controller and optimal performance can often be calculated, provided the corresponding results in the delay-free case are known Several examples of the technique are given in the present paper Some sub-optimal controllers are also discussed

Optimal nonlinear control of systems with pure delay

The effect of plant parameter variations on systems with nominally time-optimal feedback controllers is investigated. It is assumed that the plant parameter remains constant and that the time-optimal controller calculated for the nominal plant parameter is used. For higher-order systems remarkably small deviations of the plant parameter from its nominal value are found to yield instability. A sub-optimal controller design which results in sliding motion ensures stability and yields acceptable settling time for moderato parameter variations

The sensitivity of nominally time-optimal control systems to parameter variation†

The paper deals principally with a control system which has a plant consisting of three integrators with a saturable control input. The command signal input is a step plus a ramp plus a parabola. The switching surface which minimizes integral-square-error is found, partly algebraically and partly numerically, by methods which start from Pontryagin's maximum principle. All optimum trajectories have an infinite number of switches before the origin is reached, except for two trajectories which have no switches. Some optimum trajectories have the property that the ratio of any two successive switching intervals is constant. All other optimum trajectories (apart from the two exceptional cases) converge rapidly towards these constant ratio trajectories. Thus, when finding optimum trajectories by backwards numerical computation from near the origin of the Hamiltonian system of equations, it is necessary to adjust the ‘initial’ values to be, not only small, but also close to a constant ratio trajectory. ...

A. T. Fuller

Papers

In-the-large stability of relay and saturating control systems with linear controllers

Analysis of nonlinear stochastic systems by means of the Fokker–Planck equation†

Optimal nonlinear control of systems with pure delay

The sensitivity of nominally time-optimal control systems to parameter variation†

Minimization of Integral-square-error for Non-Linear Control Systems of Third and Higher Order