Continuous second order sliding mode based finite time tracking of a fully actuated biped robot

International audienceA second order sliding mode controller is modified to form a continuous homogeneous controller. Uniform finite time stability is proved by extending the homogeneity principle of discontinuous systems to the continuous case with uniformly decaying piece-wise continuous nonhomogeneous disturbances. The modified controller is then utilised to track reference trajectories for all the joints of a fully actuated biped robot where the joint torque is modeled as the control input. The modified controller ensures the attainment of a finite settling time between two successive impacts. The main contribution of the paper is to provide straightforward and realizable engineering guidelines for reference trajectory generation and for tuning a robust finite time controller in order to achieve stable gait of a biped in the presence of an external force disturbance. Such a disturbance has destabilising effects in both continuous and impact phases. Numerical simulations of a biped robot are shown to support the theoretical results

Continuous Uniform Finite Time Stabilization of Planar Controllable Systems

Continuous homogeneous controllers are utilized in a full state feedback setting for the uniform finite time stabilization of a perturbed double integrator in the presence of uniformly decaying piecewise continuous disturbances. Semiglobal strong $\mathcal{C}^1$ Lyapunov functions are identified to establish uniform asymptotic stability of the closed-loop planar system. Uniform finite time stability is then proved by extending the homogeneity principle of discontinuous systems to the continuous case with uniformly decaying piecewise continuous nonhomogeneous disturbances. A finite upper bound on the settling time is also computed. The results extend the existing literature on homogeneity and finite time stability by both presenting uniform finite time stabilization and dealing with a broader class of nonhomogeneous disturbances for planar controllable systems while also proposing a new class of homogeneous continuous controllers

Modelling and finite time stability analysis of psoriasis pathogenesis

A new systems model of psoriasis is presented and analysed from the perspective of control theory. Cytokines are treated as actuators to the plant model that govern the cell population under the reasonable assumption that cytokine dynamics are faster than the cell population dynamics. The analysis of various equilibria is undertaken based on singular perturbation theory. Finite time stability and stabilisation has been studied in various engineering applications where the principal paradigm uses non-Lipschitz functions of the states. A comprehensive study of the finite time stability properties of the proposed psoriasis dynamics is carried out. It is demonstrated that the dynamics are finite time convergent to certain equilibrium points rather than asymptotically or exponentially convergent. This feature of finite time convergence motivates the development of a modi?ed version of the Michaelis-Menten function, frequently used in biology. This framework is used to model cytokines as fast finite time actuators

Generalized Model Predictive Static Programming and Angle-Constrained Guidance of Air-to-Ground Missiles

A new generalized model predictive static programming technique is presented for rapidly solving a class of finite-horizon nonlinear optimal control problems with hard terminal constraints. Two key features for its high computational efficiency include one-time backward integration of a small-dimensional weighting matrix dynamics, followed bya static optimization formulation that requires only a static Lagrange multiplier to update the control history. It turns out that under Euler integration and rectangular approximation of finite integrals it is equivalent to the existing model predictive static programming technique. In addition to the benchmark double integrator problem, usefulness of the proposed technique is demonstrated by solving a three-dimensional angle-constrained guidance problem for an air-to-ground missile, which demands that the missile must meet constraints on both azimuth and elevation angles at the impact point in addition to achieving near-zero miss distance, while minimizing the lateral acceleration demand throughout its flight path. Simulation studies include maneuvering ground targets along with a first-order autopilot lag. Comparison studies with classical augmented proportional navigation guidance and modern general explicit guidance lead to the conclusion that the proposed guidance is superior to both and has a larger capture region as well

Generalized model predictive static programming and its application to 3D impact angle constrained guidance of air-to-surface missiles

A new `generalized model predictive static programming (G-MPSP)' technique is presented in this paper in the continuous time framework for rapidly solving a class of finite-horizon nonlinear optimal control problems with hard terminal constraints. A key feature of the technique is backward propagation of a small-dimensional weight matrix dynamics, using which the control history gets updated. This feature, as well as the fact that it leads to a static optimization problem, are the reasons for its high computational efficiency. It has been shown that under Euler integration, it is equivalent to the existing model predictive static programming technique, which operates on a discrete-time approximation of the problem. Performance of the proposed technique is demonstrated by solving a challenging three-dimensional impact angle constrained missile guidance problem. The problem demands that the missile must meet constraints on both azimuth and elevation angles in addition to achieving near zero miss distance, while minimizing the lateral acceleration demand throughout its flight path. Both stationary and maneuvering ground targets are considered in the simulation studies. Effectiveness of the proposed guidance has been verified by considering first order autopilot lag as well as various target maneuvers

Finite time stabilization of a perturbed double integrator with unilateral constraints

A discontinuous second order sliding mode (twisting) controller is utilised in a full state feedback setting for the finite time stabilization of a perturbed double integrator in the presence of both a unilateral constraint and uniformly bounded persisting disturbances. The unilateral constraint involves rigid body inelastic impacts causing jumps in one of the state variables. Firstly, a non-smooth state transformation is employed to transform the unilaterally constrained system into a jump-free system. The transformed system is shown to be a switched homogeneous system with negative homogeneity degree where the solutions are welldefined. Secondly, a non-smooth Lyapunov function is identified to establish uniform asymptotic stability of the transformed system. The global, uniform, finite time stability is then proved by utilising the homogeneity principle of switched systems. The novelty lies in achieving finite time stabilization in the presence of jumps in one of the states without the need to analyze the Lyapunov function at the jump instants. The proposed results are of theoretical significance as they bridge non-smooth Lyapunov analysis, quasi-homogeneity and finite time stability for a class of impact mechanical systems