Convex computation of the region of attraction of polynomial control systems

We address the long-standing problem of computing the region of attraction (ROA) of a target set (e.g., a neighborhood of an equilibrium point) of a controlled nonlinear system with polynomial dynamics and semialgebraic state and input constraints. We show that the ROA can be computed by solving an infinite-dimensional convex linear programming (LP) problem over the space of measures. In turn, this problem can be solved approximately via a classical converging hierarchy of convex finite-dimensional linear matrix inequalities (LMIs). Our approach is genuinely primal in the sense that convexity of the problem of computing the ROA is an outcome of optimizing directly over system trajectories. The dual infinite-dimensional LP on nonnegative continuous functions (approximated by polynomial sum-of-squares) allows us to generate a hierarchy of semialgebraic outer approximations of the ROA at the price of solving a sequence of LMI problems with asymptotically vanishing conservatism. This sharply contrasts with the existing literature which follows an exclusively dual Lyapunov approach yielding either nonconvex bilinear matrix inequalities or conservative LMI conditions. The approach is simple and readily applicable as the outer approximations are the outcome of a single semidefinite program with no additional data required besides the problem description

Linear conic optimization for nonlinear optimal control

Infinite-dimensional linear conic formulations are described for nonlinear optimal control problems. The primal linear problem consists of finding occupation measures supported on optimal relaxed controlled trajectories, whereas the dual linear problem consists of finding the largest lower bound on the value function of the optimal control problem. Various approximation results relating the original optimal control problem and its linear conic formulations are developed. As illustrated by a couple of simple examples, these results are relevant in the context of finite-dimensional semidefinite programming relaxations used to approximate numerically the solutions of the infinite-dimensional linear conic problems.Comment: Submitted for possible inclusion as a contributed chapter in S. Ahmed, M. Anjos, T. Terlaky (Editors). Advances and Trends in Optimization with Engineering Applications. MOS-SIAM series, SIAM, Philadelphi

Minimum Restraint Functions for unbounded dynamics: general and control-polynomial systems

We consider an exit-time minimum problem with a running cost, $l\geq 0$ and unbounded controls. The occurrence of points where $l=0$ can be regarded as a transversality loss. Furthermore, since controls range over unbounded sets, the family of admissible trajectories may lack important compactness properties. In the first part of the paper we show that the existence of a $p_0$-minimum restraint function provides not only global asymptotic controllability (despite non-transversality) but also a state-dependent upper bound for the value function (provided $p_0>0$). This extends to unbounded dynamics a former result which heavily relied on the compactness of the control set. In the second part of the paper we apply the general result to the case when the system is polynomial in the control variable. Some elementary, algebraic, properties of the convex hull of vector-valued polynomials' ranges allow some simplifications of the main result, in terms of either near-affine-control systems or reduction to weak subsystems for the original dynamics.Comment: arXiv admin note: text overlap with arXiv:1503.0344

Almost Sure Stabilization for Adaptive Controls of Regime-switching LQ Systems with A Hidden Markov Chain

This work is devoted to the almost sure stabilization of adaptive control systems that involve an unknown Markov chain. The control system displays continuous dynamics represented by differential equations and discrete events given by a hidden Markov chain. Different from previous work on stabilization of adaptive controlled systems with a hidden Markov chain, where average criteria were considered, this work focuses on the almost sure stabilization or sample path stabilization of the underlying processes. Under simple conditions, it is shown that as long as the feedback controls have linear growth in the continuous component, the resulting process is regular. Moreover, by appropriate choice of the Lyapunov functions, it is shown that the adaptive system is stabilizable almost surely. As a by-product, it is also established that the controlled process is positive recurrent

A new computer method for temperature measurement based on an optimal control problem

A new computer method to measure extreme temperatures is presented. The method reduces the measurement of the unknown temperature to the solving of an optimal control problem, using a numerical computer. Based on this method, a new device for temperature measurement is built. It consists of a hardware part that includes some standard temperature sensors and it also has a software section.\ud The problem of temperature measurement, according to this new method, is mathematically modelled by means of the one-dimensional heat equation, with boundary and initial conditions, describing the heat transfer through the device.\ud \ud The principal hardware component of the new device is a rod. The variation of the temperature which is produced near one end of the rod is determined using some temperature measurements in the other end of the rod and the new computer method which is described in this work.\ud \ud This device works as an attenuator of high temperatures and as an amplifier of low temperatures. In fact, it realizes an extension of the standard working range of temperature sensors at very high and very low values.\ud \ud The mathematical model of the device and the computer method are explained in detail and some possible practical implementations and a collection of simulations are also presented

Energy-based trajectory tracking and vibration control for multilink highly flexible manipulators

In this paper, a discrete model is adopted, as proposed by Hencky for elastica based on rigid bars and lumped rotational springs, to design the control of a lightweight planar manipulator with multiple highly flexible links. This model is particularly suited to deal with nonlinear equations of motion as those associated with multilink robot arms, because it does not include any simplification due to linearization, as in the assumed modes method. The aim of the control is to track a trajectory of the end effector of the robot arm, without the onset of vibrations. To this end, an energy-based method is proposed. Numerical simulations show the effectiveness of the presented approach