Moment-Sum-Of-Squares Approach For Fast Risk Estimation In Uncertain Environments

In this paper, we address the risk estimation problem where one aims at estimating the probability of violation of safety constraints for a robot in the presence of bounded uncertainties with arbitrary probability distributions. In this problem, an unsafe set is described by level sets of polynomials that is, in general, a non-convex set. Uncertainty arises due to the probabilistic parameters of the unsafe set and probabilistic states of the robot. To solve this problem, we use a moment-based representation of probability distributions. We describe upper and lower bounds of the risk in terms of a linear weighted sum of the moments. Weights are coefficients of a univariate Chebyshev polynomial obtained by solving a sum-of-squares optimization problem in the offline step. Hence, given a finite number of moments of probability distributions, risk can be estimated in real-time. We demonstrate the performance of the provided approach by solving probabilistic collision checking problems where we aim to find the probability of collision of a robot with a non-convex obstacle in the presence of probabilistic uncertainties in the location of the robot and size, location, and geometry of the obstacle.Comment: 57th IEEE Conference on Decision and Control 201

A Probabilistic Approach to Robust Optimal Experiment Design with Chance Constraints

Accurate estimation of parameters is paramount in developing high-fidelity models for complex dynamical systems. Model-based optimal experiment design (OED) approaches enable systematic design of dynamic experiments to generate input-output data sets with high information content for parameter estimation. Standard OED approaches however face two challenges: (i) experiment design under incomplete system information due to unknown true parameters, which usually requires many iterations of OED; (ii) incapability of systematically accounting for the inherent uncertainties of complex systems, which can lead to diminished effectiveness of the designed optimal excitation signal as well as violation of system constraints. This paper presents a robust OED approach for nonlinear systems with arbitrarily-shaped time-invariant probabilistic uncertainties. Polynomial chaos is used for efficient uncertainty propagation. The distinct feature of the robust OED approach is the inclusion of chance constraints to ensure constraint satisfaction in a stochastic setting. The presented approach is demonstrated by optimal experimental design for the JAK-STAT5 signaling pathway that regulates various cellular processes in a biological cell.Comment: Submitted to ADCHEM 201

A unified framework for solving a general class of conditional and robust set-membership estimation problems

In this paper we present a unified framework for solving a general class of problems arising in the context of set-membership estimation/identification theory. More precisely, the paper aims at providing an original approach for the computation of optimal conditional and robust projection estimates in a nonlinear estimation setting where the operator relating the data and the parameter to be estimated is assumed to be a generic multivariate polynomial function and the uncertainties affecting the data are assumed to belong to semialgebraic sets. By noticing that the computation of both the conditional and the robust projection optimal estimators requires the solution to min-max optimization problems that share the same structure, we propose a unified two-stage approach based on semidefinite-relaxation techniques for solving such estimation problems. The key idea of the proposed procedure is to recognize that the optimal functional of the inner optimization problems can be approximated to any desired precision by a multivariate polynomial function by suitably exploiting recently proposed results in the field of parametric optimization. Two simulation examples are reported to show the effectiveness of the proposed approach.Comment: Accpeted for publication in the IEEE Transactions on Automatic Control (2014

Polynomial mechanics and optimal control

We describe a new algorithm for trajectory optimization of mechanical systems. Our method combines pseudo-spectral methods for function approximation with variational discretization schemes that exactly preserve conserved mechanical quantities such as momentum. We thus obtain a global discretization of the Lagrange-d'Alembert variational principle using pseudo-spectral methods. Our proposed scheme inherits the numerical convergence characteristics of spectral methods, yet preserves momentum-conservation and symplecticity after discretization. We compare this algorithm against two other established methods for two examples of underactuated mechanical systems; minimum-effort swing-up of a two-link and a three-link acrobot.Comment: Final version to EC

Accuracy of Numerical Solution to Dynamic Programming Models

Dynamic programming models with continuous state and control variables are solved approximately using numerical methods in most applications. We develop a method for measuring the accuracy of numerical solution of stochastic dynamic programming models. Using this method, we compare the accuracy of various interpolation schemes. As expected, the results show that the accuracy improves as number of nodes is increased. Comparison of Chebyshev and linear spline indicates that the linear spline may give higher maximum absolute error than Chebyshev, however, the overall performance of spline interpolation is better than Chebyshev interpolation for non-smooth functions. Two-stage grid search method of optimization is developed and examined with accuracy analysis. The results show that this method is more efficient and accurate. Accuracy is also examined by allocating a different number of nodes for each dimension. The results show that a change in node configuration may yield a more efficient and accurate solution.Research Methods/ Statistical Methods,

Computing Markov-perfect optimal policies in business-cycle models

Time inconsistency is an essential feature of many policy problems. This paper presents and compares three methods for computing Markov-perfect optimal policies in stochastic nonlinear business cycle models. The methods considered include value function iteration, generalized Euler equations, and parameterized shadow prices. In the context of a business cycle model in which a fiscal authority chooses government spending and income taxation optimally, although lacking the ability to commit, we show that the solutions obtained using value function iteration and generalized Euler equations are somewhat more accurate than that obtained using parameterized shadow prices. Among these three methods, we show that value function iteration can be applied easily, even to environments that include a risk-sensitive fiscal authority and/or inequality constraints on government spending. We show that the risk-sensitive fiscal authority lowers government spending and income taxation, reducing the disincentive to accumulate wealth that households face