Probability-guaranteed set-membership state estimation for polynomially uncertain linear time-invariant systems

2018 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting /republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other worksConventional deterministic set-membership (SM) estimation is limited to unknown-but-bounded uncertainties. In order to exploit distributional information of probabilistic uncertainties, a probability-guaranteed SM state estimation approach is proposed for uncertain linear time-invariant systems. This approach takes into account polynomial dependence on probabilistic uncertain parameters as well as additive stochastic noises. The purpose is to compute, at each time instant, a bounded set that contains the actual state with a guaranteed probability. The proposed approach relies on the extended form of an observer representation over a sliding window. For the offline observer synthesis, a polynomial-chaos-based method is proposed to minimize the averaged H2 estimation performance with respect to probabilistic uncertain parameters. It explicitly accounts for the polynomial uncertainty structure, whilst most literature relies on conservative affine or polytopic overbounding. Online state estimation restructures the extended observer form, and constructs a Gaussian mixture model to approximate the state distribution. This enables computationally efficient ellipsoidal calculus to derive SM estimates with a predefined confidence level. The proposed approach preserves time invariance of the uncertain parameters and fully exploits the polynomial uncertainty structure, to achieve tighter SM bounds. This improvement is illustrated by a numerical example with a comparison to a deterministic zonotopic method.Peer ReviewedPostprint (author's final draft

Stochastic Nonlinear Model Predictive Control with Efficient Sample Approximation of Chance Constraints

This paper presents a stochastic model predictive control approach for nonlinear systems subject to time-invariant probabilistic uncertainties in model parameters and initial conditions. The stochastic optimal control problem entails a cost function in terms of expected values and higher moments of the states, and chance constraints that ensure probabilistic constraint satisfaction. The generalized polynomial chaos framework is used to propagate the time-invariant stochastic uncertainties through the nonlinear system dynamics, and to efficiently sample from the probability densities of the states to approximate the satisfaction probability of the chance constraints. To increase computational efficiency by avoiding excessive sampling, a statistical analysis is proposed to systematically determine a-priori the least conservative constraint tightening required at a given sample size to guarantee a desired feasibility probability of the sample-approximated chance constraint optimization problem. In addition, a method is presented for sample-based approximation of the analytic gradients of the chance constraints, which increases the optimization efficiency significantly. The proposed stochastic nonlinear model predictive control approach is applicable to a broad class of nonlinear systems with the sufficient condition that each term is analytic with respect to the states, and separable with respect to the inputs, states and parameters. The closed-loop performance of the proposed approach is evaluated using the Williams-Otto reactor with seven states, and ten uncertain parameters and initial conditions. The results demonstrate the efficiency of the approach for real-time stochastic model predictive control and its capability to systematically account for probabilistic uncertainties in contrast to a nonlinear model predictive control approaches.Comment: Submitted to Journal of Process Contro

Probabilistic Robustness Analysis with Aerospace Applications

This thesis develops theoretical and computational methods for the robustness analysis of uncertain systems. The considered systems are linearized and depend rationally on random parameters with an associated probability distribution. The uncertainty is tackled by applying a polynomial chaos expansion (PCE), a series expansion for random variables similar to the well-known Fourier series for periodic time signals. We consider the linear perturbations around a system's operating point, i.e., reference trajectory, both from a probabilistic and worst-case point of view. A chief contribution is the polynomial chaos series expansion of uncertain linear systems in linear fractional representation (LFR). This leads to significant computational benefits when analyzing the probabilistic perturbations around a system's reference trajectory. The series expansion of uncertain interconnections in LFR further delivers important theoretical insights. For instance, it is shown that the PCE of rational parameter-dependent linear systems in LFR is equivalent to applying Gaussian quadrature for numerical integration. We further approximate the worst-case performance of uncertain linear systems with respect to quadratic performance metrics. This is achieved by approximately solving the underlying parametric Riccati differential equation after applying a polynomial chaos series expansion. The utility of the proposed probabilistic robustness analysis is demonstrated on the example of an industry-sized autolanding system for an Airbus A330 aircraft. Mean and standard deviation of the stochastic perturbations are quantified efficiently by applying a PCE to a linearization of the system along the nominal approach trajectory. Random uncertainty in the aerodynamic coefficients and mass parameters are considered, as well as atmospheric turbulence and static wind shear. The approximate worst-case analysis is compared with Monte Carlo simulations of the complete nonlinear model. The methods proposed throughout the thesis rapidly provide analysis results in good agreement with the Monte Carlo benchmark, at reduced computational cost

Polynomial Chaos Approximation of the Quadratic Performance of Uncertain Time-Varying Linear Systems

This paper presents a novel approach to robustness analysis based on quadratic performance metrics of uncertain time-varying systems. The considered time-varying systems are assumed to be linear and defined over a finite time horizon. The uncertainties are described in the form of real-valued random variables with a known probability distribution. The quadratic performance problem for this class of systems can be posed as a parametric Riccati differential equation (RDE). A new approach based on polynomial chaos expansion is proposed that can approximately solve the resulting parametric RDE and, thus, provide an approximation of the quadratic performance. Moreover, it is shown that for a zeroth order expansion this approximation is in fact a lower bound to the actual quadratic performance. The effectiveness of the approach is demonstrated on the example of a worst-case performance analysis of a space launcher during its atmospheric ascent

Numerical Methods for Uncertainty Quantification in Gas Network Simulation

When modeling the gas flow through a network, some elements such as pressure control valves can cause kinks in the solution. In this thesis we modify the method of simplex stochastic collocation such that it is applicable to functions with kinks. First, we derive a system of partial differential and algebraic equations describing the gas flow through different elements of a network. Restricting the gas flow to an isothermal and stationary one, the solution can be determined analytically. After introducing some common methods for the forward propagation of uncertainty, we present the method of simplex stochastic collocation to approximate functions of uncertain parameters. By utilizing the information whether a pressure regulator is active or not in the current simulation, we improve the method such that we can prove algebraic convergence rates for functions with kinks. Moreover, we derive two new error estimators for an adaptive refinement and show that multiple refinements are possible. Conclusively, several numerical results for a real gas network are presented and compared with standard methods to demonstrate the significantly better convergence results

Probabilistic Methods for Model Validation

This dissertation develops a probabilistic method for validation and verification (V&V) of uncertain nonlinear systems. Existing systems-control literature on model and controller V&V either deal with linear systems with norm-bounded uncertainties,or consider nonlinear systems in set-based and moment based framework. These existing methods deal with model invalidation or falsification, rather than assessing the quality of a model with respect to measured data. In this dissertation, an axiomatic framework for model validation is proposed in probabilistically relaxed sense, that instead of simply invalidating a model, seeks to quantify the "degree of validation". To develop this framework, novel algorithms for uncertainty propagation have been proposed for both deterministic and stochastic nonlinear systems in continuous time. For the deterministic flow, we compute the time-varying joint probability density functions over the state space, by solving the Liouville equation via method-of-characteristics. For the stochastic flow, we propose an approximation algorithm that combines the method-of-characteristics solution of Liouville equation with the Karhunen-Lo eve expansion of process noise, thus enabling an indirect solution of Fokker-Planck equation, governing the evolution of joint probability density functions. The efficacy of these algorithms are demonstrated for risk assessment in Mars entry-descent-landing, and for nonlinear estimation. Next, the V&V problem is formulated in terms of Monge-Kantorovich optimal transport, naturally giving rise to a metric, called Wasserstein metric, on the space of probability densities. It is shown that the resulting computation leads to solving a linear program at each time of measurement availability, and computational complexity results for the same are derived. Probabilistic guarantees in average and worst case sense, are given for the validation oracle resulting from the proposed method. The framework is demonstrated for nonlinear robustness veri cation of F-16 flight controllers, subject to probabilistic uncertainties. Frequency domain interpretations for the proposed framework are derived for linear systems, and its connections with existing nonlinear model validation methods are pointed out. In particular, we show that the asymptotic Wasserstein gap between two single-output linear time invariant systems excited by Gaussian white noise, is the difference between their average gains, up to a scaling by the strength of the input noise. A geometric interpretation of this result allows us to propose an intrinsic normalization of the Wasserstein gap, which in turn allows us to compare it with classical systems-theoretic metrics like v-gap. Next, it is shown that the optimal transport map can be used to automatically refine the model. This model refinement formulation leads to solving a non-smooth convex optimization problem. Examples are given to demonstrate how proximal operator splitting based computation enables numerically solving the same. This method is applied for nite-time feedback control of probability density functions, and for data driven modeling of dynamical systems

WIND POWER PROBABILISTIC PREDICTION AND UNCERTAINTY MODELING FOR OPERATION OF LARGE-SCALE POWER SYSTEMS

Over the last decade, large scale renewable energy generation has been integrated into power systems. Wind power generation is known as a widely-used and interesting kind of renewable energy generation around the world. However, the high uncertainty of wind power generation leads to some unavoidable error in wind power prediction process; consequently, it makes the optimal operation and control of power systems very challenging. Since wind power prediction error cannot be entirely removed, providing accurate models for wind power uncertainty can assist power system operators in mitigating its negative effects on decision making conditions. There are efficient ways to show the wind power uncertainty, (i) accurate wind power prediction error probability distribution modeling in the form of probability density functions and (ii) construction of reliable and sharp prediction intervals. Construction of accurate probability density functions and high-quality prediction intervals are difficult because wind power time series is non-stationary. In addition, incorporation of probability density functions and prediction intervals in power systems’ decision-making problems are challenging. In this thesis, the goal is to propose comprehensive frameworks for wind power uncertainty modeling in the form of both probability density functions and prediction intervals and incorporation of each model in power systems’ decision-making problems such as look-ahead economic dispatch. To accurately quantify the uncertainty of wind power generation, different approaches are studied, and a comprehensive framework is then proposed to construct the probability density functions using a mixture of beta kernels. The framework outperforms benchmarks because it can validly capture the actual features of wind power probability density function such as main mass, boundaries, high skewness, and fat tails from the wind power sample moments. Also, using the proposed framework, a generic convex model is proposed for chance-constrained look-ahead economic dispatch problems. It allows power system operators to use piecewise linearization techniques to convert the problem to a mixed-integer linear programming problem. Numerical simulations using IEEE 118-bus test system show that compared with widely used sequential linear programming approaches, the proposed mixed-integer linear programming model leads to less system’s total cost. A framework based on the concept of bandwidth selection for a new and flexible kernel density estimator is proposed for construction of prediction intervals. Unlike previous related works, the proposed framework uses neither a cost function-based optimization problem nor point prediction results; rather, a diffusion-based kernel density estimator is utilized to achieve high-quality prediction intervals for non-stationary wind power time series. The proposed prediction interval construction framework is also founded based on a parallel computing procedure to promote the computational efficiency for practical applications in power systems. Simulation results demonstrate the high performance of the proposed framework compared to well-known conventional benchmarks such as bootstrap extreme learning machine, lower upper bound estimation, quantile regression, auto-regressive integrated moving average, and linear programming-based quantile regression. Finally, a new adjustable robust optimization approach is used to incorporate the constructed prediction intervals with the proposed fuzzy and adaptive diffusion estimator-based prediction interval construction framework. However, to accurately model the correlation and dependence structure of wind farms, especially in high dimensional cases, C-Vine copula models are used for prediction interval construction. The simulation results show that uncertainty modeling using C-Vine copula can lead the system operators to get more realistic sense about the level of overall uncertainty in the system, and consequently more conservative results for energy and reserve scheduling are obtained