Ellipsoidal bounds on state trajectories for discrete-time systems with linear fractional uncertainties

Computation of exact ellipsoidal bounds on the state trajectories of discrete-time linear systems that have time-varying or time-invariant linear fractional parameter uncertainties and ellipsoidal uncertainty in the initial state is known to be NP-hard. This paper proposes three algorithms to compute ellipsoidal bounds on such a state trajectory set and discusses the tradeoffs between computational complexity and conservatism of the algorithms. The approach employs linear matrix inequalities to determine an initial estimate of the ellipsoid that is refined by the subsequent application of the skewed structured singular value ν. Numerical examples are used to illustrate the application of the proposed algorithms and to compare the differences between them, where small conservatism for the tightest bounds is observed.Institute for Advanced Computing Applications and Technologie

State bounds estimation for nonlinear systems using μ-analysis

Developing state bound estimation algorithms for nonlinear systems has been of high importance in robustness analysis of dynamic systems. For many cases, Monte-Carlo simulation might be the only tool to estimate these bounds for a general type of nonlinear systems. The required number of simulations for a tight bound, however, would be very large and it might be impossible to complete within a given computation time. β-formulation for state bounds transforms the bound estimation problem to a singularity problem and the singular problem is solved using a randomised optimisation approach. The performance of the algorithms are demonstrated by a simple discrete system; large-scale biological systems; and a hybrid system

Verification of system properties of polynomial systems using discrete-time approximations and set-based analysis

Magdeburg, Univ., Fak. für Elektrotechnik und Informationstechnik, Diss., 2015von Philipp Rumschinsk

Robust Nonlinear Model Predictive Control using Polynomial Chaos Expansions

The performance of model predictive controllers (MPCs) is largely dependent on the accuracy of the model predictions as compared to the actual plant outputs. Irrespective of the model used, first-principles (FP) or empirical, plant-model mismatch is unavoidable. Consequently, model based controllers must be robust to mismatch between the model predictions and the actual process behavior. Controllers that are not robust may result in poor closed loop response and even instability. Model uncertainty can generally be formulated into two broader forms, parametric uncertainty and unstructured uncertainty. Most of the current robust nonlinear MPC have been based on FP-model where only robustness to bounded disturbances rather than parametric uncertainty has been addressed. Systematically accounting for parametric uncertainty in the robust design has been difficult in FP-models due to varying forms in which uncertain parameters occur in the models. To address parametric uncertainty robustness tests based on Structured Singular Value (SSV) and Linear Matrix Inequalities (LMI) have been proposed previously, however these algorithms tend to be conservative because they consider worst-case scenarios and they are also computationally expensive. For instance the SSV calculation is NP-hard and as a result it is not suitable for fast computations. This provides motivation to work on robust control algorithms addressing both parametric and unstructured uncertainty with fast computation times. To facilitate the design of robust controllers which can be computed fast, empirical models are used in which parametric uncertainty is propagated using Polynomial Chaos Expansion (PCE) of parameters. PCE assists in speeding up the computations by providing an analytical expression for the L^2-norm of model predictions while also eliminating the need to design for the worst-case scenario which results in conservatism. Another way of speeding up computations in MPC algorithms is by grouping subsets of available the inputs and outputs into subsystems and by controlling each of the subsystems by MPC controllers of lower dimensions. This latter approach, referred in the literature as Distributed MPC, has been tackled by different strategies involving different degrees of coordination between subsystems but it has not been studied in terms of robustness to model error. Based on the above considerations the current work investigates different robustness aspects of predictive control algorithms for nonlinear processes with special emphasis on the following three situations, i) a nonlinear predictive control based on a Volterra series model where the uncertain parameters are formulated as PCE’s, ii) The application of a PCE-based approach to control and optimization of bioreactors where the model is based on dynamic flux metabolic models, and iii) A Robust Distributed MPC with a robust estimator that is needed to account for the interactions between sub-systems in distributed control

Dynamic operability assessment : a mathematical programming approach based on Q-parametrization

Bibliography: pages 197-208.The ability of a process plant to guarantee high product quality, in terms of low variability, is emerging as a defining feature when distinguishing between alternative suppliers. The extent to which this can be achieved is termed a plant's dynamic operability and is a function of both the plant design and the control system design. In the limit, however, the closedloop performance is determined by the properties inherent in the plant. This realization of the interrelationship between a plant design and its achievable closed-loop performance has motivated research toward systematic techniques for screening inherently inferior designs. Pioneering research in the early 1980's identified right-half-plane transmission zeros, time delays, input constraints and model uncertainty as factors that limit the achievable closedloop performance of a process. Quantifying the performance-limiting effect of combinations of these factors has proven to be a challenging problem, as reflected in the literature. It is the aim of this thesis to develop a systematic procedure for dynamic operability assessment in the presence of combinations of performance-limiting factors. The approach adopted in this thesis is based on the Q-parametrization of stabilizing linear feedback controllers and involves posing dynamic operability assessment as a mathematical programming problet? In the proposed formulation, a convex objective function, reflecting a measure of closed-loop performance, is optimized over all stable Q, subject. to a set of constraints on the closed-loop behavior, which for many specifications of interest is convex. A discrete-time formulation is chosen so as to allow for the convenient hand.ling of time delays and time-domain constraints. An important feature of the approach is that, due to the convexity, global optimality is guaranteed. Furthermore, the fact that Q parametrizes all stabilizing linear feedback controllers implies that the performance at the optimum represents the best possible performance for any such controller. The results are thus not biased by controller type or tuning, apart from the requirement that the controller be linear

Orbit Uncertainty Propagation with Separated Representations

In light of recent collisions and an increasing population of objects in Earth orbit, the space situational awareness community has significant motivation to develop novel and effective methods of predicting the behavior of object states under the presence of uncertainty. Unfortunately, approaches to uncertainty quantification often make simplifying assumptions in order to reduce computation cost. This thesis proposes the method of separated representations (SR) as an efficient and accurate approach to uncertainty quantification. The properties of an orthogonal polynomial basis and a uni-directional least squares regression approach allow for the theoretical computation cost of SR to remain low when compared to Monte Carlo or other surrogate methods. Specifically, SR does not suffer from the curse of dimensionality, where computation cost increases exponentially with respect to input dimension. Benefits of this low computation cost are shown in a series of low Earth orbit test cases, where SR is used to accurately approximate non-Gaussian posterior distribution functions. Here, the dimension of the problem is increased from 6 to 20 without incurring significantly more computation time. Taking advantage of a large input dimension, this research presents a global sensitivity analysis computed via SR, which affords a more nuanced analysis of a previously examined case in the literature. By considering design variables, SR is formulated to perform optimization under uncertainty. A novel method that utilizes a Brent optimizer to create training data at unique times of closest approach is devised and implemented in order to detect low probability collision events. This methodology is leveraged to design an optimal avoidance maneuver, which would be intractable when using traditional Monte Carlo. Lastly, a multi-element algorithm is formulated and presented to estimate solutions that are challenging for unmodified SR. This multi-element SR leads to orders of magnitude in accuracy improvement when considering the ability of unmodified SR to approximate discontinuous, multimodal, or diffuse solutions