A probabilistic interpretation of set-membership filtering: application to polynomial systems through polytopic bounding

Set-membership estimation is usually formulated in the context of set-valued calculus and no probabilistic calculations are necessary. In this paper, we show that set-membership estimation can be equivalently formulated in the probabilistic setting by employing sets of probability measures. Inference in set-membership estimation is thus carried out by computing expectations with respect to the updated set of probability measures P as in the probabilistic case. In particular, it is shown that inference can be performed by solving a particular semi-infinite linear programming problem, which is a special case of the truncated moment problem in which only the zero-th order moment is known (i.e., the support). By writing the dual of the above semi-infinite linear programming problem, it is shown that, if the nonlinearities in the measurement and process equations are polynomial and if the bounding sets for initial state, process and measurement noises are described by polynomial inequalities, then an approximation of this semi-infinite linear programming problem can efficiently be obtained by using the theory of sum-of-squares polynomial optimization. We then derive a smart greedy procedure to compute a polytopic outer-approximation of the true membership-set, by computing the minimum-volume polytope that outer-bounds the set that includes all the means computed with respect to P

Computation of the Structured Singular Value via Moment LMI Relaxations

The Structured Singular Value (SSV) provides a powerful tool to test robust stability and performance of feedback systems subject to structured uncertainties. Unfortunately, computing the SSV is an NP-hard problem, and the polynomial-time algorithms available in the literature are only able to provide, except for some special cases, upper and lower bounds on the exact value of the SSV. In this work, we present a new algorithm to compute an upper bound on the SSV in case of mixed real/complex uncertainties. The underlying idea of the developed approach is to formulate the SSV computation as a (nonconvex) polynomial optimization problem, which is relaxed into a sequence of convex optimization problems through moment-based relaxation techniques. Two heuristics to compute a lower bound on the SSV are also discussed. The analyzed numerical examples show that the developed approach provides tighter bounds than the ones computed by the algorithms implemented in the Robust Control Toolbox in Matlab, and it provides, in most of the cases, coincident lower and upper bounds on the structured singular value

Direct data-driven control of constrained linear parameter-varying systems: A hierarchical approach

In many nonlinear control problems, the plant can be accurately described by a linear model whose operating point depends on some measurable variables, called scheduling signals. When such a linear parameter-varying (LPV) model of the open-loop plant needs to be derived from a set of data, several issues arise in terms of parameterization, estimation, and validation of the model before designing the controller. Moreover, the way modeling errors affect the closed-loop performance is still largely unknown in the LPV context. In this paper, a direct data-driven control method is proposed to design LPV controllers directly from data without deriving a model of the plant. The main idea of the approach is to use a hierarchical control architecture, where the inner controller is designed to match a simple and a-priori specified closed-loop behavior. Then, an outer model predictive controller is synthesized to handle input/output constraints and to enhance the performance of the inner loop. The effectiveness of the approach is illustrated by means of a simulation and an experimental example. Practical implementation issues are also discussed.Comment: Preliminary version of the paper "Direct data-driven control of constrained systems" published in the IEEE Transactions on Control Systems Technolog

Computation of the Structured Singular Value via Moment LMI Relaxations

The Structured Singular Value (SSV) provides a powerful tool to test robust stability and performance of feedback systems subject to structured uncertainties. Unfortunately, computing the SSV is an NP-hard problem, and the polynomial-time algorithms available in the literature are only able to provide, except for some special cases, upper and lower bounds on the exact value of the SSV. In this work, we present a new algorithm to compute an upper bound on the SSV in case of mixed real/complex uncertainties. The underlying idea of the developed approach is to formulate the SSV computation as a (nonconvex) polynomial optimization problem, which is relaxed into a sequence of convex optimization problems through moment-based relaxation techniques. Two heuristics to compute a lower bound on the SSV are also discussed. The analyzed numerical examples show that the developed approach provides tighter bounds than the ones computed by the algorithms implemented in the Robust Control Toolbox in Matlab, and it provides, in most of the cases, coincident lower and upper bounds on the structured singular value

Performance-oriented model learning for data-driven MPC design

Model Predictive Control (MPC) is an enabling technology in applications requiring controlling physical processes in an optimized way under constraints on inputs and outputs. However, in MPC closed-loop performance is pushed to the limits only if the plant under control is accurately modeled; otherwise, robust architectures need to be employed, at the price of reduced performance due to worst-case conservative assumptions. In this paper, instead of adapting the controller to handle uncertainty, we adapt the learning procedure so that the prediction model is selected to provide the best closed-loop performance. More specifically, we apply for the first time the above "identification for control" rationale to hierarchical MPC using data-driven methods and Bayesian optimization.Comment: Accepted for publication in the IEEE Control Systems Letters (L-CSS

An SDP approach for l0-minimization: application to ARX model segmentation

Abstract Minimizing the ℓ 0 -seminorm of a vector under convex constraints is a combinatorial (NP-hard) problem. Replacement of the ℓ 0 -seminorm with the ℓ 1 -norm is a commonly used approach to compute an approximate solution of the original ℓ 0 -minimization problem by means of convex programming. In the theory of compressive sensing, the condition that the sensing matrix satisfies the Restricted Isometry Property (RIP) is a sufficient condition to guarantee that the solution of the ℓ 1 -approximated problem is equal to the solution of the original ℓ 0 -minimization problem. However, the evaluation of the conservativeness of the ℓ 1 -relaxation approaches is recognized to be a difficult task in case the {RIP} is not satisfied. In this paper, we present an alternative approach to minimize the ℓ 0 -norm of a vector under given constraints. In particular, we show that an ℓ 0 -minimization problem can be relaxed into a sequence of semidefinite programming problems, whose solutions are guaranteed to converge to the optimizer (if unique) of the original combinatorial problem also in case the {RIP} is not satisfied. Segmentation of {ARX} models is then discussed in order to show, through a relevant problem in system identification, that the proposed approach outperforms the ℓ 1 -based relaxation in detecting piece-wise constant parameter changes in the estimated model

LPV model order selection in an LS-SVM setting

In parametric identification of Linear Parameter-Varying (LPV) systems, the scheduling dependencies of the model coefficients are commonly parameterized in terms of linear combinations of a-priori selected basis functions. Such functions need to be adequately chosen, e.g., on the basis of some first-principles or expert's knowledge of the system, in order to capture the unknown dependencies of the model coefficient functions on the scheduling variable and, at the same time, to achieve a low-variance of the model estimate by limiting the number of parameters to be identified. This problem together with the well-known model order selection problem (in terms of number of input lags, output lags and input delay of the model structure) in system identification can be interpreted as a trade-off between bias and variance of the resulting model estimate. The problem of basis function selection can be avoided by using a non-parametric estimator of the coefficient functions in terms of a recently proposed Least-Square Support-Vector-Machine (LS-SVM) approach. However, the selection of the model order still appears to be an open problem in the identification of LPV systems via the LS-SVM method. In this paper, we propose a novel reformulation of the LPV LS-SVM approach, which, besides of the non-parametric estimation of the coefficient functions, achieves data-driven model order selection via convex optimization. The properties of the introduced approach are illustrated via a simulation example

Data-driven LPV modeling of continuous pulp digesters

In this technical report, the LPV-IO identification techniques described in Kauven et al. [2013] (Chapter 5) are applied in order to estimate an LPV model of a continuous pulp digester. The pulp digester simulator (described in Modén [2011]) has been provided by ABB for benchmark studies as part of its participation in the EU project Autoprofi