A review of convex approaches for control, observation and safety of linear parameter varying and Takagi-Sugeno systems

This paper provides a review about the concept of convex systems based on Takagi-Sugeno, linear parameter varying (LPV) and quasi-LPV modeling. These paradigms are capable of hiding the nonlinearities by means of an equivalent description which uses a set of linear models interpolated by appropriately defined weighing functions. Convex systems have become very popular since they allow applying extended linear techniques based on linear matrix inequalities (LMIs) to complex nonlinear systems. This survey aims at providing the reader with a significant overview of the existing LMI-based techniques for convex systems in the fields of control, observation and safety. Firstly, a detailed review of stability, feedback, tracking and model predictive control (MPC) convex controllers is considered. Secondly, the problem of state estimation is addressed through the design of proportional, proportional-integral, unknown input and descriptor observers. Finally, safety of convex systems is discussed by describing popular techniques for fault diagnosis and fault tolerant control (FTC).Peer ReviewedPostprint (published version

<i>H</i>₂ and mixed <i>H</i>₂/<i>H</i>_∞ Stabilization and Disturbance Attenuation for Differential Linear Repetitive Processes

Repetitive processes are a distinct class of two-dimensional systems (i.e., information propagation in two independent directions) of both systems theoretic and applications interest. A systems theory for them cannot be obtained by direct extension of existing techniques from standard (termed 1-D here) or, in many cases, two-dimensional (2-D) systems theory. Here, we give new results towards the development of such a theory in H2 and mixed H2/H∞ settings. These results are for the sub-class of so-called differential linear repetitive processes and focus on the fundamental problems of stabilization and disturbance attenuation

Convex Optimization In Identification Of Stable Non-Linear State Space Models

A new framework for nonlinear system identification is presented in terms of optimal fitting of stable nonlinear state space equations to input/output/state data, with a performance objective defined as a measure of robustness of the simulation error with respect to equation errors. Basic definitions and analytical results are presented. The utility of the method is illustrated on a simple simulation example as well as experimental recordings from a live neuron.Comment: 9 pages, 2 figure, elaboration of same-title paper in 49th IEEE Conference on Decision and Contro

A Unified Analysis of Stochastic Optimization Methods Using Jump System Theory and Quadratic Constraints

We develop a simple routine unifying the analysis of several important recently-developed stochastic optimization methods including SAGA, Finito, and stochastic dual coordinate ascent (SDCA). First, we show an intrinsic connection between stochastic optimization methods and dynamic jump systems, and propose a general jump system model for stochastic optimization methods. Our proposed model recovers SAGA, SDCA, Finito, and SAG as special cases. Then we combine jump system theory with several simple quadratic inequalities to derive sufficient conditions for convergence rate certifications of the proposed jump system model under various assumptions (with or without individual convexity, etc). The derived conditions are linear matrix inequalities (LMIs) whose sizes roughly scale with the size of the training set. We make use of the symmetry in the stochastic optimization methods and reduce these LMIs to some equivalent small LMIs whose sizes are at most 3 by 3. We solve these small LMIs to provide analytical proofs of new convergence rates for SAGA, Finito and SDCA (with or without individual convexity). We also explain why our proposed LMI fails in analyzing SAG. We reveal a key difference between SAG and other methods, and briefly discuss how to extend our LMI analysis for SAG. An advantage of our approach is that the proposed analysis can be automated for a large class of stochastic methods under various assumptions (with or without individual convexity, etc).Comment: To Appear in Proceedings of the Annual Conference on Learning Theory (COLT) 201

Identification of stable models via nonparametric prediction error methods

A new Bayesian approach to linear system identification has been proposed in a series of recent papers. The main idea is to frame linear system identification as predictor estimation in an infinite dimensional space, with the aid of regularization/Bayesian techniques. This approach guarantees the identification of stable predictors based on the prediction error minimization. Unluckily, the stability of the predictors does not guarantee the stability of the impulse response of the system. In this paper we propose and compare various techniques to address this issue. Simulations results comparing these techniques will be provided.Comment: number of pages = 6, number of figures =

Efficient Off-Policy Q-Learning for Data-Based Discrete-Time LQR Problems

This paper introduces and analyzes an improved Q-learning algorithm for discrete-time linear time-invariant systems. The proposed method does not require any knowledge of the system dynamics, and it enjoys significant efficiency advantages over other data-based optimal control methods in the literature. This algorithm can be fully executed off-line, as it does not require to apply the current estimate of the optimal input to the system as in on-policy algorithms. It is shown that a persistently exciting input, defined from an easily tested matrix rank condition, guarantees the convergence of the algorithm. A data-based method is proposed to design the initial stabilizing feedback gain that the algorithm requires. Robustness of the algorithm in the presence of noisy measurements is analyzed. We compare the proposed algorithm in simulation to different direct and indirect data-based control design methods.Comment: 12 pages, journal articl

An LMI Framework for Contraction-based Nonlinear Control Design by Derivatives of Gaussian Process Regression

Contraction theory formulates the analysis of nonlinear systems in terms of Jacobian matrices. Although this provides the potential to develop a linear matrix inequality (LMI) framework for nonlinear control design, conditions are imposed not on controllers but on their partial derivatives, which makes control design challenging. In this paper, we illustrate this so-called integrability problem can be solved by a non-standard use of Gaussian process regression (GPR) for parameterizing controllers and then establish an LMI framework of contraction-based control design for nonlinear discrete-time systems, as an easy-to-implement tool. Later on, we consider the case where the drift vector fields are unknown and employ GPR for functional fitting as its standard use. GPR describes learning errors in terms of probability, and thus we further discuss how to incorporate stochastic learning errors into the proposed LMI framework

Measurements, Models, Systems and Design

531 s.

Fuzzy control turns 50: 10 years later

In 2015, we celebrate the 50th anniversary of Fuzzy Sets, ten years after the main milestones regarding its applications in fuzzy control in their 40th birthday were reviewed in FSS, see [1]. Ten years is at the same time a long period and short time thinking to the inner dynamics of research. This paper, presented for these 50 years of Fuzzy Sets is taking into account both thoughts. A first part presents a quick recap of the history of fuzzy control: from model-free design, based on human reasoning to quasi-LPV (Linear Parameter Varying) model-based control design via some milestones, and key applications. The second part shows where we arrived and what the improvements are since the milestone of the first 40 years. A last part is devoted to discussion and possible future research topics.Guerra, T.; Sala, A.; Tanaka, K. (2015). Fuzzy control turns 50: 10 years later. Fuzzy Sets and Systems. 281:162-182. doi:10.1016/j.fss.2015.05.005S16218228