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

Fuzzy Bilevel Optimization

In the dissertation the solution approaches for different fuzzy optimization problems are presented. The single-level optimization problem with fuzzy objective is solved by its reformulation into a biobjective optimization problem. A special attention is given to the computation of the membership function of the fuzzy solution of the fuzzy optimization problem in the linear case. Necessary and sufficient optimality conditions of the the convex nonlinear fuzzy optimization problem are derived in differentiable and nondifferentiable cases. A fuzzy optimization problem with both fuzzy objectives and constraints is also investigated in the thesis in the linear case. These solution approaches are applied to fuzzy bilevel optimization problems. In the case of bilevel optimization problem with fuzzy objective functions, two algorithms are presented and compared using an illustrative example. For the case of fuzzy linear bilevel optimization problem with both fuzzy objectives and constraints k-th best algorithm is adopted.:1 Introduction 1 1.1 Why optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Fuzziness as a concept . . . . . . . . . . . . . . . . . . . . .. . . . . . . 2 1.3 Bilevel problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Preliminaries 11 2.1 Fuzzy sets and fuzzy numbers . . . . . . . . . . . . . . . . . . . . . 11 2.2 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Fuzzy order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Fuzzy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17 3 Optimization problem with fuzzy objective 19 3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Solution method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Local optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.4 Existence of an optimal solution . . . . . . . . . . . . . . . . . . . . 25 4 Linear optimization with fuzzy objective 27 4.1 Main approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.4 Membership function value . . . . . . . . . . . . . . . . . . . . . . . . 34 4.4.1 Special case of triangular fuzzy numbers . . . . . . . . . . . . 36 4.4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 5 Optimality conditions 47 5.1 Differentiable fuzzy optimization problem . . . . . . . . . . .. . . . 48 5.1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.1.2 Necessary optimality conditions . . . . . . . . . . . . . . . . . . .. 49 5.1.3 Suffcient optimality conditions . . . . . . . . . . . . . . . . . . . . . . 49 5.2 Nondifferentiable fuzzy optimization problem . . . . . . . . . . . . 51 5.2.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.2.2 Necessary optimality conditions . . . . . . . . . . . . . . . . . . . 52 5.2.3 Suffcient optimality conditions . . . . . . . . . . . . . . . . . . . . . . 54 5.2.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6 Fuzzy linear optimization problem over fuzzy polytope 59 6.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.2 The fuzzy polytope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63 6.3 Formulation and solution method . . . . . . . . . . . . . . . . . . .. . 65 6.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7 Bilevel optimization with fuzzy objectives 73 7.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 7.2 Solution approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74 7.3 Yager index approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7.4 Algorithm I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 7.5 Membership function approach . . . . . . . . . . . . . . . . . . . . . . .78 7.6 Algorithm II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80 7.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8 Linear fuzzy bilevel optimization (with fuzzy objectives and constraints) 87 8.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8.2 Solution approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 8.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 8.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 9 Conclusions 95 Bibliography 9

Gain-scheduled H∞ control via parameter-dependent Lyapunov functions

Synthesising a gain-scheduled output feedback H∞ controller via parameter-dependent Lyapunov functions for linear parameter-varying (LPV) plant models involves solving an infinite number of linear matrix inequalities (LMIs). In practice, for affine LPV models, a finite number of LMIs can be achieved using convexifying techniques. This paper proposes an alternative approach to achieve a finite number of LMIs. By simple manipulations on the bounded real lemma inequality, a symmetric matrix polytope inequality can be formed. Hence, the LMIs need only to be evaluated at all vertices of such a symmetric matrix polytope. In addition, a construction technique of the intermediate controller variables is also proposed as an affine matrix-valued function in the polytopic coordinates of the scheduled parameters. Computational results on a numerical example using the approach were compared with those from a multi-convexity approach in order to demonstrate the impacts of the approach on parameter-dependent Lyapunov-based stability and performance analysis. Furthermore, numerical simulation results show the effectiveness of these proposed techniques

On the decomposition of Generalized Additive Independence models

The GAI (Generalized Additive Independence) model proposed by Fishburn is a generalization of the additive utility model, which need not satisfy mutual preferential independence. Its great generality makes however its application and study difficult. We consider a significant subclass of GAI models, namely the discrete 2-additive GAI models, and provide for this class a decomposition into nonnegative monotone terms. This decomposition allows a reduction from exponential to quadratic complexity in any optimization problem involving discrete 2-additive models, making them usable in practice

A new dominance intensity method to deal with ordinal information about a DM's preferences within MAVT

Dominance measuring methods are a new approach to deal with complex decision-making problems with imprecise information. These methods are based on the computation of pairwise dominance values and exploit the information in the dominance matrix in dirent ways to derive measures of dominance intensity and rank the alternatives under consideration. In this paper we propose a new dominance measuring method to deal with ordinal information about decision-maker preferences in both weights and component utilities. It takes advantage of the centroid of the polytope delimited by ordinal information and builds triangular fuzzy numbers whose distances to the crisp value 0 constitute the basis for the de?nition of a dominance intensity measure. Monte Carlo simulation techniques have been used to compare the performance of this method with other existing approaches

State-Dependent Dynamic Tube MPC: A Novel Tube MPC Method with a Fuzzy Model of Disturbances

Most real-world systems are affected by external disturbances, which may be impossible or costly to measure. For instance, when autonomous robots move in dusty environments, the perception of their sensors is disturbed. Moreover, uneven terrains can cause ground robots to deviate from their planned trajectories. Thus, learning the external disturbances and incorporating this knowledge into the future predictions in decision-making can significantly contribute to improved performance. Our core idea is to learn the external disturbances that vary with the states of the system, and to incorporate this knowledge into a novel formulation for robust tube model predictive control (TMPC). Robust TMPC provides robustness to bounded disturbances considering the known (fixed) upper bound of the disturbances, but it does not consider the dynamics of the disturbances. This can lead to highly conservative solutions. We propose a new dynamic version of robust TMPC (with proven robust stability), called state-dependent dynamic TMPC (SDD-TMPC), which incorporates the dynamics of the disturbances into the decision-making of TMPC. In order to learn the dynamics of the disturbances as a function of the system states, a fuzzy model is proposed. We compare the performance of SDD-TMPC, MPC, and TMPC via simulations, in designed search-and-rescue scenarios. The results show that, while remaining robust to bounded external disturbances, SDD-TMPC generates less conservative solutions and remains feasible in more cases, compared to TMPC.Comment: 39 pages, 16 figures, 4 tables, 2 appendices, to be submitted to "international journal of robust and nonlinear control", [40] from paper cites our code to be submitted

A fuzzy hybrid sequential design strategy for global surrogate modeling of high-dimensional computer experiments

Complex real-world systems can accurately be modeled by simulations. Evaluating high-fidelity simulators can take several days, making them impractical for use in optimization, design space exploration, and analysis. Often, these simulators are approximated by relatively simple math known as a surrogate model. The data points to construct this model are simulator evaluations meaning the choice of these points is crucial: each additional data point can be very expensive in terms of computing time. Sequential design strategies offer a huge advantage over one-shot experimental design because information gathered from previous data points can be used in the process of determining new data points. Previously, LOLA-Voronoi was presented as a hybrid sequential design method which balances exploration and exploitation: the former involves selecting data points in unexplored regions of the design space, while the latter suggests adding data points in interesting regions which were previously discovered. Although this approach is very successful in terms of the required number of data points to build an accurate surrogate model, it is computationally intensive. This paper presents a new approach to the exploitation component of the algorithm based on fuzzy logic. The new approach has the same desirable properties as the old method but is less complex, especially when applied to high-dimensional problems. Experiments on several test problems show the new approach is a lot faster, without losing robustness or requiring additional samples to obtain similar model accuracy

Linking objective and subjective modeling in engineering design through arc-elastic dominance

Engineering design in mechanics is a complex activity taking into account both objective modeling processes derived from physical analysis and designers’ subjective reasoning. This paper introduces arc-elastic dominance as a suitable concept for ranking design solutions according to a combination of objective and subjective models. Objective models lead to the aggregation of information derived from physics, economics or eco-environmental analysis into a performance indicator. Subjective models result in a confidence indicator for the solutions’ feasibility. Arc-elastic dominant design solutions achieve an optimal compromise between gain in performance and degradation in confidence. Due to the definition of arc-elasticity, this compromise value is expressive and easy for designers to interpret despite the difference in the nature of the objective and subjective models. From the investigation of arc-elasticity mathematical properties, a filtering algorithm of Pareto-efficient solutions is proposed and illustrated through a design knowledge modeling framework. This framework notably takes into account Harrington’s desirability functions and Derringer’s aggregation method. It is carried out through the re-design of a geothermal air conditioning system