Piecewise Affine Research Articles

Variational autoencoder for the identification of piecewise models

The paper presents a variational autoencoder (VAE) tailored for the identification of hybrid piecewise models in input-output form. We show that using a specialized autoencoder structure, the latent space can provide an interpretable representation in terms of the modes of the underlying hybrid system. In particular, we use categorical encoding of the discrete latent variables whose distribution is approximated via the encoder neural network, characterizing a partition of the regressor space, while the decoder consists of a set of neural networks, each corresponding to a local submodel of the piecewise hybrid system. By employing variational Bayesian framework for inference, the constitutive terms of the evidence lower bound (ELBO) are derived analytically with the chosen VAE architecture. The ELBO loss consists of a reconstruction error term and a regularization term over the latent modes. This loss is optimized in order to train the encoder-decoder networks concurrently via back-propagation. The developed framework is not restricted to simple piecewise affine (PWA) models and it can be straightforwardly extended to general class of piecewise non-linear systems over non-polyhedral domains.

Parametric Piecewise-affine Approximation of Nonlinear Systems: A Cut-Based Approach

Piecewise-affine (PWA) approximations are widely used among hybrid modeling frameworks as a way to increase computational efficiency in nonlinear control and optimization problems. A variety of approaches to construct PWA approximations have been proposed, most of which are tailored to specific application areas by using some prior knowledge of the system in their assumptions and/or steps. In this paper, a parametric method is proposed to identify PWA approximations of nonlinear systems, without any prior knowledge of their dynamics or application requirements. The algorithm defines the regions parametrically using hyperplanes to cut the domain, and increases the number of regions iteratively until a user-defined error tolerance criterion is met. General remarks are given on the algorithm's implementation and a case study is provided to illustrate its application to vehicle dynamics.

Convergence of iterates in nonlinear Perron-Frobenius theory

Let $ C $ be a closed cone with nonempty interior $ {{C}}^\circ $ in a Banach space. Let $ f:{{C}}^\circ \rightarrow {{C}}^\circ $ be an order-preserving subhomogeneous function with a fixed point in $ {{C}}^\circ $. We introduce a condition which guarantees that the iterates $ f^k(x) $ converge to a fixed point for all $ x \in {{C}}^\circ $. This condition generalizes the notion of type K order-preserving for maps on $ \mathbb{R}^n_{>0} $. We also prove that when iterates converge to a fixed point, the rate of convergence is always R-linear in two special cases: for piecewise affine maps and also for order-preserving, homogeneous, analytic, multiplicatively convex functions on $ \mathbb{R}^n_{>0} $. This later category includes the maps associated with the homogeneous eigenvalue problem for nonnegative tensors.

Active Model Discrimination for Piecewise Affine Inclusion Systems

This paper proposes a novel set-membership active model discrimination (AMD) algorithm for actively separating/discriminating among a set of piecewise Affine inclusion systems with bounded noise. Specifically, to overcome the difficulties of dealing the integer variables in the lower/inner level of the associated bilevel optimization problem that stem from the mapping of the piecewise inclusions and subregions, we propose an alternative reformulation that moves the integer variables into the higher/outer level. This reformulation allows us to leverage Karush-Kuhn-Tucker (KKT) conditions to obtain an equivalent single level mixed-integer linear programming (MILP) problem. Moreover, in contrast to standard AMD algorithms that strictly enforces model separation, we propose a slight modification/extension that separates as many models as possible when a strict separation of all models is not possible.

A stochastic adaptive dynamics model for bacterial populations with mutation, dormancy and transfer

This paper introduces a stochastic adaptive dynamics model for the interplay of several crucial traits and mechanisms in bacterial evolution, namely dormancy, horizontal gene transfer (HGT), mutation and competition. In particular, it combines the recent model of Champagnat, Meleard and Tran (2021) involving HGT with the model for competition-induced dormancy of Blath and Tobias (2020). Our main result is a convergence theorem which describes the evolution of the different traits in the population on a `doubly logarithmic scale' as piece-wise affine functions. Interestingly, even for a relatively small trait space, the limiting process exhibits a non-monotone dependence of the success of the dormancy trait on the dormancy initiation probability. Further, the model establishes a new `approximate coexistence regime' for multiple traits that has not been observed in previous literature.

Data-Driven Design of Explicit Predictive Controllers With Structural Priors

In this paper, we propose a data-driven approach to derive explicit predictive control laws. The key idea of the presented strategy is to exploit the prior knowledge that the optimal solution is a piece-wise affine controller. As the proposed method allows us to automatically retrieve also a model of the closed-loop system, we show that we can apply classical Lyapunov techniques to perform a prior stability check for safe controller deployment. The effectiveness of the proposed strategy is assessed on a benchmark simulation example, through which we also discuss the use of regularization and preprocessing techniques to handle the presence of noise.

Метод триангуляции для приближенного решения вариационных задач нелинейной теории упругости

A variational problem for the minimum of the stored energy functional is considered in the framework of the nonlinear theory of elasticity, taking into account admissible deformations. An algorithm for solving this problem is proposed, based on the use of a polygonal partition of the computational domain by the Delaunay triangulation method. Conditions for the convergence of the method to a local minimum in the class of piecewise affine mappings are found.

Model-Based Reinforcement Learning via Stochastic Hybrid Models

Optimal control of general nonlinear systems is a central challenge in automation. Enabled by powerful function approximators, data-driven approaches to control have recently successfully tackled challenging applications. However, such methods often obscure the structure of dynamics and control behind black-box over-parameterized representations, thus limiting our ability to understand closed-loop behavior. This article adopts a hybrid-system view of nonlinear modeling and control that lends an explicit hierarchical structure to the problem and breaks down complex dynamics into simpler localized units. We consider a sequence modeling paradigm that captures the temporal structure of the data and derive an expectation-maximization (EM) algorithm that automatically decomposes nonlinear dynamics into stochastic piecewise affine models with nonlinear transition boundaries. Furthermore, we show that these time-series models naturally admit a closed-loop extension that we use to extract local polynomial feedback controllers from nonlinear experts via behavioral cloning. Finally, we introduce a novel hybrid relative entropy policy search (Hb-REPS) technique that incorporates the hierarchical nature of hybrid models and optimizes a set of time-invariant piecewise feedback controllers derived from a piecewise polynomial approximation of a global state-value function.

H∞ Control of Scalar Nonlinear Systems

The £2 disturbance attenuation control or the nonlinear H∞ control of a nonlinear system requires solving the Hamilton-Jacobi-Isaacs (HJI) partial differential inequality, which is generally a difficult undertaking. Though there are inverse optimal approaches, they have limited utilization from a practical point of view. In this work, a novel approach of state feedback H∞ control is proposed for scalar nonlinear systems using piecewise affine bounds of scalar-valued nonlinear functions. An LMI based systematic procedure is proposed. To illustrate the proposed method, an example of a scalar nonlinear system possessing significant nonlinearity has been considered. To demonstrate the effectiveness of the proposed scheme, it is compared with Linear Parameter Varying (LPV) based state feedback H∞ control, using quasi-LPV models based on Taylor series expansion.

Mixed logical dynamical modeling and model predictive control of piecewise affine systems with dead zone constraints

Mixed logical dynamical modeling and model predictive control of piecewise affine systems with dead zone constraints

Event-triggered control for discrete-time piecewise affine systems

This work addresses the event-triggered control (ETC) of discrete-time piecewise affine systems. We propose a method to design a triggering strategy relying on an implicit representation of piecewise affine systems. Thanks to this implicit representation based on ramp functions, we propose a partition-dependent piecewise quadratic functions to define the trigger criterion and use a piecewise quadratic Lyapunov function candidate to derive conditions to certify the global exponential stability of the origin under the ETC strategy. Since the stability conditions can be expressed as linear matrix inequalities constraints, we propose a convex optimization solution to design the triggering function parameters and to compute the Lyapunov function to ensure the closed-loop stability and a reduction on the control updates. The approach is illustrated by numerical examples.

Hybrid model predictive control of a nonlinear three-tank system based on the proposed compact form of piecewise affine model

Hybrid model predictive control of a nonlinear three-tank system based on the proposed compact form of piecewise affine model

A novel stochastic model for flexible unit commitment of off-grid microgrids

This work proposes a novel two stage stochastic optimization approach for the Energy Management System (EMS) of MicroGrids (MG). It combines the scenario-based uncertainty characterization with the use of piecewise affine correction rules for the recourse decision. These rules are used by the EMS to set the commitment status of the programmable generators and to correct the storage management as a function of the realization of the uncertainty, measured in real-time. The novel stochastic model is integrated in a hierarchical EMS, based on two control layers: the first one employs the proposed stochastic approach to determine the daily strategic scheduling of the MG, and the second layer optimizes the real-time dispatch.. The proposed EMS is applied to a real-world case study of a rural MG. Results indicate that the proposed EMS outperforms state-of-the-art optimization approaches in terms ofservice reliability (99.8%) and fuel efficiency. Moreover, rolling horizon simulations of the proposed stochastic model showed 100% reliability, and 30% of fuel cost savings with respects to state-of-the-art methods. The novel EMS is then deployed on a laboratory-scale microgrid (Multi-Goods MicroGrid Laboratory, MG2lab), demonstrating secure and economic operations. A comparative analysis is made with respect to deterministic and standard stochastic two-stage approaches; the proposed solution outperforms the other models during real operations, with savings of fuel cost up to almost 10%.

An implicit representation for the analysis of piecewise affine discrete-time systems

This paper presents a new framework for stability assessment of discrete-time piecewise affine systems. An implicit representation for piecewise functions based on ramp nonlinearities is proposed. Instead of the usual sector inequalities adopted in the study of Lurie systems, we directly exploit properties of the ramp function, which are given by a particular set of identities and inequalities. These properties are then used to derive stability conditions associated to piecewise quadratic Lyapunov functions. These functions are implicitly defined from quadratic forms involving ramp functions. These conditions can be cast as LMIs and their numerical solution is illustrated by examples highlighting the advantages of the proposed method over existing stability analysis methods for PWA systems.

An optimized excitation signal design for identification of PWA model and application to automotive throttles

In the automotive application for Electronic Control Units (ECU), dynamic models with high-precision to simulate mechatronic actuators are required for various analysis and design tasks like Hardware-in-the-Loop (HiL) simulation. It is known that the modeling of systems with significant friction effects is not accessible. The piecewise-affine (PWA) model has recently attracted considerable interest due to its high approximation capability for modeling systems with friction. Because the model quality depends intensely on the test signal design, a model-free optimized excitation signal design for identifying PWA models is presented in this paper. The presented method is demonstrated with an automotive throttle, and a PWA model is identified with a four-step identification approach. The result shows that the optimization method can achieve a better model quality, which is sufficient for the HiL simulation.

Signal Approximations Based on Nonlinear and Optimal Piecewise Affine Functions

Signal Approximations Based on Nonlinear and Optimal Piecewise Affine Functions

A piecewise ellipsoidal reachable set estimation method for continuous bimodal piecewise affine systems

In this work, the issue of estimation of reachable sets in continuous bimodal piecewise affine systems is studied. A new method is proposed, in the framework of ellipsoidal bounding, using piecewise quadratic Lyapunov functions. Although bimodal piecewise affine systems can be seen as a special class of affine hybrid systems, reachability methods developed for affine hybrid systems might be inappropriately complex for bimodal dynamics. This work goes in the direction of exploiting the dynamical structure of the system to propose a simpler approach. More specifically, because of the piecewise nature of the Lyapunov function, we first derive conditions to ensure that a given quadratic function is positive on half spaces. Then, we exploit the property of bimodal piecewise quadratic functions being continuous on a given hyperplane. Finally, linear matrix characterizations of the estimate of the reachable set are derived.

Output-only identification of self-excited systems using discrete-time Lur'e models with application to a gas-turbine combustor

A self-excited system is a nonlinear system with the property that a constant input yields a bounded, nonconvergent response. Nonlinear identification of self-excited systems is considered using a Lur'e model structure, where a linear model is connected in feedback with a nonlinear feedback function. To facilitate identification, the nonlinear feedback function is assumed to be continuous and piecewise affine (CPA). The present paper uses least-squares optimisation to estimate the coefficients of the linear dynamics and the slope vector of the CPA nonlinearity, as well as mixed-integer optimisation to estimate the order of the linear dynamics and the breakpoints of the CPA function. The proposed identification technique requires only output data, and thus no measurement of the constant input is required. This technique is illustrated on a diverse collection of low-dimensional numerical examples as well as data from a gas-turbine combustor.

On Constructing a Piecewise Affine Stabilizer for a Nonlinear System

On Constructing a Piecewise Affine Stabilizer for a Nonlinear System

Multilevel Selective Harmonic Modulation via Optimal Control

We consider the Selective Harmonic Modulation (SHM) problem, consisting in the design of a staircase control signal with some prescribed frequency components. In this work, we propose a novel methodology to address SHM as an optimal control problem in which the admissible controls are piecewise constant functions, taking values only in a given finite set. In order to fulfill this constraint, we introduce a cost functional with piecewise affine penalization for the control, which, by means of Pontryagin’s maximum principle, makes the optimal control have the desired staircase form. Moreover, the addition of the penalization term for the control provides uniqueness and continuity of the solution with respect to the target frequencies. Another advantage of our approach is that the number of switching angles and the waveform need not be determined a priori. Indeed, the solution to the optimal control problem is the entire control signal, and therefore, it determines the waveform and the location of the switches. We also provide numerical examples in which the SHM problem is solved by means of our approach.