Piecewise Affine Model Research Articles

On convergence for hybrid models of gene regulatory networks under polytopic uncertainties: a Lyapunov approach

Hybrid models of genetic regulatory networks allow for a simpler analysis with respect to fully detailed quantitative models, still maintaining the main dynamical features of interest. In this paper we consider a piecewise affine model of a genetic regulatory network, in which the parameters describing the production function are affected by polytopic uncertainties. In the first part of the paper, after recalling how the problem of finding a Lyapunov function is solved in the nominal case, we present the considered polytopic uncertain system and then, after describing how to deal with sliding mode solutions, we prove a result of existence of a parameter dependent Lyapunov function subject to the solution of a feasibility linear matrix inequalities problem. In the second part of the paper, based on the previously described Lyapunov function, we are able to determine a set of domains where the system is guaranteed to converge, with the exception of a zero measure set of times, independently from the uncertainty realization. Finally a three nodes network example shows the validity of the results.

Study on Multi-Model Soft Sensor Modeling Method and Its Model Optimization for the Fermentation Process of Pichia pastoris.

The problems that the key biomass variables in Pichia pastoris fermentation process are difficult measure in real time; this paper mainly proposes a multi-model soft sensor modeling method based on the piecewise affine (PWA) modeling method, which is optimized by particle swarm optimization (PSO) with an improved compression factor (ICF). Firstly, the false nearest neighbor method was used to determine the order of the PWA model. Secondly, the ICF-PSO algorithm was proposed to cooperatively optimize the number of PWA models and the parameters of each local model. Finally, a least squares support vector machine was adopted to determine the scope of action of each local model. Simulation results show that the proposed ICF-PSO-PWA multi-model soft sensor modeling method accurately approximated the nonlinear features of Pichia pastoris fermentation, and the model prediction accuracy is improved by 4.4884% compared with the weighted least squares vector regression model optimized by PSO.

Machine learning based model linearization of a wind turbine for power regulation

ABSTRACT Wind turbine systems exhibit highly nonlinear dynamics influenced by the aerodynamic torque induced in the wind turbine blades and thrust force on the turbine structure due to the wind flow. This paper presents a system identification approach to approximate the nonlinear wind turbine model. A clustering-based piecewise affine system identification technique is utilized to construct an affine multiple-model that is valid for the power regulation region of a wind turbine. A comprehensive study is performed to validate the accuracy and performance of the developed model. The piecewise affine model identified in this paper can be widely used for advanced control systems design and the security assessment of the power grid.

Piecewise affine modeling and hybrid optimal control of intelligent vehicle longitudinal dynamics for velocity regulation

In this paper, a novel piecewise affine (PWA) modeling framework and a hybrid optimal control scheme based on the developed PWA model for the intelligent vehicle longitudinal dynamics are proposed. The proposed methodology approximates the nonlinear characteristics of the system major components, i.e. the engine, the automatic transmission and the tire, using the PWA representation method firstly. Due to the affine submodel switching behaviors of the PWA model and the fact that the intelligent vehicle must be worked in two discrete modes (driving and braking) for regulating the longitudinal velocity automatically, the control procedure of the intelligent vehicle longitudinal dynamics actually shows a hybrid nature with both continuous variables and discrete events. Thus, to further solve the system hybrid control problem, the intelligent vehicle longitudinal dynamics whose major components are modelled in the PWA form is further transformed into a computational mixed logical dynamical (MLD) model in this work, based on which a hybrid model predictive control (HMPC) strategy, which can simultaneously calculate the switching sequences of the intelligent vehicle longitudinal working modes (binary control inputs) and the throttle angle and braking pressure (continuous control inputs) during autonomous velocity regulation, is designed by solving a mixed-integer quadratic programming (MIQP) problem. Simulation and experimental results are finally provided to verify the superior control performance of the designed hybrid controller in longitudinal velocity regulation under typical driving conditions.

Inferring Switched Nonlinear DynamicalSystems

AbstractIdentification of dynamical and hybrid systems using trajectory data is an important way to construct models for complex systems where derivation from first principles is too difficult. In this paper, we study the identification problem for switched dynamical systems with polynomial ODEs. This is a difficult problem as it combines estimating coefficients for nonlinear dynamics and determining boundaries between modes. We propose two different algorithms for this problem, depending on whether to perform prior segmentation of trajectories. For methods with prior segmentation, we present a heuristic segmentation algorithm and a way to classify themodes using clustering. Formethods without prior segmentation, we extend identification techniques for piecewise affine models to our problem. To estimate derivatives along the given trajectories, we use Linear MultistepMethods. Finally, we propose a way to evaluate an identified model by computing a relative difference between the predicted and actual derivatives. Based on this evaluation method, we perform experiments on five switched dynamical systems with different parameters, for a total of twenty cases. We also compare with three baseline methods: clustering with DBSCAN, standard optimization methods in SciPy and identification of ARX models in Matlab, as well as with state-of-the-art identification method for piecewise affine models. The experiments show that our two methods perform better across a wide range of situations.

A Novel PWA Lateral Dynamics Modeling Method and Switched T-S Observer Design for Vehicle Sideslip Angle Estimation

In this article, a novel vehicle sideslip angle estimation method is proposed using the Takagi–Sugeno (T–S) observer based on a piecewise affine (PWA) lateral dynamics model. Since conventional lateral dynamics models have complex structures and nonlinearities, difficulties in sideslip angle observer design and online implementation are inevitable. To address nonlinearities, the PWA and T–S fuzzy modeling approaches are applied in this article. First, a nominal PWA model is obtained using the nominal tyre load and tyre-road friction coefficient. An explicit relationship between the PWA model parameters and the tyre load, tyre-road friction coefficient is deduced based on the Dugoff tyre model. Then, a novel PWA lateral dynamics model is obtained considering a varying tyre load and tyre-road friction coefficient. Due to the residual nonlinear terms, this PWA model is further converted into a T–S fuzzy model. Finally, a switched T–S observer is designed as it is computationally simple and efficient enough for online estimation. To validate the effectiveness of the proposed approach, experiments are conducted on both low- and high-friction coefficient roads. Experimental results show that the proposed PWA modeling technique is effective and the T–S observer can estimate the sideslip angle well under both stable and unstable steering maneuvers.

Topology-induced dynamics in a network of synthetic oscillators with piecewise affine approximation.

In synthetic biology approaches, minimal systems are used to reproduce complex molecular mechanisms that appear in the core functioning of multi-cellular organisms. In this paper, we study a piecewise affine model of a synthetic two-gene oscillator and prove existence and stability of a periodic solution for all parameters in a given region. Motivated by the synchronization of circadian clocks in a cluster of cells, we next consider a network of N identical oscillators under diffusive coupling to investigate the effect of the topology of interactions in the network's dynamics. Our results show that both all-to-all and one-to-all coupling topologies may introduce new stable steady states in addition to the expected periodic orbit. Both topologies admit an upper bound on the coupling parameter that prevents the generation of new steady states. However, this upper bound is independent of the number of oscillators in the network and less conservative for the one-to-all topology.

A Semi-Supervised Learning Approach for Identification of Piecewise Affine Systems

Piecewise affine (PWA) models are attractive frameworks that can represent various hybrid systems with local affine submodels and polyhedral regions due to their universal approximation properties. The PWA identification problem amounts to estimating both the submodel parameters and the polyhedral partitions from data. In this paper, we propose a novel approach to address the identification problem of PWA systems such that the number of submodels, parameters of submodels, and the polyhedral partitions are obtained. In particular, a cluster-based algorithm is designed to acquire the number of submodels, the initial labeled data set, and initial parameters corresponding to each submodel. Additionally, we develop a modified self-training support vector machine algorithm to simultaneously identify the hyperplanes and parameter of each submodel with the ouputs of the cluster-based algorithm. The proposed algorithm is computationally efficient for region estimation and able to accomplish this task with only a small quantity of classified regression vectors. The effectiveness of the proposed identification approach is illustrated via simulation results.

New trends in modeling and control of hybrid systems

New trends in modeling and control of hybrid systems

A novel model predictive control for a piecewise affine class of hybrid system with repetitive disturbance

The present investigation addresses an innovative method based on explicit form of the model predictive control (EMPC) for a constrained Piecewise affine (PWA) class of hybrid systems, considering repetitive disturbance. This model of hybrid systems is investigated due to the fact that PWA modeling structure can approximate nonlinear systems via various operating points, and also because the simulation of PWA models are easy. With EMPC, the problem of optimization is solved in an offline way only once. Unlike conventional EMPC, the process information of the past and the data which are predicted are applied in the proposed strategy. This is the first time that in this study, the investigators adopt an approach in which these predicted data are weighted by another optimization problem (OP) and this weighted predicted sequence along with the past information of the process as an updating control input formula. In fact, two separate OPs are solved simultaneously at each step of proposed EMPC. The first one is linked with calculating the control input from the constrained cost function of EMPC algorithm and the second one concerns finding the optimal weighting factors in order to minimize the error signal, i.e. the difference between the reference path and the output signal at each optimization step of EMPC strategy. The precision of the proposed method is extremely dependent on the accuracy of the process model, so iterative learning control (ILC) algorithm is applied to protecting the process model against the periodic disturbances. These mathematical analyses are proven and validated by simulation results.

Optimal low-level control strategies for a high-performance hybrid electric power unit

In this paper we present models and optimization algorithms to compute the optimal low-level control strategies for hybrid electric powertrains. Specifically, we study the minimum-fuel operation of a turbocharged internal combustion engine coupled to an electrical energy recovery system, consisting of a battery and two motors connected to the turbocharger and to the wheels, respectively. First, we combine physics-based modeling approaches with neural networks to identify a piecewise affine model of the power unit accounting for the internal dynamics of the engine, and formulate the minimum-fuel control problem for a given driving cycle. Second, we parse the control problem to a mixed-integer linear program that can be solved with off-the-shelf optimization algorithms that guarantee global optimality of the solution. Finally, we validate our model against a high fidelity nonlinear simulator and showcase the presented framework with a case-study for racing applications. Our results show that cylinder deactivation and turbocharger electrification can decrease fuel consumption up to 4% and 8%, respectively.

Identification of a piecewise affine model for the tire cornering characteristics based on experimental data

Tire cornering characteristics have significant influence on vehicle lateral dynamics control. Unlike traditional tire mechanics models which are established based on the research experience or the mechanics mechanism, in this study, a novel experimental data-driven modeling approach is presented to model the tire cornering characteristics based on piecewise affine (PWA) identification method. In this approach, the highly nonlinear dynamic of the tire cornering characteristics is well approximated by several affine submodels acting on different regions. To obtain the experimental data which can accurately reflect the tire cornering characteristics, the tire tests are firstly carried out through a high-performance flat-plate test bench. On this basis, the PWA identification of the tire cornering characteristics is composed of the data clustering, the parameter estimation of the affine submodels and the calculation of the hyperplane coefficient matrices. The simulation results of the PWA identification model are finally compared with the experimental data to illustrate that the identified model has high accuracy in approximating the tire nonlinear cornering characteristics under wide range driving conditions.

Rao-Blackwellized sampling for batch and recursive Bayesian inference of Piecewise Affine models

This paper addresses batch (offline) and recursive (online) Bayesian inference of Piecewise Affine (PWA) regression models. By exploiting the particular structure of PWA models, efficient Rao-Blackwellized Monte Carlo sampling algorithms are developed to approximate the joint posterior distribution of the model parameters. Only the marginal posterior of the parameters used to describe the regressor-space partition is approximated, either in a batch mode using a Metropolis–Hastings Markov-Chain Monte Carlo (MCMC) sampler, or sequentially using particle filters, while the conditional distribution of the other model parameters is computed analytically. Probability distributions for the predicted outputs given new test inputs are derived and modifications of the proposed approaches to address maximum-a-posteriori estimate are discussed. The performance of the proposed algorithms is shown via a numerical example and through a benchmark case study on data-driven modelling of the electronic component placement process in a pick-and-place machine.

A PWA model identification method for nonlinear systems using hierarchical clustering based on the gap metric

A piecewise affine (PWA) model identification method for nonlinear systems using hierarchical clustering based on the gap metric is proposed. The model parameter estimation is realized by clustering input-output data according to the local models. We initially introduce the gap metric to analyze the similarity between the local models from the perspective of the system, which distinguishes the proposed method from other identification methods that only focus on data features. To determine the optimal number of submodels, the hierarchical clustering aimed at the identification error minimization is addressed. Furthermore, Softmax regression is adopted to completely partition the valid region of a PWA model. Particle swarm optimization (PSO) algorithm is applied to simultaneously update the partition boundaries and model parameters in order to avoid the mismatch between them. Case studies on the multivariable pH neutralization process demonstrate that the proposed method achieves more accurate and stable identification.

A PWA model identification method based on optimal operating region partition with the output-error minimization for nonlinear systems

When piecewise affine (PWA) model-based control methods are applied to nonlinear systems, the first question is how to get sub-models and corresponding operating regions. Motivated by the fact that the operating region of each sub-model is an important component of a PWA model and the parameters of a sub-model are strongly coupled with the operating region, a new PWA model identification method based on optimal operating region partition with the output-error minimization for nonlinear systems is initiated. Firstly, construct local data sets from input-output data and get local models by using the least square (LS) method. Secondly, cluster local models according to the feature vectors and identify the parameter vectors of sub-models by weighted least squares (WLS) method. Thirdly, get the initial operating region partition by using a normalized exponential function, which is to partition the operating space completely. Finally, simultaneously determine the optimal parameter vectors of sub-models and the optimal operating region partition underlying the output-error minimization, which is executed by particle swarm optimization (PSO) algorithm. Simulation results demonstrate that the proposed method can improve model accuracy compared with two existing methods.

A Semi-Continuous PWA Model Based Optimal Control Method for Nonlinear Systems

To alleviate the mode mismatch of multiple model methods for nonlinear systems when completely discrete dynamical equations are adopted, a semi-continuous piecewise affine (SCPWA) model based optimal control method is proposed. Firstly, a SCPWA model is constructed where modes evolve in continuous time and continuous states evolve in discrete time. Thanks to this model, a piecewise affine (PWA) system can switch at any time instant whereas mode switching only occurs at sample instants when a completely discrete PWA model is adopted, which improves the prediction accuracy of multi-models. Secondly, the switching condition is relaxed such that operating subspaces have overlaps and switching condition parameters are introduced. As a consequence, an optimal control problem with fixed mode switching sequence is established. Finally, a SCPWA model based model predictive control (MPC) policy is designed for nonlinear systems. The convergence of the MPC algorithm is proved. Compared with widely used mixed logical dynamic (MLD) model based methods, the proposed method not only alleviates mode mismatch, but also lightens the computing burden, hence improves the control performance and reduces the computation time. Some numerical examples are provided as well to show the efficiency of the method.

Hybrid modelling and predictive control of utility-scale variable-speed variable-pitch wind turbines

Wind energy has proven to be the highest reliable source of renewables due to the maturity of the technology. Wind turbine (WT) systems are complex systems with undergoing development process that requires innovative methods of design and control. In this paper, WTs are studied through hybrid systems framework. Hybrid models of WT are extracted with representable dynamics from nonlinear complex design code. The WT model is formulated into a mixed-logical dynamical model (MLD) and a piecewise affine (PWA) model. Then, a receding horizon control strategy is applied to WT hybrid model resulting in a hybrid model predictive control (HMPC) with mixed-integer programming (MIP) problem. The performance of the proposed controller is compared against the baseline controller within a simulation environment for the National Renewable Energy Laboratory (NREL) 5MW benchmark WT as a case study. The analysis and investigation of HMPC highlight its capability as a potential tool for exploiting the control objectives of WT systems.

Modeling and Control of a Planetary Compound System Under External Magnetic Load

Abstract Many experimental setups that are used in characterizing gear and rotor dynamics employ noncontact hysteresis brakes to control the load torque as opposed to employing a clutch-type brake mechanism. Although hysteresis brakes are highly reliable maintenance-free torque resisting instruments, the presence of minor cogging torques within the brake is shown to physically mask the gear dynamics by causing a bistable region in the speed response function that is otherwise nonexistent. This paper investigates the dynamic characteristics of this experimental arrangement in detail and lays out a simulation-based tuning method for employing robust-adaptive sliding mode control (RASMC) to improve the speed response function of the gear drive. A global dynamic model is constructed from a set of piecewise-affine models that are experimentally validated over their applicable dynamic ranges. A proportional–integral (PI) controller is further developed and numerically tuned based on the global dynamics and is shown to compare well with the RASMC performance, testifying to the high fidelity of affine experimental models under limited information of the underlying analytical dynamics.

Machine learning–based piecewise affine model of wind turbines during maximum power point tracking

AbstractIn this paper, a discrete‐time piecewise affine (PWA) model of a wind turbine during Maximum Power Point Tracking (MPPT) region is identified. A clustering‐based identification method is utilized to create PWA maps for nonlinear aerodynamic torque and thrust force functions. This method exploits the combined use of clustering, pattern recognition, and parameter identification techniques. The well‐known K‐means clustering method is employed along with a perceptron‐based multiclassifier for pattern recognition and the least squared technique for parameter estimation. The identified maps are approximated the nonlinear static functions of the dynamic model of the wind turbine. Characteristics of a 5‐MW wind turbine are considered and the resulting model, which consists of 25 subregions is compared with the nonlinear dynamic model. Two test cases are studied in order to validate the presented model. Simulation results demonstrate the effectiveness and accuracy of the PWA model such that the response of the identified PWA model is fitted well to the nonlinear one. The PWA model identified in this paper can be widely used for advanced control systems design and long‐term performance and security assessment of the power grid.

On Convergence for Piecewise Affine Models of Gene Regulatory Networks via a Lyapunov Approach

In this paper, we analyse convergence and stability properties for piecewise affine models of gene regulatory networks, by means of piecewise quadratic Lyapunov functions. In the first part, we present the piecewise affine model of a gene regulatory network then, after proving a result on the existence of a minimal, attractive, and positively invariant box, we describe a linear matrix inequalitie framework to construct a Lyapunov function for the system. In the second part of the work, after a characterization of the system solutions when considering also isolated Zeno behavior of the trajectories, a monotonicity property of the so found Lyapunov function is formally proved, together with a result on the convergence of the trajectories, in the sense of measure. Finally, we present two examples, showing the applicability and utility of the main theoretical results previously explained.