Piecewise Nonlinear Model Research Articles

Piecewise non-linear model predictive control with bounded disturbance

This work addresses the problem of robust [Formula: see text] model predictive control for constrained piecewise non-linear systems corrupted by norm-bounded disturbances for the first time. In this approach, the system can have different operating points with different subregions. In each subregion, the system is made up of an affine model disturbed by an additive non-linear term that is locally Lipschitz. The employment of the piecewise non-linear model leads to a non-convex optimization problem, which is far more difficult to solve. It is also much more challenging to develop the [Formula: see text] model predictive control for piecewise non-linear systems. The proposed method introduces a control strategy in the form of a convex optimization problem subject to linear matrix inequality constraints, with the ability to minimize the L2 gain between the disturbance input and the controlled output. The proposed controller guarantees system stability with a prescribed [Formula: see text] disturbance attenuation level under switching between subregions. Simulations on a highly non-linear chemical process are conducted to demonstrate the efficacy of the proposed approach.

Non‐linear activation function approximation using a REMEZ algorithm

Here a more accurate piecewise approximation (PWA) scheme for non-linear activation function is proposed. It utilizes a precision-controlled recursive algorithm to predict a sub-range; after that, the REMEZ algorithm is used to find the corresponding approximation function. The PWA realized in three ways: using first-order functions, that is, piecewise linear model, second-order functions (piecewise non-linear model), and hybrid-order model (a mixture of first-order and second-order functions). The hybrid-order approximation employs the second-order derivative of non-linear activation function to decide the linear and non-linear sub-regions, correspondingly the first-order and second-order functions are predicted, respectively. The accuracy is compared to the present state-of-the-art approximation schemes. The multi-layer perceptron model is designed to implement XOR-gate, and it uses an approximate activation function. The hardware utilization is measured using the TSMC 0.18-μm library with the Synopsys Design Compiler. Result reveals that the proposed approximation scheme efficiently approximates the non-linear activation functions.

Multi-Rate Control Design Leveraging Control Barrier Functions and Model Predictive Control Policies

In this letter we present a multi-rate control architecture for safety critical systems. We consider a high level planner and a low level controller which operate at different frequencies. This multi-rate behavior is described by a piecewise nonlinear model which evolves on a continuous and a discrete level. First, we present sufficient conditions which guarantee recursive constraint satisfaction for the closed-loop system. Afterwards, we propose a control design methodology which leverages Control Barrier Functions (CBFs) for low level control and Model Predictive Control (MPC) policies for high level planning. The control barrier function is designed using the full nonlinear dynamical model and the MPC is based on a simplified planning model. When the nonlinear system is control affine and the high level planning model is linear, the control actions are computed by solving convex optimization problems at each level of the hierarchy. Finally, we show the effectiveness of the proposed strategy on a simulation example, where the low level control action is updated at a higher frequency than the high level command.

Ileal digestibility and endogenous protein losses of milk replacers based on whey proteins alone or with an enzyme-treated soybean meal in young dairy calves

Our objective was to measure and compare apparent ileal digestibility, standard ileal digestibility, and true ileal digestibility of crude protein (CP) and amino acids (AA) in milk replacers (MR) containing all milk proteins (WPC) or with 50% of the CP provided by an enzyme-treated soybean meal (ESBM). A T-cannula was placed in the ileum of 9 Holstein calves (8 males and 1 freemartin female) at approximately 15 d of age. After 2 wk postsurgery, calves were randomly assigned to a 3 × 3 replicated Latin square with 5-d periods. Calves were fed twice daily at a rate of 2% (dry matter) of body weight (1.25 kg/d on average), reconstituted to 15% solids, and adjusted weekly. No starter was offered to minimize rumen development. Digesta samples were collected continuously during 12 h on d 4 and 5 of each period. Basal endogenous losses of AA and CP were estimated by feeding an N-free MR to each calf during 1 period. Total endogenous losses (basal + specific; ENDtotal) were estimated by multivariate regression of the chi-squared distances between digesta and reference protein AA profiles. Ileal digesta pH with the ESBM diet was lower than that with the WPC diet. According to the piecewise nonlinear model of pH fluctuation, digesta pH during ESBM decreased more slowly after feeding and reached its nadir later than with the WPC diet. Diet did not affect average daily gain, but calves on the ESBM diet showed a bigger increment of withers height and lower mean fecal scores. The basal endogenous losses of AA and CP were 13.9 ± 1.1 and 22.4 ± 1.1 g/kg of dry matter intake, respectively. The estimated ENDtotal of AA and CP was higher with ESBM than with WPC. Accordingly, apparent ileal digestibility and standard ileal digestibility of most AA, CP, and total AA were lower or tended to be lower with ESBM. However, true ileal digestibility did not differ between diets for CP and all AA except Ala and Ile, which were greater with WPC, and Arg, which tended to be greater with ESBM. In agreement with the estimated differences in ENDtotal, we found that flows of digesta DNA and crude mucin were greater with ESBM. Substitution of 50% of the protein from whey with enzymatically treated soybean meal did not affect major nutrient digestibility or calf growth and even improved fecal consistency. Adjusting digestibilities of CP and AA in MR by endogenous protein losses is crucial when comparing bioavailability of alternative proteins and milk proteins.

Piecewise Nonlinear Dynamic Modeling for Gear Transmissions With Rotary Inertia and Backlash

Establishing an accurate dynamic model for gear transmission is challenging because of the complex dynamic mechanical behavior of the gear transmission, particularly under difficult working conditions such as heavy loads and frequent starts and stops. This feature significantly increases the difficulty level of the dynamic analysis, control, and real-time fault diagnosis of such a gear transmission, given that sufficiently accurate dynamic models are indispensable for them. Thus, this study proposes a piecewise nonlinear model for gear transmissions by considering rotary inertia, backlash, and friction in control applications under difficult working conditions. The model parameters of gear transmissions can be calculated by using analytical calculation formulas. The experiments are conducted for the gear transmission system of a heavy-duty manipulator in start–stop and continuous rotation conditions. The experimental results show that the proposed piecewise nonlinear model is in good agreement with the practical dynamic gear transmission behavior, which is feasible for the control applications of gear transmissions under difficult working conditions.

Damage mechanism of composite cemented backfill based on complex defects influence

AbstractIn order to reduce mining waste discharge and realize efficient application of tailing material in underground backfilling, the characteristics and mechanism of backfill damage is analyzed, and also the composed behavior and failure characteristic of cemented backfill. Five types of backfills were tested under uniaxial compression loading in SHT4206 electro‐hydraulic servo testing system. Based on the experimental results, the damage constitutive equations of cemented backfills with five cement−tailing ratios were proposed on the basis of damage mechanics and then validated. Research shows that the damage growth rate of backfills with lower cement−tailing is lower before peak value of stress, but it grows more rapidly after that. In addition, four backfill combination schemes were designed for mechanical test. Combining with research results on fracture characteristic of different backfills, a defects coupled constitutive model based on continuum damage mechanics was established. Research shows that the piecewise nonlinear model can well embodies effect of complex defects in backfills on the stress‐strain curve. Accuracy of the model is well verified by measured curve of backfill composite members. The results of this work provide a scientific basis for failure process prediction of backfills and reasonable matching design of framed filling.

Multiplicative non-Gaussian noise and additive Gaussian white noise induced transition in a piecewise nonlinear model

We study the transition problems in a piecewise nonlinear model induced by correlated multiplicative non-Gaussian noise and additive Gaussian white noise. Firstly, applying the path integral approach, the unified colored noise approximation, the analytical expression of the steady-state probability density function (SPD) is derived. Then the change regulation of the SPD is analyzed with the change of the strength and relevance of multiplicative noise and additive noise. From numerical computations we obtain some new nonlinear phenomena: the transition can be induced by the cross-correlation strength between noises, the non-Gaussian noise intensity and the Gaussian noise intensity as well as the non-Gaussian noise deviation parameter. This indicates that the effect of the non-Gaussian noise intensity on SPD is the same as that of the Gaussian noise intensity. Moreover, we also find the correlation time of the non-Gaussian noise can not induce the transition.

Stochastic resonance in a piecewise nonlinear model driven by multiplicative non-Gaussian noise and additive white noise

The phenomenon of stochastic resonance (SR) in a piecewise nonlinear model driven by a periodic signal and correlated noises for the cases of a multiplicative non-Gaussian noise and an additive Gaussian white noise is investigated. Applying the path integral approach, the unified colored noise approximation and the two-state model theory, the analytical expression of the signal-to-noise ratio (SNR) is derived. It is found that conventional stochastic resonance exists in this system. From numerical computations we obtain that: (i) As a function of the non-Gaussian noise intensity, the SNR is increased when the non-Gaussian noise deviation parameter q is increased. (ii) As a function of the Gaussian noise intensity, the SNR is decreased when q is increased. This demonstrates that the effect of the non-Gaussian noise on SNR is different from that of the Gaussian noise in this system. Moreover, we further discuss the effect of the correlation time of the non-Gaussian noise, cross-correlation strength, the amplitude and frequency of the periodic signal on SR.

A piecewise nonlinear model for a production system under maintenance, trade credit and limited warehouse space

Small and medium size manufacturers exist commonly in developing countries. Limited warehouse space is a characteristic in small and medium size manufacturers. In today’s business environment, many upstream companies offer a credit period to downstream companies. In production system, a manufacturer usually adopts the system maintenance when the system is in the out-of-control state. Therefore, this paper considers a production-inventory model for a manufacturer under system maintenance, trade credit and limited warehouse space. The objective is to determine the optimal production run time to minimise the total cost. We develop an algorithm based on several theorems for solving the problem described. We provide several examples to illustrate the solution procedure and discuss how system parameters affect the manufacturer’s decision behaviour. Computational analysis demonstrates that the results of the proposed model are consistent with economic insights.

Active fault tolerant control of piecewise affine systems with reference tracking and input constraints

SUMMARYAn active fault tolerant control (AFTC) method is proposed for discrete‐time piecewise affine (PWA) systems. Only actuator faults are considered. The AFTC framework contains a supervisory scheme, which selects a suitable controller in a set of controllers such that the stability and an acceptable performance of the faulty system are held. The design of the supervisory scheme is not considered here. The set of controllers is composed of a normal controller for the fault‐free case, an active fault detection and isolation controller for isolation and identification of the faults, and a set of passive fault tolerant controllers (PFTCs) modules designed to be robust against a set of actuator faults. In this research, the piecewise nonlinear model is approximated by a PWA system. The PFTCs are state feedback laws. Each one is robust against a fixed set of actuator faults and is able to track the reference signal while the control inputs are bounded. The PFTC problem is transformed into a feasibility problem of a set of LMIs. The method is applied on a large‐scale live‐stock ventilation model. Copyright © 2013 John Wiley & Sons, Ltd.

Comparative study between the PNL method and a MN in modelling fatigue of composite materials

ABSTRACTThe aim of this study was to compare the results obtained using the piecewise nonlinear model with those of a modular network (MN) architecture in modelling fatigue behaviour. We used a database consisting of ten materials derived from the literature and represented by the following acronyms: C10, C12, DD16, HTA‐913, IM7–977, MAT(0)2, T800–5245, T800H‐3631, QQ1 and QQ1T. The results obtained by MN were compared with those of the PNL. The results for both the PNL and MN model were satisfactory for the fatigue behaviour of glass fiber‐based composite materials, but for carbon fiber‐based composites only MN showed satisfactory results.

Calcium wave propagation in chains of endothelial cells with nonlinear reaction dynamics: Green’s function approach

We present a Green's function-based perturbative approach to solving nonlinear reaction-diffusion problems in networks of endothelial cells. We focus on a single component (Ca2+), piecewise nonlinear model of endoplasmic calcium dynamics and trans-membrane diffusion. The decoupling between nonlinear reaction dynamics and the linear diffusion enables the calculation of the diffusion part of the Green's function for network of cells with nontrivial topologies. We verify analytically and then numerically that our approach leads to the known transition from propagation of calcium front to failure of propagation when the diffusion rate is varied relative to the reaction rates. We then derive the Green's function for a semi-infinite chain of cells with various boundary conditions. We show that the calcium dynamics of cells in the vicinity of the end of the semi-infinite chain is strongly dependent on the boundary conditions. The behavior of the semi-infinite chain with absorbing boundary conditions, a simple model of a multicellular structure with an end in contact with the extracellular matrix, suggests behavioral differentiation between cells at the end and cells embedded within the chain.

Use of Piecewise Nonlinear Models to Estimate Variable Size‐Related Mortality Rates

AbstractI present a method and examples of a piecewise function in nonlinear regression to estimate size‐related differential mortality using age‐ and length‐based models for data collected from a Tennessee River (Alabama) population of blue catfish Ictalurus furcatus. The model is similar to catch curve regression but computes two slope coefficients that can be used to estimate mortality; these slopes meet at a joint (or knot) that estimates the age or length at which differences in mortality occur. The model is described as loge(Ni) = b0 − (b1 + b2)ti, where Ni is the number of fish collected for the ith age or length interval; ti is the age or relative age of the ith age or length interval; b0 is the intercept coefficient; b1 is the slope coefficient and is equal to b*1 when ti is less than or equal to the knot and 0 otherwise; b2 is a second slope coefficient and is equal to b*1 − b*2 when ti is greater than the knot and 0 otherwise; and b*1 and b*2 are estimated by the model. For these data, fish ranged in age from 3 to 25 years and ranged in total length (TL) from 301 to 1,291 mm. With this piecewise nonlinear approach, the fit between age or length and loge(Ni) improved over that determined with linear catch curves. For both age‐ and length‐based structure data, annual mortality for small blue catfish ranged from 0.41 to 0.49 for younger and smaller fish (age ≤ 10 years; TL ≤ 684 mm). For larger, older fish (age = 11–25 years; TL > 685 mm), annual mortality estimates ranged from 0.08 to 0.16. Observations of the fishery indicated that anglers harvested smaller blue catfish (<562 mm TL); harvest of larger fish was nil, which probably accounted for the size‐related differences in mortality. I provide a computer program to derive estimates of size‐related mortality using piecewise nonlinear regression.

Piecewise nonlinear model for financial time series forecasting with artificial neural networks

This study proposes a piecewise nonlinear model based on the segmentation of financial time series. The basic concept of proposed model is to obtain intervals divided by change points, to identify them as change-point groups, and to use them in the forecasting model. The proposed model consists of two stages. The first stage detects successive change points in time series dataset and forecasts change-point groups with backpropagation neural networks (BPNs). In this stage, the following three change-point detection methods are applied and compared: the parametric method, the nonparametric approach, and the model-based approach. The next stage forecasts the final output with BPN using the groups. This study applies the proposed model to interest rate forecasting and examines three different models based on various change point detection methods. The experimental result shows that the proposed models outperforms conventional neural network model.

Analyzing stock market tick data using piecewise nonlinear model

Trading in stock market indices has gained unprecedented popularity in major financial markets around the world. However, the prediction of stock price index is a very difficult problem because of the complexity of the stock market data. This study proposes stock trading model based on chaotic analysis and piecewise nonlinear model. The core component of the model is composed of four phases: The first phase determines time-lag size in input variables using chaotic analysis. The second phase detects successive change-points in the stock market data and the third phase forecasts the change-point group with backpropagation neural networks (BPNs). The final phase forecasts the output with BPN. The experimental results are encouraging and show the usefulness of the proposed model with respect to profitability.