Static Nonlinearity Research Articles

Output feedback model predictive control for Hammerstein model with bounded disturbance

This paper proposes two model predictive control (MPC) methods for a Hammerstein model with both unmeasurable state and bounded disturbance. First, a dynamic output feedback MPC is designed such that the dynamics of the state estimation error is decoupled from that of the estimated state. Hence, the separation principle holds and a part of the controller parameters can be designed off-line. Second, a case is considered where Hammerstein nonlinearity is not exactly inverted, that is, a reserved static nonlinearity exists in the closed-loop system. This reserved nonlinearity is merged into a polytopic description in combination with the linear dynamic part. The control move is then parameterized as a feedback law followed by the approximate inverse of Hammerstein nonlinearity. The recursive feasibility and closed-loop stability of both approaches are guaranteed. Numerical examples are given in order to show effectiveness of the proposed approaches.

Nonlinear Stability Analysis of the Classical Nested PI Control of Voltage Sourced Inverters

This note provides the first nonlinear analysis of the industry standard “partial decoupling plus nested PI loops” control of voltage sourced inverters. In spite of its enormous popularity, to date only linearization-based tools are available to carry out the analysis, which are unable to deal with large-signal stability and fail to provide estimates of the domain of attraction of the desired equilibrium. Instrumental to establish our result is the representation of the closed-loop dynamics in a suitable Lure-like representation, that is, a forward system in closed-loop with a static nonlinearity. The stability analysis is then done by generating an adequate Popov multiplier. Comparison with respect to linearization is discussed together with numerical results demonstrating non-conservativeness of the proposed conditions.

A Non-Parametric Approach for Identification of Parameter Varying Hammerstein Systems

This paper presents a novel method for non-parametric identification of parameter-varying (PV) Hammerstein systems where the parameters of both the static nonlinearity and the linear dynamics change with a scheduling variable. The proposed method estimates the individual elements of the PV Hammerstein system by describing the static nonlinear element using a PV Chebychev basis expansion and the linear dynamic element as a non-parametric PV impulse response function with Laguerre basis expansion. The method was validated using Monte-Carlo simulations of a PV Hammerstein model of ankle reflex stiffness during large movements. Results demonstrated that the method is simple, effective, and robust; it accurately identified the PV Hammerstein system in the presence of relatively large colored, time-varying measurement noise (average SNR of 15dB). These results demonstrate the two main contributions of the method: (1) It accurately and precisely estimates both the linear and nonlinear elements of the PV Hammerstein cascade as they vary with a scheduling variable; and (2) Models identified with the method accurately predict the response of the PV Hammerstein system to novel scheduling variable trajectories. To our knowledge, no other Hammerstein method can achieve these.

Adaptive Hammerstein Filtering via Recursive Non-Convex Projection

Hammerstein model with a static nonlinearity followed by a linear filter is commonly used in numerous applications. This paper focuses on adaptive filtering techniques for parameter identification of Hammerstein systems and output prediction of nonlinear systems. By formulating the underlying filtering problem as a recursive bilinear least-squares optimization with the non-convex feasible region constraint, we develop a recursive non-convex projected least-squares (RncPLS) algorithm based on alternating direction method of multipliers (ADMM). The RncPLS algorithm alternates between implementing ridge regression and projecting on the non-convex feasible set, which successively refines the system parameters. The convergence and accuracy properties of the proposed RncPLS algorithm are theoretically investigated. Moreover, extensive simulation results in the context of system identification, nonlinear predication, and acoustic echo cancellation, are also included to demonstrate the performance characteristics of the proposed algorithm.

Hybrid Predistorter for Broadband Power Amplifiers Linearization With Relaxed DAC Speed in the Signal Transmit Path

This paper proposes a novel predistorter architecture suitable for hybrid baseband/RF implementation. The distortion function is split into a memory polynomial and an AM/AM only look-up table (LUT). The memory polynomial DPD is to be applied in the baseband digital domain whereas the LUT predistorter is suitable for low complexity implementation in either the baseband digital or the RF analog domains. The proposed model was compared to benchmark models using 20MHz and 40MHz wide 5G test signals, and found to be superior in terms of adjacent channel leakage ratio performance and complexity. Experimental results showed that the model can achieve ACLR levels of better than 50dBC with a relatively low number of coefficients. The proposed model requires around 20% less coefficients than the reverse twin-nonlinear two-box model, and achieves slightly better ACLR while easing the analog implementation of the static nonlinearity predistortion function. Most importantly, the low-complexity hybrid implementation of the proposed model can drastically reduce the speed requirements in the digital to analog converter of the signal transmit path by up to 80% when compared to traditional DPD systems.

A multilayer-multiplexer network processing scheme based on the dendritic integration in a single neuron.

Advances in neuronal studies suggest that a single neuron can perform integration functions previously associated only with neuronal networks. Here, we proposed a dendritic abstraction employing a dynamic thresholding function that models the spatiotemporal dendritic integration process of a CA3 pyramidal neuron. First, we developed an input-output quantification process that considers the natural neuronal response and the full range of dendritic dynamics. We analyzed the IO curves and demonstrated that dendritic integration is branch-specific and dynamic rather than the commonly employed static nonlinearity. Second, we completed the integration model by creating a dendritic abstraction incorporating the spatiotemporal characteristics of the dendrites. Furthermore, we predicted the dendritic activity in each dendritic layer and the corresponding somatic firing activity by employing the dendritic abstraction in a multilayer-multiplexer information processing scheme comparable to a neuronal network. The subthreshold activity influences the suprathreshold regions via its dynamic threshold, a parameter that is dependent not only on the driving force but also on the number of activated synapses along the dendritic branch. An individual dendritic branch performs multiple integration modes by shifting from supralinear to linear then to sublinear. The abstraction includes synaptic input location-dependent voltage delay and decay, time-dependent linear summation, and dynamic thresholding function. The proposed dendritic abstraction can be used to create multilayer-multiplexer neurons that consider the spatiotemporal properties of the dendrites and with greater computational capacity than the conventional schemes.

Adaptive compound control based on generalized Bouc-Wen inverse hysteresis modeling in piezoelectric actuators.

This paper focuses on the study of the dynamic hysteresis compensation and control of piezoelectric actuators so as to improve the swing accuracy of the piezoelectric fast steering mirror mechanism in the photoelectric compound-axis control system. Moreover, in view of the rate dependence and asymmetry of piezoelectric hysteresis, and the complex inversion process of the generalized Bouc-Wen hysteresis model, the Hammerstein dynamic inverse hysteresis model of the piezoelectric actuator is established. To be specific, the static nonlinearity and rate dependence of the piezoelectric inverse hysteresis are represented by the generalized Bouc-Wen inverse model and the auto-regressive exogenous model, respectively, and the parameters of the model are identified by the adaptive beetle swarm optimization algorithm. In the process of the open-loop feedforward compensation, the dynamic positioning accuracy of the piezoelectric actuator is greatly affected by various disturbances and the uncertainty of the hysteresis compensation model. In this context, a compound control strategy that combines the feedforward compensation with the single-neuron adaptive proportion-integration-differentiation control is proposed based on the Hammerstein dynamic inverse hysteresis model of the piezoelectric actuator. The experimental results verify the effectiveness and superiority of the proposed control strategy.

Saddle-Point Convergence of Constrained Primal-Dual Dynamics

In this letter, we consider saddle-point convergence of primal-dual dynamics for inequality constrained convex optimization problems. By considering the primal-dual dynamics as the interconnection of (i) a gradient system and (ii) a nonlinear controller with incremental sector-bounded static nonlinearity, we establish asymptotic stability by invoking the classical notions of Lyapunov stability, the invariance principle, along with some regularity assumptions. For the special case of strongly convex problem with affine constraints, we prove global exponential convergence. As compared to existing techniques in the literature, the proposed approach offers simple and transparent tuning guideline for robust exponential stability of the saddle-point dynamics and allows for establishing an upper bound on the exponential decay rate.

Integral-Type Edge-Event- and Edge-Self-Triggered Synchronization to Multi-Agent Systems with Lur’e Nonlinear Dynamics

This paper proposes event- and self-triggered control strategies to achieve distributed synchronization for multiple Lur’e systems with unknown static nonlinearities. Firstly, the integral-type edge-event-triggered function is designed here without Zeno behaviors. Compared to the traditional event-triggered schemes, the considered algorithm has the advantages of reducing controller update frequency and sensor energy consumption. Then, the integral-type self-triggered is further investigated, which implements discontinuous monitoring and discontinuous agent listening. Finally, numerical simulations verified the effectiveness and superiority of our policies.

Mixed-mode Method Used for Pt100 Static Transfer Function Linearization

Abstract Pt100 is a resistance temperature detector characterized by a relatively linear resistance/temperature relationship in a narrow temperature range. However, the Pt100 sensor shows a certain degree of static transfer function nonlinearity of 4.42 % in the range between −200 °C and 850 °C, which is unacceptable for some applications. As a solution to this problem, a mixed-mode linearization method based on a special dual-stage piecewise linear ADC design is proposed in this paper. The first stage of the proposed dual-stage piecewise linear ADC is performed with a low-complex and low-power flash ADC of a novel sequential design. The novelty of the proposed sequential design is reflected in the fact that the number of employed comparators is equal to the flash ADC resolution. The second stage is performed with a delta-sigma ADC with a differential input and differential reference. Using the 6-bit flash ADC of novel design and the 24-bit delta-sigma ADC, the nonlinearity error is reduced to 2.6·10−3 %, in the range between −200 °C and 850 °C. Two more ranges are examined, and the following results are obtained: in the range between 0 °C and 500 °C, the nonlinearity error is reduced from 1.99 % to 5·10−4 %, while in the range between −50 °C and 150 °C, the nonlinearity error is reduced from 0.755 % to 2.15·10−4 %.

Identification of Hammerstein–Wiener models with hysteresis front nonlinearities

This paper deals with the identification of Hammerstein–Wiener models. The novelty lies in the fact that the front nonlinear block is allowed to be the memory of hysteresis type. The latter is any rate-independent hysteresis operator, that is characterised by closed limit cycles when excited by periodic input signals called ‘loading-unloading’ (or ‘increasing-decreasing’). Designing appropriate ‘loading-unloading’ periodic inputs will prove to be crucial in the proposed identification method. By using such signals, the intermediary linear subsystem is made transparent and the hysteresis front block is equivalent to two static nonlinearities (i.e. the lateral borders of the limit cycle). Then, the input and output signals turn out to be related by two composed functions. A decomposition algorithm is then resorted to extract the back nonlinearity and both lateral borders of the front hysteresis operator. Finally, the impulse response of the intermediary linear subsystem is determined using a least-squares like estimator. The quality of the estimated model is checked using a theoretical analysis and simulation results.

A novel LSSVM-L Hammerstein model structure for system identification and nonlinear model predictive control of CSTR servo and regulatory control

AbstractA continuous stirred tank reactor (CSTR) servo and the regulatory control problem are challenging because of their highly non-linear nature, frequent changes in operating points, and frequent disturbances. System identification is one of the important steps in the CSTR model-based control design. In earlier work, a non-linear system model comprises a linear subsystem followed by static nonlinearities and represented with Laguerre filters followed by the LSSVM (least squares support vector machines). This model structure solves linear dynamics first and then associated nonlinearities. Unlike earlier works, the proposed LSSVM-L (least squares support vector machines and Laguerre filters) Hammerstein model structure solves the nonlinearities associated with the non-linear system first and then linear dynamics. Thus, the proposed Hammerstein’s model structure deals with the nonlinearities before affecting the entire system, decreasing the model complexity and providing a simple model structure. This new Hammerstein model is stable, precise, and simple to implement and provides the CSTR model with a good model fit%. Simulation studies illustrate the benefit and effectiveness of the proposed LSSVM-L Hammerstein model and its efficacy as a non-linear model predictive controller for the servo and regulatory control problem.

Case Studies on Finite Element Modeling of Welded Joints

Abstract Joints made out of tubular steel members are often used in industrial constructions, like offshore, trusses or tree-shaped columns. The joint area is the weakest point in a truss structure. Beside the normal static behaviour many non-linearities due to the geometry or to the welding process have to take into account. Steel hollow sections are often used. Much research was made on welded nodes so far, but mostly it is valid for limited geometries or load cases. A review of the main aspects in designing welded nodes are given. The second part is about a study of two aspects. The first aim is to investigate the influence of the joint angle on the resistance of a T-joint made out of Circular-Hollow-Sections or Squared-Hollow-Sections under a tensional load. To archive the most realistic results, the second aspect, a numerical study on the finite element types and the element shape functions will be made.

An Enhanced Equivalent Circuit Model With Real-Time Parameter Identification for Battery State-of-Charge Estimation

This article introduces an efficient modeling approach based on the Wiener structure to reinforce the capacity of classical equivalent circuit models (ECMs) in capturing the nonlinearities of lithium-ion (Li-ion) batteries. The proposed block-oriented modeling architecture is composed of a simple linear ECM followed by a static output nonlinearity block, which helps achieving a superior nonlinear mapping property while maintaining the real-time efficiency. The observability of the established battery model is analytically proven. This article also introduces an efficient parameter estimator based on extended-kernel iterative recursive least squares algorithm for real-time estimation of the parameters of the proposed Wiener model. The proposed approach is applied for state-of-charge (SoC) estimation of 3.4-Ah 3.6-V nickel-manganese-cobalt-based Li-ion cells using the extended Kalman filter (EKF). The results show about 1.5% improvement in SoC estimation accuracy compared with the EKF algorithm based on the second-order ECM. A series of real-time tests are also carried out to demonstrate the computational efficiency of the proposed method.

The influence of materials on the behaviour of joints with multiple bar connections

Joints made out of tubular steel members are often used in industrial constructions, like offshore, trusses or tree-shaped columns. The joint area is the weakest point in a truss structure. Beside the normal static behaviour many non-linearities due to the geometry or to the welding process have to be taken into account. Steel hollow sections are often used. Much research was made on welded nodes so far. To increase the resistance of a joint it is an option to vary the material choice. Many variations were tested in the past. It is possible to change the overall material of a joint or using composite structures, like a concrete filled column. Or the steel of the welding line can get an additional material component. Another aspect is to reinforce the joint for example by a stiffened ring. In this paper a state-of-the-art review of the influence of different materials to the resistance of welded nodes is given.

Joint tracking coefficients and the time delay of a nonstationary linear system, preceded by a static nonlinearity

SummaryThe paper considers a block‐oriented nonlinear Hammerstein system that consists of a linear block with real‐valued time‐varying unknown coefficients and a pure time delay, preceded by the static nonlinearity. The joint tracking of linear block coefficients and the time delay while processing observation samples is under consideration. The difficulty of the tracking process consists in the initial nonconvexity of the objective function to be minimized, in respect of the time delay. To obtain current estimates of the true coefficients and the time delay it is necessary to overcome the multiextremality of the objective function. A global optimization problem could be solved by changing a form of the nonconvex objective function into a convex one. The optimization problem has been subdivided into two problems that are analyzed in this article, as well. The first one consists of removal of the objective function multiextremality in respect of the time delay, while hunting for the global extremum. Therefore, the well‐known approach based on the method of corrective operators, which is used to transform the multiextremal criterion into a unimodal function for the nonstationary Wiener system, is developed here for the Hammerstein one. The other problem consists of the applicability of unified adaptive algorithms, used for Wiener systems in the case of Hammerstein systems. The recursive parametric identification technique, based on the adaptive algorithm procedures, is developed here for the nonstationary block‐oriented Hammerstein systems, too. The applicability of algorithms is supported by various simulation tests solved by computer for distinct inputs and for two static nonlinearities.

Robust Cooperative Control of Isolated AC Microgrids Subject to Unreliable Communications: A Low-Gain Feedback Approach

Inherent nonlinearities and uncertainties are inseparable parts of the real-life engineering systems, which make theoretical analyzes and control system designs more challenging. Usually, most of the physical parts in control systems such as controllers and actuators are subjected to saturation. On the other hand, uncertainties are ubiquitous in the physical systems, which demand some necessities in the design of the control system. Additionally, unreliability in communication networks is another practical constraint in multiagent systems, which is required to be considered. In this article, based on the mentioned issues, a distributed robust algorithm is proposed for the semiglobal stabilization of the voltage and frequency of isolated (islanded) ac microgrids considering parameters uncertainties, static nonlinearity of actuators, and unreliability in the communication network. In this regard, by employing an adaptive low-gain feedback protocol, the robust performance of the closed-loop system is guaranteed. Finally, to evaluate the proposed control strategy’s performance, digital time-domain simulations are carried out on a test microgrid system in MATLAB/Simulink environment, and the results are compared with several previously reported methods. Simulation results and comparison with previous works reveal the proposed method’s effectiveness and accuracy in regulating the microgrid voltage/frequency and providing accurate proportional active power sharing.

Identification of static nonlinearities by sinusoidal excitation with variable DC offsets.

When identifying nonlinear systems with input-output measurements, a suitable test signal must be selected. Nonlinear systems are almost always in a cascade with linear systems, i.e., a Wiener-Hammerstein type system cascade. A suitable test signal is preferably less influenced by the linear systems and is therefore sinusoidal, if time-varying signals are required for the measurement principle, e.g., for induction or vibration measurements. Then, a sinusoidal excitation with different DC offsets is a suitable signal to analyze a static nonlinear system in a Wiener-Hammerstein type cascade by measuring the cascade output at higher harmonics of the input frequency in a steady state, e.g., by using sensitive lock-in techniques. To calculate the cascade output given the input signal or to reconstruct the static nonlinear system also given the output signal, the transfer function of the DC offset at the nonlinear system input to the higher harmonics at the nonlinear system output is required. Those transfer functions are calculated here with emphasis on the first harmonic component. The reconstruction of a static nonlinear system is demonstrated in a simple simulation scenario by inverse filtering, i.e., deconvolution, with the derived transfer function. It is pointed out that a commonly made small signal assumption to the test signal is bypassed with the deconvolution method, which can lead to more precise measurements in applications due to a higher signal-to-noise ratio at the cascade output.

Hammerstein Box-Jenkins System Identification of the Cascaded Tanks Benchmark System

A common process control application is the cascaded two-tank system, where the level is controlled in the second tank. A nonlinear system identification approach is presented in this work to predict the model structure parameters that minimize the difference between the estimated and measured data, using benchmark datasets. The general suggested structure consists of a static nonlinearity in cascade with a linear dynamic filter in addition to colored noise element. A one-step ahead prediction error-based technique is proposed to estimate the model. The model is identified using a separable least squares optimization, where only the parameters that appear nonlinearly in the output of the predictor are solved using a modified Levenberg–Marquardt iterative optimization approach, while the rest are fitted using simple least squares after each iteration. Finally, MATLAB simulation examples using benchmark data are included.

Nonfragile H ∞ Stabilizing Nonlinear Systems Described by Multivariable Hammerstein Models

This paper presents the problem of robust and nonfragile stabilization of nonlinear systems described by multivariable Hammerstein models. The objective is focused on the design of a nonfragile feedback controller such that the resulting closed-loop system is globally asymptotically stable with robust H ∞ disturbance attenuation in spite of controller gain variations. First, the parameters of linear and nonlinear blocks characterizing the multivariable Hammerstein model structure are separately estimated by using a subspace identification algorithm. Second, approximate inverse nonlinear functions of polynomial form are proposed to deal with nonbijective invertible nonlinearities. Thereafter, the Takagi–Sugeno model representation is used to decompose the composition of the static nonlinearities and their approximate inverses in series with the linear subspace dynamic submodel into linear fuzzy parts. Besides, sufficient stability conditions for the robust and nonfragile controller synthesis based on quadratic Lyapunov function, H ∞ criterion, and linear matrix inequality approach are provided. Finally, a numerical example based on twin rotor multi-input multi-output system is considered to demonstrate the effectiveness.