Observer Design Problem Research Articles

Observer Design for Fractional-Order Polynomial Fuzzy Systems Depending on a Parameter

For fractional-order systems, observer design is remarkable for the estimation of unavailable states from measurable outputs. In addition, the nonlinear dynamics and the presence of parameters that can vary over different operating conditions or time, such as load or temperature, increase the complexity of the observer design. In view of the aforementioned factors, this paper investigates the observer design problem for a class of Fractional-Order Polynomial Fuzzy Systems (FORPSs) depending on a parameter. The Caputo–Hadamard derivative is considered in this study. First, we prove the practical Mittag-Leffler stability, using the Lyapunov methods, for the general case of Caputo–Hadamard Fractional-Order Systems (CHFOSs) depending on a parameter. Secondly, based on this stability theory, we design an observer for the considered class of FORPSs. The state estimation error is ensured to be practically generalized Mittag-Leffler stable by solving Sum Of Squares (SOSs) conditions using the developed SOSTOOLS.

Interval observers for hybrid continuous-time stationary systems

The problem of interval observer design for hybrid continuous-time stationary systems under external disturbances and measurement noise is studied. It is assumed that continuous-time dynamic of such systems is described by linear or nonlinear differential state equations with linear function of output. System parameters depend on system states and are switched based on the control system with finite set of states. The relations allowing designing hybrid interval observer of minimal dimension estimating the set of admissible values of the prescribed linear vector function of the system states are derived. To solve the problem, algebra of partitions and linear algebra are used. Theoretical results are illustrated by example.

Indirect adaptive observer control (I-AOC) design for truck–trailer model based on T–S fuzzy system with unknown nonlinear function

Tracking is a crucial problem for nonlinear systems as it ensures stability and enables the system to accurately follow a desired reference signal. Using Takagi–Sugeno (T–S) fuzzy models, this paper addresses the problem of fuzzy observer and control design for a class of nonlinear systems. The Takagi–Sugeno (T–S) fuzzy models can represent nonlinear systems because it is a universal approximation. Firstly, the T–S fuzzy modeling is applied to get the dynamics of an observational system in order to estimate the unmeasurable states of an unknown nonlinear system. There are various kinds of nonlinear systems that can be modeled using T–S fuzzy systems by combining the input state variables linearly. Secondly, the T–S fuzzy systems can handle unknown states as well as parameters known to the indirect adaptive fuzzy observer. A simple feedback method is used to implement the proposed controller. As a result, the feedback linearization method allows for solving the singularity problem without using any additional algorithms. A fuzzy model representation of the observation system comprises parameters and a feedback gain. The Lyapunov function and Lipschitz conditions are used in constructing the adaptive law. This method is then illustrated by an illustrative example to prove its effectiveness with different kinds of nonlinear functions. A well-designed controller is effective and its performance index minimizes network utilization—this factor is particularly significant when applied to wireless communication systems.

Partial-Neurons-Based Proportional-Integral Observer Design for Artificial Neural Networks: A Multiple Description Encoding Scheme.

This article is concerned with a new partial-neurons-based proportional-integral observer (PIO) design problem for a class of artificial neural networks (ANNs) subject to bounded disturbances. For the purpose of improving the reliability of the data transmission, the multiple description encoding mechanisms are exploited to encode the measurement data into two identically important descriptions, and the encoded data are then transmitted to the decoders via two individual communication channels susceptible to packet dropouts, where Bernoulli-distributed stochastic variables are utilized to characterize the random occurrence of the packet dropouts. An explicit relationship is discovered that quantifies the influences of the packet dropouts on the decoding accuracy, and a sufficient condition is provided to assess the boundedness of the estimation error dynamics. Furthermore, the desired PIO parameters are calculated by solving two optimization problems based on two metrics (i.e., the smallest ultimate bound and the fastest decay rate) characterizing the estimation performance. Finally, the applicability and advantage of the proposed PIO design strategy are verified by means of an illustrative example.

Stabilization of coupled delayed nonlinear time fractional reaction diffusion systems using sampled‐in‐space sensing and actuation

AbstractThis paper is considered with the asymptotic stabilization of coupled delayed nonlinear time fractional reaction diffusion systems (FRDSs) governed by fractional parabolic partial differential equations (PDEs) with space‐dependent coefficients under sampled‐data in space control. It is assumed that state measurements can be averaged measurements (AMs) or point measurements (PMs), and a finite number of sensing and actuation devices are located in a spaced manner along the spatial domain of the interest. With the proposed sampled‐data in space controller, the closed‐loop stability is obtained. Tuning rules of system parameters and control parameters are derived using the fractional Halanay's inequality and the fractional Lyapunov method. Subsequently, the dual problem of observer design is formulated. Fractional examples are used to valid the theoretical result. Discussions on the extension of sampled‐data boundary feedback stabilization are provided finally.

On observer compensator design for non-autonomous control semi-linear evolution equations

This paper investigates the Luenberger observer design problem for non-autonomous control semilinear evolution equations with disturbances in Banach spaces. Then, the practicalstabilization problem of the system is solved, yielding a compensator based on the Luenberger observer by using integral inequalities of the Gronwall type. Sufficient conditions of the controller and observer problem are satisfied, we show that the proposed controller with estimatedstate feedback from the proposed practical Luenberger observer will achieve global practical stabilization. We develop novel ideas and techniques, which present the further development of mathematical control theory. Furthermore, an example is given to show the applicability of our theoretical results.

Interval observer design for nonlinear discrete‐time dynamic systems

SummaryThe problem of interval observer design for systems described by nonlinear models with uncertainties is considered. The problem is solved based on special mathematical technique (the algebra of functions) which allows obtaining solution for systems containing non‐smooth nonlinearities. The relations to design interval observer insensitive or having minimal sensitivity to the disturbance and estimating the prescribed nonlinear function of the state vector of the original system are derived. Theoretical results are illustrated by example.

Interval estimation in discrete-time linear systems with parametric uncertainties

The problem of interval observer design for discrete-time linear systems under the external disturbances, measurement noise, and parametric uncertainties is studied. The relation allowing designing the interval observer of minimal dimension estimating the set of admissible values of the specified linear vector function of the system state are derived. Theoretical results are illustrated by the example.

Interval Observer Design for Fault Diagnosis

The problem of interval observer design for systems described by linear models under the external disturbances is considered. The problem is solved based on the reduced-order model of the original system of minimal dimension. The relations allowing designing interval observer insensitive or having minimal sensitivity to the disturbance are obtained. The designed interval observers are used to solve the problem of fault diagnosis. Theoretical results are illustrated by example.

Stability of conformable fractional time-varying linear systems

In this paper, we investigate the exponential and asymptotic stability analysis of conformable fractional linear time-varying systems. We propose sufficient conditions for testing the asymptotic and exponential stability of conformable differential systems by using the notion of stable functions and differential Lyapunov inequalities. Furthermore, we apply the developed approach to the problem of conformable stabilisation and observer design for conformable fractional linear time-varying systems.

Observer Design for a Lorenz Oscillator

This article addresses the design problem of observers for the oscillator, focusing on the nonlinearities present in the model, which arise from the product between components of the state vector. These nonlinearities are particularly interesting because they allow obtaining an error dynamics that can be used to study the convergence properties of the observation scheme. In this sense, the design of a full-order observer is proposed, which is characterized by being a replica of the original system, with the addition of a correction term. Additionally, a numerical evaluation of the presented results is performed.

Observer-based safety monitoring of nonlinear dynamical systems with neural networks via quadratic constraint approach

The safety monitoring for nonlinear dynamical systems with embedded neural network components is addressed in this paper. The interval-observer-based safety monitor is developed consisting of two auxiliary neural networks derived from the neural network components of the dynamical system. Due to the presence of nonlinear activation functions in neural networks, we use quadratic constraints on the global sector to abstract the nonlinear activation functions in neural networks. By combining a quadratic constraint approach for the activation function with Lyapunov theory, the interval observer design problem is transformed into a series of quadratic and linear programming feasibility problems to make the interval observer operate with the ability to correctly estimate the system state with estimation errors within acceptable limits. The applicability of the proposed method is verified by simulation of the lateral vehicle control system.

Design of unknown input observer for discrete-time Markov jump systems with unknown input in both state equation and output equation

The problem of unknown input observer design for discrete-time nonlinear generalized Markov jump systems is studied. First, like a normal system, the whole nonlinear system is transformed into a local linear system, and then a large number of linear system theories can be applied to solve related problems. Second, in the observer design of general discrete-time Markov jump systems, only the unknown input in the state equation is usually considered. In this paper, the unknown input is considered in both the state equation and the output equation. The state estimation error system is derived by defining the error. The non-uniform Lyapunov functional is selected to stabilize the estimation error system using the Lyapunov theory. The sufficient conditions for the stability of the system are obtained and transformed into the feasibility problem of linear matrix inequality. The problem of unknown input observer design for discrete-time nonlinear generalized Markov jump systems is solved using MATLAB software. Finally, a numerical example of two rules and two modes is used to verify the effectiveness and feasibility of the proposed unknown input observer.

Unknown Input Observer Design for a Class of Semilinear Hyperbolic Systems With Dynamic Boundary Conditions

The problem of unknown input observer design is considered for coupled PDE/ODE systems subject to incremental sector bounded nonlinearities and unknown boundary inputs. Assuming available measurements at the boundary of the distributed domain, the synthesis of the unknown input observer is based on Lyapunov methods and recast in terms of some matrix inequalities. Numerical simulations support and confirm the theoretical findings, illustrating the robust estimation performances of the proposed nonlinear unknown input observer.

Adaptive Neural Network-Based Observer Design for Switched Systems With Quantized Measurements.

This study is concerned with the adaptive neural network (NN) observer design problem for continuous-time switched systems via quantized output signals. A novel NN observer is presented in which the adaptive laws are constructed using quantized measurements. Then, persistent dwell time (PDT) switching is considered in the observer design to describe fast and slow switching in a unified framework. Accurate estimations of state and actuator efficiency factor can be obtained by the proposed observer technique despite actuator degradation. Finally, a simulation example is provided to illustrate the effectiveness of the developed NN observer design approach.

Fully predefined‐time distributed observer design for second‐order strict‐feedback nonlinear systems

AbstractDistributed observer is an effective tool to deal with leader‐following control problems. This article investigates the fully predefined‐time distributed observer (FPTDO) design problem of a generalized second‐order strict‐feedback nonlinear leader system. Firstly, given the system parameters and the upper bounds of the system nonlinearities, an FPTDO is proposed to provide the state estimates. Secondly, considering that the time‐varying system parameters and nonlinear functions of the leader system are unknown to each follower, an FPTDO is developed to estimate the time‐varying system parameters, nonlinearities and full state of the leader. The proposed FPTDOs exhibit the following advantages: (1) the observation errors converge to zero within a finite time that is predefined by only one design parameter independent of the initial conditions; (2) the designs of the FPTDOs do not require any information of the communication topology (Laplacian matrix); and (3) the exact models of the nonlinear functions are not required by each follower. Finally, simulation examples are given to demonstrate the effectiveness of the proposed FPTDOs.

On the sample complexity of observers for unknown linear systems with biased dynamics estimations

AbstractObservers have broad applications in power systems, whereas observer models are hard to obtain when the system is unknown. This article considers the observer design problems for unknown noisy linear time‐invariant systems based on biased dynamics estimations. Unlike the unbiased methods, biased methods have stronger generalization ability, which benefits obtaining stable estimations for noisy systems. However, the biased estimation's influence on observer design still needs to be investigated. To analyze the influence, we exploit the impact of estimation bias‐variance trade‐off to observer design. Specifically, we propose a support vector regression (SVR) based estimator to provide biased estimations for the system identification of unknown linear systems. The sample complexity results of SVR with bias‐variance trade‐offs are analyzed and used for observer design and performance analysis. Then, a stable observer gain design algorithm is developed based on biased estimation. The observation performance is evaluated by the mean square observation error, which is shown to be adjustable by tuning the trade‐off between bias and variance, thus achieving higher scalability than the unbiased methods. Finally, observing performance analysis demonstrates the influence of the bias‐variance trade‐off for the observer. Extensive simulation validations are conducted to verify the computed estimation error and performance optimality with different bias‐variance trade‐offs and noise settings.

Proportional-Integral Observer Design for Multirate-Networked Systems Under Constrained Bit Rate: An Encoding-Decoding Mechanism.

In this article, the proportional-integral observer design problem is studied for a class of multirate networked systems subject to constrained bit rate. The sensor sampling period is allowed to be different from the system updating period and, to facilitate the observer design, the underlying multirate system is cast into a general single-rate one by resorting to the lifting technique. In order to curb the communication burden and promote the data security, the encoding-decoding procedure is implemented on the sensor-to-observer channel to convert the measurement signals into binary codewords. A sufficient condition is first proposed to reveal the fundamental relationship between the bit-rate constraints and the decoding accuracy, and then the exponentially ultimate boundedness of the error dynamics is assessed with the aid of the Lyapunov method. Subsequently, the desired observer gains are determined by solving two optimization problems with the aim to achieve two distinct performance indices, namely, the smallest ultimate bound and the fastest decay rate. Finally, the validity of the developed observer design approach is thoroughly demonstrated via the simulation examples.

Interval Observer Design for Discrete-Time Nonlinear Dynamic Systems

The paper considers the problem of interval observer design for nonlinear dynamic systems described by discrete-time models under external disturbances, measurement noise, and parametric uncertainties. The problem is to design the observer with fewer dimensions than that of the original system; such an observer must generate upper and lower bounds of admissible values of the prescribed nonlinear function of the original system state vector. To solve the problem, special mathematical tool is used. The advantage of this tool is that it allows studying the systems described by models with non-smoo th nonlinearities. To construct interval observer, the reduced-order model of the original system insensitive or having minimal sensitivity to the disturbances is designed. The designing procedure is based on two algorithms: the first one is intended to design the model of minimal sensitivity; the second one is used to reduce the dimension of the model. The rules are formulated to ensure stability based on the prescribed set of the desirable eigenvalues and feedback. The interval observer consists of two subsystems: the first one generates the lower bound, the second one the upper bound. The relations describing both subsystems are given. To construct such an observer in the nonlinear case, the terms of positive and negative influence of variables describing the model are introduced. These terms allow finding out how the upper and lower bounds of these variables will appear in the interval observer. The conditions under which the observer can be designed are given. The theoretical results are illustrated by an example of three tank system. Simulation results based on the package Matlab show the effectiveness of the developed theory.

Observer design for an infectious disease PDE model considering reinfection

This paper is concerned with the observer design problem for a system composed of two Kermack–McKendrick equations considering reinfection. The system is described by two first-order hyperbolic equations that have a time lag in the nonlocal boundary condition. The element of time lag can be expressed by a transport equation. As a result, the system with one delay is equivalently written by a 3 × 3 hyperbolic equations system. In this paper, we construct observers using a backstepping methodology of PDEs. Especially, decoupling transformations are used. Numerical experimental results are also given.