Output Injection Research Articles

Switched Observer Design for a Class of Non-Linear Systems

This paper is concerned with the switched observer design for a class of systems subject to locally Lipschitz non-linearities. By performing a suitable description of the estimation error dynamics into a linear parameter varying (LPV) system representation, sufficient conditions for the existence of a switching output injection gain are proposed such that the asymptotic stability of the estimation error is guaranteed. These conditions can be conveniently expressed by means of linear matrix inequalities (LMIs), which are easily computationally tractable. A numerical example is provided to show the favorable performance achieved by the proposed observer, which can be applied to a large class of non-linear systems.

State-of-Charge and State-of-Health Estimation in Li-Ion Batteries Using Cascade Electrochemical Model-Based Sliding-Mode Observers

This paper proposes a cascade approach based on a sliding mode observer (SMO) for estimating the state of charge (SoC) and state of health (SoH) of lithium-ion (Li-ion) batteries using a single particle model (SPM). After convergence, the observation error signal of the current node in the cascade observer is generated from the output injection signal of the previous node’s observer. The current node’s observer generates its output injection signal, leading to its convergence. This sequential process accurately determines the observed values of each node using only the battery’s current and voltage. Subsequently, the SoC and SoH are estimated using observations of lithium-ion concentrations on the surface and inside the battery anode. The accuracy of this approach is validated using Dynamic Stress Test (DST) and Federal Urban Driving Scheme (FUDS) experimental data. A comparative analysis with conventional SMO and Extended Kalman Filter (EKF) algorithms demonstrates the approach’s effectiveness and superior performance, confirming its practical applicability.

Design of unknown input integral sliding‐mode discrete observer for linear systems

AbstractThis paper considers the problem of the observer design to estimate the state and reconstruct the unknown inputs for linear discrete‐time systems in the presence of uncertainties. By appropriate use of system transformation and the integral term in the sliding‐surface design, we propose a novel output injection term in its structure to achieve convergence of the sliding‐manifold. The dimension of the designed observer is independent of the number of additive uncertainties in the model of the plant. A numerical simulation demonstrates the design procedure and compares the proposed observer with the standard proportional integral observer (PIO) design.

State reconstruction for stochastic nonlinear systems with unknown local nonlinearities via output injection

This paper addresses state estimation for dynamical systems involving localized unknown nonlinearities. Direct application of linear state estimation techniques, e.g., the Kalman filter, would yield erroneous state estimates. Existing approaches in the literature either assume or estimate the nonlinearities. Alternatively, the present paper proposes to reject the unknown nonlinearities as if they were unknown disturbances. By applying an existing disturbance rejection technique, the need to know or to estimate the nonlinearities is avoided. The efficiency of the proposed method is demonstrated through numerical simulations on a nonlinear mechanical system.

Integral barrier Lyapunov functions based sliding mode observer‐controller for nonlinear systems with performance constraints

AbstractThis article presents an observer‐controller structure for nonlinear systems with external disturbances, uncertainties and performance constraints. Firstly, considering the unmeasurable state issue, a sliding mode observer is designed to estimate the unknown state. Then an integral barrier Lyapunov function (iBLF) is established to ensure that the observer estimation errors are held within the performance constraints. With the objective of utilizing iBLF, an equivalent output injection technique is adopted. Combining with backstepping technique, an observer‐controller scheme is given. The controller can hold all signals in the closed‐loop system bounded with the help of iBLF, and the tracking error will converge to a bounded tight set without violating the corresponding constraints. The effectiveness of the proposed strategy is clarified via a vessel heading control example.

Actuator Anomaly Detection in Linear Parabolic Distributed Parameter Cyber-Physical Systems

Distributed parameter cyber-physical systems (DPCPSs) are characterized by two features: 1) they exhibit spatio-temporal (distributed parameter) dynamics and 2) there exist cyber components in the form of communication channels between the physical plant and control system. DPCPSs are vulnerable to anomalies such as nonmalicious physical faults and malicious cyber-attacks. Although the safety and security aspects of CPSs modeled by ordinary differential equations (ODEs) have been extensively explored during the past decade, security of DPCPSs has not received its due attention despite its safety-critical nature. In this work, we explore anomaly diagnostics in DPCPSs from a system theoretic viewpoint. Specifically, we focus on DPCPSs modeled by linear parabolic partial differential equations (PDEs) subject to physical faults and cyber-attacks in the actuation channel. First, we explore the detectability of malicious anomalies (i.e., cyber-attacks) and derive conditions for stealthy attacks in the context of output-feedback-based monitoring. Such stealthy attacks essentially illustrate the fundamental limitations of feedback-based monitoring algorithms given perfect adversarial knowledge. Subsequently, focusing on the anomalies that are detectable, we develop a design framework for anomaly detection algorithms based on output injection observers. Such detection algorithm explicitly considers the conflicting objectives of robustness to uncertainties and sensitivity to anomalies in their design. Finally, theoretical analysis and simulation studies are performed to illustrate the effectiveness of the proposed approach.

A novel approach to state estimation of HIV infection dynamics using fixed-time fractional order observer

This paper presents a novel approach to designing a fixed-time fractional order observer for estimating the states of the dynamic model of human immunodeficiency virus (HIV) infection. The proposed approach combines output injection terminal sliding mode and RBF neural network strategies to achieve a robust and efficient estimation of the states of the HIV model within a fixed time frame. The main contributions of this work are the introduction of an output injection observer that ensures the stability of the error system along with a novel nonlinear sliding surface that guarantees the fixed-time error convergence to the neighborhood of zero. Moreover, the closed-loop scheme of the observer design is proven to be bounded, and the fixed-time stability of the observer error is obtained using the fractional Lyapunov stability approach. Simulation results show that the proposed fixed-time fractional order observer design provides accurate and efficient estimation of the states of the HIV model.

Incipient Fault Detection and Reconstruction Using an Adaptive Sliding-Mode Observer for the Actuators of Fixed-Wing Aircraft

This paper proposes a new method of detection and reconstruction of the incipient fault of fixed-wing actuators based on an adaptive sliding-mode observer. First, a mathematical model of a fixed-wing aircraft is derived under certain assumptions, considering nonlinear terms and system disturbances. Second, by introducing a nonsingular coordinate transformation, the incipient faults are separated from the disturbances. For a subsystem with no disturbances, the Luenberger observer can estimate the incipient fault. For a subsystem with disturbances, the sliding-mode observer is robust against these disturbances. The Lyapunov stability theory guarantees dynamic error convergence and system stability. The evaluation function was designed to realize residual evaluation and threshold judgment. Third, based on the concept of equivalent output injection, an adaptive sliding-mode observer method is proposed to reconstruct actuator faults precisely under the condition of the uncertain system structure. The design steps of the proposed reconstruction method are introduced in the form of a linear matrix inequality problem, which provides an effective method for calculating the design parameters. Finally, the simulation results of the De Havilland DHC-2 “Beaver” aircraft demonstrate the correctness and effectiveness of the proposed method.

Bayesian monitoring of substructures under unknown interface assumption

Structural Health Monitoring (SHM) enables assessing in-service structures’ performance by localizing structural anomaly instances immediately after their occurrence. Typical SHM approaches monitor the entire structural spatial domain aggravating the required density and cost of instrumentation. Further, with model-based approaches, the entire structural domain is needed to be defined with high dimensional, compute-intensive models rendering the SHM approaches ill-posed and slow especially when the instrumentation is limited and system observability is compromised. Moreover, in absence of high-fidelity models, oversimplification and subsequent model inaccuracies may lead to inaccurate estimation and possibly false alarms even if a subdomain is modeled inaccurately, e.g. support boundaries. To mitigate such issues, stand-alone monitoring focusing only on a subdomain of interest may be a computationally cheaper and prompt approach while being substantially robust to false alarms. Typically, such stand-alone substructure monitoring approaches demand extensive measurement of the interface, which can be a challenge in real-life applications. This paper presents a novel filtering-based online time domain approach for estimating substructure parameters without the need to measure or estimate the substructure interfaces. The proposed component-wise estimation is stand-alone so that the health estimation of the complete structural domain can be undertaken in parallel and later coupled through post-processing. The requirement of the interface measurement has been alleviated by employing an output injection approach. The proposal has been validated on a numerical beam structure subjected to arbitrary forces and subsequently, the sensitivity against noise and damage severity of the proposal has been investigated. Finally, the proposal is validated on a real beam to illustrate its real-life applicability and significance.

Distributed estimation for multi‐agent systems with relative measurements and quantized communication: A feedback quadratic framework

SummaryIn this paper, the feedback quadratic distributed state estimation problem is investigated for a class of linear discrete time‐varying multi‐agent systems subject to non‐Gaussian noises. For the multi‐agent systems, each agent has access to not only the local measurements but the relative states to its adjacent agents as well. Due to the bandwidth constraints of the digital communication networks, the signals are quantized and then exchanged between two adjacent agents, where the probabilistic uniform quantizations are taken into account. The purpose of the addressed problem is to design a novel quadratic distributed estimator for each agent based on the local information and the predictions received from the neighbors. In particular, an output injection term is introduced to handle the unstable systems. By means of the state/measurement augmentation approach, the underlying system is transformed into an augmented one, which aggregates the original vector and its second‐order Kronecker power. Accordingly, a distributed estimator is constructed such that an upper bound is ensured for the estimation error covariance and the suboptimal estimator parameters, which minimize such an upper bound, are subsequently calculated in terms of the solutions to certain matrix difference equations. Finally, an illustrative example is provided to verify the effectiveness and superiority of the proposed quadratic estimation scheme.

Subdomain Fault Isolation for Linear Parameter Varying Systems through Coupled Marginalized Particle and Kitanidis Filters

Typically, for linear parameter varying systems, which can potentially get influenced by spatio-temporal external parameters, possible changes in their eigenstructure are not easy to be attributed conclusively to system faults or spatio-temporal parametric variations. Such spatio-temporal variations can although be estimated alongside, yet at the cost of making the estimable system dimension disproportionately large. Such augmented system dimension can thereby jeopardize tracking of the system evolution, either due to computational constraints or due to insufficient measurement channels (ill-posedness). This paper proposes a localized estimation approach wherein only a subdomain of the entire system is considered which reduces the dimension of the estimated model within manageable limits. To focus on the subdomain properties without knowledge of the rest of the model parameterization, a robust algorithm is developed through output injection using a simpler and sub-optimal version of Kitanidis filtering approach to induce robustness in the system parameter estimation against the boundary measurements. Finally, the subdomain model is estimated employing a marginalized filtering approach wherein a particle filter is employed for estimating both the eigenstructure and the controlling parameter while an ensemble Kalman filter estimates the states. The approach is demonstrated with the help of a mechanical system under spatial variation in temperature for which subdomain isolation necessitates the interface to be measured. In the context of the numerical application, the induced fault is due to damage, and the mechanical model is controlled and parameterized by the internal temperature, whose variations can be significant due to substantial external thermal variations inducing significant variations in the dynamic properties.

Experimental parameter estimation methodology based on equivalent output injection

This work proposes a methodology to estimate parameters for linear and nonlinear dynamical systems, with partial state measurement, that satisfy the property of parameter linearity. This methodology is experimental, offline, and recursive. It uses discontinuous state observers to estimate all state variables and the disturbance terms needed in the estimation processes. Because the equivalent output injection corresponds to the disturbances produced by the parameter uncertainties, the methodology allows us to obtain the best parameter estimation by minimising an index related to the power of the equivalent output injection; a smaller value represents a better estimation. With this parameter estimation, we can establish a model that facilitates the design and implementation of many control algorithms, including robust controllers. We validate the methodology through numerical simulations and experiments with linear, nonlinear, and discontinuous systems. Based on the experimental results, we conclude that the proposed algorithm's performance is better than other methodologies.

Transient Improvement of Model-Reference Adaptive Control and Parameter Convergence Rate Analysis

The parameter convergence analysis of a modified model-reference adaptive control (M-MRAC) algorithm is presented. The plants considered are continuous with relative degree one. It is verified that the M-MRAC scheme approaches the MRAC scheme when a control design gain increases. Moreover, before this gain reaches high values, the M-MRAC improves the update parameters’ convergence. This analysis is performed by noting a clear connection between the M-MRAC and the conventional MRAC by adding a control output injection term. The methodology developed allows for estimating the parameter convergence rate, and simulation results illustrate the improvement in the transient behavior and the parameter convergence attained with the analyzed adaptive schemes.

Approximation of infinite-dimensional observer-based state feedback for systems with boundary control and observation

Infinite-dimensional linear systems with unbounded input and output operators are considered. For the purpose of finite-dimensional observer-based state feedback, an observer approximation scheme will be developed which can be directly combined with existing late-lumping controllers and observer output injection gains. It relies on a decomposition of the feedback gain, resp. observer output injection gain, into a bounded and an unbounded part. Based on a perturbation result, the spectrum-determined growth condition is established, for the closed loop.

Stabilisation in distribution by delay feedback controls for hybrid stochastic delay differential equations

This article aims to design a linear delay feedback control to stabilise an unstable hybrid stochastic delay differential equation in distribution. Under the global Lipschitz condition, sufficient criteria are established to guarantee the stability of the controlled system. Then LMI techniques are employed to design the control law in two structure forms: state feedback and output injection.

Exponential output consensus tracking for multi-agent systems with uncertain local output function via sampled communication

In this paper, the output consensus tracking problem is investigated for heterogeneous multi-agent systems (MASs) with asynchronous sampled communication, where only local output of the leader is required and the local output function contains an unknown parameter. A distributed adaptive observer is designed for recovering the leader’s state and unknown parameter, which has three features: (i) two predictors are contained in the observer to recover the output injection and the consensus error separately; (ii) by designing proper gain matrices, the state estimation is decomposed into the detectable part merely using the sampled local output of the leader and the undetectable part only using sampled information of the neighbors; (iii) the negative effect of the parameter estimation errors is offset by adding complementary terms with coupled auxiliary variables into the observer. Based on the observer, the control scheme is provided to realize the exponential output consensus tracking. It is shown that the maximal sampling period for the follower to receive local output from the leader is independent of the neighbors.

Threshold design for fault detection with first order sliding mode observers

Sliding Mode Observer (SMO) based methods have been extensively used for Fault Estimation (FE). However, the fault detection (FD) problem for these SMO based FE methods has not been completely solved. In this paper a robust threshold on the so-called Equivalent Output Injection (EOI) is presented which enables FD for systems with measurement noise and unmatched uncertainties. This threshold is applicable to a large class of existing SMO based FE methods, and its applicability can easily be verified. Theoretical guarantees on the detection performance of this threshold are provided, and further demonstrated via a simulation study.

Event-triggered interval-based state estimator for continuous-time linear systems

In this paper we propose an interval-based state estimator for continuous-time linear systems with discrete-time measurements using an event-triggered mechanism and an explicit reachability method. An output injection method combined with a state variables permutation procedure are applied to design the robust estimator. In addition, the convergence of the proposed set-membership state estimator and the existence of a lower bound on the inter-event times are shown. Throughout a numerical example, the performance of this estimator are illustrated and compared to related works.

On output stabilizability of differential–algebraic equations

We study output stabilizability of differential–algebraic systems. Two different notions of stability, an asymptotic one and one based on the L q norm of the output are compared and shown be equivalent. To introduce the use of the main concepts of output injection and the Kalman observability decomposition, we begin with a short proof of the result for the special case of ordinary differential equations. These results can then be generalized to differential–algebraic equations, which requires some further attention, especially with respect to observability. Equivalence of the two output stabilizabilities is worth knowing in its own right, and has applications for example in the context of optimal control.

A 1.12–1.91 mW/GHz 2.46–4.92 GHz Cascaded Clock Multiplier in 65 nm CMOS

We present a low-power and low jitter two-stage 2.46&#x2013;4.92-GHz clock multiplier using a 38.4-MHz reference clock. The proposed clock multiplier implements an <inline-formula> <tex-math notation="LaTeX">$8\times $ </tex-math></inline-formula> clock multiplication with a delay-locked loop and an edge combiner (EC) in the first stage. The regulated supply of the voltage-controlled delay line and EC within the delay-locked loop limits the first-stage clock multiplication voltage sensitivity. An in-depth phase noise analysis of the first stage with the proposed phase domain modeling and spur analysis in the EC helps low-power clock multiplier design. The first-stage output injection locks a pseudo-differential ring oscillator embedded in a frequency tracking loop, thereby achieving a <inline-formula> <tex-math notation="LaTeX">$64\times $ </tex-math></inline-formula>&#x2013;<inline-formula> <tex-math notation="LaTeX">$128\times $ </tex-math></inline-formula> clock multiplication in the second stage. In collaboration with the simulated phase noise from sources, a system-level phase noise modeling defines the design specifications of the two stages for minimum output jitter in a given power budget. Fabricated in a 65-nm CMOS process, the first-stage clock multiplier achieves an integrated jitter 761 fs_rms at 307.2 MHz while consuming 2.5 mW. The mismatch and offset-induced systematic jitter is calibrated, giving &#x2212;53.4-dBc reference spur at the first-stage output. The second-stage injection-locked clock multiplier adds low random jitter to the first stage with total output jitter 825 fs_rms at 4.92 GHz, &#x2212;28.2-dBc reference spur, and 3-mW power consumption.