Reduced-order Observer Design Research Articles

Design of reduced-order observer based steady state optimal linear-quadratic feedback controllers

In this paper we formulate and solve a problem encountered in engineering applications when a linear-quadratic (LQ) optimal feedback controller uses state estimates obtained via a reduced-order observer. Due to the use of state estimates instead of the actual state variables, the optimal quadratic performance is degraded in a pretty complex manner. In the paper, we show how to find the exact formula for the optimal performance degradation (optimal performance loss) in linear time invariant systems for the steady state case (infinite horizon optimization problem). The optimal performance loss is obtained in terms of solution of a reduced-order algebraic Lyapunov equation whose dimension is equal to the dimension of the reduced-order observer. The quantities that impact the performance criterion loss are identified. Practical examples (an inverted pendulum on a cart and an aircraft) show that the optimal performance loss can be very significant in some applications, and even very high in the presented linear-quadratic optimal control of an inverted pendulum problem. We have shown, using the derived formula, how the optimal performance loss can be considerably reduced by properly choosing the reduced-order observer initial conditions via the least square method.

Dynamic output feedback based fault tolerant control for continuous-time switched affine systems via reduced-order observer

In this study, the issue on dynamic output feedback fault-tolerant controller design is addressed for a class of continuous-time switched affine systems in the presence of actuator faults. The existence of affine terms gives rise to the lack of a common equilibrium point and further makes it more difficult to propose the desired output-feedback-based fault-tolerant control scheme. Firstly, the actuator fault estimation is achieved through a reduced-order observer design approach. Then, a kind of dynamic output feedback controller is constructed to perform the fault tolerance goal by utilizing fault estimation information. Moreover, the mode-dependent average dwell time switching constraint is borrowed to avoid the chattering owing to the high-frequency switching. The sufficient conditions with less conservative are derived to guarantee the practical exponential stability of closed-loop system with a prescribed L∞ gain. Finally, both the effectiveness and validity of the designed fault-tolerant control strategy are illustrated via a case study of a buck-boost DC-DC converter.

Data-driven reduced-order unknown-input observers

In this paper we propose a data-driven approach to the design of reduced-order unknown-input observers (rUIOs). We first recall the model-based solution, by assuming a problem set-up slightly different from those traditionally adopted in the literature, in order to be able to easily adapt it to the data-driven scenario. Necessary and sufficient conditions for the existence of a reduced-order unknown-input observer, whose matrices can be derived from a sufficiently rich set of collected historical data, are first derived and then proved to be equivalent to the ones obtained in the model-based framework. Finally, a numerical example is presented, to validate the effectiveness of the proposed scheme.

Optimal design of Luenberger reduced-order observer with low sensitivity for linear multivariable systems

For the linear multivariable systems, by combining both merits of orthogonal-function approach and evolutionary optimization, in this paper, a new method is presented for designing a Luenberger reduced-order observer to solve the low-sensitivity design issue for physical system parameter deviation and simultaneously to minimize a measurement of the quadratic performance for reducing state transient estimation error. Two given examples illustrate the effectiveness of the presented new low-sensitivity design approach on state estimation performance. From the given examples, it shows that the estimated state errors are not sensitive to system parameter deviation and have the asymptotical convergence property. Besides, the performances are apparently superior to those without considering low-sensitivity design means.

Fault Estimation Observers for the Vehicle Suspension with a Varying Chassis Mass

This paper proposes a descriptor Nonlinear Parameter Varying (NLPV) observer for damper fault estimation in vehicle semi-active suspension systems, here considering the highly practical case of a varying sprung mass, which is an open problem. This mass and the control input are treated as known scheduling parameters for the design and implementation of the observers, thus making the structure adaptive. This novel modeling leads to parameter-dependent dynamics and output matrices in the nonlinear parameter-varying formulation. To solve this problem, we extend an existing reduced-order observer design method (following descriptor system modeling) to this new case. Both polytopic and grid-based methods are then considered to compute and implement the problem solutions. The designs are then assessed using frequency-domain analysis and realistic time-domain simulations.

Neural-Network-Based Fuzzy Observer With Data-Driven Uncertainty Identification for Vehicle Dynamics Estimation Under Extreme Driving Conditions: Theory and Experimental Results

We present a neural network based Takagi-Sugeno (TS) fuzzy observer to estimate the lateral speed (or sideslip angle) of nonlinear vehicle dynamics subject to modeling uncertainties and unknown inputs. To this end, we first propose a TS fuzzy reduced-order observer design, which can be implemented with low computation effort, for nonlinear systems. The stability and robustness of the observer scheme against the modeling uncertainty is guaranteed by the filtering method. A data-driven approach is proposed to construct feedforward neural networks (NNs) for uncertainty approximation. This data-driven approach exploits a specific sliding mode observer (SMO) to identify the model uncertainty data from the collected training data. The NN-based uncertainty approximation is incorporated into the TS fuzzy observer structure to mitigate the effect of uncertainty and improve the estimation quality. Via Lyapunov's stability theory, design conditions of both the TS fuzzy reduced-order observer for dynamics estimation and the SMO for uncertainty identification are derived in terms of linear matrix inequalities. Experimental results obtained with the INSA autonomous vehicle on a real test track demonstrate the effectiveness of the proposed TS fuzzy observer under various driving scenarios. Performance comparisons are also performed to illustrate the interest of using NN-based uncertainty approximation for the new reduced-order observer scheme, especially under extreme driving conditions.

Prescribed-Time Unknown Input Observers Design by Using Periodic Delayed Output With Application to Fault Estimation

This article considers the design of prescribed-time unknown input observers (PTUIOs) for linear systems, i.e., observers that estimate the state of a linear system with unknown inputs at a prescribed finite time. To this end, the generalized inverse is used first to transform a form amenable for observer design. Then, PTUIOs are designed by using the periodic delayed output so that their estimation errors converge to zero at a prescribed time. Both full-order and reduced-order PTUIOs are considered. For the reduced-order observer design, an alternative approach is also adopted, based on a transformation to a generalization of the normal form. In addition, the proposed PTUIOs are applied to estimate faults. Finally, two examples are given to illustrate the effectiveness of the proposed approaches.

Reduced-Order Observer Design for Robot Manipulators

This letter investigates the design of reduced-order observers for robot manipulators. Observer stability conditions are obtained based on a Lyapunov analysis. The proposed observer is enhanced with a hybrid scheme that may adjust the gains to cope with possible unbounded velocities of the robot joints. Thanks to such hybrid strategy, the observer works accurately both for robots driven by open-loop controllers and by output feedback controllers. Numerical simulations illustrate the efficacy of the reduced-order observer in several scenarios, including a comparison with the performance of a classical full-order observer.

Reduced-order observer design for fault diagnosis of Boolean control networks

In this paper, we propose an approach to design reduced-order observer of Boolean control networks (BCNs) for fault diagnosis by applying the semi-tensor product (STP) of matrices. At first, for observability analysis a computationally efficient approach based on the relational coarsest partition problem (RCP) is proposed. Then a necessary and sufficient condition for fault detectability analysis is introduced. After that an approach is introduced to design a reduced-order observer for fault detection. If fault occurrence has been detected, then an approach is proposed to solve fault isolation problem of BCNs. The basic idea is to separate dynamics of the faults into different independent subsystems by using graph theory. For each subsystem a residual generator is constructed based on reduced-order observer by getting rid of indistinguishable states. By theoretical analysis and simulation study it is shown that the approach proposed in the paper for fault isolation requires lower computational effort than the conventional scheme of a bank of full-order observers.

Parametric design of reduced-order functional observers for linear time-varying delay systems

This paper investigates the design of reduced-order functional observer for linear time-invariant systems (LTI) with time-varying delay. Firstly, the sufficient condition for the existence of reduced-order delay-free functional observers is obtained. Secondly, by solving generalized Sylvester matrix equations (GSEs), completely parameterized formulas of the coefficient matrices of reduced-order function observer are established. The free parameter matrices in the formulas provide the design degrees of the freedom and improve the convergence rate of the error system. Finally, two numerical and a practical examples are provided to illustrate the effectiveness of the proposed approach.

An Adaptive Output Feedback Controller for Boost Converter

The main contribution of this paper is to propose an adaptive reduced-order state observer for boost converter to reconstruct the inductor current and load conductance. Note that the unknown parameter appears in the output dynamics, which poses a detectability obstacle, imposing a more stringent requirement on the system behavior. As a result, the design of an adaptive reduced-order state observer is more challenging. In this paper, using the dynamic extension technique, we transform the state observation into the parameter estimation. Constructing the parameter observer, the current and load conductance can be estimated. Introducing the estimated terms to a saturated PI passivity-based control, an adaptive output feedback saturated controller is presented. To assess the control performance, the simulation and experimental results are given.

Adaptive output-Feedback asymptotic tracking control for a class of nonlinear systems with actuator failure

This study deals with the output-feedback asymptotic tracking control problem for a class of nonlinear strict-feedback systems with actuator loss of effectiveness failure. To handle with the output-feedback control issue in the presences of nonlinearities, a new reduced-order observer design is presented, by utilizing the dynamic gain technique, which not only eliminates the limitation that the Lipchitz coefficients are required to be known in the existing output-feedback results, but makes full use of the measurable information. Furthermore, a new failure compensation mechanism is proposed to erase the effect of actuator failure, by introducing a cubic absolute-value Lyapunov function method and a novel (σ,σf)-modification technique. Compared with the existing output-feedback failure compensation results, our proposed method can not only relax the assumption requirement on nonlinear function, i.e., the nonlinear function with respect to output y can be extended to the nonlinear one with respect to state variable χ¯i in the means of asymptotic tracking, but also avoid the issue that the estimate for actuator efficiency indicator drifts to a large value suddenly. Further, within the framework of backstepping design, a new high-gain reduced-order observer based adaptive output-feedback failure compensation control is developed. Then, with the aid of Lyapunov analysis method, it is shown that all the signals in the closed-loop system are globally bounded, and the system output can asymptotically track a given reference signal. Finally, a simulation example is given to illustrate the efficiency of the proposed techniques.

Natural Speed Observer for Nonsalient AC Motors

This letter addresses experimental validation of the reduced-order natural speed observer design for position sensorless drive with nonsalient permanent magnet synchronous motor. The natural speed observer and the active flux estimator are connected in a cascaded fashion, which results in a simple sensorless algorithm that needs only to tune one bandwidth parameter for flux estimation and one bandwidth parameter for speed observation. Experimental results of high speed reversal test, zero speed stopping test, and slow speed reversal test are included, where the practice of applying nonzero <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d$</tex-math></inline-formula> -axis current at zero speed has shown to be effective for loaded zero speed stopping test, but it causes zero-speed locked-up at slow speed reversal with acceleration rate of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\rm \text{50}\,r/\text{min}/s$</tex-math></inline-formula> . Four remedies are compared to improve the slow speed reversal test and the proposed method gives almost ramp actual speed waveform, nondiverging <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$q$</tex-math></inline-formula> -axis current and smooth transition in position waveform during slow zero speed crossing. A new dynamic expression of the active flux is proposed and with the aid of the active flux concept, the proposed sensorless algorithm is also applicable to various types of ac motors.

Reduced Order Observer Based Pole Placement Design for Inverted Pendulum on Cart

The inverted pendulum is an under actuated system and unstable system without a controller. In this paper modelling of an inverted pendulum is done using the Euler-Lagrange equation for stabilization of it. The controller gain is evaluated through state feedback and reduced-order observer design techniques and the result for the different initial conditions is compared. The state feedback controller is designed by Pole- placement technique for different desired pole locations. The simulation of the inverted pendulum based on reduced order pole placement design has been done on MATLAB/SIMULINK. It has been observed from the simulation result that the angular velocity and cart speed tracks the system response for different initial conditions by varying the desired pole location for the left-hand plane of the s-plane. In general, if some of the systems are unknown and the other state is known we can design using a re-duced-order observer for any physical system.

Autoencoder-Based Reduced Order Observer Design for a Class of Diffusion-Convection-Reaction Systems

The application of autoencoders in combination with Dynamic Mode Decomposition for control (DMDc) and reduced order observer design as well as Kalman Filter design is discussed for low order state reconstruction of a class of scalar linear diffusion-convection-reaction systems. The general idea and conceptual approaches are developed following recent results on machine-learning based identification of the Koopman operator using autoencoders and DMDc for finite-dimensional discrete-time system identification. The resulting linear reduced order model is combined with a classical Kalman Filter for state reconstruction with minimum error covariance as well as a reduced order observer with very low computational and memory demands. The performance of the two schemes is evaluated and compared in terms of the approximated L2 error norm in a numerical simulation study. It turns out, that for the evaluated case study the reduced-order scheme achieves comparable performance with significantly less computational load.

Reduced-order interval observer design for continuous-time descriptor LPV systems with uncertainties

In this paper, a reduced-order interval observer (R-IO) design technique is developed for continuous-time descriptor linear-parameter-varying systems with unknown-but-bounded uncertainties. First, by introducing an intermediate variable, the R-IO design amounts to estimating the linear functional of the system states. Then, with a parameter-dependent Luenberger-like structure, the R-IO existence condition is eventually formulated into a set of differential-algebraic equations, not involving any linear transformation process. Further, a parametric solution to such an R-IO is derived through solving the equation set, which clearly shows the design degrees of freedom. Finally, the correctness of the proposed results is verified by a numerical system and an electrical circuit system.

Parametric design to reduced-order functional observer for linear time-varying systems

This article studies the parametric design of reduced-order functional observer (ROFO) for linear time-varying (LTV) systems. Firstly, existence conditions of the ROFO are deduced based on the differentiable nonsingular transformation. Then, depending on the solution of the generalized Sylvester equation (GSE), a series of fully parameterized expressions of observer coefficient matrices are established, and a parametric design flow is given. Using this method, the observer can be constructed under the expected convergence speed of the observation error. Finally, two numerical examples are given to verify the correctness and effectiveness of this method and also the aircraft control problem.

Bipartite consensus of multi-agent systems with reduced-order observer-based distributed control protocols

This paper considers the bipartite consensus problem for multi-agent systems, in which both bipartite leaderless consensus and bipartite tracking problems are investigated under signed digraph. Two distributed output feedback control laws are proposed based on reduced-order observer design, respectively, for solving the bipartite leaderless consensus and bipartite tracking consensus problems. Some sufficient conditions are obtained for achieving the bipartite consensus of the multi-agent systems, by using the algebraic Riccati equation design and Jordan decomposition technique. Finally, two simulation examples are presented to illustrate the effectiveness of the obtained theoretical results.

Consensus-Based Distributed Reduced-Order Observer Design for LTI Systems.

In this article, we refocus on the distributed observer construction of a continuous-time linear time-invariant (LTI) system, which is called the target system, by using a network of observers to measure the output of the target system. Each observer can access only a part of the component information of the output of the target system, but the consensus-based communication among them can make it possible for each observer to estimate the full state vector of the target system asymptotically. The main objective of this article is to simplify the distributed reduced-order observer design for the LTI system on the basis of the consensus communication pattern. For observers interacting on a directed graph, we first address the problem of the distributed reduced-order observer design for the detectable target system and provide sufficient conditions involving the topology information to guarantee the existence of the distributed reduced-order observer. Then, the dependence on the topology information in the sufficient conditions will be eliminated by using the adaptive strategy and so that a completely distributed reduced-order observer can be designed for the target system. Finally, some numerical simulations are proposed to verify the theoretical results.

Observers for rectangular descriptor systems with output nonlinearities: application to secure communications and microcontroller implementation

This work considers full and reduced-order observer design for rectangular descriptor systems with application to secure communications. The output of the system is assumed to have a nonlinear term coupled with the linear part, a case that is often overlooked in the literature. The observer design is feasible under some algebraic conditions and the feasibility of a linear matrix inequality. The results are showcased through the application of secure communications, for the safe transmission and estimation of an information signal and also an encrypted image. Also, a microcontroller implementation of the master system is performed, which is the first step towards a full realisation of the design.