Parameter Estimation-Based Extended Observer for Linear Systems with Polynomial Overparametrization

We consider a class of uncertain linear time-invariant overparametrized systems affected by bounded disturbances, which are described by a known exosystem with unknown initial conditions. For such systems an exponentially stable extended adaptive observer is proposed, which, unlike known solutions, simultaneously: (i) allows one to reconstruct original (physical) states of the system represented in arbitrarily chosen state-space form rather than virtual states of the observer canonical form; (ii) ensures convergence of the state observation error to zero under extremely weak requirement of the regressor finite excitation; (iii) does not include Luenberger correction gain and forms states estimate using algebraic rather than differential equation; (iv) additionally reconstructs the unmeasured external disturbance. Illustrative simulations support obtained theoretical results.Comment: 6 pages, 2 figure

Exact asymptotic estimation of unknown parameters of regression equations with additive perturbations

Most identification methods of unknown parameters of linear regression equations (LRE) ensure only boundedness of a parametric error in the presence of additive perturbations, which is almost always unacceptable for practical scenarios. In this paper, a new identification law is proposed to overcome this drawback and guarantee asymptotic convergence of the unknown parameters estimation error to zero in case the mentioned additive perturbation meets special averaging conditions. Theoretical results are illustrated by numerical simulations.Comment: 6 pages, 6 figure

Monotonous Parameter Estimation of One Class of Nonlinearly Parameterized Regressions without Overparameterization

The estimation law of unknown parameters vector ${\theta}$ is proposed for one class of nonlinearly parametrized regression equations $y\left( t \right) = \Omega \left( t \right)\Theta \left( \theta \right)$. We restrict our attention to parametrizations that are widely obtained in practical scenarios when polynomials in $\theta$ are used to form $\Theta \left( \theta \right)$. For them we introduce a new 'linearizability' assumption that a mapping from overparametrized vector of parameters $\Theta \left( \theta \right)$ to original one $\theta$ exists in terms of standard algebraic functions. Under such assumption and weak requirement of the regressor finite excitation, on the basis of dynamic regressor extension and mixing technique we propose a procedure to reduce the nonlinear regression equation to the linear parameterization without application of singularity causing operations and the need to identify the overparametrized parameters vector. As a result, an estimation law with exponential convergence rate is derived, which, unlike known solutions, (i) does not require a strict P-monotonicity condition to be met and a priori information about $\theta$ to be known, (ii) ensures elementwise monotonicity for the parameter error vector. The effectiveness of our approach is illustrated with both academic example and 2-DOF robot manipulator control problem.Comment: 7 pages, 2 figure

Parameter Estimation-Based States Reconstruction of Uncertain Linear Systems with Overparameterization and Unknown Additive Perturbations

The problem of state reconstruction is considered for uncertain linear time-invariant systems with overparametrization, arbitrary state-space matrices and unknown additive perturbation described by an exosystem. A novel adaptive observer is proposed to solve it, which, unlike known solutions, simultaneously: (i) reconstructs the physical state of the original system rather than the virtual state of its observer canonical form, (ii) ensures exponential convergence of the reconstruction error to zero when the condition of finite excitation is satisfied, (iii) is applicable to systems, in which mentioned perturbation is generated by an exosystem with fully uncertain constant parameters. The proposed solution uses a recently published parametrization of uncertain linear systems with unknown additive perturbations, the dynamic regressor extension and mixing procedure, as well as a method of physical states reconstruction developed by the authors. Detailed analysis for stability and convergence has been provided along with simulation results to validate the results of the theoretical analysis.Comment: 7 pages, 3 figures. arXiv admin note: text overlap with arXiv:2302.1370

Exponentially Stable Adaptive Observation for Systems Parameterized by Unknown Physical Parameters

The method to design exponentially stable adaptive observers is proposed for linear time-invariant systems parameterized by unknown physical parameters. Unlike existing adaptive solutions, the system state-space matrices A, B are not restricted to be represented in the observer canonical form to implement the observer. The original system description is used instead, and, consequently, the original state vector is obtained. The class of systems for which the method is applicable is identified via three assumptions related to: (i) the boundedness of a control signal and all system trajectories, (ii) the identifiability of the physical parameters of A and B from the numerator and denominator polynomials of a system input/output transfer function and (iii) the complete observability of system states. In case they are met and the regressor is finitely exciting, the proposed adaptive observer, which is based on the known GPEBO and DREM procedures, ensures exponential convergence of both system parameters and states estimates to their true values. Detailed analysis for stability and convergence has been provided along with simulation results to validate the developed theory.Comment: 8 pages, 2 figure

Unknown Piecewise Constant Parameters Identification with Exponential Rate of Convergence

The scope of this research is the identification of unknown piecewise constant parameters of linear regression equation under the finite excitation condition. Compared to the known methods, to make the computational burden lower, only one model to identify all switching states of the regression is used in the developed procedure with the following two-fold contribution. First of all, we propose a new truly online estimation algorithm based on a well-known DREM approach to detect switching time and preserve time alertness with adjustable detection delay. Secondly, despite the fact that a switching signal function is unknown, the adaptive law is derived that provides global exponential convergence of the regression parameters to their true values in case the regressor is finitely exciting somewhere inside the time interval between two consecutive parameters switches. The robustness of the proposed identification procedure to the influence of external disturbances is analytically proved. Its effectiveness is demonstrated via numerical experiments, in which both abstract regressions and a second-order plant model are used.Comment: 31 pages, 12 figure

Regression Filtration with Resetting to Provide Exponential Convergence of MRAC for Plants with Jump Change of Unknown Parameters

This paper proposes a new method to provide the exponential convergence of both the parameter and tracking errors of the composite adaptive control system without the requirement of the regressor persistent excitation (PE). Instead, the composite adaptation law obtained in this paper requires the regressor to be finitely exciting (FE) to guarantee the above-mentioned properties. Unlike known solutions, not only does it relax the PE requirement, but also it functions effectively under the condition of a jump change of the plant uncertainty parameters. To derive such an adaptation law, an integral filter of regressor with damping and resetting is proposed. It provides the required properties of the control system, and its output signal is bounded even when its input is subjected to noise and disturbances. A rigorous analytical proof of all mentioned properties of the developed adaptation law is presented. Such law is compared with the known composite ones relaxing the PE requirement. The wing-rock problem is used for the modeling of the developed composite MRAC system. The obtained results fully support the theoretical analysis and demonstrate the advantages of the proposed method.Comment: 12 pages, 3 figure

Relaxation of condition for convergence of dynamic regressor extension and mixing procedure

A generalization of the dynamic regressor extension and mixing procedure is proposed. First of all, it relaxes the requirement of the regressor finite excitation, which is known to be the condition for the mentioned procedure convergence. Secondly, if the weaker requirement of the regressor semi-finite-excitation is met, it guarantees the uniform ultimate boundedness of the parameter error and elementwise monotonicity for transients of some parameters to be identified.Comment: 21 pages. In Russia

Exponentially Stable Adaptive Optimal Control of Uncertain LTI Systems

A novel method of an adaptive linear quadratic (LQ) regulation of uncertain continuous linear time-invariant systems is proposed. Such an approach is based on the direct self-tuning regulators design framework and the exponentially stable adaptive control technique developed earlier by the authors. Unlike the known solutions, a procedure is proposed to obtain a non-overparametrized regression equation (RE) with respect to the unknown controller parameters from an initial RE of the LQ-based reference tracking control system. On the basis of such result, an adaptive law is proposed, which under mild regressor finite excitation condition provides monotonous convergence of the LQ-controller parameters to an adjustable set of their true values, which bound is defined only by the machine precision. Using the Lyapunov-based analysis, it is proved that the mentioned law guarantees the exponential stability of the closed-loop adaptive optimal control system. The simulation examples are provided to validate the theoretical contributions.Comment: 37 pages, 7 figure