Adaptive parameter identification of linear SISO systems with unknown time-delay☆
Introduction
Time-delay is usually unavoidable in control systems, which may lead to a sluggish response or even trigger instability [1]. The control design for time-delay systems mainly assumes a precise model (in particular for time-delay parameter). In this sense, online parameter identification of time-delay systems is an important and challenging topic [2]. An intuitive method is to approximate unknown time-delay by a rational transfer function (e.g. pade approximation) [3], where more parameters are introduced and the approximation error for a large time-delay may degrade the identification performance. In [4], a direct identification method has been developed based on step responses, and the relay-based method has been proposed for some low-order time-delay systems in [5]. These approaches are implemented offline and sensitive to noise. Another category has been rooted in adaptive identification: in [6] and the subsequent work [7], the identifiability analysis of SISO time-delay systems has been first studied, and then an adaptive identification was proposed based on the distributed delay flavored (DDF) procedure, where the time-delay parameter is determined in terms of an identifier that includes more fictitious delay terms than the actual system. Adaptive estimation for pure time-delay was also studied in [8], where the input signal and its derivative are all used. In [9], a nonlinear optimization was used in the adaptive identification, where the solution of the resulting min–max problem is not trivial. Recently, an algebraic identification method has been presented [10] using algebraic operations with annihilation and integration. Taking the merit of variable structure observers, several algorithms for delay identification were presented [11], where the derivatives of the input and/or output are assumed to be measurable and available. This assumption may be stringent for practical applications.
An alternative idea, polynomial identification [12], was developed to identify the time-delay and rational transfer function parameters. However, the complexity of the practical implementation for high-order systems limits its application. Although the least-squares (LS) algorithm [13] has been well-recognized in the identification of linearly parametric systems, it cannot be directly used for time-delay systems because the time-delay usually imposes nonlinearity in the formulation and thus is difficult to be presented as an explicit parameter in a linear-in-parameterized form. To address this issue, in our previous work [14], a nonlinear least-squares algorithm was developed to simultaneously identify the time-delay and other parameters of linear SISO systems. This idea has also been extended to nonlinear time-delay systems by means of neural networks. [15]. However, the adaptive laws proposed in [14] are derived based on the gradient error embedded in a nonlinear optimization problem, and thus the parameter estimation convergence has not been proved. Moreover, a stringent assumption that the system relative degree should be larger than two is imposed in [14] so that the applicability of the method is limited.
In this paper, we focus on the full parameter identification of linear continuous-time SISO systems with unknown time-delay and further improve our previous work [14] by proposing novel adaptive laws. It is assumed that the structure of the underlying system (e.g., system order) is known and only system parameters and time-delay are unknown. We first apply the Taylor series expansion to reformulate the time-delay system into a parameterized form, where the time-delay term can be explicitly indicated in a linear-in-parameterized formula. Then appropriate parameter estimation error information is derived, which is used in the design of adaptive laws. Consequently, the parameter convergence can be proved by means of Lyapunov theory under the conventional PE condition that can be online tested, and the improved performance (e.g., fast convergence) is achieved because of the inclusion of this parameter estimation error. In contrast to [12], [14], the assumption on the system relative degree is successfully removed so that the identification for general high-order systems is straightforward. Finally, by incorporating the sliding mode technique into the design and synthesis of adaptive laws, an alternative parameter identification scheme is studied guaranteeing finite-time (FT) error convergence. Compared to other methods (e.g., [7], [8], [11]), the time-delay parameter can be directly estimated and the derivatives of the input and output are not required, i.e., only the input and output measurements are used. Comparative simulations are provided to verify the characteristics and the improved performance of the proposed algorithms.
The paper is organized as follows. Section 2 presents the problem formulation and Section 3 introduces the adaptive parameter identification scheme. FT parameter identification is studied in Section 4. Simulation examples are provided in Section 5 and some conclusions are given in Section 6.
Section snippets
Problem formulation
Consider the following linear continuous-time SISO system with an input time-delay where are the system input and output, respectively, is an unknown bounded disturbances, the system order (e.g.,) is a prior known, and and are unknown parameters and is the unknown time-delay. The objective of this paper is to estimate and using only the input and output .
Taking the
Adaptive parameter identification
To estimate the unknown parameter vector , we define the auxiliary regressor matrix and vector as where is a design parameter. From (10), one may find that and can be reformulated as the filtered variable of time-varying system dynamics and in terms of a stable system , respectively.
Subsequently, one can obtain the solution of (10) as
Finite-time parameter identification
In this section, the idea proposed in Section 3 is further improved to obtain finite-time parameter estimation. For this purpose, the parameter estimation is updated by where is the adaptive learning gain.
The identifier dynamics are governed by the functional differential equation (22) with a discontinuous right-hand side [19]. The meaning of this equation remains conventional beyond the discontinuity manifold , whereas along this manifold it is defined in the sense of
Simulation
Example 1 Consider an SISO time-delay system studied in [14] as where and are the parameters to be estimated. For fair comparison, the input signal is taken as the sum of two square waves as [14]: one is with amplitude 3 and frequency 1 rad/s and the other one is with amplitude 1.5 and frequency 2 rad/s. The polynomial and the parameters used in the simulation are determined via a
Conclusion
This paper is concerned with online parameter identification of linear SISO time-delay systems. All unknown parameters including time-delay are estimated simultaneously in terms of a novel design of adaptive laws based on the parameter estimation error, and finite-time identification error convergence can be achieved by introducing a sliding mode term in the adaptation. The structure of the underlying system (e.g., system order) is known, while the derivatives of input and output are not
Acknowledgments
The author would like to thank the editors and anonymous reviewers for the constructive comments that helped to improve this paper.
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