An Adaptive Model Uncertainty Estimator Using Delayed State-Based Model-Free Control

Baek, Seungmin, et al. "An Adaptive Model Uncertainty Estimator Using Delayed State-Based Model-Free Control and Its Application to Robot Manipulators." IEEE/ASME Transactions on Mechatronics (2022).

I. Motivation

Despite many successful time-delay estimation-based control (TDC) applications, it has been long acknowledged that TDE suffers a fatal estimation error (i.e., the "time-delay estimation (TDE) error"), especially when the system experiences abrupt changes caused by friction, external forces, payloads, or changes in trajectory.

Because the aforementioned control schemes employ a conventional TDE technique using previous samples as current estimates, the performance limitation of the former greatly depends on those of the latter. Hence, the TDE technique affects its control gains, which creates many limitations for practical applications. This tradeoff causes serious problems when intermittent disturbances or uncertainties occur, and because estimation by TDE techniques is poorly achieved, it becomes very difficult to find appropriate gains.

(Contributions) In this article, we propose an innovative model-free control (MFC) algorithm using an adaptive model uncertainty estimator (AMUE) that provides stable torque input while allowing more precise control, even in the presence of instantaneous disturbances, such as friction, payload, or trajectory changes. The proposed algorithm achieves better tracking performance by considering not only the one-sample delayed signal but also its gradient with an adaptive gain.

II. TDC: Basic Principles and Issues

A. Dynamics of an -DOF Robot Manipulator

The dynamical equation of a robot manipulator is as follows: where is an unknown or unavailable term, given as Property 1: is a symmetric and positive-definite matrix that is uniformly bounded in if there exist positive constants, and , such that Property 2: The Coriolis/centripetal vector is uniformly bounded if the velocity is uniformly bounded, and the following equation holds: meaning that is skew-symmetric.

Property 3: he friction vector is represented as follows:

B. Design of Time-Delay Control

By using a constant diagonal matrix , the robot dynamics can be rewritten as where is treated as a disturbance to the nominal model dynamics.

We design the following control law to compensate the disturbance: is often measured at the preceding sampling time point using the TDE technique as follows: with one sampling period . The we have the following error dynamics: For the tracking error et to converge, the TDE error should be upper-bounded as with a constant , with the following well-known stability condition: When the TDE error becomes instantaneously large, owing to friction, payload, or trajectory changes, the tracking performance is degraded if the constant gain is used.

C. Issues With

• For the initial parameter tuning, is set to be small to satisfy stability. If is too small, the TDE error is amplified. Hence, the corresponding tracking error becomes large.
• On the contrary, if ​ is set to be large within the range of the condition, the noise may be amplified; hence, extreme discontinuity of input torque (i.e., input chattering) occurs, which has an undesirable effect on hardware.

III. Controller Design

To design an adaptive gain that can effectively deal with the tracking error, we introduce the concept of a sliding variable as

A. Adaptive Model Uncertainty Estimator

Together with a one-sample delayed measurement, the disturbances and unmodeled dynamics, , are adaptively computed by using the gradient of the measurement as follows: The adaptive structure of is proposed using is is a nonnegative constant, which is designed to ensure stability.

With the definitions , we have the control input as:

We obtain the following error dynamics:

B. Stability Analysis

To guarantee the stability of the proposed AMUE-MFC, consider a Lyapunov function which is guaranteed to be positive for nonzero since . Taking the time derivative, we have According to the stability condition, the estimation error of the proposed estimator is bounded by a certain positive . Then, we have If the switching gain is chosen to be , we have If reaches or becomes steady, the second term in the above equation disappears, and hence, it becomes negative. Otherwise, we have In fact, in this paper, the following relationship is used when proving the stability: It may be wrong.

Then, we have Hence, the time derivative of the Lyapunov function is always negative for nonzero , which means that the sliding variable is attracted to the sliding manifold. Therefore, the closed-loop system is guaranteed to be asymptotically stable on the sliding surface, .