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This work proposes an active disturbance rejection approach for the establishment of a sliding mode control strategy in fault-tolerant operations. The core of the proposed active disturbance rejection assistance is a Generalized Proportional Integral (GPI) observer which is in charge of the active estimation of lumped nonlinear endogenous and exogenous disturbance inputs related to the creation of local sliding regimes with limited control authority. Possibilities are explored for the GPI observer assisted sliding mode control in fault-tolerant schemes. Convincing improvements are presented with respect to classical sliding mode control strategies. As a collateral advantage, the observer-based control architecture offers the possibility of chattering reduction given that a significant part of the control signal is of the continuous type. The case study considers a classical DC motor control affected by actuator faults, parametric failures, and perturbations. Experimental results and comparisons with other established sliding mode controller design methodologies, which validate the proposed approach, are provided.

The main challenge of the Fault-Tolerant Control is to guarantee high performance and reliability in the most adverse operations such as the presence of perturbations, disturbances, dynamic miss-modeling, and actuator faults among others. In general, the techniques employed on the Fault-tolerant Control (FTC) can be classified into active and passive. Active FTC is characterized by the controller reconfiguration assisted by fault detection and isolation (FDI) schemes [

The robust characteristics of the sliding mode technique provide a natural environment for the use of such techniques on passive FTC schemes. This technique has been properly used in different control schemes and assisted by other effective control strategies which have shown proper performance under fault-tolerant operations (see [

Even though the performance of the aforementioned control proposals is accurate, there are still complexities in the design that are a consequence of dealing with the system faults and disturbances separately, on the one hand, and, on the other hand, the need for precise knowledge of the system model.

In the active disturbance rejection control (ADRC) philosophy, system fault and disturbances can be dealt with unitedly rendering a simplified linear control structure based on a simplified model like the classical passive fault-tolerant scheme. From the ADRC point of view, the disturbances must be rejected in an active manner, so the control system actively produces accurate estimates and reduces the causes of the output errors. ADRC as a potential solution has been explored in several domains of control engineering (see [

We are interested in a proper local sliding mode creation with the aid of a GPI disturbance observer. In the establishment of the slide surface, unknown inputs (state dependent or external) impact the correct evolution of the sliding regime demanding greater bound of the control input; when the sliding surface dynamics include an active disturbance cancellation of the influence of that kind of unknown inputs, the required switching input amplitude can be decreased. Furthermore, risk for deviations from the sliding surface, due to unexpected control input saturations, is practically avoided. The proposed GPI observer can be related to either the system dynamics or sliding surface dynamics disturbance inputs; in both cases, it is possible to correctly design a suitable assisted sliding mode control law with fault-tolerant capabilities.

It is assumed that the effect of additive state-dependent and exogenous nonlinearities, that affect the sliding mode regime, may be approximately but accurately canceled from the nonlinear system behavior via the injection of a precise and exogenously generated time-varying signal.

In this work we propose an approach of passive fault-tolerant control based on a classic sliding mode controller assisted by a GPI observer under the context of the active disturbance rejection. This scheme has been validated with the control of a DC motor subject to perturbations in the load torque, actuator faults, and parametric failures.

This paper is organized as follows. Section

It is possible to assist the creation of a sliding mode regime for a wide variety of sliding mode control strategies. The idea is to inject a continuous term via a suitably defined observer, in an active fashion, at the controller stage to ensure the correct establishment or continuation of the sliding mode regime.

The objective of the proposed fault-tolerant control design is to accurately track a desired reference trajectory, even in the presence of the unknown disturbances caused by actuator faults, parameter uncertainty, the presence of unmodeled state-dependent nonlinearities, or the combination of these previous cases with the presence of uncertain exogenous time-varying signals.

This is explained in this section by using a GPI observer-based sliding mode controller. From this point of view, all those terms are considered as a single, lumped, unstructured, time-varying disturbance term. In the establishment of the sliding mode control law, it is necessary to have an estimation of the related disturbance term. Two main benefits of using this strategy can be highlighted: (1) GPI observers allow the estimation of the state of the system, the related disturbance function, and a certain number of its time derivatives; (2) the control law is composed of a discontinuous term plus a continuous injection provided by the GPI observer. The amplitude of the switching part (

In the following section, two approaches for the creation of the sliding mode regimes assisted by GPI observers, suitable for fault-tolerant operation, are explained.

It should be noted that our approach is not the only possibility; it is merely a preferred approach with ease of analysis, (e.g., it is possible to propose a GPI observer assisted strategy of high-order sliding mode).

In this subsection a conventional first order sliding mode control is appropriately adapted by a GPI observer. Consider the following

The state-dependent unperturbed sliding surface dynamics are characterized by

Actuator faults, exogenous disturbances, modeled and non modeled internal dynamics, and possible parameter variation can be treated as an equivalent additive lumped disturbance function,

The amplitude,

The disturbance function,

We assume that

The key observation for the robust operation of the proposed sliding regimes is based on the accurate, yet approximate, on-line estimation of the scalar uncertain disturbance function

An extended state representation can be proposed to cope with the disturbance function estimation

The disturbance function estimation is given by the following GPI observer.

Letting

The corresponding estimation error vector is defined as

Hence, all trajectories

Under all the previous assumptions, the discontinuous active disturbance rejection feedback controller

The observer-based control law renders the following closed loop sliding surface dynamics:

Consider the following Lyapunov function candidate:

Differentiating the Lyapunov function (

In the previous subsection, the power of the GPI observer injections for a proper establishment and development of a first order sliding mode regimen was demonstrated. In this subsection, the GPI observer is used in a wider perspective allowing both sliding surface coordinate function (

Consider the nonlinear, scalar, differentially flat system

For a given smooth control input function,

It is desired to drive the flat output

Regarding controlled system (

The disturbance function

It is assumed that a solution

Let

Assumption

With reference to simplified system (

Now, the GPI observer for the state,

The estimation error vector,

Suppose that all previous assumptions are valid. Let the coefficients,

This problem has already been proposed with slightly different notation in [

Consequently with Theorem

Regarding controlled systems (

An estimated version of the previous surface can be given by

The sliding surface dynamics of

On the other hand from the GPI Observer we have

According to (

Now, consider the following Lyapunov function candidate:

Differentiating the Lyapunov function (

The system used for the experimental comparison of the proposed control strategies is a mechatronic system composed of two directly coupled DC motors. The first motor (also called the main-motor) acts as the system to be controlled. The second motor (also called the load-motor) generates perturbation loads to the main-motor. Hence, the proposed control strategies are applied to control the angular speed of the main-motor, while the load-motor acts as the load-torque perturbation generator by means of a current control loop. Figure

General scheme of the experimental setup.

The most common faults in DC-motor drives can be classified as actuator, parametric, and sensor faults [

parametric fault due to a change in the resistance of DC-motor armature and

actuator fault due to level shift of the DC-bus voltage that feeds the full-bridge drive of the main-motor.

Consider the following dynamic model describing a DC-motor controlled by armature voltage

The parameters

By rewriting and lumping together some terms of (

The problem is as follows. Consider a DC-motor described by the dynamics presented in (

The following approximation concerning the internal model of the disturbance function

It is defined as an estimation error:

Given the previously described uncertain model (

By defining the tracking error as

A modified version of the sliding surface that uses estimates of

Applying time derivative to (

Finally, the following discontinuous feedback control law is considered:

In this section, we describe the experiments that were carried out to assess the performance of the proposed GPI observer assisted sliding mode control (SMC+GPIobs) against the classic sliding mode control (SMC) applied to a mechatronic system affected by perturbations and faults. First, the experimental setup is described; then two different operation cases are exposed and analyzed under a tracking problem: system with perturbations and system with faults.

The designed controllers were implemented in a MATLAB xPC Target environment using a sampling period of 0.1 ms on a computer equipped with a Pentium D processor. The connection between the mechatronic system and each controller was performed by two National Instruments PCI-6024E data acquisition cards. A PWM output at 8000 Hz and a digital output were both used to command each DC-motor full-bridge driver, one PWM input was used to read the main-motor encoder frequency, two digital outputs were used for enabling/disabling the faults, and an analog input was used to read the load-motor current sensor output.

Fault 1 consists of increasing the armature resistance of the main-motor; when this fault is enabled, the armature resistance of the motor is increased by

The parameters of the controller are defined as follows:

Figure

Time response of the control systems under the load-torque perturbations. From top to bottom: speed control response, control signal, and load-torque perturbation.

The plots of (a), (c), and (e) in the first column of Figure

On the other hand, the proposed GPI observer assisted SMC was capable of rejecting the perturbations and the tracking performance was maintained. In this case, the switching gain

Figure

Time response of the control systems under faults 1 and 2. Fault 1 is applied at

Figure

Figures

Figure

Time response of the control systems (no faults and no perturbations) under variations in the switching gain

In this paper, an extension of Generalized Proportional Integral observer-based control has been proposed to the problem of robust creation of sliding regimes for nonlinear single-input single-output systems, with limited switching control input authority for fault-tolerant operation. The approach considers the use of a GPI observer for the accurate (linear) estimation of nonlinear endogenous, as well as exogenous, disturbance inputs affecting the existence of local sliding regimes on a given smooth sliding manifold. Active, on-line disturbance estimation and subsequent cancellation of state-dependent and time-dependent disturbances, significantly contribute to reducing the required switching control amplitude needed to sustain a sliding regime. As an additional bonus it experimented a chattering reduction.

It was shown through the experimental tests that the proposed GPI observer assisted sliding mode control strategy is capable of maintaining the sliding regime even under hard operating conditions such as system uncertainties, perturbations, actuator faults, parametric failures, and small switching control input authority. This demonstrates the robustness of the proposed strategy accomplished by a simple linear GPI observer-based control working on an ADRC paradigm.

It was experimentally probed that (a) the proposed GPI observer-based SMC strategy allows reducing the chattering in the controlled variable (limited by the lumped perturbation estimation error) and (b) the proposed strategy forces the system to approximately keep its nominal performance in the presence of perturbations, faults, and uncertainties.

The authors gratefully acknowledge the support for the realization of this work from the following institutions: Universidad Nacional de Colombia, Cinvestav-IPN, and UPIITA-IPN.