# Self-repairing Adaptive PID Control for Plants with Sensor Failures

- DOI
- 10.2991/jrnal.2018.5.2.8How to use a DOI?
- Keywords
- PID control; Adaptive control; Fault-tolerant control; Sensor failure
- Abstract
This paper presents a new design method for a self-repairing adaptive PID control system. The control system has the adaptive adjusting mechanisms for the PID gains, and also can detect the sensor failure by self-test using the integrator in the PID controller. Hence, for plants with unknown parameters, self-repairing control can be successfully attained. Furthermore, in this paper, the control performances are theoretically analyzed.

- Copyright
- Copyright © 2018, the Authors. Published by Atlantis Press.
- Open Access
- This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

## 1. Introduction

Against sensor failure, several types of self-repairing control systems (SRCS) have been developed in the previous works. The SRCS can detect failure by self-testing using the internal signals, and automatically replace the failed sensor with the healthy backup to maintain its stability. They belong to active fault-tolerant control (AFTC) based on dynamic redundancy^{1}. The main difference (feature point) from the conventional AFTC is that the fault detector is quite simple, and its structure does not depend on the mathematical model of the plant.^{2,3,4} For example, the integrator in the PID controller can be utilized as fault detector for the SRCS^{5}.

However, in most existing SRCSs, roughly estimated parameters of plants are required to construct the controller. This paper presents a new design method for a self-repairing adaptive PID control system. The control system has the adaptive adjusting mechanisms for the PID gains, and also can detect the sensor failure by self-test using the integrator in the PID controller. Thus, no *a priori* information about the plant is needed to attain the SRC. The proposed SRCS is expected to be widely utilized as fault-tolerant PID control.

Furthermore, in this paper, the control stability, failure detection and sensor-recovery are theoretically analyzed. In addition, the numerical simulation is explored to confirm the effectiveness of the proposed adaptive SRCS.

Throughout this paper, with *x* ∈ ℝ, we define the “sgn” function by

## 2. Problem Statement

Consider the following linear time invariant system:

*y*∈ ℝ is the actual output,

*u*∈ ℝ is the input, and

**∈ ℝ**

*z*^{n}

^{−1}is the state of the plant. Here, assume that the plant is minimum-phase, that is, all eigenvalues of

**∈ ℝ**

*F*^{(n}

^{−1)×(}

^{ n}

^{−1)}lie in ℂ

^{−}.

For measurement of the output *y*, the two sensors are exploited; one is the primary sensor #1 and the other is the backup sensor #2 for occasion of failure. Then, the measured feedback signal can be represented as

*y*

_{i}∈ ℝ,

*i*= 1, 2 is the output of the sensor #

*i*, and

*t*

_{D}∈ ℝ is the detection time whose details will be determined later. If the sensors are healthy, then the measured signals are equivalent to the actual output, that is, we have

*y*

_{i}=

*y*. Unfortunately, the primary sensor fails in the following way.

*t*

_{F}∈ ℝ

^{+}is the unknown failure time, and

*φ*∈ ℝ is the unknown stuck value.

The problem to be considered here, is to construct the active fault-tolerant PID control system for plant with unknown parameters and sensor failures (3), that can automatically replace the failed sensor with the backup to maintain the control stability.

## 3. Basin Design of the Control System

First of all, the adaptive PID controller is designed by

*v*∈ ℝ is the output of the integrator:

*τ*∈ ℝ

^{+}, which is introduced for detection of failure. The fault detection using the signal

*v*will be discussed in the next section.

In the above adaptive controller, the PID gains, *k*_{P}: ℝ^{+} → ℝ^{+} , *k*_{I}: ℝ^{+} → ℝ and *k*_{D}: ℝ^{+} → ℝ are adaptively tuned as follows.

*σ*∈ ℝ

^{+}is an any small constant, and

*γ*∈ ℝ

^{+}is a positive constant. Also, a small constant

*δ*∈ ℝ

^{+}is introduced as a scaling factor. These are given by designers.

Here, we shall analyze the control stability on the time period [0, *t*_{F}) where the sensor is healthy, i.e., *y*_{s} = *y*. Now, define the augmented signal *ɛ*: ℝ^{+} → ℝ by

*ɛ*,

**,**

*z**v*obey the following equations.

Consider the positive definite function *S*: ℝ^{+} → ℝ^{+} :

**∈ ℝ**

*P*^{(n}

^{−1)×(}

^{n}

^{−1)}is the positive definite matrix which satisfies

*F*^{T}

*P*^{+}

**= −2**

*PF***for any positive definite**

*Q***∈ ℝ**

*Q*^{(n}

^{−1)×(}

^{n}

^{−1)}.

Taking the time derivative of *S*, gives

**,**

*Q**α*

_{i}

*>*0,

*i*= 1,2,3. Then, we have

*t*

_{F}).

The block diagram of the proposed control system is illustrated in Fig. 1.

## 4. Failure Detection & Repairing

This section shows the concrete detection method utilizing the integrator (5).

From the above discussion, on the time period [0, *t*_{F}), there exists Γ ∈ ℝ^{+} such that

*φ*= 0, then the output of the integrator can be represented by

*v*is unbounded with respect to

*t*. Hence, the inequality (14) hold no longer. By using this unstable behavior, we can find failure, and define the detection time

*t*

_{D}by

*φ*≠ 0. Then, the solution (

*v*,

*k*

_{I}) obeys

*γ*,

*σ*and

*τ*are selected so that

*v*approaches to the equilibrium point, it hits the threshold Γ. Therefore, the detection time

*t*

_{D}defined by (16) exists when

*φ*≠ 0. Failure detection can be successfully achieved. After detection, the failed sensor is replaced with the backup, and the control stability can be maintained.

## 5. Numerical Simulation

To confirm the effectiveness of the proposed method, the numerical simulation is explored.

Consider the following unstable plant:

For the above plant, we construct the adaptive PID controller based on (4) and (5). By trial and error, the design parameters are selected as follows.

The simulation results are shown in Figs. 2 and 3.

In Fig. 2, the measured output *y*_{s}, the actual output *y*(top) and the absolute value of *v*(bottom) are shown.

Also, in Fig. 3, the three adaptive gains, *k*_{p} (top), *k*_{I} (middle) and *k*_{D} (bottom) are shown.

From these results, it is clear that the failure (17) can be found and the failed sensor is replaced at *t*_{D} ≅ 26[s]. Furthermore, the control system can be well stabilized, and the actual output converges to a very small ball before and after the failure.

## 6. Conclusions

In this paper, a new adaptive PID control system has been developed that has self-repairing function against sensor failures, and the concrete failure detection is shown by using the integrator in the adaptive PID controller. The applications to MIMO case, nonlinear systems and so on are still left in the future works.

## Acknowledgment

This work is supported by JSPS KAKENHI Grant Number JP16K06429.

## References

### Cite this article

TY - JOUR AU - Masanori Takahashi PY - 2018 DA - 2018/09/30 TI - Self-repairing Adaptive PID Control for Plants with Sensor Failures JO - Journal of Robotics, Networking and Artificial Life SP - 110 EP - 113 VL - 5 IS - 2 SN - 2352-6386 UR - https://doi.org/10.2991/jrnal.2018.5.2.8 DO - 10.2991/jrnal.2018.5.2.8 ID - Takahashi2018 ER -