Self-repairing Adaptive PID Control for Plants with Sensor Failures

Masanori Takahashi

doi:10.2991/jrnal.2018.5.2.8

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Volume 5, Issue 2, September 2018, Pages 110 - 113

Self-repairing Adaptive PID Control for Plants with Sensor Failures

Authors

Masanori Takahashimasataka@ktmail.tokai-u.jp

Department of Electrical Engineering and Computer Science, Tokai University, 9-1-1 Toroku, Higashi-ku, Kumamoto, 862-8652, Japan

Available Online 30 September 2018.

DOI: 10.2991/jrnal.2018.5.2.8 How 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 redundancy¹. 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 SRCS⁵.

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

sgn[x]={1(x≥0)−1(x<0)

Notice that this is slightly different from the ordinary one.

2. Problem Statement

Consider the following linear time invariant system:

(1)y˙=ay+bu+hTzz˙=Fz+gy

where y ∈ ℝ is the actual output, u ∈ ℝ is the input, and z ∈ ℝⁿ⁻¹ is the state of the plant. Here, assume that the plant is minimum-phase, that is, all eigenvalues of F ∈ ℝ ⁽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

(2)yS(t)={y1(t)(t≤tD)y2(t)(t>tD)

where 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.

(3)y1(t)=φ, t≥tF

where 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

(4)u=kP(−ys+v)−kDy˙S

where v ∈ ℝ is the output of the integrator:

(5)v˙=−kIyS−τsgn[yS]

with a constant 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.

(6)k˙P=−σkP+γ(−yS+v)2k˙I=−σkI−γyS(yS−2v)k˙D=−σkD−δγy˙S(−yS+v)

where 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

(7)ε:=−yS+v

Then, the signals, ɛ, z, v obey the following equations.

(8)ε˙=(11+bkD*){−(bkP*−a−kI*+bkD*kI*)ε+bΔPε−ΔI(ε−v)−(kI*+a−bkD*kI*)v−bΔDy˙−hTz−(1−bkD*)τsgn[y]}z˙=Fz−gε+gvv˙=−kI*v+kI*ε−ΔI(ε−v)−τsgn[y]

where ΔP:=kP*−kP , ΔI:=kI*−kI and ΔD:=kD*−kD , and kP*∈ℝ+ , kI*∈ℝ+ and kD*∈ℝ+ are the ideal gains for PID control.

Consider the positive definite function S: ℝ⁺ → ℝ⁺ :

(9)S=12{(1+bkD*)ε2+zTPz+v2+bγΔP2+1γΔI2+bδγΔD2}

where P ∈ ℝ ⁽ⁿ^−1)×(ⁿ⁻¹⁾ is the positive definite matrix which satisfies F^TP ⁺ PF = −2Q for any positive definite Q ∈ ℝ ⁽ⁿ^−1)×(ⁿ⁻¹⁾.

Taking the time derivative of S, gives

(10)S˙=−(bkP*−a−kI*+bkD*kI*)ε2+bΔPε2−ΔI(ε−v)ε−bΔDy˙ε−(kI*+a−bkD*kI*)vε−hTzε−(1−bkD*)τsgn[y]ε−zTQz−zTPgε+zTPgv−kI*v2+kI*εv−ΔI(ε−v)v−τsgn[y]v−bγΔPk˙P−1γΔIk˙I−bδγΔDk˙D

Furthermore, from (6), it follows that

(11)S˙≤−12α1ε2−12α2‖z‖2−12α3v2−bσ2γΔP2−σ2γΔI2−bσ2δγΔD2+β

where

α1=bkP*+bkD*kI*−3|a|−2kI*−‖h‖2−‖Pg‖2α2=2λmin[Q]−3α3=(1−bkD*kI*)kI*−|a|−‖Pg‖2β=τ22(1bkP*+bkD*+1kI*)+bσ2γ(kP*)2+bσ2γ(kI*)2+bσ2δγ(kD*)2

Here, choose Q, kP* , kI* and kD* so that α_i > 0, i = 1,2,3. Then, we have

(12)S˙≤−αS+β,α=min[α11+bkD*,α2λmin[P],α3,σ]

which yields,

(13)S(t)≤S(0)exp(−αt)+βα,t∈[0,tF)

Hence, from (9), it is verified that all the signals in the adaptive control system are bounded on [0, 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

(14)|v(t)|<Γ, t∈[0,tF)

However, if the sensor fails, and φ = 0, then the output of the integrator can be represented by

(15)v=v(tF)−∫tFtτ dt=v(tF)−τ(t−tF)

Clearly, 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

(16)tD:min{t||v(t)|≥Γ}

Next, we consider the faulty situation where φ ≠ 0. Then, the solution (v, k_I) obeys

v˙=−kIφ−τsgn[φ]k˙I=−σkI−γφ(φ−2v)

Hence, the equilibrium point ( v¯ , k¯I ), is given by

v¯=φ2−στ2γφ2sgn[φ]k¯I=−τφsgn[φ]

Now, we suppose that Γ, γ, σ and τ are selected so that

Γ<|φ2−στ2γφ2|

Then, even if 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:

y˙=y+2u+z, y(0)=1z˙=−z+y, z(0)=−0.5

The failure scenario is supposed that

(17)tF=25[s], φ=y(tF)

In the simulation, they are assumed to be unknown.

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.

σ=0.01, γ=1, δ=0.1

The other parameters for failure detection are given by

τ=1, Γ=1

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

1.R Isermann, R Schwarz, and S Stolzl, Fault-tolerant, Drive-by-wire systems, IEEE Control Systems Magazine, Vol. 22, No. 5, 2002, pp. 64-81.

2.Y Zhang and J Jiang, Biographical review on reconfigurable fault-tolerant control systems, Annual Reviews in Control, Vol. 32, No. 2, 2008, pp. 229-252.

3.K Zhang, B Jiang, and V Cocquempot, Fast adaptive fault estimation and accommodation for nonlinear time-varying delay systems, Asian Journal of Control, Vol. 11, No. 6, 2009, pp. 643-652.

4.Y Wan, D Zhou, SJ Qin, and H Wang, Active fault- tolerant control for a class of nonlinear systems with sensor faults, Int. J. of Control, Automation and Systems, Vol. 6, No. 3, 2008, pp. 339-350.

5.M Takahashi, Self-repairing PI/PID control against sensor failures, Int. J. Innovative Computing, Information and Control, Vol. 12, No. 1, Feb 2016, pp. 193-202.

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Journal: Journal of Robotics, Networking and Artificial Life
Volume-Issue: 5 - 2
Pages: 110 - 113
Publication Date: 2018/09/30
ISSN (Online): 2352-6386
ISSN (Print): 2405-9021
DOI: 10.2991/jrnal.2018.5.2.8 How to use a DOI?
Open Access: This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

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ris enw bib

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  -

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