Journal of Robotics, Networking and Artificial Life, Vol. 5, No. 2 (September 2018) 110-113___________________________________________________________________________________________________________Self-repairing Adaptive PID Control forPlants with Sensor Failures
Masanori Takahashi

Department of Electrical Engineering and Computer Science, Tokai University
9-1-1 Toroku, Higashi-ku, Kumamoto, 862-8652, Japan
E-mail: masataka@ktmail.tokai-u.jp

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 theintegrator in the PID controller. Hence, for plants with unknown parameters, self-repairing control can be successfullyattained. Furthermore, in this paper, the control performances are theoretically analyzed.
Keywords: PID control, Adaptive control, Fault-tolerant control, Sensor failure.

test using the integrator in the PID controller. Thus, no apriori information about the plant is needed to attain theSRC. The proposed SRCS is expected to be widelyutilized as fault-tolerant PID control.

1. Introduction
Against sensor failure, several types of self-repairingcontrol systems (SRCS) have been developed in theprevious works. The SRCS can detect failure by selftesting using the internal signals, and automaticallyreplace the failed sensor with the healthy backup tomaintain 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 ofthe plant.2,3,4 For example, the integrator in the PID
controller can be utilized as fault detector for the SRCS5.

Furthermore, in this paper, the control stability, failure
detection and sensor-recovery are theoretically analyzed.
In addition, the numerical simulation is explored toconfirm the effectiveness of the proposed adaptive SRCS.
Throughout this paper, with 𝑥 ∈ ℝ, we define the “sgn”
function by
1𝑥 0
sgn 𝑥
1𝑥 0
Notice that this is slightly different from the ordinary one.

However, in most existing SRCSs, roughly estimatedparameters of plants are required to construct thecontroller. This paper presents a new design method fora self-repairing adaptive PID control system. The control
system has the adaptive adjusting mechanisms for thePID gains, and also can detect the sensor failure by self-2. Problem Statement
Consider the following linear time invariant system:𝑦 𝑎𝑦 𝑏𝑢 𝒉 𝒛𝒛 𝑭𝒛 𝒈𝑦(1)

Copyright © 2018, the Authors. Published by Atlantis Press.
This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
110

Journal of Robotics, Networking and Artificial Life, Vol. 5, No. 2 (September 2018) 110-113___________________________________________________________________________________________________________where 𝑦 ∈ ℝ is the actual output, 𝑢 ∈ ℝ is the input,and 𝒛 ∈ ℝ
is the state of the plant. Here, assume thatthe plant is minimum-phase, that is, all eigenvalues of𝑭∈ℝ
lie in ℂ .

introduced as a scaling factor. These are given bydesigners.
Here, we shall analyze the control stability on the time
period 0, 𝑡 where the sensor is healthy, i.e., 𝑦𝑦.
Now, define the augmented signal 𝜀: ℝ → ℝ by𝜀:𝑦𝑣(7)Then, the signals, 𝜀, 𝒛, 𝑣 obey the following equations.

For measurement of the output 𝑦, the two sensors areexploited; one is the primary sensor #1 and the other isthe backup sensor #2 for occasion of failure. Then, themeasured feedback signal can be represented as𝑦 𝑡 𝑡 𝑡(2)𝑦 𝑡
𝑦 𝑡 𝑡 𝑡1
1 𝑏𝑘 ∗

𝜀where 𝑦 ∈ ℝ, 𝑖 1, 2 is the output of the sensor #𝑖, and𝑡 ∈ ℝ is the detection time whose details will bedetermined later. If the sensors are healthy, then themeasured signals are equivalent to the actual output, thatis, we have 𝑦𝑦. Unfortunately, the primary sensor
fails in the following way.
𝑦 𝑡
(3)
𝜑, 𝑡 𝑡where 𝑡 ∈ ℝ is the unknown failure time, and 𝜑 ∈ ℝis the unknown stuck value.

𝒛𝑣

𝑏𝑘 ∗𝑎𝑘∗𝑎𝑏𝑘 ∗ 𝑘 ∗ 𝑣

Δ 𝜀𝑣

𝑘∗𝒉 𝒛

1

𝑏𝑘 ∗ 𝜏sgn 𝑦𝑭𝒛𝒈𝜀𝑘∗𝑣

𝑏𝑘 ∗ 𝑘 ∗ 𝜀𝑏Δ 𝜀𝑏Δ 𝑦𝒈𝑣
𝑘∗𝜀Δ 𝜀𝑣∗𝜏sgn 𝑦(8)∗where Δ : 𝑘
𝑘 , Δ: 𝑘𝑘 and Δ : 𝑘 ∗𝑘 , and 𝑘 ∗ ∈ ℝ , 𝑘 ∗ ∈ ℝ and 𝑘 ∗ ∈ ℝ are the ideal
gains for PID control.
Consider the positive definite function 𝑆: ℝ → ℝ :𝑆The problem to be considered here, is to construct theactive fault-tolerant PID control system for plant withunknown parameters and sensor failures (3), that canautomatically replace the failed sensor with the backup tomaintain the control stability.

112𝑏𝑘 ∗ 𝜀𝒛 𝑷𝒛𝑏Δ𝛾𝑣

1Δ𝛾𝑏
Δ
𝛿𝛾(9)

where 𝑷 ∈ ℝ
which satisfies 𝑭 𝑷
definite 𝑸 ∈ ℝ

3. Basin Design of the Control System
First of all, the adaptive PID controller is designed by𝑢 𝑘𝑣
𝑘 𝑦
4𝑦where 𝑣 ∈ ℝ is the output of the integrator:
𝜏sgn 𝑦5𝑣
𝑘𝑦with a constant 𝜏 ∈ ℝ , which is introduced for detectionof failure. The fault detection using the signal 𝑣 will be
discussed in the next section.

is the positive definite matrix
𝑷𝑭2𝑸 for any positive.

Taking the time derivative of 𝑆, gives
𝑆𝑏𝑘 ∗ 𝑎 𝑘 ∗ 𝑏𝑘 ∗ 𝑘 ∗ 𝜀𝑏Δ 𝜀𝑘Δ 𝜀

∗

1𝑎

∗

𝑣 𝜀𝑏Δ 𝑦𝜀∗𝒉 𝒛𝜀𝑏𝑘 𝑘 𝑣𝜀𝑏𝑘 ∗ 𝜏sgn 𝑦 𝜀𝒛 𝑸𝒛 𝒛 𝑷𝒈𝜀 𝒛 𝑷𝒈𝑣

In the above adaptive controller, the PID gains, 𝑘 : ℝ →ℝ , 𝑘 : ℝ → ℝ and 𝑘 : ℝ → ℝ are adaptively tuned
as follows.
𝑘𝜎𝑘𝛾 𝑦𝑣
𝑘𝜎𝑘𝛾𝑦 𝑦2𝑣
𝑘𝜎𝑘𝛿𝛾𝑦𝑣(6)𝑦where 𝜎 ∈ ℝ is an any small constant, and 𝛾 ∈ ℝ is apositive constant. Also, a small constant 𝛿 ∈ ℝ is𝑘∗𝑣
𝑏Δ 𝑘𝛾𝑘 ∗ 𝜀𝑣

Δ 𝜀1Δ𝑘𝛾𝑣 𝑣

𝜏sgn 𝑦 𝑣𝑏Δ 𝑘𝛿𝛾

Furthermore, from (6), it follows that
111𝑆𝛼 𝜀𝛼 ‖𝒛‖
𝛼 𝑣
222𝑏𝜎𝜎𝑏𝜎ΔΔ
Δ2𝛾2𝛾2𝛿𝛾

111

10𝛽

11

Journal of Robotics, Networking and Artificial Life, Vol. 5, No. 2 (September 2018) 110-113___________________________________________________________________________________________________________where𝑏𝑘 ∗𝛼𝑏𝑘 ∗ 𝑘 ∗𝛼2𝜆𝑸𝛼1𝑏𝑘 ∗ 𝑘 ∗ 𝑘 ∗𝛽1𝜏2 𝑏𝑘 ∗3|𝑎|2𝑘 ∗‖𝒉‖3𝑏𝑘 ∗𝑏𝜎 ∗𝑘2𝛾|𝑎|From the above discussion, on the time period 0, 𝑡 ,there exists Γ ∈ ℝ such that|𝑣 𝑡 | Γ, 𝑡 ∈ 0, 𝑡
(14)
However, if the sensor fails, and 𝜑 0, then the outputof the integrator can be represented by‖𝑷𝒈‖

1𝑘∗𝑏𝜎 ∗𝑘2𝛾𝑏𝜎 ∗𝑘2𝛿𝛾Here, choose 𝑸, 𝑘 ∗ , 𝑘 ∗ and 𝑘 ∗ so that 𝛼0, 𝑖Then, we have𝑆
𝛼𝑆 𝛽,𝛼𝛼𝛼 min,,𝛼 ,𝜎1 𝑏𝑘 ∗ 𝜆𝑷
which yields,𝑆 𝑡This section shows the concrete detection methodutilizing the integrator (5).

‖𝑷𝒈‖

𝑆 0 exp

𝛼𝑡𝑣

1,2,3.

𝛽, 𝑡 ∈ 0, 𝑡𝛼𝑣 𝑡𝜏 dt𝑣 𝑡

𝜏 𝑡𝑡

15

Clearly, 𝑣 is unbounded with respect to 𝑡 . Hence, theinequality (14) hold no longer. By using this unstablebehavior, we can find failure, and define the detection
time 𝑡 by𝑡 : min 𝑡 | |𝑣 𝑡 | Γ16

1213

Next, we consider the faulty situation where 𝜑Hence, from (9), it is verified that all the signals in theadaptive control system are bounded on 0, 𝑡 .

0. Then,the solution 𝑣, 𝑘 obeys𝑣𝑘 𝜑 𝜏sgn 𝜑𝑘𝜎𝑘𝛾𝜑 𝜑 2𝑣
Hence, the equilibrium point 𝑣̅ , 𝑘 , is given by𝜎𝜏𝜑sgn 𝜑𝑣̅
2 2𝛾𝜑𝜏𝑘sgn 𝜑𝜑Now, we suppose that Γ, 𝛾, 𝜎 and 𝜏 are selected so that𝜑𝜎𝜏Γ
2 2𝛾𝜑The block diagram of the proposed control system isillustrated in Fig. 1.

Then, even if 𝑣 approaches to the equilibrium point, ithits the threshold Γ. Therefore, the detection time 𝑡
defined by (16) exists when 𝜑 0. Failure detection canbe successfully achieved. After detection, the failedsensor is replaced with the backup, and the control
stability can be maintained.

5.

Numerical Simulation

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

Fig. 1. Block diagram of the proposed adaptive PID control
system with self-repairing function.

Consider the following unstable plant:
4.

Failure Detection & Repairing

𝑦𝑧

112

𝑦𝑧

2𝑢 𝑧, 𝑦 0𝑦,𝑧 010.5

Journal of Robotics, Networking and Artificial Life, Vol. 5, No. 2 (September 2018) 110-113___________________________________________________________________________________________________________The failure scenario is supposed that𝑡
25 𝑠 , 𝜑 𝑦 𝑡(17)
In the simulation, they are assumed to be unknown.

Also, in Fig. 3, the three adaptive gains, 𝑘 (top), 𝑘(middle) and 𝑘 (bottom) are shown.
From these results, it is clear that the failure (17) can befound and the failed sensor is replaced at 𝑡 ≅ 26 [s].
Furthermore, the control system can be well stabilized,
and the actual output converges to a very small ballbefore and after the failure.

For the above plant, we construct the adaptive PID
controller based on (4) and (5). By trial and error, thedesign parameters are selected as follows.
𝜎 0.01, 𝛾 1, 𝛿 0.1
The other parameters for failure detection are given by𝜏 1, Γ 1

6.

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

𝑦Conclusions

In this paper, a new adaptive PID control system has beendeveloped 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.

𝑦Acknowledgment[s]This work is supported by JSPS KAKENHI Grant
Number JP16K06429.
Sensorreplaced
here

References|𝑣|

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2. Y. Zhang and J. Jiang: Biographical review onreconfigurable fault-tolerant control systems, Annual
Reviews in Control, Vo. 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
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[s]Fig. 2. Simulation results: the measured output, the actual
output and the absolute value of the the integrator output.

𝑘[s]𝑘[s]𝑘[s]Fig. 3. Simulation results: the three adaptive PID gains.

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

113