Journal of Robotics, Networking and Artificial Life

Volume 7, Issue 3, December 2020, Pages 160 - 164

Self-repairing Control against Actuator Failures Using a Spiking Neuron Model

Authors
Masanori Takahashi*
Department of Electrical Engineering & Computer Science, Tokai University, 9-1-1 Toroku Kumamoto, 862-8652, Japan
Corresponding Author
Masanori Takahashi
Received 9 November 2019, Accepted 21 May 2020, Available Online 14 September 2020.
DOI
https://doi.org/10.2991/jrnal.k.200909.004How to use a DOI?
Keywords
Self-repairing control, actuator failure, fault detection, dynamic redundancy, spiking neuron model
Abstract

This paper presents a new Self-repairing Control System (SRCS) for plants with actuator failures. The proposed SRCS uses the well-known Izhikevich spiking neuron model as a fault detector. When the actuator fails, the neuron model is excited and then spikes occur. Thus, counting up spikes makes it possible to find failures. Compared with the existing active fault-tolerant control systems, the SRCS has the following advantages: one can not only set a maximum detection time in advance, but also construct a simple control system whose structure is independent of the mathematical model of the plant. To confirm the effectiveness of the SRCS, this paper shows theoretical performance analysis and numerical simulation.

Copyright
© 2020 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

In the previous works [1,2], several types of the Self-repairing Control Systems (SRCSs) have been developed as one of active Fault-tolerant Control Systems (FTCSs). The SRCS can automatically detect a failure and replace the failed component with the healthy backup so as to recover the stability of the control system. Compared with existing active FTCSs, the SRCS has the following advantages: (1) the maximum time of detection can be prescribed arbitrarily in advance, and (2) the structure of the control system does not depend on the mathematical model of the plant. Thus, early and robust FTC can be accomplished even if the plants have uncertainties.

Unfortunately, the conventional SRCSs have utilized an unstable detection filter [1] which is not suitable for the concept of the strong stability [3]. Recently, as a remedy, the well-known Izhikevich spiking neuron model [4] is used as a fault detector. A faulty signal in the control loop excites the neuron model. Hence, just counting up the number of spiking waves makes it possible to find failures. Because the boundedness of all the signals in the neuron model is always guaranteed, the requirement of the strong stability can be satisfied. Of course, the advantages of the original SRCSs are retained. Moreover, by utilizing the spiking neuron model, setting a threshold for fault detection is no longer needed. However, only an issue of sensor failure has been considered. A problem of actuator failure has not been solved.

In this paper, the SRCS using the spiking neuron model is modified to tolerate actuator failure. Furthermore, the theoretical analysis on stability and the mechanism for fault detection are shown. In addition, the effectiveness of the SRCS is confirmed through numerical simulation.

Throughout this paper, let 𝕉 , 𝕉+ and 𝕅 denote the sets of real numbers, nonnegative real numbers and natural numbers including zero, respectively. In addition, with x∈𝕉 , define the “sgn” function by

sgn[x]={1(x≥0)-1(x<0)

2. PROBLEM STATEMENT

Consider a linear time invariant system of the form:

ΣP:y˙=ay+bu+hTz      z˙=Fz+gy (1)
where, y∈𝕉 is the output, u:𝕉+→𝕉 is the actual control input, z∈𝕉n-1 is the state, and n∈𝕅 is the order of the plant. Here, assume that the high frequency gain b∈𝕉 is positive. Moreover, F∈𝕉(n-1)×(n-1) is supposed to be a stable matrix (i.e., all eigenvalues lie in the left half complex plane).

For occasion of failure, the two actuators are prepared. One is the primary actuator #1, and the other is the backup #2. Then, the actual control input can be expressed as follows.

u(t)={u1(t)(t≤tD)u2(t)(t>tD) (2)
where, tD∈𝕉+ is the detection time, and its detail will be discussed later. Each ui∈𝕉,i∈{1,2} is the output of the actuator #i. Obviously, in healthy case, we have ui = uc, where uc:𝕉+→𝕉 is the designed control input. Based on dynamic redundancy (2), the primary actuator #1 is usually utilized, but it is switched to the backup when the failure is detected.

The failure scenario to be considered here, is expressed as follows:

u1(t)=φ,t≥tF (3)
where, tF∈𝕉+ is the unknown failure time, and φ∈𝕉 is the unknown stuck value. Such a failure occurs when the actuator gets stuck.

The problem is to design the SRCS, which can replace the failed actuator with the backup so as to maintain the stability and guarantee the convergence property of y:

limt→∞sup|y(t)|≤λ (4)
for arbitrarily given λ∈𝕉+ .

3. A CONTROL SYSTEM WITH A DETECTION FILTER

First of all, the detection filter is introduced based on the spiking neuron model [4,5]

ΣD:v˙=sgn[y](θ2v2+θ0-ηw)+θ1v+p(y+y3)w˙=ε(γ1sgn[y]v+γ0-w) (5)
if sgn[y]v≥vT then{v←sgn[y]vRw←w+wR (6)
where, θ0∈𝕉,θ1∈𝕉,θ2∈𝕉+,γ0∈𝕉,γ1∈𝕉 and η∈𝕉+ are designed parameters, whose details will be discussed in the next section. Also, vT∈𝕉+ is the threshold for the “auxiliary resetting” (for spiking). To avoid degradation of stability by spikes, the resetting rule (6) should be invalid before the steady state.

Next, the high-gain feedback controller is designed by

ΣC:uc=-p2b(y+v+y3+v3) (7)
where, p∈𝕉+ is the feedback gain to stabilize both the plant and the detection filter.

Then, the following lemma stands.

Lemma 1.

Consider the control system constructed by (1)–(7). But, the resetting rule (6) is supposed to be invalid. If there is no failure, then all of the signals in the control system are bounded. Furthermore, regarding convergence of the plant output, the inequality (4) holds.

Proof.

Suppose that there is no failure, i.e., u = uc. Now, define the new variable: s:𝕉+→𝕉 by

s:=y+v (8)

From (1), (5), (7) and (8), it is shown that

s˙=-(p2-p-a)s-p2s3-(p-a)v-pv3+3(p2-p)s2v-3(p2-p)sv2+hTz+sgn[y]θ2v2+θ1+sgn[y]θ0-sgn[y]ηw (9)
z˙=Fz+gs-gv (10)
v˙=-pv-pv3+sgn[y]θ2v2+θ1v+sgn[y]θ0-sgn[y]ηw+ps+ps3-3ps2v+3psv2 (11)

Consider the positive definite function V:𝕉+→𝕉+ as,

V:=12{s2+δ1zTPz+v2+δ2w2} (12)
where, P∈𝕉(n-1)×(n-1) is the positive definite matrix which satisfies FT P + PT F = −2Q for any positive definite Q∈𝕉(n-1)×(n-1) . Also, δ1∈𝕉+ and δ2∈𝕉+ are small positive constants. Taking the time derivative of V gives
V˙=-(p2-p-a)s2-p2s4-(p-a)sv-psv3       +3(p2-p)s3v-3(p2-p)s2v2+hTzs       +sgn[y]θ2sv2+θ1s+sgn[y]θ0s-sgn[y]ηws       -δ1zTQz+δ1zTPgs+δ1zTPgv       -pv2-pv4+sgn[y]θ2v3+θ1v2       +sgn[y]θ0v-sgn[y]ηwv       +psv+ps3v-3ps2v2+3psv3       +δ2εγ1sgn[y]vw+δ2εγ0w-δ2εw2 (13)

Assume that p is chosen so that

p2-p-a>0 (14)

Then, the time derivative of V can be evaluated as

V˙≤-α1s2-α2s4-δ1α3‖z‖2-α4v2-α5v4   -12δ2εw2+2δ2εγ02+δ3 (15)
where, δ3∈𝕉+ is a small positive constant, and
α1=p2-p-a-|a|2-34δ1‖h‖2-3θ124δ3-3θ024δ3-2η2δ2ε-34δ1‖Pg‖2α2=14p2-p-|θ2|4α3=λmin[Q]-1α4=p-|a|2+34δ1‖Pg‖2-|θ2|-|θ1|-3θ024δ3-2η2δ2ε-2δ2εγ12α5=14p-|θ2|

Choose sufficiently large p. Then, αi > 0∀i. Hence, from (15), it follows that

V˙(t)≤-αV(t)+β,t∈[0,tF) (16)
where
α=min{2α1,2α2λmax[P],2α3,ε}β=2δ2εγ02+δ3 (17)

Solving the differential inequality (16), the following inequality can be obtained.

V(t)≤V(0)e-αt+βα,t∈[0,tF) (18)

Therefore, if no failure occurs, that is, tF = ∞, then all the signals in the control system are bounded. Moreover, taking |y| ≤ |s| + |v| into consideration, it follows that

limt→∞sup|y(t)|≤limt→∞sup2V(t)≤22βα (19)

It is clear that 2 22β/α<λ for sufficiently small δ2 and δ3. This means that the inequality (4) holds if no failure occurs. The proof is completed.

Regarding the filtered signal v, from (18), it follows that

limt→∞sup|v(t)|≤λ (20)

Enable the resetting rule (6) with the threshold vT greater than λ, in the steady state. Then, as long as the actuator is healthy, no spike occurs and so the stability and convergence property (4) can be ensured.

4. ACTUATOR FAILURE DETECTION

In this section, the real-time failure detection is shown using the detection filter constructed by (5) and (6).

For preparation, refer to the Izhikevich neuron model [4] of the same scale and parameters as (5):

v˙=θ2v2+θ1v+θ0-ηww˙=ε(γ1v+γ0-w) (21)
where, θ0, θ1, θ2, γ0, γ1 and η are supposed to be chosen such that the bursting appears as shown in Figure 1.

Figure 1

The bursting pattern of the spiking neuron model.

In the figure, tB∈𝕉+ represents the bursting time, and it can be arbitrarily shortened by setting the parameters. Furthermore, nR∈𝕅 is the number of spikes in the pattern, and it also can be prescribed by parameters. The figure shows an example of ηR = 5.

Now, consider the case when the actuator gets stuck. Assume that tB is set so small that there exists a finite time tE ≥ tF defined by

tE=inf{T≥tF|sgn[y(s)]=sgn[y(T)], T≤s<T+tB} (22)

Because of continuity of y and boundedness of y˙ on a finite time interval, the sign of y does not change in a neighborhood of the failure time tF. This implies that tE = tF for sufficiently small tB.

In the case of sgn [y(tE)] = 1, from (5), the behaviors of the signals in the detection filter obey

v˙=θ2v2+θ1v+θ0-ηw+p(y+y3)w˙=ε(γ1v+γ0-w) (23)

In the above equations, the effect of the control input uc gets lost because the actuator fails. Hence, the bursting occurs in the signal v.

By comparing (21) with (23), it is regarded that the term p(y + y3) > 0 is additionally injected to the spiking model (21). Generally, a larger stimulus causes more spikes. Therefore, the bursting becomes more frequently than the pattern of (21) on the time period [tE, tE + tB).

In the case of sgn [y(tE)] = −1, the negative spikes of the same bursting pattern as above is induced.

Next, consider the case when there is no failure. Then the spiking of detection filter does not occur because the filtered signal v is suppressed smaller than vT .

Consequently, from the above discussion, the bursting pattern appears only when the actuator fails. Thus, by just counting the number of the spikes, the failure can be found. Specifically, the detection time tD is defined by

tD:=min{t|cR(t)≥nR} (24)
where, cR∈𝕅 is the counted number of the spikes in the filtered signal v.

From (24), it follows that tD ≤ tE + tB. Hence, by setting tB small, the maximum detection time can be shortened arbitrarily.

After replacing the failed actuator, the boundedness of all the signals in the control system are guaranteed again, and the plant output can converge to the small region, that is, the inequality (4) holds.

The overall control system is illustrated in Figure 2.

Figure 2

Block diagram of the SRCS against actuator failures.

Remark.

If there does not exist finite tE, that is, tE = ∞, then the failure may not be found theoretically. As a remedy, it is recommended to monitor the plant output y for checking the convergence property. As long as the inequality (4) holds, the control objective can be achieved regardless of presence of failures.

5. NUMERICAL EXAMPLES

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

Consider the following unstable plant.

y˙=-y+u+z,y(0)=1z˙=-2z+y,z(0)=-1 (24)

The failure scenario is supposed that

tF=25[s],φ=u(tF) (25)

For the above plant, the parameters for the detection filter are selected as θ0 = −0.06, θ1 = −0.6, θ2 = 4, γ  = δ = 1, ε = 0.02, γ0  = −6, γ1  = 20. Also, the parameters for resetting are vT = 1, vR = 0.2, wR = 2. The prespecified number of the spikes in the bursting pattern is supposed to be five per second, that is, nR = 5 within almost tB = 1 [s]. At last, the controller gain is chosen as p = 6.

The simulation results are shown in Figure 3 where the plant output y (top), the filtered signal v (middle), and the actual input u (bottom) are shown.

Figure 3

Simulation results: the plant output (top), the filtered signal (middle) and the actual input (bottom).

From the simulation results, the failed actuator can be replaced at the detection time tD ≅ 26 [s]. Also, the plant output y converges to a very small ball before and after the failure.

6. CONCLUSION

This paper has presented the new SRCS that can find actuator failure by using the Izhikevich neuron model. From theoretical and numerical analysis, it is shown that fault-tolerant control can be accomplished.

In the previous work [5], the basic idea of the SRCS using the spiking neuron model has already been proposed for plants with sensor failures. The main differences from the previous version are as follows: the actuator failures can be tolerated, and in constructing the detection filter, the time derivative of the plant output y is not utilized. Thus, the control system not only becomes simple but also has more robustness with respect to component failures and noises in the plant output.

CONFLICTS OF INTEREST

The author declares no conflicts of interest.

ACKNOWLEDGMENT

This work is supported by JSPS KAKENHI Grant Number JP16K06429.

AUTHOR INTRODUCTION

Dr. Masanori Takahashi

He received his B. Eng., M. Eng., and D. Eng. degrees from Kumamoto University, Japan in 1992, 1994 and 1998 respectively. He is currently a Professor with the Department of Electrical Engineering and Computer Science, Tokai University, Japan. His research interests are in the area of fault tolerant control, fault detection and adaptive control.

Journal
Journal of Robotics, Networking and Artificial Life
Volume-Issue
7 - 3
Pages
160 - 164
Publication Date
2020/09
ISSN (Online)
2352-6386
ISSN (Print)
2405-9021
DOI
https://doi.org/10.2991/jrnal.k.200909.004How to use a DOI?
Copyright
© 2020 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Masanori Takahashi
PY  - 2020
DA  - 2020/09
TI  - Self-repairing Control against Actuator Failures Using a Spiking Neuron Model
JO  - Journal of Robotics, Networking and Artificial Life
SP  - 160
EP  - 164
VL  - 7
IS  - 3
SN  - 2352-6386
UR  - https://doi.org/10.2991/jrnal.k.200909.004
DO  - https://doi.org/10.2991/jrnal.k.200909.004
ID  - Takahashi2020
ER  -