Journal of Robotics, Networking and Artificial Life
Vol. 7(3); December (2020), pp. 160–164
DOI: https://doi.org/10.2991/jrnal.k.200909.004; ISSN 2405-9021; eISSN 2352-6386
https://www.atlantis-press.com/journals/jrnal

Research Article

Self-repairing Control against Actuator Failures
Using a Spiking Neuron Model
Masanori Takahashi*
Department of Electrical Engineering & Computer Science, Tokai University, 9-1-1 Toroku Kumamoto, 862-8652, Japan

ARTICLE INFOABSTRACT

Article History

This paper presents a new Self-repairing Control System (SRCS) for plants with actuator failures. The proposed SRCS uses thewell-known Izhikevich spiking neuron model as a fault detector. When the actuator fails, the neuron model is excited and thenspikes 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 constructa simple control system whose structure is independent of the mathematical model of the plant. To confirm the effectiveness ofthe SRCS, this paper shows theoretical performance analysis and numerical simulation.

Received 09 November 2019
Accepted 21 May 2020
Keywords
Self-repairing control
actuator failurefault detection
dynamic redundancy
spiking neuron model© 2020 The Authors. Published by Atlantis Press B.V.
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 thehealthy 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 prescribedarbitrarily in advance, and (2) the structure of the control systemdoes not depend on the mathematical model of the plant. Thus,
early and robust FTC can be accomplished even if the plants haveuncertainties.
Unfortunately, the conventional SRCSs have utilized an unstabledetection 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 modelis always guaranteed, the requirement of the strong stability canbe satisfied. Of course, the advantages of the original SRCSs areretained. Moreover, by utilizing the spiking neuron model, setting athreshold 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.

the effectiveness of the SRCS is confirmed through numericalsimulation.
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 1 (x ≥ 0)
sgn[x] = 
-1 (x < 0)

2. PROBLEM STATEMENT
Consider a linear time invariant system of the form:		

å P : y = ay + bu + hT z
(1)z = Fz + gy

where, y ∈ is the output, u :  + ®  is the actual control input,
n-1
is the state, and n ∈ is the order of the plant. Here,z ∈assume that the high frequency gain b ∈ is positive. Moreover,( n- 1 )´( n- 1)FÎ
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 theprimary actuator #1, and the other is the backup #2. Then, theactual control input can be expressed as follows.

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,

u1 (t ) (t ≤ t D ) (2)		  u(t ) = 
u2 (t ) (t > t D )Email: masataka@ktmail.tokai-u.jp

where, t D Î  + is the detection time, and its detail will be discussed later. Each ui ∈ , i ∈{1, 2} is the output of the actuator #i.

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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 thebackup when the failure is detected.

v = - pv - pv 3 + sgn[ y]q 2v 2 + q 1v+ sgn[ y]q 0 - sgn[ y]hw		(11)The failure scenario to be considered here, is expressed as follows:

Consider the positive definite function V :  + ®  + as,		    u1 (t ) = j , t ³ t F (3)where, t F Î  is the unknown failure time, and j Î 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:+ ps + ps 3 - 3 ps 2v + 3 psv 2		 V :={}

where, P Î (n- 1) ´ (n- 1) is the positive definite matrix which satisfies
FT P + PT F = −2Q for any positive definite Q Î (n -1) ´ (n -1) . Also,d 1 Î  + and d 2 Î  + are small positive constants. Taking the time
derivative of V gives			 limsup y(t ) £ l (4)V = - ( p2 - p - a)s 2 - p2 s 4 - ( p - a)sv - psv 3t ®¥+ 3( p2 - p)s 3v - 3( p2 - p)s 2v 2 + hT zs

for arbitrarily given l Î  +.

+ sgn[ y]q 2 sv 2 + q 1s + sgn[ y]q 0 s - sgn[ y]hws- d 1z T Qz + d 1z T Pgs + d 1z T Pgv

3. A CONTROL SYSTEM WITH A
DETECTION FILTERå D : v = sgn[ y](q 2v 2 + q 0 - hw ) + q 1v + p( y + y 3 )(5)w = e (g 1sgn[ y]v + g 0 - w )ïìv ¬ sgn[ y]v R	   if sgn[ y]v ³ vT then í w ¬ w + w (6)Rîï+where, q 0 Î R,q 1 Î R , q 2 Î R + , g 0 Î R, g 1 Î Rand h Î  are desig­
ned parameters, whose details will be discussed in the next section.
Also, vT Î  + 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 byp2( y + v + y 3 + v 3 ) (7)bwhere, p Î  + is the feedback gain to stabilize both the plant andthe detection filter.

+ psv + ps 3v - 3 ps 2v 2 + 3 psv 3+ d 2eg 1sgn[ y]vw + d 2eg 0w - d 2ew 2Assume that p is chosen so that			 p2 - p - a > 0 (14)Then, the time derivative of V can be evaluated asV £ -a 1s 2 - a 2 s 4 - d 1a 3 ||z||2 -a 4v 2 - a 5v 41	  (15)- d 2ew 2 + 2d 2eg 02 + d 3
2
where, d 3 Î  + is a small positive constant, anda 1 = p2 - p - a -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.

3q 2 3q 2 2h 23|a |3||h|| 2- 1 - 0 ||Pg ||224d 1
4d 3 4d 3 d 2e 4d 1

q1 2p - p- 2
44a 3 = lmin[Q] - 1a2 =a4 = p-Then, the following lemma stands.

(13)- pv 2 - pv 4 + sgn[ y]q 2v 3 + q 1v 2+ sgn[ y]q 0v - sgn[ y]hwv

First of all, the detection filter is introduced based on the spiking
neuron model [4,5]		 å C : uc = -1 2s + d 1z T Pz + v 2 + d 2w 2 (12)2a5 =3q 2 2h 2|a |3- 2d 2eg 12+||Pg ||2 - q 2 - q 1 - 0 2 4d 1
4d 3 d 2e

1p - q2
4

Proof. Suppose that there is no failure, i.e., u = uc. Now, define thenew variable: s :  + ®  byChoose sufficiently large p. Then, ai > 0∀i. Hence, from (15), it
follows that		 V (t ) £ -aV (t ) + b , t Î[0, t ) (16)			 s := y + v (8)whereFrom (1), (5), (7) and (8), it is shown thats = - ( p2 - p - a)s - p2 s 3 - ( p - a)v - pv 3	 + 3( p2 - p)s 2v - 3( p2 - p)sv 2 + hT zFì
ü
2a 2a = min í2a 1 ,,2a 3 , e ýlmax [P ]îþ(9)	    b = 2d 2eg 02 + d 3 (17)z = Fz + gs - gv (10)	 Solving the differential inequality (16), the following inequality canbe obtained.

+ sgn[ y]q 2v 2 + q 1 + sgn[ y]q 0 - sgn[ y]hw �

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162		 V (t ) £ V (0)e-a t + b , t Î[0 , t ) (18)Fa
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.
Because of continuity of y and boundedness of y on a finite timeinterval, the sign of y does not change in a neighborhood of thefailure 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
23	    lim sup y(t ) £ lim sup 2 V (t ) £ 2 2 b (19)t ®¥t ®¥a	    v = q 2v + q 1v + q 0 - hw + p( y + y ) (23)w = e (g 1v + g 0 - w )It is clear that 2 2 b / a < l for sufficiently small d2 and d3. Thismeans that the inequality (4) holds if no failure occurs. The proof
is completed.

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.

Regarding the filtered signal v, from (18), it follows that			 limsup v(t ) £ l (20)t ®¥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).

Enable the resetting rule (6) with the threshold vT greater thanλ, in the steady state. Then, as long as the actuator is healthy, nospike occurs and so the stability and convergence property (4) canbe ensured.

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

4. ACTUATOR FAILURE DETECTION

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

In this section, the real-time failure detection is shown using thedetection filter constructed by (5) and (6).
For preparation, refer to the Izhikevich neuron model [4] of thesame scale and parameters as (5):v = q 2v 2 + q 1v + q 0 - hw (21)		  w = e (g 1v + g 0 - w )where, q0, q1, q2, g0, g1 and h are supposed to be chosen such that the
bursting appears as shown in Figure 1.
In the figure, t B Î  represents the bursting time, and it can bearbitrarily shortened by setting the parameters. Furthermore,nR Î is the number of spikes in the pattern, and it also can beprescribed by parameters. The figure shows an example of hR = 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 byNext, consider the case when there is no failure. Then the spikingof detection filter does not occur because the filtered signal v issuppressed smaller than vT .

{}t D := min t|cR (t ) ≥ nR (24)		  where, cR Î is the counted number of the spikes in the filteredsignal v.
From (24), it follows that tD ≤ tE + tB. Hence, by setting tB small, themaximum 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 canconverge to the small region, that is, the inequality (4) holds.
The overall control system is illustrated in Figure 2.
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   t E = inf {T ³ t F | sgn[ y(s)] = sgn[ y(T )], T £ s <T + t B } (22)Figure 1 | The bursting pattern of the spiking neuron model.

Figure 2 | Block diagram of the SRCS against actuator failures.

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From the simulation results, the failed actuator can be replaced atthe detection time tD ≅ 26 [s]. Also, the plant output y converges toa 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 andnumerical analysis, it is shown that fault-tolerant control can beaccomplished.

Figure 3 | Simulation results: the plant output (top), the filtered signal(middle) and the actual input (bottom).

to monitor the plant output y for checking the convergence property. As long as the inequality (4) holds, the control objective can beachieved regardless of presence of failures.

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 notutilized. 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.

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

ACKNOWLEDGMENT

Consider the following unstable plant.

This work is supported by JSPS KAKENHI Grant Number
JP16K06429.

y = - y + u + z, y(0) = 1		  (24)z = -2z + y, z (0) = -1

REFERENCES

The failure scenario is supposed that[1]		  t F = 25[s], j = u(t F ) (25)[2]

For the above plant, the parameters for the detection filter
are selected as q0 = −0.06, q1 = −0.6, q2 = 4, g = d = 1, e = 0.02,
g0 = −6, g1 = 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 withinalmost 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.

[3]
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AUTHOR INTRODUCTION
Dr. Masanori Takahashi
He received his B. Eng., M. Eng., andD. Eng. degrees from Kumamoto
University, Japan in 1992, 1994 and1998 respectively. He is currently aProfessor with the Department ofElectrical Engineering and Computer
Science, Tokai University, Japan. His
research interests are in the area offault tolerant control, fault detection
and adaptive control.