Journal of Robotics, Networking and Artificial Life
Vol. 6(2); September (2019), pp. 105–108
DOI: https://doi.org/10.2991/jrnal.k.190828.007; ISSN 2405-9021; eISSN 2352-6386
https://www.atlantis-press.com/journals/jrnal

Research Article

Izhikevich Model-based Self-repairing Control for Plants
with Sensor Failures and Disturbances
Masanori Takahashi
Department of Electrical Engineering and Computer Science, Tokai University, 9-1-1 Toroku, Higashi-ku, Kumamoto 862-865, Japan

ARTICLE INFO
Article History
Received 10 November 2018
Accepted 20 November 2018
Keywords
Self-repairing control
sensor failurefault detection
dynamic redundancy
spiking neuron model

ABSTRACTIn the previous works, several types of the Self-Repairing Control Systems (SRCS) have been developed against unknown sensor
failures. The SRCS can automatically detect failures, and replace failed sensors with healthy backups so as to maintain the systemstability. This paper presents a new SRCS, whose detection filter is constructed based on the spiking neuron model proposed by
Izhikevich. Just counting up the number of spikes in the filtered signal makes it possible to find the sensor failure promptly. Also,in this paper, the robustness with respect to disturbances is theoretically analyzed, and it is shown that SRC can be accomplishedin the presence of unknown disturbances.
© 2019 The Authors. Published by Atlantis Press SARL.
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
Stabilities of feedback control systems are guaranteed under theassumption that the feedback loops are healthy. Obviously, if justone loop has a failed sensor, then the control system would lose
its stability. Hence, detection for sensor failures plays an important role in maintaining the control system. In the previous works,
several types of the Self-Repairing Control Systems (SRCS) havebeen developed against unknown sensor failures [1,2]. The SRCS
can automatically detect failures, and replace failed sensors withhealthy backups so as to maintain the system stability. Comparedwith existing active fault tolerant controls, the SRCS has the­following advantages: (1) the detection filter has a simple structurethat does not depend on the mathematical model of the plant, and(2) the maximum time for detection can be specified arbitrarily
in advance, i.e., early fault detection can be attained. However,
because unstable detection filters have been used [1], the conventional SRCS have been contrary to the concept of the strong­stability, which claims that control systems should be constructedby stable elements [3].
In this paper, for the SRCS against sensor failures, a new design
method for the detection filter is presented based on the simple
spiking neuron model by Izhikevich [4], and also a concrete failure
detection by counting the number of spikes in the filtered signal isshown. This method satisfies strong stability concept, because thefiltered signal is always bounded. Furthermore, this paper shows
the high-gain feedback controller stabilizing both the plant and thedetection filter. It is shown that the overall control system has highrobustness with respect to disturbances.
Email: masataka@ktmail.tokai-u.jp

Throughout this paper, with x Î , we define the ‘sgn’ function byì 1 (x ³ 0)
sgn[x] = íî -1 (x < 0)Notice that this is slightly different from the ordinary one.

2. SPIKING NEURON MODEL
The spiking neuron model proposed by Izhikevich [4] is ­represented
as [Equations (1) and (2)]:2		 v = 0.04v + 5v + 140 − w + I (1)w = e (g v − w )ì v ¬ vR		 if v ³ 30 [mV] then í w ¬ w + w (2)R
îwhere v Î  is the membrane potential of the neuron, w Î  is therecovery variable, I Î  is the stimulus, and e Î  and g Î  are
the parameters for the recovery dynamics. The auxiliary after-spike
resetting is expressed by (2), where vR Î  and wR Î  are the reset
values.
Figure 1 shows the well-known ‘bursting’ pattern with the parameters e = 0.02, g = 0.2, vR = −50, wR = 2, the initial values v(0) = −50,w(0) = 2 and the stimulus I = 10 [mV]. In both figures, ‘A’ indicatesthe first point where the auxiliary resetting (2) is performed.
Clearly, it is shown that each spike is shaped by the auxiliary
resetting. This will be used for the fault detection of the proposed
SRCS.

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M. Takahashi / Journal of Robotics, Networking and Artificial Life 6(2) 105–108

where tF Î + is the unknown failure time, and j Î  is theunknown stuck value.
The aim of this paper is to design the SRCS, which can replace thefailed sensor with the backup to maintain the control stability andguarantee the convergence property of y:y(t ) £ l (6)			 limsupt ®¥for arbitrarily given l Î +.

4. CONTROL SYSTEM DESIGNFirst of all, the detection filter is introduced based on the spiking
neuron model expressed by (7) and (8).
S D : v = sgn[ y S ] (q 2v 2 + q 0 - hw ) + q 1v + y S + py S
(7)w = e (g 1sgn[ y S ] v + g 0 - w )ìv ¬ sgn[ y S ] v R
if sgn[ y S ] v ³ vT then íî w ¬ w + w R (8)Figure 1 | A ‘bursting’ pattern by Izhikevich spiking neuron model: thetime history (top) and the trajectory (v, w) of the v – w plane (bottom).

3. PROBLEM STATEMENT
Consider the following linear time invariant system with unknown
disturbances.
S P : y = ay + bu + hT z + d y		(3)z = Fz + g y + d z
where y Î  is the actual output, u:  →  is the control input, andz Î n−1 is the state. Also, dy Î  and dz Î n−1 are unknown but
bounded disturbances. Here, it is assumed that the high frequency
gain b Î  is positive, and that F Î (n−1)×(n−1) is the stable matrix(i.e., all eigenvalues lie in −).
+For measurement of the output y, two sensors are prepared. One is
the primary sensor #1, and the other is the backup #2 for occasionof failure. Then, the feedback signal ys: + →  can be expressed asfollows [Equation (4)].
		 y S (t ) =  y1 (t ) (t ≤ t D ) (4) y2 (t ) (t > t D )where tD Î + is the detection time, whose details will be discussed
later. Each yi Î , i Î {1,2} is the output of the sensor #i. The
healthy sensor output is yi = y. Based on dynamic redundancy (4),
we usually use the primary sensor #1, but switch to the backup #2
when the failure of the primary one is detected.
The failure scenario to be considered here, is expressed as follows[Equation (5)]:		    y1 (t ) = j, t ≥ t F (5)where q0 Î , q1 Î , q2 Î , g0 Î , g1 Î , s Î + and h Î +
are design parameters. Also, vT Î  is the threshold for the auxiliary resetting. These parameters can be selected by scaling of the
Izhikevich model parameters (see Section 6). For stability, the rule(8) of the resetting should be invalid before steady state, and thenbecome valid after a sufficiently long control time.
Comparing the detection filter with the Izhikevich model, we canfind that the part of the output feedback, y S + py S in (7) corresponds to the stimulus I in (1). Furthermore, because ys takes negative values, the sign function sgn[ys] is introduced.
Next, the high-gain feedback controller is designed by
		 S C : u =1- p( y S + v ) - y S3 - v 3 (9)b{}where p Î + is the feedback gain to stabilize the plant and thedetection filter. The nonlinear part, − y S3 − v 3 is introduced tohandle the nonlinearities in the filter (7).
The overall control system is illustrated in Figure 2.
Here, consider the case when the sensor is healthy, i.e., ys = y. From(3), (7), and (9), the behavior of the overall control system without
the auxiliary resetting (8) obeysy = −( p − a) y − pv − y 3 − v 3 + hT z + d yz = Fz + g y + d z
23v
	    = −( p − q 1 )v + sgn[ y]q 2v − v
(10)+ sgn[ y](q 0 − hw ) + ay + hT z + d yw = e (g 1sgn[ y]v + g 0 − w )Define the positive definite function S : + → + as [Equation (11)]		 S : ={}1 2
y +d zT Pz + v 2 + dw 2 (11)2M. Takahashi / Journal of Robotics, Networking and Artificial Life 6(2) 105–108

107

For sufficiently small d (sufficiently large p), we have 2db / a < l.
This means that the inequality (6) holds if no failure occurs.
With the same analysis, it can be verified that the filtered signal v
also remains in a small region of radius l, i.e., lim sup v £ l . Sett ®¥the threshold as vT > l. Make the rule (8) of the resetting valid after
a sufficiently long time. Then, spikes occur only during the failure,and no spike is induced whenever the sensor is healthy.

5. FAULT DETECTION
This section shows the concrete detection method using the detection filter given by (7) and (8).
Consider the case when the failure occurs. Then, the detection
filter is represented as [Equation (17)]Figure 2 | The block diagram of the proposed SRCS with the Izhikevich
spiking neuron model.

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).
Moreover, small constants d > 0 and d > 0 are introduced only for
analysis, i.e., these are not design parameters.
Taking the time derivative of S gives [Equation (12)]11S ≤ −  a 1 y 2 + da 2  z 2  +a 3v 2 + dew 2  + db (12)

2
2
where
2

2
0

q
1  h   Pg 
- a2 dddda 2 = 2lmin[Q] - 3
2h 1 1a 3 = p - 1 - 2q 1 - 2q 22 - -  h 2 -deg 12
de d dödeg 021æ
+  P 2 dz2 ÷
b = ç 1 + d y2 +2èdøa 1 = p - 2a -2and d y ≥ d y (t ) , dz ≥  d z (t ) . Choosing sufficiently large p, we
have ai > 0, i = 1, 2, 3. Hence, from (12), it follows that		 S(t ) £ -aS(t ) + db ,t Î[0, t F ) (13)

where
ì
a2
1 ü
		 a = min ía 1 , l [ P ] ,sa 3 , 2 e ý (14)maxîþSolving the differential inequality (13), we have [Equation (15)]		 S(t ) £ S(0)e -a t +db,
at Î[0, t F ) (15)Therefore, we can conclude that all the signals in the control system
are bounded in spite of existence of the disturbances. Moreover, if
no sensor fails, that is, tF = ∞, then we have [Equation (16)]	    limsup y(t ) £ limsup 2S(t ) £
t ®¥

t ®¥

2db
(16)
a

	   

v = sgn[j ](q 2v 2 + q 0 − hw ) + q 1v + pj
(17)w = e (g 1sgn[j ]v + g 0 − w )This is similar to the Izhikevich model (1). Refer to the ‘bursting’ asmentioned in Section 2. Roughly speaking, when the two nullclinesin (1) are apart from each other (see the dashed blue lines, v = 0
and w = 0 in Figure 1), the bursting pattern appears. Here, supposethat the parameters in (7) are chosen so that the following two nullclines are apart with a sufficient distance and never intersect.
0 = q 2v 2 + q 1v + q 0 − hw
		  
(18)
0 = g 1v + g 0 − w
Then, from (8) and (17), it is shown that in the filtered signal v, thebursting pattern appears just like Figure 1. Hence, by monitoring
the spikes in v, the failure can be detected. Thus, the detection time
tD is defined by{}t D := min t |cR (t ) ≥ nR (19)		  where cR Î  is the number of the spikes in v, that is counted in realtime, and nR Î  is the specified minimum number of spikes. Notethat it is possible to count up the spikes from the number of timesof the auxiliary resetting (8) in real time.
Because the occurrence time of the bursting can be arbitrarily
shortened by selecting the parameters for the detection filter, thedetection time can be hastened.
After replacing the failed sensor, the boundedness of all the signals
in the control system are guaranteed again, and the convergence (6)can be obtained.

6. NUMERICAL EXAMPLES
To confirm the effectiveness of the proposed method, the numerical simulation is explored.
Consider the following plant with disturbances.
y = − y + u + z + 0.5 + sin(0.01t )		 z = −2z + y + cos(0.1t )The initial values are y(0) = 1 and z(0) = −1.

M. Takahashi / Journal of Robotics, Networking and Artificial Life 6(2) 105–108

108

Taking preliminary simulation results into account, the rule (8) of
the auxiliary resetting is made valid from the beginning. Of course,no spike occurs whenever the sensor is healthy.
The simulation results are shown in Figure 3. In this figure, themeasured output yS, the actual output y (top) and the filteredsignal v (bottom) are shown. From this result, it is clear that thecontrol system can be well stabilized in spite of the existence of the
disturbances, and the actual output y converges to a very small ballbefore and after the failure. The SRC can be accomplished, and thefailed sensor is replaced at tD @ 26 [s], i.e., early fault detection canbe achieved by using the spiking neuron model.

7. CONCLUSION

Figure 3 | Simulation results: the measured output and the actual output(top) and the filtered signal (bottom).

In this example, the radius l of the small region (6) to which theoutput y converges, is given by l = 0.01.
The failure scenario (5) is supposed thatt F = 25[s], j = y (t F )By the appropriate scaling of the bursting pattern shown in
Figure 1, the parameters for the detection filter are selected asfollows.

		

q 0 = 0.04, q 1 = −0.6, q 2 = 4, h = 0.01,
e = 0.02, g 0 = −6, g 1 = 20,
vT = 1 > l, v R = 0.2, w R = 2

These are chosen so that the bursting pattern of more than fivespikes per second can be observed during sensor failure. Hence, byselecting nR = 5, we can find the failure within almost 1 s. Thus, thedetection time can be arbitrarily shortened.
To stabilize the plant and the detection filter mentioned above, bytrial and error, the controller parameter is chosen asp=5

Author Introduction
Dr. Masanori Takahashi
He received his B. Eng., M. Eng. and
D. 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.

In this paper, the new SRCS has been developed that has the detection filter based on the spiking neuron model by Izhikevich. In thismethod, the sensor failure can be detected by counting the spikes
in the filtered signal. The applications to nonlinear systems withnoise, multiple-input and multiple output (MIMO) systems and soon are still left in the future works.

CONFLICTS OF INTEREST
The authors declare they have no conflicts of interest.

ACKNOWLEDGMENT
This work is supported by JSPS KAKENHI Grant Number
JP16K06429.

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