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

2. PROBLEM STATEMENT

Consider a linear time invariant system of the form:

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.

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 u_i = u_c, 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:

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:

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]

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

where, p∈𝕉+ is the feedback gain to stabilize both the plant and the detection filter.

Then, the following lemma stands.

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

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

where, P∈𝕉(n-1)×(n-1) is the positive definite matrix which satisfies F^T P + P^T 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

Assume that p is chosen so that

Then, the time derivative of V can be evaluated as

where, δ3∈𝕉+ is a small positive constant, and

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

where

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

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

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

Enable the resetting rule (6) with the threshold v_T 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):

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 η_R = 5.

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

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 t_F. This implies that t_E = t_F for sufficiently small t_B.

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

In the above equations, the effect of the control input u_c 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 [t_E, t_E + t_B).

In the case of sgn [y(t_E)] = −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 v_T .

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 t_D is defined by

where, cR∈𝕅 is the counted number of the spikes in the filtered signal v.

From (24), it follows that t_D ≤ t_E + t_B. Hence, by setting t_B 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.

5. NUMERICAL EXAMPLES

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

Consider the following unstable plant.

The failure scenario is supposed that

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 v_T = 1, v_R = 0.2, w_R = 2. The prespecified number of the spikes in the bursting pattern is supposed to be five per second, that is, n_R = 5 within almost t_B = 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 t_D ≅ 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.