Progressive Optimal Fault-Tolerant Control Combining Active and Passive Control Manners

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Progressive Optimal Fault-Tolerant Control Combining Active and Passive Control Manners

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Submitted:

15 March 2024

Posted:

15 March 2024

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Abstract

This study develops a progressive optimal fault-tolerant control method based on insufficient fault information. By combining passive and active fault-tolerant control manners during the process of fault diagnosis, insufficient fault information is fully used, and optimal fault-tolerant control effect is achieved. In addition, the fault-tolerant control method based on guaranteed robust cost control is introduced. The proposed progressive optimal fault-tolerant control method considers two aspects. First, as the amount of fault information continually increases, the performance index of the progressive optimal fault-tolerant controller improves. Second, at each moment, based on the corresponding insufficient fault information and prior knowledge, optimal fault-tolerant control is achieved according to current fault information. The process of progressive optimal fault-tolerant control converges to active fault-tolerant control when the fault is completely identified and the optimal fault-tolerant controller is no longer reconfigured until no more useful fault information can be provided. Furthermore, a progressive optimal fault-tolerant control algorithm based on the grid segmentation in the parameter uncertainty domain and the selection of different auxiliary center points is introduced. The simulation results verify the feasibility of the proposed algorithm and the validity of the proposed theory.

Keywords:

Subject: Engineering - Control and Systems Engineering

1. Introduction

In industrial systems, fault diagnosis (FD) and fault-tolerant control (FTC) are closely related. In recent years, there have been many studies on fault diagnosis and fault-tolerant control in various fields, including aerospace [1,2,3], power systems [4,5,6], high-speed rail [7,8,9,10,11], and satellites [12,13]. Based on previous research, fault diagnosis will inevitably develop in a faster and more accurate direction in the future. Obtaining fast and accurate fault information has been an essential requirement of active fault-tolerant control (AFTC) to ensure safe system operation; otherwise, the AFTC performance cannot be guaranteed, and even its implementation cannot be guaranteed. As well known, passive fault-tolerant control (PFTC) has been relatively conservative. However, unlike the AFTC, the PFTC does not require real-time fault information, which represents its inherent advantage. Traditional fault diagnosis has predominantly been mostly based on mathematical models [14,15,16]; however, model-based methods for fault diagnosis require an accurate system model, and the majority of these methods can only provide only two types of results (i.e., fault undiagnosed or fault diagnosed). Unfortunately, such approaches do not provide continuous fault information which is insufficient but crucial for effective fault-tolerant control. In recent years, there have been many studies on non-model-based methods (e.g., data-driven methods for fault diagnosis), especially artificial intelligence-based algorithms [17,18,19]. Fault diagnosis by artificial intelligence-based algorithms, such as neural networks, have a common problem that the behavior inside the “black box” is difficult to determine [20]. Thus, both model-based and non-model-based fault diagnosis methods have their own advantages and disadvantages. However, how to develop a more accurate and faster fault diagnosis method has still been a challenge.

When fault information cannot be rapidly and accurately obtained, a fault-tolerant control strategy which performs the PFTC before the fault is fully identified and then switch to the AFTC after the fault has been completely diagnosed seems to be a good choice. This idea has been proposed in [21]. However, its main disadvantage is the failure to utilize incomplete fault information, which is still valuable. Mechanically combining two fault-tolerant control methods is not ideal due to their respective shortcomings [22,23,24]. Therefore, it would be a perfect solution if the advantages of AFTC and PFTC can be combined under the condition of insufficient fault information. Existing research has indicated that the PFTC and AFTC are typically applied independently, with fewer studies exploring the combination of these two methods. Although the hybrid fault-tolerant control [25] combines the AFTC and the PFTC to a certain extent, these two control types are used separately depending on whether a fault has been fully diagnosed. In [26], a fault-tolerant control system (FTCS) design based on imprecise fault identification and robust reconfigurable control is proposed. This method reduces the time delay between the onset of a fault and the controller reconfiguration so that the system stability after the fault occurrence can be recovered rapidly. However, this method mainly emphasizes system stability rather than performance optimization. Moreover, the control object of this method does not have generality. In [27], actuator faults were considered as additive faults, and a combined passive-active FTC method based on reliable control is proposed, achieving a balance between performance and complexity. However, the predefined control laws were obtained offline rather than online by designing bottom-up extensible controllers with a minimal acceptable configuration, and the nominal performance remained at a sub-optimal level after a fault.

Although the above-mentioned methods combining the PFTC and AFTC methods are most mechanical, they have been valuable, but insufficient fault information has not been fully used. Moreover, the concept of progressive performance optimization has not been reflected, according to which, as fault information increases, the fault-tolerant control effect improves. In view of that, this study examines how to use insufficient fault information fully and combine the PFTC and AFTC manners efficiently to achieve optimal performance. This idea has been partly introduced in [28].

The comparison of the existing fault diagnosis methods has indicated that the parameter interval algorithm shows superiority over the other algorithms [29]. The parameter interval algorithm can continuously obtain increasingly accurate fault information during the fault diagnosis process. The smaller the parameter interval, the more accurate the fault identification. From the moment of fault occurrence to the moment of complete fault identification, the obtained fault information can be fully used to reconfigure the controller to ensure an optimal fault-tolerant control performance.

The pursuit of achieving a robust and optimal control effect while addressing the limitations of practical methods and ensuring system stability, even at the expense of some system performance, has long been a research focus [30,31,32,33]. Notably, Xue’s study [34] on robust and optimal control, which considers both robustness and the control system’s effectiveness, holds significant reference value. Since a system fault can be viewed as system uncertainty, Xue’s research result on robust and optimal control can be applied to the field of fault-tolerant control for faulty systems.

The process of a progressive optimal fault-tolerant control method combines the PFTC and AFTC manners, as introduced in this article. When a fault occurs, the maximum uncertainty domain could be determined based on the prior knowledge. Moreover, the more fault information is obtained, the smaller the uncertainty domain of the faulty system. The progressive optimal fault-tolerant control method based on robust guaranteed cost fault-tolerant control, has been used to reconfigure the controller and ensure the optimal fault-tolerant control effect with the improving fault information. When a fault is completely identified the process of progressive optimal fault-tolerant control converges to active fault-tolerant control and the optimal fault-tolerant controller is no longer reconfigured until no more useful fault information can be provided. The essence of the progressive optimal fault-tolerant control method lies in combining active and passive fault-tolerant control manners by using the continuously improving fault information.

The rest of this article is structured as follows. In Section 2, the necessary preliminaries and the problem formulation of progressive optimal fault-tolerant control are provided. Section 3 explores a progressive optimal fault-tolerant control method in a linear uncertain system. A case study is presented in Section 4. Finally, Section 5 concludes this article.

2. Preliminaries and Problem Formulation

A system fault can be considered as a deviation of the system parameters [35]. Therefore, a faulty system can be modeled as an uncertain dynamic system with parameter uncertainty. The area where an actual value point of the system parameter vector might exist is called the uncertainty domain.

An uncertain dynamic system is defined by (1), and its state feedback is given by (2).

{\begin{matrix} \dot{x} = f (x, θ, u) \\ y = C x \\ θ = θ_{0} + Δ θ \end{matrix}

(1)

u = - K x

(2)

In (1) and (2),

x \in R^{n}

represents the system’s state parameter,

u \in R^{p}

represents the input, and

y \in R

represents the output, respectively;

f (•)

is a non-linear function of

x

and

u

parameterized by a vector

θ

;

C

represents the output matrix with a proper dimension;

Δ θ

denotes the uncertainty of the parameter vector related to the uncertainty domain

Ω

, i.e.,

θ_{0} + Δ θ \in Ω

. In this study, it is assumed that the uncertainty domain

Ω

surrounds the nominal value

θ_{0}

of the system parameter vector

θ

, and

K \in R^{q}

is the controller parameter vector.

The selection of the controller parameter vector

K

is called controller configuration. It is assumed that this selection is related to the object function (3).

J = F (x, u)

(3)

Furthermore, assume that

p_{1} (x, u), p_{2} (x, u), ... p_{r} (x, u) ... p_{g} (x, u)

are

g

parameters of a closed-loop system with certain constraint conditions. These

g

parameters can take eigenvalues of the closed-loop system (1) or other values, depending on the application context. The constraint condition of the controller parameter vector

K

can be defined by (4).

p_{r} (x, u) \in Λ_{r}, \forall r = 1, 2, ..., v, \forall K \in Ψ (Ω), \forall θ \in Ω

(4)

The constraint condition (4) implies a set of crucial indexes that should be satisfied and represents the basic constraint condition of the controller parameter selection. In (4),

Λ_{r}

represents a certain domain in a complex plane. For instance, if

p_{r}

is an eigenvalue, then

Λ_{r}

can be a left-half s plane. The closed-loop system (1) is considered to have good stability if condition (4) is satisfied.

Definition 1:

All values of controller vector

K

under constraint condition (4) form a feasible domain

Ψ (Ω)

corresponding to an uncertainty domain

Ω

[28].

Then, the objective of controller configuration is that the closed-loop system (1) satisfies the following condition:

\min J = \min F (x, u)

(5)

where

\min J

can be analytic or non-analytic expression; for instance, in the general case, it can be a minimizing operation of a quadratic function of system state variables. Alternatively, it could be described non-analytically, such as “the controller is the simplest to obtain”.

For the state feedback controller (2), (6) is selected as one of the constraint conditions.

J \leq J^{*}

(6)

In (6),

J^{*}

is a positive number and denotes the upper bound of the performance index

J

For the closed-loop system (1) and performance index (3), all controller parameter values

K

corresponding to the uncertainty domain

Ω

that satisfy condition (6) form as feasible domain

Ψ (Ω)

. Therefore, the progressive optimal fault-tolerant control is discussed in the feasible domain

Ψ (Ω)

corresponding to the uncertainty domain

Ω

Each time with the narrowing of the uncertainty domain

Ω

of a fault, i.e., the fault information becomes increasingly sufficient, intuitively, there exists the following relation:

Ω \supset Ω_{1} \supset Ω_{2} \supset ... \supset Ω_{i} \supset ... \supset Ω_{j}

, where

Ω_{i}

denotes the uncertainty domain at the

i th

moment. And the

j th

moment is after the

i th

moment, if

j > i

. This indicates that the uncertainty domain

Ω

of a fault and its narrowed sub-domains exhibit the nested property.

With the continuous increase and improvement of fault information, the uncertainty domain of the fault shrinks.

To illustrate the progressive optimal fault-tolerant control method, we first introduce Lemma 1.

Lemma 1:

For an uncertain dynamic system (1), the smaller the range of uncertainty domain

Ω

of a fault, the lower the upper bound of the performance index.

Proof

Consider an arbitrary sub-domain

Ω_{i}

of an uncertainty domain

Ω

of a fault. For uncertain system (1), suppose that the upper bounds

J^{*} (Ω)

and

J^{*} (Ω_{i})

of performance index corresponding to

Ω

and

Ω_{i}

, respectively, satisfy the following condition:

J^{*} (Ω_{i}) > J^{*} (Ω)

(7)

As long as the actual system parameter value satisfies

θ \in Ω

, it holds that

J \leq J^{*} (Ω)

. Under the condition of

Ω_{i} \subset Ω

, the actual system parameter value

θ

locates in

Ω_{i}

, but it also locates in

Ω

simultaneously due to the nested property. Therefore, the performance index

J

corresponding to

Ω_{i}

satisfies the condition of

J \leq J^{*} (Ω)

according to (6). Thus, the upper bound of the performance index

J^{*} (Ω_{i})

corresponding to

Ω_{i}

satisfies the condition of

J^{*} (Ω_{i}) \leq J^{*} (Ω)

, and (7) is not true.

Hence the proof.

Based on Lemma 1,

J^{*} (Ω_{i}) \leq J^{*} (Ω)

is valid, and in accordance with the nested property, when the range of the uncertainty domain is narrowing (i.e.,

Ω_{j} \subset Ω_{i}

), then it holds that

J^{*} (Ω_{j}) \leq J^{*} (Ω_{i}), j \leq i

(8)

It should be noted that

J^{*} (Ω_{j}) = J^{*} (Ω_{i})

indicates that regardless of the sub-domain where an actual system parameter value can be located, the upper bound of the performance index will not change. That further means the fault has been identified or the fault diagnosis procedure cannot provide more useful fault information.

Definition 2:

With each narrowing of the uncertainty domain

Ω_{i} \subset Ω_{j} \subset Ω

of a fault depending on the progressively sufficient fault information, the controller with the minimum upper bound of the performance index can be defined as follows:

u = K_{i} x

(9)

K_{i} = \arg \min (J^{*} (Ω_{i})), \forall θ \in Ω_{i}

(10)

(11)

Controller

K_{i}

that satisfies (10) and (11) corresponding to uncertainty sub-domain

Ω_{i}

of a fault represents a progressive optimal fault-tolerant controller, and the whole control process is progressive optimal fault-tolerant control.

Theorem 1:

When dynamic system (1) satisfies the following three conditions in a different and continuously narrowing uncertainty domain

Ω_{i}

of a fault,

1): $J (Ω_{i}) \leq J^{*} (Ω_{i}), \forall K_{i} \in Ψ (Ω_{i}), Ω \supset Ω_{1} \supset Ω_{2} \supset ... \supset Ω_{j} \supset ... \supset Ω_{i}, \forall j < i, \forall j = 1, 2, 3...$ ;
2): $K_{i} = \arg \min (J^{*} (Ω_{i}))$ ;
3): $\min (J^{*} (Ω_{i})) \leq \min (J^{*} (Ω_{j}))$ ;

then system (1) is a progressive optimal fault-tolerant control system, where

Ψ (Ω_{i})

is the feasible domain formed by controller parameter vectors

K_{i}

that satisfy constraint condition 1) for the uncertainty domain

Ω_{i}

Proof of Theorem 1.

According to Definition 2, with the narrowing of uncertainty sub-domain

Ω_{i}

of a fault, a progressive optimal fault-tolerant controller

K_{i}

is currently optimal with

\min (J^{*} (Ω_{i}))

When the uncertainty sub-domain

Ω_{i}

of a fault decreases with the gradually improving fault information, in accordance with Lemma 1 and the nested property of the uncertainty domain, the upper bound of the performance index decreases, i.e.,

J^{*} (Ω_{i}) \leq J^{*} (Ω)

(12)

Then, it holds that

\min (J^{*} (Ω)) \geq \min (J^{*} (Ω_{j})) \geq \min (J^{*} (Ω_{i})), Ω \supset Ω_{1} \supset Ω_{2} \supset ... \supset Ω_{j} \supset ... \supset Ω_{i}, \forall j = 1, 2, 3...

(13)

Hence the proof.

From (13), it’s obvious that the narrower the uncertainty domain of a fault, the better the control effect achieved during the process of progressive optimal fault-tolerant control. In the current uncertainty domain, a fault-tolerant controller is optimal with a minimum upper bound

\min (J^{*} (Ω_{i}))

of the performance index. The progressive optimal fault-tolerant control is performed until the fault is fully identified or the diagnosis process cannot provide more useful fault information.

3. Progressive Optimal Fault-Tolerant Control in A Linear Uncertain System

Consider a linear system defined as follows:

\dot{x} = A x + B u, y = C x

(14)

Assume that there is a parameter fault in a linear uncertain system (14), which can be expressed by

\dot{x} = (A + Δ A) x + (B + Δ B) u, y = C x

(15)

where

A

and

B

represent the nominal system matrix and control matrix, respectively, and

A \in R^{n \times n}, B \in R^{n \times m}

;

C

is the output matrix. The possible deviation domains of the faulty parameters are considered to be uncertainty domains;

Δ A

and

Δ B

denote the parameter uncertainties caused by a fault of the controlled object and actuator, respectively, and these two types of fault are reflected in changes in the matrices

A

and

B

Δ A

and

Δ B

denote uncertain real-value matrices with appropriate dimension. According to [34], it can be written that

Δ A = M_{1} F_{1} (t) E_{1}

(16)

Δ B = M_{2} F_{2} (t) E_{2}

(17)

‖ Δ A ‖ \leq α, ‖ Δ B ‖ \leq β

(18)

where

M_{1}, M_{2} \in R^{n \times r}, E_{1} \in R^{q \times n}, E_{2} \in R^{q \times m}

, and they are all rational real matrices;

α

and

β

are known scalars that means

Δ A

and

Δ B

are norm bounded;

F_{1} (t), F_{2} (t) \in R^{r \times q}

are uncertainty function matrices that represent the time degeneration of a parameter fault.

Assume that matrices

F_{1} (t), F_{2} (t)

belong to a set

Θ

as defined below [34]:

Θ = {F_{z} (t) | F_{z}^{T} (t) F_{z} (t) \leq I, \forall z = 1, 2, \forall t}

(19)

Consider a progressive optimal fault-tolerant control method for a linear uncertain system, as discussed below. With the constraint condition of guaranteed robust cost control, the progressive optimal fault-tolerant control methods is achieved by searching for a feasible domain on the uncertainty domain of the fault.

3.1. Progressive Optimal Fault-Tolerant Control from the Perspective of Guaranteed Robust Cost Control

According to Theorem 8.3.2 in [34], which defines that for system (15) and performance index (20), the sufficient and necessary condition for a linear state feedback controller (21) to make a closed-loop system (15) guaranteed robust cost is that there exists a symmetric matrix

X

, matrix

Y

, and a suitable constant

ε > 0

that make the linear matrix inequality (22) hold.

J = E {\int_{0}^{\infty} (x^{T} Q x + u^{T} R u) d t}

(20)

u = - K x = - Y X^{- 1} x

(21)

[\begin{matrix} ψ & D & X E_{1}^{T} - Y^{T} E_{2}^{T} & \begin{matrix} X & Y^{T} \end{matrix} \\ * & - ε^{- 1} I & 0 & \begin{matrix} 0 & 0 \end{matrix} \\ * & * & - ε I & \begin{matrix} 0 & 0 \end{matrix} \\ \begin{matrix} * \\ * \end{matrix} & \begin{matrix} * \\ * \end{matrix} & \begin{matrix} * \\ * \end{matrix} & \begin{matrix} \begin{matrix} - Q^{- 1} \\ * \end{matrix} & \begin{matrix} 0 \\ - R^{- 1} \end{matrix} \end{matrix} \end{matrix}] < 0

(22)

where:

X

= H^{- 1}

;

Y

= K H^{- 1}

;

ψ

= A_{0} X + X A_{0}^{T} - B_{0} Y - Y^{T} B_{0}^{T}

;

I

is a unit matrix;

*

is a transpose matrix with the corresponding term.

Furthermore, the corresponding upper bound of performance index (20) is defined by

J \leq J^{*} = tr (H)

(23)

From above, there is an implicit precondition that the uncertainty domain surrounds the normal system parameter value, that is, the nominal parameter value is used to design a guaranteed robust cost controller, as shown in Figure 1.

As more fault information becomes available, the uncertainty domain of a fault where the value point of the system parameter vector can be located will become narrower. According to Theorem 1 of progressive optimal fault-tolerant control, the minimum upper bound of performance index (23) continuously decreases until a fault is fully identified or no more fault information can be provided. Furthermore, due to the currently insufficient fault information, the fault-tolerant controller should be currently optimal.

Obviously, after a fault occurs, a system must deviate from the nominal state and if the nominal parameter value is used to design a progressive optimal fault-tolerant controller, the fault-tolerant control can be conservative or even invalid. Therefore, in this study, domain segmentation is performed for the uncertainty domain of a fault to obtain the auxiliary center point to design a progressive optimal fault-tolerant controller.

At each time, uncertainty domain

Ω_{i}

of a fault can be determined according to the current and insufficient fault information. Then, domain segmentation is performed on uncertainty domain

Ω_{i}

of the fault. Each sub-domain

Ω_{i w}, w = 1, 2, 3, ... s, i = 1, 2, 3...

of uncertainty domain

Ω_{i}

satisfies the condition of

Ω_{i} = Ω_{i 1} \cup Ω_{i 2} \cup ... \cup Ω_{i w} \cup ... \cup Ω_{i s}, w = 1, 2, 3... s, i = 1, 2, 3...

, where

s

represents the number of sub-domains. Then, the center point for each sub-domain

Ω_{i w}

is selected as an auxiliary center point.

For the auxiliary center points, the farthest distance from each auxiliary center point to the boundary of the current uncertainty domain of a fault is used as the maximum uncertainty magnitude of that point. At each segmentation part for

Ω_{i}

, there is a different uncertainty domain

ϒ_{i w}, w = 1, 2, 3... s, i = 1, 2, 3...

for each auxiliary center point for each sub-domain

Ω_{i w}

. Meanwhile,

s

is also the number of auxiliary center points. For instance, for a rectangle uncertainty domain

Ω_{1}

in Figure 2, the grid segmentation is performed on the uncertainty domain, dividing it into four uncertainty sub-domains

Ω_{1 w}, w = 1, 2, 3, 4

. Next, the center point

T_{w}, w = 1, 2, 3, 4

is selected for each uncertainty sub-domain

Ω_{1 w}, w = 1, 2, 3, 4

as an auxiliary center point. Then, the length

l_{w}, w = 4, 3, 2, 1

from the auxiliary center point

T_{w}, w = 1, 2, 3, 4

to the farthest boundary point

O_{w}, w = 4, 3, 2, 1

in the whole rectangle uncertainty domain is denoted as the maximum amplitude of uncertainty domain

ϒ_{1 w}, w = 1, 2, 3, 4

. Furthermore, controller (21) is designed with the corresponding auxiliary center for each uncertainty domain

ϒ_{1 w}, w = 1, 2, 3, 4

. Finally, the controllers for all uncertainty domains

ϒ_{1 w}, w = 1, 2, 3, 4

form the feasible domain

Ψ (Ω_{1})

Theorem 2.

For the guaranteed robust cost controller vectors

K

designed for each uncertainty domain

ϒ_{i w}, w = 1, 2, 3... s, i = 1, 2, 3...

, controller with the minimum upper bound of performance index represents a progressive optimal fault-tolerant controller.

K = \arg \min (H) = \arg \min (tr (H)) = Y X^{- 1}

(24)

Proof of Theorem 2.

It is obvious that controller

K

in (21) designed for each uncertainty domain

ϒ_{i w}, w = 1, 2, 3... s, i = 1, 2, 3...

is also feasible for

Ω_{i w}, w = 1, 2, 3... s, i = 1, 2, 3...

due to

Ω_{i} = ϒ_{i 1} \cap ϒ_{i 2} \cap ... \cap ϒ_{i w} \cap ... \cap ϒ_{i s}, w = 1, 2, 3, ... s, i = 1, 2, 3...

. Namely, controllers

K

in (21) corresponding to the auxiliary center points of

ϒ_{i w}, w = 1, 2, 3... s, i = 1, 2, 3...

form a feasible domain

Ψ (Ω_{i})

Ω_{i}

. Thus, controller (24) with the minimum upper bound of performance index (23) in the feasible domain

Ψ (Ω_{i})

denotes the current progressive optimal fault-tolerant controller.

Hence the proof.

As the uncertainty domain of a fault decreases with the progressively sufficient fault information, the aim is to find a progressive optimal fault-tolerant controller corresponding to (24) in the feasible domain

Ψ (Ω_{i})

to achieve progressive optimal fault-tolerant control. The progressive optimal fault-tolerant control process based on a guaranteed robust cost control considers both stability and performance simultaneously.

3.2. Progressive Optimal Fault-Tolerant Algorithm

From above, it is necessary to segment the uncertainty domain of a fault and set the auxiliary center point to design a progressive optimal fault-tolerant control algorithm according to the aforementioned control method. Furthermore, to determine the center point for each segmented domain as an auxiliary center point more easily, grid segmentation is selected as a division method for the uncertainty domains of a fault. The number of grids to be divided is determined according to the specific uncertainty domain. Then, the center points of each grid are regarded as auxiliary center points to design a controller. For the rectangle uncertainty domain

Ω

of a fault, as shown in Figure 3, the uncertainty domain

Ω

is divided into four grids, denoted by

Ω_{1}, Ω_{2}, Ω_{3}

and

Ω_{4}

. The grid center points

D_{1} - D_{4}

of each grid are used as auxiliary center points.

The pseudo-code of the progressive optimal fault-tolerant control algorithm is presented below.

Progressive optimal fault-tolerant control algorithm from the perspective of guaranteed robust cost control

Input:

A, B

for

i

from the first time to the

n th

time with an increment of 1
for

w

form 1 to

s

with an increment of 1
Assure uncertainty domain

Ω_{i}

of a fault and perform the grid segmentation on

Ω_{i}

, dividing it into

s

uncertainty sub-domains denoted by

Ω_{i w}, i = 1, 2, 3, ..., n, w = 1, 2, 3, ..., s, i = 1, 2, 3...

. Use the grid center points of the grids as auxiliary center points and the farthest distance from the center points of each grid to the boundary of

Ω_{i}

is used as a maximum uncertainty magnitude of that point. Determine

Δ A_{i w}

and

Δ B_{i w}

of each uncertainty domain

ϒ_{i w}, w = 1, 2, 3... s, i = 1, 2, 3...

and realize the singular value decomposition of

Δ A_{i w} = M_{i w} F_{i w} (t) E_{i w}

Δ B_{i w} = M_{i w} F_{i w} (t) E_{i w}

;
Solve linear matrix inequality (22) for each grid.
All guaranteed robust cost controllers (21) for each grid form a feasible domain

Ψ (Ω_{i})

Ω_{i}

. Find the controller

K

satisfying (24) in

Ψ (Ω_{i})

as the current progressive optimal fault-tolerant controller.
end for
if no more useful fault information is provided
end for
end if
return controller

K

end for

4. Simulations

In the simulation part, progressive optimal fault-tolerant control of a DC motor with the state space model is considered [36],

\begin{array}{l} \dot{x} = (A + Δ A) x + B u \\ y = C x \end{array}

(25)

with

x = {[\begin{matrix} i_{a} & ω \end{matrix}]}^{T}

and

Δ A

being the uncertainty caused by the parameter fault.

\begin{array}{l} A = [\begin{matrix} - \frac{R_{a}}{L_{a}} & - \frac{K_{v}}{L_{a}} \\ \frac{K_{m}}{J_{i}} & - \frac{G}{J_{i}} \end{matrix}], B = [\begin{matrix} \frac{1}{L_{a}} \\ 0 \end{matrix}], \\ D = [\begin{matrix} 0 \\ 1 \end{matrix}], C = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}], Δ A = [\begin{matrix} 0 & - \frac{σ_{v}}{L_{a}} \\ \frac{σ_{m}}{J_{i}} & 0 \end{matrix}] \end{array}

(26)

u = K x = [k_{1}, k_{2}] x

(27)

where

i_{a}

ω

, and

v_{a}

denote the armature current, angular velocity, and armature voltage, respectively.

R_{a}

is the armature resistance and

L_{a}

is the inductance of the DC motor.

K_{v}

and

K_{m}

are the voltage and motor constants, which are supposed to have parameter variations of

| σ_{k v} | \leq 2

and

| σ_{k m} | \leq 2

due to the fault, respectively.

J_{i}

is the moment of inertia, and

G

is the friction coefficient.

K \in R^{1 \times 2}

is the controller parameter vector.

The purpose of the simulation is to regulate the output error of

y

, which represented the error of armature current and angular velocity, to be near zero under a fault. The normal parameter values used in the simulations are presented in Table 1.

The faulty system (28) is considered.

A_{f} = [\begin{matrix} - \frac{R_{a}}{L_{a}} & - \frac{K_{v} - 0.4}{L_{a}} \\ \frac{K_{m} + 0.6}{J_{i}} & - \frac{G}{J_{i}} \end{matrix}], B = [\begin{matrix} \frac{1}{L_{a}} \\ 0 \end{matrix}], D = [\begin{matrix} 0 \\ 1 \end{matrix}], C = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]

(28)

Then, Algorithm is performed below:

Case 1: After a parametric fault occurred, the maximum uncertainty domain could be assured according to prior knowledge and

Δ A_{1} = [\begin{matrix} 0 & - \frac{| 2 |}{L_{a}} \\ \frac{| 2 |}{J_{i}} & 0 \end{matrix}]

σ_{k v} = 0, σ_{k v} = 2, σ_{k v} = - 2, σ_{k m} = 0, σ_{k m} = 2, σ_{k m} = - 2

are used to perform grid segmentation on the uncertainty domain, as shown in The center points of the grids,

H_{1} - H_{4}

, are used as auxiliary center points. The optimization performance index

\min (J^{*})

corresponding to the feasible domain and the progressive optimal fault-tolerant controller are shown in Table 2.

The fault-tolerant control result of Case 1 obtained using fault-tolerant controllers

K_{1}

K_{2}

K_{3}

, and

K_{4}

is shown in Figure 5. Based on Table 2, the progressive optimal fault-tolerant controller is

K_{1}

, with the minimum upper bound of optimization performance index of

\min (J_{1}^{*}) = 15.2337

comparing to

K_{2}

with

\min (J_{1}^{*}) = 36.4675

K_{3}

with

\min (J_{1}^{*}) = 32.9878

and

K_{4}

with

\min (J_{1}^{*}) = 34.5717

. As shown in Figure 5, controller

K_{1}

performs better with a smaller overshot and better comprehensive performance, than controllers

K_{2}

K_{3}

, and

K_{4}

Case 2: In this case, it is assumed that the uncertainty domain narrowed with the increase of the fault information amount and

Δ A_{2} = [\begin{matrix} 0 & - \frac{| 1 |}{L_{a}} \\ \frac{| 1 |}{J_{i}} & 0 \end{matrix}]

. Furthermore,

σ_{k v} = 0, σ_{k v} = 1, σ_{k v} = - 1, σ_{k m} = 0, σ_{k m} = 1, σ_{k m} = - 1

are used to perform grid segmentation on the uncertainty domain, as shown in Figure 6. The center points of the grids,

D_{1} - D_{4}

, are used as auxiliary center points. The optimization performance index

\min (J^{*})

corresponding to the feasible domain and the progressive optimal fault-tolerant controller are shown in Table 3.

The fault-tolerant control result of Case 2 obtained using fault-tolerant controllers

K_{5}

K_{6}

K_{7}

, and

K_{8}

is shown in Figure 7. Based on Table 3, the progressive optimal fault-tolerant controller is

K_{5}

, with the minimum upper bound of optimization performance index of

\min (J_{2}^{*}) = 9.7222

comparing to

K_{6}

with

\min (J_{2}^{*}) = 18.8913

K_{7}

with

\min (J_{2}^{*}) = 22.6618

and

K_{8}

with

\min (J_{2}^{*}) = 44.2438

. As shown in Figure 7, controller

K_{1}

performs better with a smaller overshot and better comprehensive performance, than controllers

K_{6}

K_{7}

, and

K_{8}

Case 3: In this case, it is assumed that the uncertainty domain decreases with the increase in the fault information amount, and a fault parameter has been identified, that is,

σ_{3 v} = 0

. Then, it is obtained that

Δ A_{3} = [\begin{matrix} 0 & 0 \\ \frac{| 0.8 |}{J_{i}} & 0 \end{matrix}]

. Furthermore,

σ_{k v} = 0.4, σ_{k m} = 0, σ_{k m} = 0.8, σ_{k m} = - 0.8

are used to perform grid segmentation on the uncertainty domain, as shown in Figure 8. The center points of the grids,

G_{1}

and

G_{2}

, are used as auxiliary center points. The optimization performance index

\min (J^{*})

corresponding to the feasible domain and the progressive optimal fault-tolerant controller are shown in Table 4.

The fault-tolerant control result of Case 3 obtained using fault-tolerant controllers

K_{9}

and

K_{10}

is shown in Figure 9. Based on Table 4, the progressive optimal fault-tolerant controller is

K_{9}

, with the minimum upper bound of the performance index of

\min (J_{3}^{*}) = 6.8313

compare to

K_{10}

with

\min (J_{3}^{*}) = 10.2335

. As shown in Figure 9, controller

K_{9}

performs better than controller

K_{10}

, having smaller overshot and better comprehensive performance.

According to the data presents in Table 4, Table 5 and Table 6, the minimum upper bound

\min (J^{*})

of the performance index decreases as the uncertainty domain becomes narrower, it is

\min (J_{1} *) = 15.2337

for

Δ A_{1} = [\begin{matrix} 0 & - \frac{| 2 |}{L_{a}} \\ \frac{| 2 |}{J_{i}} & 0 \end{matrix}]

\min (J_{1} *) = 9.7222

for

Δ A_{2} = [\begin{matrix} 0 & - \frac{| 1 |}{L_{a}} \\ \frac{| 1 |}{J_{i}} & 0 \end{matrix}]

, and

\min (J_{1} *) = 6.83134

for

Δ A_{3} = [\begin{matrix} 0 & 0 \\ \frac{| 0.8 |}{J_{i}} & 0 \end{matrix}]

, which met the theory of progressive optimal fault-tolerant control. After each grid segmentation of the uncertainty domain, the progressive optimal fault-tolerant controller could be obtained. In addition, the corresponding progressive optimal fault-tolerant controller is currently optimal before the uncertainty domain stops narrowing, that is, before the useful fault information amount stops increasing or the fault is identified. The above simulation results also verify the feasibility of the algorithm.

5. Conclusion

This paper studies the progressive optimal fault-tolerant control method combining the AFTC and PFTC manners by fully using insufficient fault information. In this study, a system fault is considered as system uncertainty. The progressive optimal fault-tolerant control method based on guaranteed robust cost control is proposed. The proposed method addresses two aspects. First, as the domain of parameter uncertainty of a fault becomes narrower, the fault-tolerant effect improves. Second, at each time, based on the uncertainty domain of the corresponding fault information, an currently optimal fault-tolerant controller is determined. In the process of progressive optimal fault-tolerant control, the optimal fault-tolerant controller is no longer reconfigured until no more useful fault information can be provided. Finally, the process of progressive optimal fault-tolerant control converges to active fault-tolerant control while the fault is completely identified. A progressive optimal fault-tolerant control algorithm is introduced, and it’s based on grid segmentation of the uncertainty domains of a fault and the selection of auxiliary center points. The proposed method is validated by the theoretical analysis and simulation. The proposed method has potential application value in practical control systems. In future work, attention will be focused on exploring progressive optimal fault-tolerant control with weaker conservatism and lower computational complexity.

Author Contributions

Conceptualization, Z.L. and B.D.; methodology, Z.L. and D.D.; software, D.D.; validation, D.D.; formal analysis, Z.L. and D.D.; investigation, D.D.; resources, Z.L.; data curation, D.D.; writing—original draft preparation, D.D.; writing—review and editing, Z.L. and D.D.; visualization, D.D.; supervision, Z.L.; project administration, Z.L.; funding acquisition, Z.L. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by National Natural Science Foundation of China, grant number 61963009.

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. The uncertainty domain diagram.

Figure 2. The diagram of the uncertainty domain determination.

Figure 3. An example of grid segmentation.

Figure 4. The grid segmentation in Case 1.

Figure 5. The control result of Case 1.

Figure 6. The grid segmentation in Case 2.

Figure 7. The control result of Case 2.

Figure 8. The grid segmentation in Case 3.

Figure 9. The control result of Case 3.

Table 1. Parameter values [37].

Parameter	Value
	1.2 Ω
	0.05 mH
	0.6
	0.6
	0.1352
G	0.3

Table 2. The progressive optimal fault-tolerant controller parameters in case 1.

Table 3. The progressive optimal fault-tolerant controller parameters in case 2.

Table 4. The progressive optimal fault-tolerant controller parameters in case 3.

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Progressive Optimal Fault-Tolerant Control Combining Active and Passive Control Manners

Abstract

1. Introduction

2. Preliminaries and Problem Formulation

3. Progressive Optimal Fault-Tolerant Control in A Linear Uncertain System

3.1. Progressive Optimal Fault-Tolerant Control from the Perspective of Guaranteed Robust Cost Control

3.2. Progressive Optimal Fault-Tolerant Algorithm

4. Simulations

5. Conclusion

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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