CN106647584A - Fault tolerant control method of four-rotor-wing aircraft based on optimal sliding mode - Google Patents

Abstract

The invention discloses an active fault-tolerant control method of a quadrotor aircraft based on an optimal sliding model. Considering the time lag caused by wireless network transmission, an optimal fault-tolerant control method is proposed for quadrotor aircraft with actuator failure, combining optimal control and sliding mode control. A sliding mode surface with time-delay compensation is designed, and the sufficient conditions for the asymptotic stability of the ideal sliding mode are given by using the principle of linear matrix inequality. fault-tolerant controller. The method of the present invention can eliminate the influence of the time lag by constructing the sliding mode surface of the time lag compensation, and can make the control law of the nominal system optimal by designing the quadratic optimal performance index, effectively improving the performance of the quadrotor aircraft. The control accuracy can provide a basis for the design of fault-tolerant controllers for complex quadrotors with actuator failures. The invention is used for passive fault-tolerant control of quadrotor aircraft with constant time delay.

Description

A fault-tolerant control method for quadrotor aircraft based on optimal sliding mode

technical field

The invention relates to a fault-tolerant control method of a quadrotor aircraft based on an optimal sliding mode, and belongs to the field of aircraft fault diagnosis and fault-tolerant control.

Background technique

There are many types of helicopters, mainly including: single-rotor helicopters, dual-rotor helicopters (tandem, transverse) and quadrotor helicopters. Among them, the quadrotor helicopter, as a new branch of helicopter development, is different in structure and flight principle from traditional helicopters. It has two sets of propellers, front and rear, and left and right. The lift force can be changed by changing the propeller, and then the position and attitude can be changed. It can easily achieve vertical takeoff and landing, hovering, lateral and longitudinal flight and other actions. Compared with conventional layout helicopters, the structure is simpler, and the anti-torque torque generated by the four rotors can cancel each other out, without special anti-torque propeller moment. At the same time, quadrotor helicopters have the advantages of small size, light weight, low cost, flexible use, and good concealment, so they are widely used in military and civilian fields. Quadrotor helicopter is a complex controlled object, which has various complex problems such as multiple input and multiple output, nonlinearity, strong coupling, time delay, etc., and it will inevitably encounter wind disturbance and engine vibration during flight. And other uncertain factors, coupled with the lack of human real-time control, once the helicopter fails, it will cause catastrophic consequences. Therefore, the fault-tolerant controller needs to have a strong fault-tolerant ability in the case of time-delay and uncertainty in the system.

At present, the fault-tolerant control methods for quadrotor aircraft are mainly divided into active fault-tolerant control and passive fault-tolerant control. Active fault-tolerant control ensures the stability of the system after a fault occurs through fault adjustment or signal reconstruction. This method is flexible in design and strong in fault tolerance. However, the controller structure is relatively complex, and clear fault information needs to be obtained, and the system design cost is relatively high. Compared with active fault-tolerant control, passive fault-tolerant control uses the robustness of the controller itself without changing the controller structure and parameters. To make the closed-loop system insensitive to certain faults, so that the system can still operate under the original performance index after a fault occurs. This control method is relatively simple in design and low in cost, and does not need to know clear fault information, especially It is more appropriate to use passive fault-tolerant control in nonlinear systems containing multiple uncertain factors.

Since the sliding mode of sliding mode control is fully adaptive to system parameter perturbation and external disturbance, it is very suitable for dealing with the problem of passive fault-tolerant control of quadrotor helicopter flight control system. Its control is discontinuous. During the control process, the structure of the closed-loop system is constantly changing, forcing the system state to move along the pre-designed sliding surface, gradually "sliding" to the state equilibrium point, that is, asymptotically stable. Its main advantage is that once the system state quantity reaches the sliding surface, the system will not be affected by parameter changes and external disturbances. Sliding mode control is widely used in the flight control system, which provides a new idea for the fault-tolerant control of the flight control system.

However, there are still many problems to be solved in the sliding mode fault-tolerant control. For example, how to improve the robustness of the sliding mode, how to reduce the approach time, and how to ensure that the ideal sliding mode is optimal. In order to ensure the robustness of the sliding mode and improve the effect of fault-tolerant control, the idea of optimal control can be introduced. Combined with sliding mode control, the optimal sliding mode control law can effectively simplify the controller and save costs.

The existing methods cannot fully consider various factors that may exist in the actual system, such as time lag, uncertainty, faults, etc., and it is difficult to have a good control effect on the complex flight control system, so the present invention has good practicability.

Contents of the invention

Purpose of the invention: Aiming at the above-mentioned prior art, a fault-tolerant control method of a quadrotor aircraft based on optimal sliding mode is proposed, which can effectively eliminate the negative effects caused by time lag, make the ideal sliding mode performance optimal, and the fault-tolerant control law can Overcome the impact of failures on the system.

Technical solution: A fault-tolerant control method for quadrotor aircraft based on optimal sliding mode, characterized in that: considering the time lag and actuator failure of quadrotor aircraft, combining optimal control and sliding mode control, an optimal fault-tolerant control method is proposed The control method enables the aircraft to continue to fly safely after an actuator failure occurs, and to ensure good flight quality. According to the obtained model parameters of the aircraft, an integral sliding mode surface with time-delay compensation is designed to eliminate the influence of time-delay, and the quadratic optimal performance index is designed for the nominal system to obtain the optimal ideal sliding mode, and then Design the corresponding sliding mode control law, and finally constitute the optimal fault-tolerant controller. Including the following specific steps:

Step 1) establishes the mathematical model of quadrotor aircraft:

where A∈R ^n×n , A _d ∈ R ^n×n , B∈R ^n×m , C∈R ^p×n , x∈R ⁿ are the state variables of the system, ΔA(t) and ΔA _d (t) is the modeling uncertainty, x(t-τ) represents the state variable with time lag, u(t)∈R ^m is the control input of the system, and f(x,t)∈R ⁿ represents the actuator failure.

Step 2) For the above quadrotor flight control system with time lag and actuator failure, the optimal sliding mode design of the nominal system is carried out:

The nominal system of system (1) is:

In the nominal system (2), let u=u ₀ , and then define the quadratic optimal performance index as follows:

Here Q∈R ^n×n is a positive semi-definite state weight matrix, and R∈R ^n×m is a positive definite weight matrix.

According to the N iteration method, the approximate solution of the optimal control law is:

Among them, the matrix P is the positive definite solution of the following Riccati equation:

PA+A ^T P-PBR ^- 1B ^T P+Q＝0 (4)

and is the sum of the first n solutions of a set of differential equations. Control law (3) can guarantee the robustness of the whole nominal system.

Step 3) On the basis of step 1) and step 2), construct an integral sliding surface with time-delay compensation:

Among them, the matrix G∈R ^m×n satisfies GB non-singularity (because the matrix B has full rank, the choice of matrix G here is not unique). K=R ^-1 B ^T P∈R ^m×n is a constant matrix to be designed, which can be obtained by solving the linear matrix inequality (5).

It can be proved that if there is a matrix Y∈R ^m×n , a positive definite matrix X∈R ^n×n and normal constants ε ₁ , ε ₂ , ε ₃ such that the linear matrix inequality (5) holds:

Then the standard sliding mode is asymptotically stable.

in

Step 4) Construct the discontinuous sliding mode control law, so that the state trajectory of the time-delay system with faults and uncertainties is the same as the nominal system trajectory.

According to the design method of sliding mode control, the fault-tolerant controller is designed as follows:

u=u _con +u _dis , (6)

Among them, u _con is the continuous part of the sliding mode control law, and the discontinuous part u _dis is used to maintain the ideal sliding mode of the system on the sliding mode surface.

Step 4.1) The linear part of the fault-tolerant controller can be determined by the equivalent optimal control method. Due to the particularity of the sliding surface structure in step 3), the linear part of the controller is designed as follows:

Step 4.2) Design the discontinuous control part:

The design of the discontinuous part of the control law needs to know the upper bound of the uncertainty and the fault. The upper bound of the uncertainty is known, but the fault information is unknown, which is also in line with the actual situation. We can define two adaptive quantities to estimate unknown parameters online:

Then the discontinuous part of the fault-tolerant control law is:

where η is a small normal constant.

Combining equations (7) and (9), the complete optimal sliding mode fault-tolerant control law can be obtained as follows:

Step 5) According to the flight state of the quadrotor aircraft, select appropriate parameters to complete its fault-tolerant control. Beneficial effects: the present invention proposes a fault-tolerant control method for quadrotor aircraft based on the optimal sliding mode, constructs a sliding mode surface with time-delay compensation, designs the quadratic optimal performance index according to the optimal control idea, and obtains the optimal Optimal ideal sliding mode, combined with sliding mode control design method, finally constitutes a complete optimal sliding mode fault-tolerant controller.

Has the following advantages:

(1) By designing an integral sliding surface with time-delay compensation, the influence of time-delay is effectively eliminated;

(2) Using the linear matrix inequality to give the upper limit value of the time lag to ensure the asymptotic stability of the system, fully considering the time lag phenomenon that may exist in the actual flight process of the quadrotor aircraft, so that the design of the controller has better practicability;

(3) Design the quadratic optimal performance index to ensure that the ideal sliding mode of the system is the best in performance;

(4) Introduce the method of adaptive boundary estimation to estimate the magnitude of the actuator fault of the quadrotor aircraft, and the fault-tolerant control law constantly changes the parameters, making the system less conservative and the control effect better.

As a fault-tolerant control method of the quadrotor aircraft, the method used in the invention has certain practical application value, is easy to realize, has strong fault tolerance, and can effectively improve the flight safety of the quadrotor aircraft. The method has strong operability, convenient and reliable application.

Description of drawings

Fig. 1 is a flow chart of the inventive method;

Fig. 2 is Quanser's four-rotor aircraft simulation experiment system;

Fig. 3 is a schematic diagram of a quadrotor aircraft;

Fig. 4 is a schematic block diagram of a quadrotor aircraft control system;

Figure 5 is a comparison diagram of the X-axis displacement response curve;

Figure 6 is a comparison chart of the X-axis speed response curve;

Figure 7 is a comparison diagram of the dynamic response curve of the actuator;

Fig. 8 is a comparison chart of ideal sliding modal response curves;

Figure 9 is a simulink simulation diagram.

detailed description

The present invention will be further explained below in conjunction with the accompanying drawings.

As shown in Figure 1, considering the time lag and actuator failure of the quadrotor aircraft, combined with optimal control and sliding mode control, an optimal fault-tolerant control method is proposed, so that the aircraft can continue to fly safely after the actuator failure occurs, and Ensure good flight quality. According to the obtained model parameters of the aircraft, an integral sliding mode surface with time-delay compensation is designed to eliminate the influence of time-delay, and the quadratic optimal performance index is designed for the nominal system to obtain the optimal ideal sliding mode, and then Design the corresponding sliding mode control law, and finally constitute the optimal fault-tolerant controller. Including the following specific steps:

Step 1) establishes the mathematical model of quadrotor aircraft:

where A∈R ^n×n , A _d ∈ R ^n×n , B∈R ^n×m , C∈R ^p×n , x∈R ⁿ are the state variables of the system, ΔA(t) and ΔA _d (t) is the modeling uncertainty, x(t-τ) represents the state variable with time lag, u(t)∈R ^m is the control input of the system, and f(x,t)∈R ⁿ represents the actuator failure.

Step 2) For the above quadrotor flight control system with time lag and actuator failure, the optimal sliding mode design of the nominal system is carried out:

The nominal system of system (1) is:

In the nominal system (2), let u=u ₀ , and then define the quadratic optimal performance index as follows:

Here Q∈R ^n×n is a semi-positive definite state weight matrix, and R∈R ^m×m is a positive definite weight matrix.

According to the N iteration method, the approximate solution of the optimal control law is:

Among them, the matrix P is the positive definite solution of the following Riccati equation:

PA+A ^T P-PBR ^-1 B ^T P+Q＝0 (4)

and is the sum of the first n solutions of a set of differential equations. Control law (3) can guarantee the robustness of the whole nominal system.

Step 3) On the basis of step 1) and step 2), construct an integral sliding surface with time-delay compensation:

Among them, the matrix G∈R ^m×n satisfies GB non-singularity (because the matrix B has full rank, the choice of matrix G here is not unique).

K=R ^-1 B ^T P∈R ^m×n is a constant matrix to be designed, which can be obtained by solving the linear matrix inequality (5).

It can be proved that if there is a matrix Y∈R ^m×n , a positive definite matrix X∈R ^n×n and normal constants ε ₁ , ε ₂ , ε ₃ such that the linear matrix inequality (5) holds:

Then the standard sliding mode is asymptotically stable.

in

Step 4) Construct the discontinuous sliding mode control law, so that the state trajectory of the time-delay system with faults and uncertainties is the same as the nominal system trajectory.

According to the design method of sliding mode control, the fault-tolerant controller is designed as follows:

u=u _con +u _dis , (6)

Among them, u _con is the continuous part of the sliding mode control law, and the discontinuous part u _dis is used to maintain the ideal sliding mode of the system on the sliding mode surface.

Step 4.1) The linear part of the fault-tolerant controller can be determined by the equivalent optimal control method. Due to the particularity of the sliding surface structure in step 3), the linear part of the controller is designed as follows:

Step 4.2) Design the discontinuous control part:

The design of the discontinuous part of the control law needs to know the upper bound of the uncertainty and the fault. The upper bound of the uncertainty is known, but the fault information is unknown, which is also in line with the actual situation. We can define two adaptive quantities to estimate unknown parameters online:

Then the discontinuous part of the fault-tolerant control law is:

where η is a small normal constant.

Combining equations (7) and (9), the complete optimal sliding mode fault-tolerant control law can be obtained as follows:

Step 5) According to the flight state of the quadrotor aircraft, select appropriate parameters to complete its fault-tolerant control.

The above is only a preferred embodiment of the present invention, it should be pointed out that, for those of ordinary skill in the art, without departing from the principle of the present invention, some improvements and modifications can also be made, and these improvements and modifications can also be made. Should be regarded as the protection scope of the present invention,

The effectiveness of the implementation scheme is illustrated below with actual case simulation.

The semi-physical simulation platform of Qball-X4 quadrotor aircraft produced by Canada Quanser Company is used as the specific algorithm experiment simulation object. Figure 2 is Quanser's quadrotor aircraft simulation experiment system, and Figure 3 is a schematic diagram of the quadrotor aircraft's attitude movement. The simulation experiment system is composed of a ground control station, a camera positioning system and an aircraft. The main control machine communicates with each unmanned tool through a wireless local area network, and mainly performs positioning and mission planning for the system. Once the control algorithm design of the entire control system is completed, the control station can only play a positioning role, so as to conduct research on the autonomous control of unmanned tools and the coordinated control among multiple tools. The system realizes spatial three-dimensional positioning through six infrared cameras to obtain the required parameters.

The mathematical model of the quadrotor aircraft is as follows:

Among them, each coefficient matrix is as follows:

In the simulation experiment, it can be found through hardware measurement that the time delay caused by wireless transmission is generally 80-120 milliseconds. In order to prove the effectiveness of the method proposed in this chapter, it is assumed that the time delay is τ = 1s.

In the simulation process, the model of the controlled system is built with matlab simulink, and the control law and fault type can be easily modified.

Assume that the system has a sudden failure of the following form at t=11s:

Take the state vector of the system at the initial moment as:

x ₀ ＝[x ₁ x ₂ x ₃ ] ^T ＝[1 1 1.1] ^T

According to the method of the invention, fault-tolerant control is performed on a quadrotor aircraft in which actuator failure occurs. According to step 1)-step 5), the values of the undetermined parameters are as follows: the sliding surface coefficient matrix G=[0 0 1], the upper bound of uncertainty a=0.8602, a _d =0.5, and the state feedback coefficient matrix is obtained by solving K = [1.6263 0.5438 4.9179].

Figure 5-Figure 8 shows the results of fault-tolerant control. Figures 5-7 are the response curves of displacement, velocity, and actuator dynamics in the X-axis direction, respectively, and are compared with methods without time-delay processing. Figure 8 is a comparison curve between optimal fault-tolerant control and traditional sliding mode control.

It can be seen from Fig. 5-Fig. 7 that when the actuator fails in the system, the X-axis displacement and speed of the aircraft fluctuate obviously, but under the fault-tolerant control of the present invention, they can all tend to be stable in a short period of time, and the response The speed is fast and the overshoot is small, which means that the aircraft can still maintain the original flight state after the system fails, avoiding accidents and maintaining good flight quality. From the comparison of the ideal sliding mode curves in Fig. 8, it can be seen that compared with the traditional sliding mode fault-tolerant control, the method of the present invention can obtain the optimal ideal sliding mode. Therefore, the fault-tolerant control law can well guarantee the control accuracy and safety of the aircraft.

Claims (1)

1. A kind of fault-tolerant control method of quadrotor aircraft based on optimal sliding mode, it is characterized in that: consider that quadrotor aircraft has time lag and actuator failure, combines optimal control and sliding mode control, proposes an optimal fault-tolerant control The method enables the aircraft to continue to fly safely after an actuator failure occurs, and to ensure good flight quality. According to the obtained model parameters of the aircraft, an integral sliding mode surface with time-delay compensation is designed to eliminate the influence of time-delay, and the quadratic optimal performance index is designed for the nominal system to obtain the optimal ideal sliding mode, and then Design the corresponding sliding mode control law, and finally constitute the optimal fault-tolerant controller. Including the following specific steps: Step 1) establishes the mathematical model of quadrotor aircraft: $\left\{\begin{array}{c}\stackrel{<span\; class="notranslate">\; \&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&Delta;</span>}\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&Delta;A</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>\\ \mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; C</span>}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$ where A∈R ^n×n , A _d ∈ R ^n×n , B∈R ^n×m , C∈R ^p×n , x∈R ⁿ are the state variables of the system, ΔA(t) and ΔA _d (t) is the modeling uncertainty, x(t-τ) represents the state variable with time lag, u(t)∈R ^m is the control input of the system, and f(x,t)∈R ⁿ represents the actuator failure. Step 2) For the above quadrotor flight control system with time lag and actuator failure, the optimal sliding mode design of the nominal system is carried out: The nominal system of system (1) is: In the nominal system (2), let u=u ₀ , and then define the quadratic optimal performance index as follows: $\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; \&infin;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; Q</span>}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}^{\mathrm{<span\; class="notranslate">\; T</span>}}{\mathrm{<span\; class="notranslate">\; Ru</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; t</span>}$ Here Q∈R ^n×n is a semi-positive definite state weight matrix, and R∈R ^m×m is a positive definite weight matrix. According to the N iteration method, the approximate solution of the optimal control law is: ${\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; o</span>}\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; R</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; B</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; h</span>}}}_{\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$ Among them, the matrix P is the positive definite solution of the following Riccati equation: PA+A ^T P-PBR ^-1 B ^T P+Q＝0 (4) and is the sum of the first n solutions of a set of differential equations. Control law (3) can guarantee the robustness of the whole nominal system. Step 3) On the basis of step 1) and step 2), construct an integral sliding surface with time-delay compensation: $\mathrm{<span\; class="notranslate">\; \&sigma;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; G</span>}{<span\; class="notranslate">\; \&Integral;</span>}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; t</span>}}<span\; class="notranslate">\; \&lsqb;</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; B</span>}\mathrm{<span\; class="notranslate">\; K</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; BR</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; B</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}{\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; h</span>}}}_{\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&rsqb;</span>\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}$ Among them, the matrix G∈R ^m×n satisfies GB non-singularity (because the matrix B has full rank, the choice of matrix G here is not unique). K=R ^-1 B ^T P∈R ^m×n is a constant matrix to be designed, which can be obtained by solving the linear matrix inequality (5). It can be proved that if there is a matrix Y∈R ^m×n , a positive definite matrix X∈R ^n×n and normal constants ε ₁ , ε ₂ , ε ₃ such that the linear matrix inequality (5) holds: $\left[\begin{array}{ccc}{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}& {\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}& \mathrm{<span\; class="notranslate">\; P</span>}\mathrm{<span\; class="notranslate">\; D.</span>}\\ {\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}& <span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; I</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ {\mathrm{<span\; class="notranslate">\; D.</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}\mathrm{<span\; class="notranslate">\; P</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& <span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; I</span>}\end{array}\right]<span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$ Then the standard sliding mode is asymptotically stable. in Step 4) Construct the discontinuous sliding mode control law, so that the state trajectory of the time-delay system with faults and uncertainties is the same as the nominal system trajectory. According to the design method of sliding mode control, the fault-tolerant controller is designed as follows: u=u _con +u _dis , (6) Among them, u _con is the continuous part of the sliding mode control law, and the discontinuous part u _dis is used to maintain the ideal sliding mode of the system on the sliding mode surface. Step 4.1) The linear part of the fault-tolerant controller can be determined by the equivalent optimal control method. Due to the particularity of the sliding surface structure in step 3), the linear part of the controller is designed as follows: ${\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; c</span>}\mathrm{<span\; class="notranslate">\; o</span>}\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; K</span>}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; R</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; B</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}{\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; h</span>}}}_{\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$ Step 4.2) Design the discontinuous control part: The design of the discontinuous part of the control law needs to know the upper bound of the uncertainty and the fault. The upper bound of the uncertainty is known, but the fault information is unknown, which is also in line with the actual situation. We can define two adaptive quantities to estimate unknown parameters online: ${\stackrel{<span\; class="notranslate">\; \&CenterDot;</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&gamma;</span>}}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; \&sigma;</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; ,</span>{\stackrel{<span\; class="notranslate">\; \&Center\; Dot;</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&gamma;</span>}}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; \&sigma;</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$ Then the discontinuous part of the fault-tolerant control law is: ${\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; G</span>}\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; \&lsqb;</span>\mathrm{<span\; class="notranslate">\; \&eta;</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; +</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; (</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&gamma;</span>}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&gamma;</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&rsqb;</span>\frac{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; \&sigma;</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$ where η is a small normal constant. Combining equations (7) and (9), the complete optimal sliding mode fault-tolerant control law can be obtained as follows: $\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; K</span>}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; R</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; B</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}{\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; h</span>}}}_{\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; G</span>}\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; G</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; \{</span>\mathrm{<span\; class="notranslate">\; \&eta;</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&tau;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; +</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; (</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&gamma;</span>}}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; \&gamma;</span>}}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; \}</span>\frac{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}{<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; \&sigma;</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 10</span>}<span\; class="notranslate">\; )</span>$ Step 5) According to the flight state of the quadrotor aircraft, select appropriate parameters to complete its fault-tolerant control.