A robust nonlinear control method based on time-varying gain extended state observer

Abstract

The application provides a robust nonlinear control method based on a time-varying gain extended state observer, and belongs to the technical field of quadrotor aircraft control. The application mainly faces the attitude system of a quadrotor aircraft, solves the problems of unknown external disturbance, model uncertainty, unmeasurable angular velocity and "derivative peak" in the traditional linear extended state observer in the attitude tracking of the quadrotor aircraft by designing a time-varying gain extended state observer. In addition, the time-varying gain extended state observer is combined with a backstepping sliding mode control method to form a novel robust nonlinear control method, the speed and precision of the tracked attitude of the quadrotor aircraft are improved, and the effectiveness and superiority of the control method are embodied.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of quadrotor aircraft system control, and particularly relates to a robust nonlinear control method based on a time-varying gain extended state observer. BACKGROUND

[0002] With the development and progress of battery technology, motor technology, flight control technology and the like, the quadrotor aircraft has been rapidly developed. Due to the advantages of small size, simple operation, high reliability and convenience, the quadrotor aircraft has been widely applied in load transportation, city monitoring, environmental monitoring, precision agriculture and the like, and has very important civil and military values. However, the quadrotor aircraft is a complex system with nonlinearity, strong coupling and under-actuation, and the quadrotor aircraft system often has problems such as unknown external disturbance, model uncertainty and unmeasurable state in practice, which will affect the actual flight effect of the quadrotor aircraft. Therefore, these problems need to be considered in the process of designing the controller. How to design an adaptive, suitable and strong robust controller to enable the quadrotor aircraft to achieve stable control in the presence of the above problems is currently studied by many scholars. Therefore, the application proposes a robust nonlinear control method based on a time-varying gain extended state observer for the above problems. SUMMARY

[0003] In view of the problems in the background art, the application aims to provide a robust nonlinear control method based on a time-varying gain extended state observer. The application mainly faces the attitude system of the quadrotor aircraft, and the time-varying gain extended state observer is designed to observe the "total disturbance" (including unknown external disturbance and model uncertainty) and unmeasurable angular velocity of the attitude system of the quadrotor aircraft. In addition, the time-varying gain extended state observer can overcome the "derivative peak" phenomenon often occurring in the traditional linear extended state observer. The time-varying gain extended state observer is combined with backstepping sliding mode control to form a new robust nonlinear control method, and the control method has strong robustness.

[0004] To achieve the above object, the technical scheme of the application is as follows: a robust nonlinear control method based on a time-varying gain extended state observer, comprising the following steps:

[0005] Step 1: establishing a mathematical model of the attitude system of the quadrotor aircraft;

[0006] A quadrotor is a nonlinear, under-actuated, and strongly coupled complex system, whose motion model is established on the earth coordinate system and the body coordinate system; the control of the quadrotor is affected by the lift of the four rotors and the gravity of the body, and the lift of each rotor is related to the rotating speed of the rotor; in addition, the quadrotor realizes vertical motion by simultaneously increasing or decreasing the rotating speed of the four rotors, realizes pitching and rolling motion by changing the rotating speed difference of two opposite rotors, and realizes yawing motion by changing the rotating speed difference of two pairs of rotors.

[0007] The mathematical model of the attitude system of the quadrotor is as follows:

[0008]

[0009] In the above formula, θ,ψ respectively represent the roll angle, the pitch angle and the yaw angle of the quadrotor. represent the attitude angular velocity of the quadrotor. represent the attitude angular acceleration. x ,I y ,I z respectively represent the moment of inertia of the quadrotor along the three direction axes; k θ ,k ψ represent the air resistance coefficient; d1, d2, d3 represent unknown external disturbances; φ ,τ θ ,τ ψ represent the control input of the quadrotor. It is assumed that x ,I y ,I z , k θ ,k ψ are unknown constants, and is an unmeasurable state.

[0010] Step 2: Design a time-varying gain extended state observer for the attitude system of the quadrotor, and the specific process is as follows:

[0011] The attitude system (2-1) of the quadrotor is converted as follows:

[0012]

[0013] In the above formula, X3=(f1,f2,f3) T is a newly defined state variable, is a newly defined control input, The first derivative state represented as f1, f2, f3.

[0014] For (2-2), the time-varying gain extended state observer is designed as:

[0015]

[0016] In the above formula, Z1, Z2, Z3 are the observation states of X1, X2, X3 respectively; The first derivative observation state is represented. U1 represents the system control input; w0, l1(t), l2(t) represent the gain coefficients of the observer; l1(t), l2(t) represent the time-varying gain, whose expression is:

[0017]

[0018] In the above formula, w i > 0, b0 > 0, c0 > 0 are parameters to be designed, w i is the gain coefficient of the observer, b0 is the function amplitude rise time, c0 is the adjustable rate coefficient, l i (t) is smooth and strictly monotonically increasing;

[0019] Step 3: Design a backstepping sliding mode control method according to the attitude tracking error and angular velocity tracking error of the quadrotor aircraft;

[0020] First, define the attitude tracking error E1 as:

[0021] E1 = X 1d - X1 (2-5)

[0022] In the formula, is represented as the expected tracking attitude, The Lyapunov function V1 is selected as:

[0023]

[0024] Taking the derivative of V1 gives:

[0025]

[0026] Define a virtual control rate k1 > 0 is a parameter to be designed; based on the above derivation, define the angular velocity tracking error E2 as:

[0027] E2 = X 2d - X2 (2-8)

[0028] Define the sliding surface S1 = λ1E1 + E2, λ1 > 0 is a parameter to be designed; then select the Lyapunov function V2 as:

[0029]

[0030] wherein, the derivation of V2 is:

[0031]

[0032] In order to make satisfy the following conditions k2>0 is a parameter to be designed, let

[0033] The preliminary design of the quadrotor attitude system controller U1 is:

[0034]

[0035] Step 4: X2, X3 in the controller (2-11) are both unmeasurable states. In order to improve the reliability and robustness of the designed controller, X2, X3 are replaced by the observed states Z2, Z3 of the time-varying gain extended state observer respectively; the finally preliminary designed controller is rewritten as:

[0036]

[0037] Compared with the closest prior art, the present application has the beneficial effects of:

[0038] The technical scheme provided by the present application proposes a time-varying gain extended state observer for the unknown external disturbance, model uncertainty and unmeasurable angular velocity of the quadrotor, which can solve the above problems and overcome the "derivative peak" problem often occurring in the traditional linear extended state observer. The proposed robust nonlinear control method can make the quadrotor track the desired attitude well in the presence of the above problems. On the other hand, the control method provided by the present application is optimized in improving tracking accuracy, tracking speed and optimizing control input amplitude. BRIEF DESCRIPTION OF DRAWINGS

[0039] Figure 1 The structure of the quadrotor;

[0040] Figure 2 The control flowchart of the quadrotor. DETAILED DESCRIPTION

[0041] The specific embodiments of the present application will be further described in detail below with reference to the accompanying drawings.

[0042] In order to make the objects, technical solutions and advantages of the embodiments of the present application clearer, the technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are some but not all of the embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by a person of ordinary skill in the art without creative work fall within the protection scope of the present application.

[0043] Figure 1 is a structural diagram of a quadrotor aircraft. Figure 2 is a control flow diagram of a quadrotor aircraft, as Figure 2 shown, the present application discloses a robust nonlinear control method based on a time-varying gain extended state observer, and the specific steps are as follows:

[0044] Step 1: Obtain the mathematical model of the attitude system of the quadrotor aircraft according to the Newton-Euler equation, and add the assumptions of unknown external disturbance, model uncertainty and unmeasurable angular velocity in the mathematical model. The mathematical model of the attitude system of the quadrotor aircraft is:

[0045]

[0046] In the above formula, θ,ψ respectively represent the roll angle, pitch angle and yaw angle of the quadrotor aircraft. represent the attitude angular velocity of the quadrotor aircraft. represent the attitude angular acceleration. x ,I y ,I z respectively represent the rotational inertia of the quadrotor aircraft along the three direction axes; k θ ,k ψ represent the air resistance coefficient; d1,d2,d3 represent unknown external disturbance; φ ,τ θ ,τ ψ represent the control input of the quadrotor aircraft. It is assumed that x ,I y ,I z , k θ ,k ψ are unknown constants, and is an unmeasurable state.

[0047] Step 2: Design a time-varying gain extended state observer for the transformed attitude system, set part of the gain coefficients of the observer as time-varying gains, configure the poles of the observer at different places and set the gain coefficients.

[0048] First, quadrotor attitude system (3-1) is rewritten as:

[0049]

[0050] In the above formula, X3 = (f1, f2, f3) T is a newly defined state variable, is a newly defined control input, is the first derivative state represented as f1, f2, f3.

[0051] A time-varying gain extended state observer is designed for (3-2), and its mathematical model is as follows:

[0052]

[0053] In the above formula, Z1, Z2, Z3 are the observed states of X1, X2, X3 respectively; is the first derivative observed state. U1 represents the system control input; w0, l1(t), l2(t) represent the gain coefficients of the observer; l1(t), l2(t) represent the time-varying gain, whose expression is:

[0054]

[0055] In the above formula, w i > 0, b0 > 0, c0 > 0 are parameters to be designed, w i is the gain coefficient of the observer, b0 can determine the function amplitude rise time, c0 is the adjustable rate coefficient, l i (t) is smooth and strictly monotonically increasing.

[0056] Step 3: Design a backstepping sliding mode control method according to the attitude tracking error and angular velocity tracking error of the quadrotor.

[0057] First, define the attitude tracking error E1 as

[0058] E1 = X 1d - X1 (3-5)

[0059] In the formula, is the expected tracking attitude, The Lyapunov function V1 is selected as

[0060]

[0061] The derivative of V1 is

[0062]

[0063] A virtual control rate is defined next k1 > 0 is a parameter to be designed. Based on the above derivation, the angular velocity tracking error E2 is further defined as

[0064] E2 = X 2d - X2 (3-8)

[0065] The sliding surface S1 = λ1E1 + E2 is further defined, where λ1 > 0 is a parameter to be designed; and the Lyapunov function V2 is selected as

[0066]

[0067] The derivation of V2 is

[0068]

[0069] In order to make satisfy the following condition k2 > 0 is a parameter to be designed, and let

[0070]

[0071] Step 4: The observed states of the time-varying gain extended state observer are substituted for the unmeasurable states and the "total disturbance" in the preliminarily designed control method, so as to form a new type of nonlinear control method, which improves the robustness of the control method and is convenient for engineering application.

[0072] X2 and X3 in the controller (3-11) are both unmeasurable states. In order to improve the reliability and robustness of the designed controller, X2 and X3 are respectively replaced by the observed states Z2 and Z3 of the time-varying gain extended state observer. The preliminarily designed controller is finally rewritten in the following form:

[0073]

Claims

1. A robust nonlinear control method based on a time-varying gain extended state observer, comprising the following steps: Step 1: Establish a mathematical model of the quadcopter's attitude system; The mathematical model for the attitude system of a quadcopter is: (2-1)； In the above formula, These represent the roll angle, pitch angle, and yaw angle of a quadcopter, respectively. This represents the attitude angular velocity of the quadcopter. Represents attitude angular acceleration; These represent the moments of inertia of a quadcopter along three axes. This is represented by the air drag coefficient; This represents unknown external disturbances; This represents the control input of a quadcopter; assuming All are unknown constants, and It is an unmeasurable state; Step 2: Design a time-varying gain extended state observer for the attitude system of a quadcopter. The specific process is as follows: The attitude system (2-1) of the quadcopter is converted as follows: (2-2)； In the above formula, For the newly defined state variables, For the newly defined control input, , , , , Represented as The first derivative state; For (2-2), the time-varying gain extended state observer is designed as follows: (2-3)； In the above formula, They are respectively The observation status; Represented as the first derivative observation state; Represents system control input; Represents the gain coefficient of the observer; Representing the time-varying gain, its expression is: (2-4)； In the above formula, These are all parameters that need to be designed. It is the gain coefficient of the observer. To determine the rise time of the function's amplitude, It is an adjustable rate coefficient. It is smooth and strictly monotonically increasing; Step 3: Design a backstepping sliding mode control method based on the attitude tracking error and angular velocity tracking error of the quadcopter; First, define the attitude tracking error. for: (2-5)； In the formula, This represents the desired tracking pose. Choose the Lyapunov function for: (2-6)； right Taking the derivative, we get: (2-7)； Define a virtual control law , The parameters to be designed are given; based on the above derivation, the angular velocity tracking error is defined. for: (2-8) Define the sliding surface , The parameters to be designed are then selected; subsequently, the Lyapunov function is chosen. for: (2-9)； in, right Taking the derivative, we get: (2-10)； In order to make The following conditions must be met , For the parameters to be designed, let ; Preliminary design of the attitude system controller for a quadcopter for: (2-11)； Step 4: In the controller (2-11) These are all unmeasurable states; in order to improve the reliability and robustness of the designed controller, The observation states of the time-varying gain extended state observer were respectively replaced. The initial controller design was ultimately rewritten as follows: (2-12)。

Images