Fractional order robust elastic control method for unmanned aerial vehicle altitude dynamic system with uncertain parameters

Abstract

The invention discloses a fractional order robust elastic control method for an unmanned aerial vehicle altitude dynamic system with uncertain parameters. The method comprises the following steps: firstly, establishing a fractional order model of the unmanned aerial vehicle altitude dynamic system with interference; then designing a fractional order robust elastic controller based on the state observer and a feedback elastic controller; the Lyapunov stability theory is used for proving that the unmanned aerial vehicle height dynamic composite closed-loop system meets the asymptotic stability under the performance condition; and finally, the coupling problem between the observer and the controller is processed by using a linear matrix inequality technology LMI, gain matrixes of the observer and the controller are calculated and solved, and the solved gain matrixes are substituted into the altitude dynamic system of the unmanned aerial vehicle to verify the effectiveness of the designed method through analogue simulation. According to the invention, the height dynamic system of the unmanned aerial vehicle has asymptotic stability meeting performance conditions.

Description

Technical Field

[0001] This invention belongs to the field of robust control technology for UAV altitude dynamic systems, specifically relating to fractional-order control of UAV altitude dynamic systems with uncertain parameters. Robust elastic control method. Background Technology

[0002] Research on control theory based on system mechanism models has gradually become a focus of attention in the scientific and engineering communities. However, practical engineering system models often exhibit high complexity, strong coupling, and prominent nonlinear characteristics. With the continuous advancement of science and technology and the rapid development of modern industrial automation, the requirements for system accuracy are increasing, and control systems need to maintain good control performance even in the face of uncertainty and external disturbances. Unmanned aerial vehicles (UAVs), as a simple and easy-to-operate flight system, have been widely used in many fields due to their low cost and high flexibility, such as forest monitoring, marine monitoring, waterway monitoring, remote sensing mapping, urban fire fighting, and freight transport. However, due to their small size and light weight, UAVs are inevitably subject to interference from various external factors during use, making it difficult to accurately obtain state information. This leads to a decline in the system control performance of UAVs or even loss of control, posing a significant challenge to the control safety and application of UAVs. PID (Proportional, Integral, and Derivative) control is the most commonly used aircraft control method due to its simple design and ease of engineering implementation. However, when model parameters or operating conditions change, control performance is difficult to guarantee. Studies have shown that, compared with traditional integer-order PID controllers, fractional-order PID controllers can more accurately predict and adjust the changing trends of the system, achieving faster response speeds and thus better control performance under conditions of significant disturbances. The introduction of fractional-order control theory provides new ideas and approaches for solving some problems that traditional control theory cannot handle.

[0003] Fractional calculus differs from traditional calculus in that its orders are all integers. The orders of fractional calculus can be any rational, irrational, or even complex number. Integer-order differential systems are typically used to describe phenomena that can be represented by derivatives, such as the motion of objects and changes in temperature. Fractional calculus, with its wider range of orders, can more accurately describe phenomena that traditional integer-order differential systems cannot accurately represent. For example, in physics and engineering, the dynamic behavior of systems with nonlinearity and memory effects, such as springs, can be more accurately described using fractional differential equations. Fractional control systems, by incorporating fractional calculus theory, improve the accuracy and flexibility of control systems. Research shows that fractional controllers can increase the degrees of freedom in adjusting system controller parameters, thereby further improving the performance of the controlled system. In industrial automated production lines, fractional control systems are better able to adapt to complex and changing environments, such as robotic arm control. Furthermore, because fractional controllers are less sensitive to changes in system parameters and environmental disturbances, they also exhibit better robustness and stability. Direct control of fractional systems is particularly important for more accurately reflecting the dynamic characteristics of a system.

[0004] Robust control methods Performance control is the earliest method for mitigating disturbances and an important branch of modern control theory, primarily used to handle systems with uncertainties and external disturbances. Unlike traditional PID control methods, Performance control, through optimized controller design, enables the system to possess strong robustness against external disturbances and internal uncertainties, thereby ensuring system stability and performance over a wide range of disturbances and parameter variations. In recent years, with the improvement of industrial automation levels and the continuous increase in system control requirements, Performance control, due to its high adaptability to the dynamic characteristics of the system, has become the preferred method for solving complex control problems requiring high precision and high reliability. However, Implementing performance control typically involves extensive computation and complex mathematical modeling, especially in real-world systems where factors such as disturbances, nonlinear characteristics, and time delays can significantly impact performance. The design and application of performance control face significant challenges. Currently, Research on performance control mainly focuses on how to simplify controller design, improve computational efficiency, and overcome the effects of system uncertainties and disturbances. Summary of the Invention

[0005] The purpose of this invention is to provide a fractional-order method for UAV altitude dynamics systems with uncertain parameters. Robust elastic control methods enable UAV altitude dynamic systems to meet the requirements Asymptotic stability under performance conditions.

[0006] The technical solution adopted in this invention is a fractional-order method for a drone altitude dynamic system with uncertain parameters. The robust elastic control method is implemented according to the following steps: Step 1: Establish a fractional-order model of the UAV's high dynamic system with interference; Step 2: Design a fractional-order controller based on a state observer and a feedback elastic controller. Robust elastic controller; Step 3: Using Lyapunov stability theory, prove that the UAV's highly dynamic composite closed-loop system satisfies... Asymptotic stability under performance conditions; Step 4: Use the Linear Matrix Inequality (LMI) technique to handle the coupling problem between the observer and the controller, and calculate and solve for the gain matrices of the observer and the controller. Substitute the solved gain matrices into the UAV altitude dynamic system simulation to verify the effectiveness of the designed method.

[0007] The invention is further characterized in that, Step 1 is implemented in the following steps: Step 1.1, the Caputo fractional order is defined as...

[0008] in, It is greater than The smallest integer that satisfies , ,when When, the Caputo fractional derivative is defined as:

[0009] in, express On the interval Caputo fractional derivative of order 1; First, the mathematical model of the UAV altitude dynamics system considering interference factors is obtained using the Lagrange method, as shown below:

[0010] in: It is the altitude of the drone; It is the total mass of the drone; Represents gravitational acceleration; It is the control input for the drone; It is the drag coefficient of the drone; It is an externally bounded disturbance that satisfies ,and It is a bounded positive real number; Step 1.2: Transform the integer-order altitude dynamic system of the UAV into a fractional-order model, let... ,but

[0011] A second-order system of integer order can be written as a fractional-order system by choosing the fractional order. Choose intermediate variables Then there is

[0012] We obtain the following formula, that is

[0013] The fractional-order state equation of the UAV's high-dynamic system is:

[0014] Where the matrix , For drag coefficients, matrix ,matrix External interference For bounded disturbances of energy to satisfy and .

[0015] Step 2 is implemented in the following steps: Step 2.1: Discuss the altitude dynamics of an unmanned aerial vehicle (UAV) system with uncertain parameters. For performance control issues, consider studying the following model of a drone's altitude dynamic control system: (1) in, This represents the Caputo derivative. This refers to the unmeasured state of the UAV's altitude dynamics system. It is the control input of the UAV's high dynamic system. It is the measurement output of the UAV's altitude dynamics system. It is the control output of the UAV's high dynamic system. and It refers to the energy-bounded external interference existing in the highly dynamic system of the unmanned aerial vehicle (UAV), the matrix. , , , , , Given a matrix, unknown matrix , satisfy , ,in , , , Given known weight coefficients, and with uncertain parameters satisfying: , ; Step 2.2, due to the state in the UAV's high dynamic system (1) It cannot be measured directly, but is measured through the output signal of the UAV's altitude dynamic system (1). Construct a controller to ensure the stability of the UAV's altitude dynamic closed-loop system: By using the measured output of the UAV's altitude dynamic system and combining it with the system structure, establish a state observer to estimate the state values ​​of the UAV's altitude dynamic system. (2) in, State of the UAV's highly dynamic system The estimated value, The output of the state observer, The gain of the state observer; Step 2.3, Define the state estimation error as... The dynamic change of the estimation error is as follows: (3) Step 2.4, due to the state of the UAV's highly dynamic system Since it is unmeasured, the estimated state of the UAV altitude dynamic system is estimated using a state observer. Design a feedback elastic controller: (4) in, To control the gain, For feedback control gain, For feedback controller gain variation, consider two types of gain variation: multiplicative controller gain variation. Additive controller gain variation ,make , , , and It is a known matrix with appropriate dimensions, and , ; Step 2.5: Combining the state observer (2) and feedback elastic controller (4) designed in the above steps, the following UAV altitude dynamic composite closed-loop system is obtained: (5) The following enhanced UAV altitude dynamic closed-loop system was obtained by rewriting and organizing: (6) in .

[0016] In step 2, the UAV's altitude dynamic system satisfies Asymptotic stability under performance conditions must satisfy the following definition: Asymptotic stability: under disturbance Under these conditions, the UAV's highly dynamic closed-loop system is asymptotically stable, in which... ; Performance: The UAV's highly dynamic closed-loop system satisfies the inequality under zero initial conditions.

[0017] in express Gain; Step 2 leads to the following conclusion: For the UAV altitude dynamic system under parameter uncertainty and energy bounded disturbance (1), for the following parameters , , , , The UAV's highly dynamic composite closed-loop system (6) has the following requirements: Asymptotic stability under performance conditions, if some parameters exist , , , ,matrix , , , This makes the inequality hold: (7) in

[0018] If the above conditions can be solved, then it proves that the UAV's highly dynamic composite closed-loop system (6) satisfies the following conditions. Asymptotic stability under performance conditions, the conclusion holds, and the controller gain... Gain of the state observer .

[0019] Step 3 includes proving that the UAV's high-dynamic closed-loop system (6) has asymptotic stability and proving that the UAV's high-dynamic closed-loop system (7) satisfies The asymptotic stability under performance conditions consists of two parts.

[0020] Step 3 is implemented in the following steps: Prove that the UAV's highly dynamic closed-loop system (6) has asymptotic stability: The Lyapunov function is selected as follows: (8) Among them, matrix , , , Based on the definition of the Lyapunov function, we can deduce:

[0021] in, , ,right Taking the fractional derivative and combining it with formula (6), we can derive: (9) in

[0022] First, we discuss the high dynamic system of the UAV under zero initial conditions (6). Control performance and utilize performance indicators:

[0023] because ,get: (10) The above formula If true, then the inequality holds: (11) in

[0024] Secondly, under condition (7) prove Obviously, according to equation (11), we get In satisfying (12) The inequality holds true when, where

[0025] According to Yang's inequality, we obtain: (13) in

[0026] Clearly, equation (13) is a nonlinear matrix inequality. To transform a nonlinear matrix inequality into a linear inequality, we define a matrix... , and And simultaneously with the diagonal matrix After performing pre- and post-multiplication, we get: (14) in Applying Schur's lemma to the above equation, we get: (15) in

[0027] After scaling based on Young's inequality and the properties of quadratic forms, we get: (16) in

[0028] To eliminate the matrix Using Young's inequality and the properties of quadratic forms again, we get: (17) in

[0029] Obviously, the matrix Still with and The form exists and ,right Add a constraint as And it is known have to:

[0030] Combining the above inequalities, using Schur's lemma, we get: (18) in

[0031] Based on the above discussion, it is confirmed that under the initial conditions... hour, In the interval It is established internally, therefore, when At that time, we obtained: (19) The above formula is equivalent to (20) Obviously, if (7) holds, due to the parameter ,matrix , , and This is obtained through inequality (7), therefore, there exists a positive number. This makes the following inequality hold. (twenty one) definition According to (8) and (21), we get Clearly, based on Lyapunov's stability theory, the highly dynamic composite closed-loop system of the UAV... hour Therefore, the UAV's highly dynamic augmented closed-loop system (6) has asymptotic stability.

[0032] Step 3 proves that the UAV's high-dynamic closed-loop system (7) satisfies The asymptotic stability under the performance conditions is as follows: Integrating equation (20) over the interval from 0 to t, the inequality holds: (twenty two) Based on the properties of fractional calculus, we obtain: (twenty three) and (twenty four) Under the initial conditions, we obtain: (25) According to equations (24) and (25), we get Combining this expression with (22), we get: (26) make have to (27) Based on the above analysis, the UAV altitude dynamic augmentation closed-loop system meets the following requirements. Asymptotic stability under performance conditions.

[0033] Obviously, based on Lyapunov's stability theory and Theoretically, the UAV's highly dynamic closed-loop system (6) satisfies Based on performance conditions (i) and (ii), it is deduced that the UAV altitude-dynamic augmented closed-loop system (6) is asymptotically stable and satisfies Control performance.

[0034] The beneficial effect of this invention is that it designs a fractional-order [mechanism] for UAV altitude dynamic systems with uncertain parameters. A robust elastic control method is proposed to effectively control the altitude stability of a UAV during flight. This invention designs a fractional-order... The robust elastic control method, by constructing a state observer and a feedback elastic controller, observes the mixed disturbances in the unmeasured system state and can effectively suppress and attenuate the disturbances, thereby ensuring that the UAV altitude dynamic system with uncertain parameters meets the requirements. Asymptotic stability under performance conditions. Attached Figure Description

[0035] Figure 1 It is a fractional-order UAV high-dynamic system with uncertain parameters. Overall flowchart of robust elastic control method; Figure 2 It is a fractional-order UAV high-dynamic system with uncertain parameters. Block diagram of robust elastic control method; Figure 3 It is a state comparison diagram of a UAV altitude dynamic closed-loop system with uncertain parameters; Figure 4 It is the state of a UAV's highly dynamic closed-loop system with uncertain parameters. A comparison chart of its estimated value and estimation error; Figure 5 It is the state of a UAV's highly dynamic closed-loop system with uncertain parameters. A comparison chart of its estimated value and estimation error; Figure 6 It is the state of a UAV's highly dynamic closed-loop system with uncertain parameters. A comparison chart of its estimated value and estimation error; Figure 7 It is the state of a UAV's highly dynamic closed-loop system with uncertain parameters. A comparison chart of its estimated value and estimation error; Figure 8 It is the control output of a UAV altitude dynamic closed-loop system with uncertain parameters. Comparison chart. Detailed Implementation

[0036] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0037] This invention describes a fractional-order method for a drone altitude dynamics system with uncertain parameters. Robust elastic control method. Considering the high-dynamic system of an unmanned aerial vehicle (UAV), a fractional-order elastic control method is designed based on a state observer and a feedback elastic controller. Robust elastic controller, applying Lyapunov stability theory and Theoretically, it is guaranteed that the highly dynamic composite closed-loop system of the UAV has the ability to meet the requirements. Asymptotic stability under performance conditions; eliminating the coupling between the state observer and the feedback elastic controller using the linear matrix inequality (LMI) technique and obtaining the corresponding gain matrices of the observer and controller; finally, based on the actual UAV altitude dynamics system, the relevant system matrices are given, and combined with the gain matrices obtained by the LMI technique, the designed fractional-order [controller] is verified by MATLAB numerical simulation. Effectiveness of robust elastic controllers.

[0038] Example 1 The present invention relates to a fractional-order UAV altitude dynamics system with uncertain parameters. Robust elastic control method, combined with Figure 1 , Figure 2 The specific steps are as follows: Step 1: Establish a fractional-order model of the UAV's high dynamic system with interference; Step 1 is implemented in the following steps: Step 1.1, the Caputo fractional order is defined as...

[0039] in It is greater than The smallest integer that satisfies , In particular, when When, the Caputo fractional derivative is defined as:

[0040] in express On the interval The Caputo fractional derivative of order 1.

[0041] Since the equations considered in this invention are finite, only the altitude dynamics system of the UAV is considered. First, the mathematical model of the UAV altitude dynamics system considering disturbance factors is obtained through the Lagrange method, as shown below:

[0042] in: It is the altitude of the drone; It is the total mass of the drone; Represents gravitational acceleration; It is the control input for the drone; It is the drag coefficient of the drone; It is an externally bounded disturbance that satisfies ,and It is a bounded positive real number.

[0043] Step 1.2: Transform the integer-order altitude dynamic system of the UAV into a fractional-order model. Let... ,but

[0044] A second-order system of integer order can be written as a fractional-order system, by choosing the fractional order. Choose intermediate variables Then there is

[0045] Furthermore, we obtain the following formula, namely

[0046] The fractional-order state equation of the UAV's high-dynamic system is:

[0047] Where the matrix , For drag coefficients, matrix ,matrix External interference For bounded disturbances of energy to satisfy and

[0048] Step 2: Design a fractional-order controller based on a state observer and a feedback elastic controller. Robust elastic controller; Step 2 is implemented in the following steps: Step 2.1: Discuss the altitude dynamics of an unmanned aerial vehicle (UAV) system with uncertain parameters. Performance control problem. Consider studying the following model of a drone's altitude dynamic control system: (1) in This represents the Caputo derivative. This refers to the unmeasured state of the UAV's altitude dynamics system. It is the control input of the UAV's high dynamic system. It is the measurement output of the UAV's altitude dynamics system. It is the control output of the UAV's high dynamic system. and This refers to external disturbances with bounded energy within the highly dynamic system of an unmanned aerial vehicle (UAV). (Matrix) , , , , , Given a matrix, unknown matrix , satisfy , ,in , , , Given known weight coefficients, and with uncertain parameters satisfying: , .

[0049] Step 2.2, due to the state in the UAV's high dynamic system (1) It cannot be measured directly, but is measured through the output signal of the UAV's altitude dynamic system (1). Construct a controller to ensure the stability of the UAV's highly dynamic closed-loop system: First, by using the measurement output of the UAV's altitude dynamics system and combining it with the system structure of the UAV's altitude dynamics system, a state observer is established to estimate the state values ​​of the UAV's altitude dynamics system: (2) in State of the UAV's highly dynamic system The estimated value, The output of the state observer, This is the gain of the state observer.

[0050] Step 2.3, Define the state estimation error as... The dynamic change of the estimation error is as follows: (3) Step 2.4, due to the state of the UAV's highly dynamic system Since it is unmeasured, the estimated state of the UAV altitude dynamic system is estimated using a state observer. Design a feedback elastic controller: (4) in To control the gain, For feedback control gain, For feedback controller gain variation, consider two types of gain variation: multiplicative controller gain variation. Additive controller gain variation ,make ,here , , and It is a known matrix with appropriate dimensions, and , ; Step 2.5: Combining the state observer (2) and feedback elastic controller (4) designed in the above steps, the following UAV altitude dynamic composite closed-loop system is obtained: (5) The following enhanced UAV altitude dynamic closed-loop system was obtained by rewriting and organizing: (6) in

[0051] In step 2, the UAV's altitude dynamic system satisfies Asymptotic stability under performance conditions must satisfy the following definition: (Asymptotic stability) under disturbance Under these conditions, the UAV's highly dynamic closed-loop system is asymptotically stable, in which... ; ( (Performance) The highly dynamic closed-loop system of the UAV satisfies the inequality under zero initial conditions.

[0052] in express The gain.

[0053] Based on the above definition, the following conclusions can be drawn in the subsequent steps: For the UAV altitude dynamic system under parameter uncertainty and energy bounded disturbance (1), for the following parameters , , , , The UAV's highly dynamic composite closed-loop system (6) has the following requirements: Asymptotic stability under performance conditions, if some parameters exist , , , ,matrix , , , This makes the inequality hold: (7) in

[0054] If the above conditions can be solved, then it proves that the UAV's highly dynamic composite closed-loop system (6) satisfies the following conditions. The asymptotic stability under performance conditions confirms the conclusion. The controller gain... Gain of the state observer .

[0055] Step 3: Using Lyapunov stability theory, prove that the UAV's highly dynamic composite closed-loop system satisfies... Asymptotic stability under performance conditions; Step 3 is implemented in the following steps: Step 3.1: Prove that the UAV's high-dynamic closed-loop system (6) has asymptotic stability.

[0056] The Lyapunov function is selected as follows: (8) Where the matrix , , , According to the definition of the Lyapunov function, it can be deduced that...

[0057] in , ,right Taking the fractional derivative and combining it with formula (6), we can derive: (9) in

[0058] First, we discuss the high dynamic system of the UAV under zero initial conditions (6). Control performance and utilize performance indicators:

[0059] because ,get: (10) The above formula If true, then the inequality holds: (11) in

[0060] Secondly, under condition (7) prove Obviously, according to equation (11), we get In satisfying (12) The inequality holds true when, where

[0061] According to Young's inequality, we can obtain: (13) in

[0062] Clearly, equation (13) is a nonlinear matrix inequality. To transform the nonlinear matrix inequality into a linear inequality, we define a matrix... , and And simultaneously with the diagonal matrix After performing pre- and post-multiplication, we get: (14) in Applying Schur's lemma to the above equation, we get: (15) in

[0063] After scaling based on Young's inequality and the properties of quadratic forms, we get: (16) in

[0064] To eliminate the matrix Using Young's inequality and the properties of quadratic forms again, we get: (17) in

[0065] Obviously, the matrix Still with and The form exists and ,right Add a constraint as And it is known have to:

[0066] Combining the above inequalities, we can obtain using Schur's lemma: (18) in

[0067] Based on the above discussion, it is confirmed that under the initial conditions... hour, In the interval It is established internally. Therefore, when At that time, we obtained: (19) The above formula is equivalent to (20) Obviously, if (7) holds, due to the parameter ,matrix , , and This is obtained through inequality (7), therefore, there exists a positive number. This makes the following inequality hold. (twenty one) definition According to (8) and (21), we can obtain Clearly, based on Lyapunov's stability theory, the highly dynamic composite closed-loop system of the UAV... hour Therefore, the UAV's highly dynamic augmented closed-loop system (6) has asymptotic stability.

[0068] Step 3.2, prove that the UAV's high-dynamic closed-loop system (7) satisfies Asymptotic stability under performance conditions.

[0069] Integrating equation (20) over the interval from 0 to t, the inequality holds: (twenty two) Based on the properties of fractional calculus, we obtain: (twenty three) and (twenty four) Under the initial conditions, we can obtain: (25) According to equations (24) and (25), we can obtain Combining this expression with (22), we get: (26) make achievable (27) Based on the above analysis, the UAV altitude dynamic augmentation closed-loop system meets the following requirements. Asymptotic stability under performance conditions.

[0070] Obviously, based on Lyapunov's stability theory and Theoretically, the UAV's highly dynamic closed-loop system (6) satisfies Based on performance conditions (i) and (ii), it can be deduced that the UAV altitude-dynamic augmented closed-loop system (6) is asymptotically stable and satisfies Control performance.

[0071] Step 4: Use the Linear Matrix Inequality (LMI) technique to handle the coupling problem between the observer and the controller, and calculate and solve for the gain matrices of the observer and the controller. Substitute the solved gain matrices into the UAV altitude dynamic system simulation to verify the effectiveness of the designed method.

[0072] Example 2 The present invention relates to a fractional-order UAV altitude dynamics system with uncertain parameters. The robust elastic control method is implemented according to the following steps: Step 1: Establish a fractional-order model of the UAV's high dynamic system with interference; Step 2: Design a fractional-order controller based on a state observer and a feedback elastic controller. Robust elastic controller; Step 3: Using Lyapunov stability theory, prove that the UAV's highly dynamic composite closed-loop system satisfies... Asymptotic stability under performance conditions; Step 4: Use the Linear Matrix Inequality (LMI) technique to handle the coupling problem between the observer and the controller, and calculate and solve for the gain matrices of the observer and the controller. Substitute the solved gain matrices into the UAV altitude dynamic system simulation to verify the effectiveness of the designed method.

[0073] Example 3 This invention designs a fractional-order UAV altitude dynamics system with uncertain parameters based on a state observer and a feedback elastic controller. Robust elastic control method: Since the state of the UAV's high dynamic system cannot be directly measured, the controller is constructed by using the output signal of the UAV's high dynamic system. By measuring the output of the UAV's altitude dynamics system and combining it with the system's structural characteristics, a state observer is established to estimate the system's state values. The state observer aims to reconstruct the system's unobservable state information in real time by processing the measurable partial output information and known input information. This enables precise state feedback control even when some states are difficult to obtain. To better observe the operating state of this control system and eliminate the error between the state estimate and the actual dynamic system state value of the UAV's highly dynamic system, a state estimation error is defined. A feedback elastic controller is designed to adjust and eliminate errors based on the state estimates of the UAV's altitude dynamic system obtained from the state observer. A fractional-order control system for UAV high-dynamic systems is designed based on a state observer and a feedback elastic controller. Robust elastic controller; Using Lyapunov stability theory and Theoretical proofs demonstrate that the high dynamic system of unmanned aerial vehicles (UAVs) satisfies Asymptotic stability under performance conditions; The linear matrix inequality (LMI) technique is used to eliminate the coupling between the state observer and the feedback elastic controller and to obtain the corresponding gain matrices of the observer and the controller. Based on the correlation matrix given by the actual UAV altitude dynamics system and the gain matrices of the observer and controller obtained using the linear matrix inequality (LMI) technique, the fractional order algorithm designed in this invention is verified through numerical simulation using MATLAB. Effectiveness of robust elastic controllers.

[0074] Example 4 The present invention relates to a fractional-order UAV altitude dynamics system with uncertain parameters. The robust elastic control method is implemented according to the following steps: Step 1: Establish a fractional-order model of the UAV's high dynamic system with interference; Step 1 is implemented in the following steps: Step 1.1, the Caputo fractional order is defined as...

[0075] in It is greater than The smallest integer that satisfies , In particular, when When, the Caputo fractional derivative is defined as:

[0076] in express On the interval The Caputo fractional derivative of order 1.

[0077] Since the equations considered in this invention are finite, only the altitude dynamics system of the UAV is considered. First, the mathematical model of the UAV altitude dynamics system considering disturbance factors is obtained through the Lagrange method, as shown below:

[0078] in: It is the altitude of the drone; It is the total mass of the drone; Represents gravitational acceleration; It is the control input for the drone; It is the drag coefficient of the drone; It is an externally bounded disturbance that satisfies ,and It is a bounded positive real number.

[0079] Step 1.2: Transform the integer-order altitude dynamic system of the UAV into a fractional-order model. Let... ,but

[0080] A second-order system of integer order can be written as a fractional-order system, by choosing the fractional order. Choose intermediate variables Then there is

[0081] Furthermore, we obtain the following formula, namely

[0082] The fractional-order state equation of the UAV's high-dynamic system is:

[0083] Where the matrix , For drag coefficients, matrix ,matrix External interference For bounded disturbances of energy to satisfy and

[0084] Step 2: Design a fractional-order controller based on a state observer and a feedback elastic controller. Robust elastic controller; Step 3: Using Lyapunov stability theory, prove that the UAV's highly dynamic composite closed-loop system satisfies... Asymptotic stability under performance conditions; Step 4: Use the Linear Matrix Inequality (LMI) technique to handle the coupling problem between the observer and the controller, and calculate and solve for the gain matrices of the observer and the controller. Substitute the solved gain matrices into the UAV altitude dynamic system simulation to verify the effectiveness of the designed method.

[0085] Example 5 The present invention relates to a fractional-order UAV altitude dynamics system with uncertain parameters. The robust elastic control method is implemented according to the following steps: Step 1: Establish a fractional-order model of the UAV's high dynamic system with interference; Step 2: Design a fractional-order controller based on a state observer and a feedback elastic controller. Robust elastic controller; Step 3: Using Lyapunov stability theory, prove that the UAV's highly dynamic composite closed-loop system satisfies... Asymptotic stability under performance conditions; Step 3 includes two parts: proving that the UAV's high-dynamic closed-loop system (6) has asymptotic stability and proving that the UAV's high-dynamic closed-loop system (7) has asymptotic stability under the performance conditions.

[0086] Step 4: Use the Linear Matrix Inequality (LMI) technique to handle the coupling problem between the observer and controller, and calculate the gain matrices of the observer and controller. Substitute the obtained gain matrices into the UAV altitude dynamics system simulation to verify the effectiveness of the designed method. Example 6 The invention will be further described with reference to the accompanying drawings and a specific example of an unmanned aerial vehicle (UAV) altitude dynamics system. To verify the design of the fractional-order model based on the invention... The effectiveness of the robust elastic control method is evaluated, and considering flight cost and safety, a real-world UAV model is selected for simulation verification of the algorithm of this invention. The UAV model parameters are set as follows: , , .

[0087] Based on a certain UAV altitude dynamics system, the relevant matrix of the UAV altitude system is given as follows:

[0088] Other matrix choices are:

[0089] Furthermore, some uncertain functions of the controller gain variation in this UAV's highly dynamic system are selected as follows: , , and The order of the derivative of a fractional system is chosen as follows: Control performance gain set to Other parameters are set to , , Based on the given data, the results obtained by substituting into the linear matrix inequality are used to eliminate the coupling between the observers and obtain the corresponding gain matrices of the observers and controllers:

[0090] in, It is the feedback controller gain. It is the state observer gain.

[0091] To more intuitively observe the simulation results, the initial values ​​of the system in the simulation are set as follows: .

[0092] Based on the above settings, the proposed fractional order The robust elastic control method is applied to the high-dynamic system of an unmanned aerial vehicle (UAV) with parameter uncertainties and energy-bounded disturbances. Simulation results are as follows: Figure 3-8 As shown.

[0093] The state of an UAV altitude dynamic system with uncertain parameters, such as Figure 3 As shown, using the proposed fractional order After implementing a robust elastic controller, the UAV's high-dynamic system can quickly achieve the expected control performance and remains stable under disturbances and system uncertainties. The UAV's high-dynamic system possesses the following properties: Asymptotic stability under performance conditions. A comparison of the state, state estimation, and estimation error of the UAV's highly dynamic system is shown in the figure below. Figure 4 , 5 As shown in 6 and 7. Figure 8 For the control output of the UAV's altitude dynamics system, the control objective of this technology is to ensure that the UAV's altitude dynamics system possesses corresponding control performance. From Figure 8 As can be seen from the data, the control output of the UAV altitude dynamics system tends to the expected value over time, indicating that the proposed control method is effective for the UAV altitude dynamics system. Based on the above data, it can be proven that this UAV altitude dynamics system with uncertain parameters satisfies the required parameters. Asymptotic stability under performance conditions, the proposed fractional order Robust elastic control methods are effective for this UAV's highly dynamic system.

[0094] Based on the above simulation results, the fractional order proposed in this invention... Robust elastic control methods can ensure that the UAV's high-dynamic closed-loop system meets the requirements. Suppressing and attenuating interference under the premise of asymptotic stability under performance conditions solves the problem of parameter uncertainty and energy-bounded interference in the highly dynamic system of UAV.

Claims

1. Fractional order of a UAV high dynamic system with uncertain parameters Robust elastic control method, characterized in that, The step 1 is implemented according to the following steps: Step 1, a fractional order model of the unmanned aerial vehicle high dynamic system with disturbance is established; Step 2, Designing fractional order Robust elastic controller; Step 3, Lyapunov stability theory is used to prove that the unmanned aerial vehicle high dynamic compound closed-loop system meets Asymptotic stability under performance conditions; Step 4, the coupling problem between the observer and the controller is processed by using the linear matrix inequality (LMI) technique, and the gain matrix of the observer and the controller is calculated and solved, and the gain matrix solved is substituted into the unmanned aerial vehicle high dynamic system simulation to verify the effectiveness of the designed method.

2. Fractional order of the unmanned aerial vehicle high dynamic system with uncertain parameters according to claim 1 Robust elastic control method characterized in that, The step 1 is implemented according to the following steps: Step 1.1, the Caputo fractional order is defined as where is the smallest integer greater than , i.e., satisfying , when , the Caputo fractional derivative is defined as wherein denotes on the interval Caputo fractional derivative of order First, the mathematical model of the unmanned aerial vehicle high dynamic system considering the disturbance factor is obtained by the Lagrange method, as follows: wherein: is the altitude of the UAV; is the total mass of the UAV; denotes the gravitational acceleration; is the control input of the UAV; is the drag coefficient of the UAV; is an external bounded disturbance satisfying , and is a bounded positive real number; Step 1.2, converting the integer order height dynamic system of UAV into fractional order model form, let then The integer order second order system is written as a fractional order system, and the fractional order degree is selected , the intermediate variable is selected , and then The following formula is obtained, that is Then the fractional order state equation of the unmanned aerial vehicle high dynamic system is where matrix , is the drag coefficient, matrix , matrix , external disturbance is energy-bounded disturbance satisfying and .

3. Fractional order of the UAV high dynamic system with uncertain parameters according to claim 2. Robust elastic control method characterized in that, The step 2 is implemented according to the following steps: Step 2.1: Discuss the UAV altitude dynamics system with uncertain parameters. For performance control issues, consider studying the following UAV altitude dynamic control system model: （1） wherein denotes the Caputo derivative, is the unmeasured state of the UAV highly dynamic system, is the control input of the UAV highly dynamic system, is the measured output of the UAV highly dynamic system, is the control input of the UAV highly dynamic system, and is an energy-bounded external disturbance present in the UAV highly dynamic system, the matrix , , , , , is a known matrix, the uncertain matrix , satisfies , , wherein , , , is a known weight coefficient, and the uncertain parameter satisfies: , ; Step 2.2, since the states in the UAV highly dynamic system (1) are not directly measurable, the measured output signals in the UAV highly dynamic system (1) are used to estimate the states in the UAV highly dynamic system (1) Step 2.3, construct a controller to ensure the stability of the UAV highly dynamic closed-loop system: by using the measured output signals of the UAV highly dynamic system and combining the system structure of the UAV highly dynamic system, a state observer is established to estimate the state values of the UAV highly dynamic system Step 2.3, construct a controller to ensure the stability of the UAV highly dynamic closed-loop system: by using the measured output signals of the UAV highly dynamic system and combining the system structure of the UAV highly dynamic system, a state observer is established to estimate the state values of the UAV highly dynamic system （2） wherein, is an estimate of the state of the highly dynamic system of the drone, is an estimate of the state of the highly dynamic system of the drone, is an output of the state observer, is a gain of the state observer; Step 2.3, define state estimation error as , the dynamic change of the estimation error is （3） Step 2.4, since the states of the UAV highly dynamic system are not measured, the estimated states of the UAV highly dynamic system are estimated using a state observer Step 2.5, a feedback resilient controller is designed Step 2.6, the UAV highly dynamic system is controlled using the feedback resilient controller （4） wherein is a control gain, is a feedback control gain, is a feedback controller gain variation, considering two kinds of gain variations, i.e. multiplicative controller gain variation and additive controller gain variation such that , , , and are known matrices of appropriate dimensions, and , ; Step 2.5, the state observer (2) and the feedback flexible controller (4) designed in the above steps are combined to obtain the following unmanned aerial vehicle high dynamic compound closed loop system: （5） The following enhanced unmanned aerial vehicle high dynamic closed loop system is obtained by rewriting and arranging: （6） Wherein 。 4. Fractional order of the UAV high dynamic system with uncertain parameters according to claim 3. Robust elastic control method characterized in that, The step 2 unmanned aerial vehicle highly dynamic system satisfies Asymptotic stability under performance conditions requires satisfaction of the following definition: Asymptotic stability: under disturbance Under these conditions, the UAV's highly dynamic closed-loop system is asymptotically stable, in which... ; Performance: The UAV highly dynamic closed loop system meets the inequality under zero initial conditions wherein represents gain of the 5. Fractional order of the UAV high dynamic system with uncertain parameters according to claim 4. Robust elastic control method characterized in that, The step 2 has the following conclusions: For the unmanned aerial vehicle highly dynamic system (1) under parameter uncertainty and energy-bounded disturbance, for given parameters , , , , , the unmanned aerial vehicle highly dynamic compound closed-loop system (6) has asymptotic stability under the performance condition of , if there are some parameters , , , , the matrix , , , , makes the inequality hold: （7） Wherein If the above conditions can be solved, it is proved that the unmanned aerial vehicle high dynamic compound closed-loop system (6) has asymptotic stability under the performance conditions, the conclusion is established, the gain of the controller , and the gain of the state observer . .

6. Fractional order of the UAV high dynamic system with uncertain parameters according to claim 5. Robust elastic control method characterized in that, The step 3 comprises proving that the UAV altitude dynamic closed loop system (6) is asymptotically stable and proving that the UAV altitude dynamic closed loop system (7) is asymptotically stable under the performance condition two parts.

7. Fractional order of the UAV high dynamic system with uncertain parameters according to claim 6. Robust elastic control method characterized in that, The step 3 is implemented according to the following steps: It is proved that the unmanned aerial vehicle high dynamic closed loop system (6) has asymptotic stability: The Lyapunov function is selected as （8） where the matrix , , , ; according to the definition of Lyapunov function, it is concluded that in, , ,right Taking the fractional derivative and combining it with formula (6), we can derive: （9） Wherein First, we discuss the high dynamic system of the UAV under zero initial conditions (6). Control performance and utilize performance indicators: As a result , it is obtained: （10） The above formula The inequality holds immediately. （11） Wherein Second, under condition (7) we have It is shown that according to (11) we have , obviously according to (11) we have In satisfying （12） The inequality is established, wherein According to the Young inequality, the following is obtained: （13） Wherein Obviously, (13) is a nonlinear matrix inequality. To convert the nonlinear matrix inequality into a linear inequality, we define matrices , and , and simultaneously pre- and post-multiply by the diagonal matrix to obtain （14） Wherein The Schur lemma is used for the above formula to obtain: （15） Wherein According to the Young inequality and the properties of quadratic form, the following is obtained after scaling: （16） Wherein To eliminate the matrix , we use again the inequality of Young and the properties of the quadratic form to get: （17） Wherein Obviously, the matrix still exists in the form of and and , with the addition of the condition and knowing that we obtain:​​ The Schur lemma is used in combination with the above inequality to obtain: （18） Wherein From the above discussion, it is confirmed that the inequality holds at the initial condition in the interval Therefore, when the following is obtained:​ （19） The above formula is equivalent to （20） Obviously if (7) holds, since the parameters , matrix , , and are obtained by the inequality (7), therefore, there is a positive number , such that the following inequality holds （21） Definitions , according to (8) and (21), we have , obviously, based on Lyapunov stability theory, the highly dynamic compound closed-loop system of UAV is stable in Therefore, the highly dynamic augmented closed-loop system (6) of UAV is asymptotically stable.​ 8. Fractional order of the UAV high dynamic system with uncertain parameters according to claim 7. Robust elastic control method characterized in that, The step 3 proves that the UAV highly dynamic closed-loop system (7) has the asymptotic stability under the performance condition as follows: The asymptotic stability under the performance condition is as follows: The integral of formula (20) is carried out in the interval 0 to t, and the inequality is established: （22） According to the properties of fractional calculus, the following is obtained: （23） And （24） Under the initial condition, the following is obtained: （25） From equations (24) and (25) we obtain Combining this with (22) we obtain （26） Let be （27） According to the above analysis, the unmanned aerial vehicle highly dynamic augmented closed-loop system has asymptotic stability under performance conditions. performance conditions. It is clear that, based on the Lyapunov stability theory and theory, the UAV highly dynamic closed-loop system (6) satisfies the conditions (i) and (ii) of the performance, it is derived that the UAV highly dynamic augmented closed-loop system (6) is asymptotically stable and satisfies the control performance.

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