A method for designing an interference estimator based on immersion and invariant manifold theory

Abstract

The application discloses a kind of interference estimator design method based on immersion and invariant manifold theory, belong to aircraft control technical field, including: system dynamic is immersed into target dynamic, then invariant manifold is designed, so that system state and estimated state finally converge to invariant manifold in;When designing invariant manifold estimator, by introducing first-order filter, the derivative of filtered regression vector can be calculated, the nonlinear function introduced in the design of invariant manifold estimator is solved by analytical method, and then the integral obstacle problem existing in the classical invariant manifold estimator is avoided, while the projection operator and dead zone are introduced to prevent the drift of estimated parameters, and the design of invariant manifold interference estimator is completed.The updating law design of the application is independent of tracking error, and the stability of the invariant manifold estimator is not affected when the tracking error is too large due to actuator saturation, and it has strong robustness under interference.

Description

Technical Field

[0001] The present invention belongs to the technical field of aircraft control, and in particular relates to a method for designing an interference estimator based on immersion and invariant manifold theory. Background Art

[0002] Hypersonic vehicles boast high speeds, strong penetration capabilities, the ability to precisely strike time-sensitive targets beyond their defense zones, and the ability to rapidly resupply assets across land, sea, and air. Compared to conventional aircraft, their superior maneuverability makes them ineffective against modern missile defense systems, making them a key research priority for various military nations and organizations.

[0003] Air-breathing hypersonic vehicles have a highly integrated propulsion system and airframe, resulting in a strong coupling between their structure and aerodynamics. Existing research often uses adaptive backstepping control to study the cruise speed and altitude tracking control of hypersonic vehicles. Adaptive methods are used to estimate the system uncertainty, and backstepping is then used to design the control system instructions.

[0004] However, for the velocity subsystem, the norm of the estimation error tends to diverge when using classical adaptive estimation methods. In the design of a nonlinear disturbance estimator, only one disturbance needs to be estimated for each equation. However, when there is significant uncertainty in the control efficiency, the assumption of slowly varying disturbances no longer holds, weakening the estimation effectiveness of the nonlinear disturbance estimator and even affecting the stability of the control system. Therefore, the present invention proposes a disturbance estimator design method based on immersion and invariant manifolds to estimate system states and improve system robustness. Summary of the Invention

[0005] In view of this, the object of the present invention is to provide an interference estimator design method based on immersion and invariant manifold theory to solve the above technical problems.

[0006] In order to achieve the above object, the present invention provides the following technical solutions:

[0007] A disturbance estimator design method based on immersion and invariant manifold theory includes:

[0008] Step 1: Based on the theory of immersion and invariant manifold, immerse the system dynamics into the target dynamics to complete the definition and design of the invariant manifold;

[0009] Step 2: Solve the integral obstacle problem of the nonlinear function introduced in the invariant manifold definition design in step 1 using the first-order filtering method;

[0010] Step 3: Based on the projection operator and dead zone theory, the problem of estimated parameter drift is solved;

[0011] Step 4: Based on the relevant parameters calculated in steps 1 to 3, complete the design of the invariant manifold interference estimator.

[0012] Furthermore, step 1 includes:

[0013] Obtain an n-order nonlinear system:

[0014]

[0015]

[0016] Where x=[x1,x2,...,x n ] T is the system state vector, u is the control input, z i =[x1,x2,...,x i ] T , f i (·) and g i (·) is the nonlinear function to be solved;

[0017] Convert the n-order nonlinear system into a parametric expression:

[0018]

[0019]

[0020] Among them, θ i1 and θ i2 is the parameter to be solved, and is a known regression vector, when the function f i (·) and g i (·) is smooth and bounded, and the function g i The sign of (·) is known and does not change, so we get f i (·) and g i The expression of (·) is:

[0021]

[0022] Based on f i (·) and g i The expression and parameterized expression of (·) are simplified for the n-order nonlinear system to obtain a unified expression:

[0023]

[0024]

[0025]

[0026] According to the unified form expression, the invariant manifold interference estimator is derived and defined as the estimated quantity of the parameter to be solved in the i-th equation:

[0027]

[0028] Among them, ξ i is the adaptive parameter, η i is the target nonlinear function;

[0029] The definition of invariant manifold is: M={(ξ i ,x)|ξ i +η i -θ i =0};

[0030] The estimation error of the invariant manifold is:

[0031] Furthermore, step 2 includes:

[0032] A first-order filter is introduced for the regression vector, and the derivative of the regression vector after filtering is obtained as: in, is the filtered regression vector, ω if is the first-order filter cutoff frequency;

[0033] The regression vector filtering error is defined as:

[0034] when When , the estimated error equation is:

[0035]

[0036]

[0037] The pre-defined update law is obtained as:

[0038] According to the update law, the simplified estimation error equation is obtained:

[0039] According to the simplified estimation error equation, the nonlinear function is designed as follows: Among them, Γ i is the update law gain.

[0040] Furthermore, step 3 includes:

[0041] When x i When the origin is the equilibrium point, the dead zone concept is obtained:

[0042]

[0043] Among them, ε i is a positive constant, indicating the size of the dead zone;

[0044] Based on x i The origin is the equilibrium point, and The new estimation error equation is:

[0045]

[0046] Based on the new estimated error equation, the Lyapunov function is obtained: Based on the new estimated error equation, the Lyapunov function is obtained:

[0047] Derivative Lyapunov function yields:

[0048]

[0049] based on X is a positive constant, and the derivative of the transformed Lyapunov function is:

[0050]

[0051] Get the projection operator theory, where the mathematical expression of the projection operator is Proj(·) and satisfies the following equation:

[0052]

[0053] Based on the projection operator, the update law is recalculated as:

[0054]

[0055] According to the recalculated update law, the Lyapunov function is re-derived as follows:

[0056]

[0057] Among them, Γ i is the update law gain;

[0058] When the derivation of the re-derived Lyapunov function is the same as the previous Lyapunov function, the final expression of the improved update law based on the projection operator is:

[0059]

[0060] The beneficial effects of the present invention are:

[0061] Compared with the existing technology, the present invention solves the "integration barrier" problem existing in the classic invariant manifold estimator by introducing a first-order filter to solve the derivative terms of the regression matrix, and then adopts an analytical method to solve the nonlinear function in the invariant manifold estimator. Its update law design is independent of the tracking error. When the actuator is saturated and the tracking error is too large, the stability of the invariant manifold estimator is not affected, and it has strong robustness under interference.

[0062] Other advantages, objectives, and features of the present invention will be described in the following description and will be apparent to those skilled in the art to some extent, or may be taught by those skilled in the art from the practice of the present invention. The purposes and other advantages of the present invention may be realized and obtained through the structures particularly pointed out in the written description and the accompanying drawings.

[0063] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] The accompanying drawings are used to provide a further understanding of the present invention and constitute a part of the specification. Together with the embodiments of the present invention, they are used to explain the present invention and do not constitute a limitation of the present invention. In the accompanying drawings:

[0065] Figure 1 Flowchart of a method for designing an interference estimator based on immersion and invariant manifold theory in an embodiment of the present invention;

[0066] Figure 2 This is a structural block diagram of a backstepping controller in a disturbance estimator design method based on immersion and invariant manifold theory in an embodiment of the present invention;

[0067] Figure 3 This is a simulation comparison diagram of an invariant manifold estimator and a nonlinear disturbance observer in a disturbance estimator design method based on immersion and invariant manifold theory in an embodiment of the present invention. DETAILED DESCRIPTION

[0068] The preferred embodiments of the present invention are described below with reference to the accompanying drawings. It should be understood that the preferred embodiments described herein are only used to illustrate and explain the present invention, and are not used to limit the present invention.

[0069] like Figure 1 As shown, the present invention provides a disturbance estimator design method based on immersion and invariant manifold theory, which is preferably applied to the field of aircraft control technology. The disturbance estimator design based on immersion and invariant manifold theory includes:

[0070] Immerse the system dynamics into the target dynamics, and then design an invariant manifold so that the system state and the estimated state eventually converge to the invariant manifold;

[0071] When designing the invariant manifold estimator, the problem of “integration barrier” is solved by introducing a first-order filter;

[0072] The projection operator and dead zone are introduced to prevent the estimated parameters from drifting;

[0073] Immerse the system dynamics into the target dynamics, and then design an invariant manifold so that the system state and the estimated state eventually converge to the invariant manifold, including:

[0074] Consider the following nth-order nonlinear system:

[0075]

[0076]

[0077] Where x=[x1,x2,...,x n ] T is the system state vector, u is the control input, z i =[x1,x2,...,x i ] T , f i (·) and g i (·) is the nonlinear function to be solved;

[0078] Write equation (1) as a parameterized expression:

[0079]

[0080]

[0081] Among them, θ i1 and θ i2 is an unknown parameter, and Is a known regression vector. To ensure the boundedness and non-singularity of the control signal, when the function f i (·) and g i (·) is smooth and bounded, and the function g i The sign of (·) is known and does not change, so we get f i (·) and g i The expression of (·) is:

[0082]

[0083] Further simplified to the following form:

[0084]

[0085]

[0086]

[0087] It can be seen from equation (2) that all equations are expressed in a unified form; therefore, taking the i-th equation as an example, the invariant manifold estimator is derived;

[0088] Define the estimator of the unknown parameter as Among them, ξ i is the adaptive parameter, η i is the target nonlinear function;

[0089] The definition of invariant manifold is: M={(ξ i ,x)|ξ i +η i -θ i =0};

[0090] The estimation error of the invariant manifold is:

[0091] At the same time, when designing the invariant manifold estimator, the problem of “integration barrier” is solved by introducing a first-order filter, including: introducing a first-order filter for the regression vector, and obtaining the derivative of the regression vector after filtering as: in, is the filtered regression vector, ω if is the first-order filter cutoff frequency;

[0092] The regression vector filtering error is defined as:

[0093] Assume that the unknown parameter θ i is slowly changing, when When , the estimated error equation is:

[0094]

[0095] Among them, x i is the system state quantity mentioned above;

[0096] The pre-defined update law is obtained as:

[0097] According to the update law, the simplified estimation error equation is obtained:

[0098] According to the simplified estimation error equation, the nonlinear function is designed as follows: Among them, Γ i is the update law gain or a positive definite diagonal matrix;

[0099] Based on the above, it can be seen that the solution of the nonlinear function is not unique. In order to prevent the estimated parameters from drifting, the projection operator and dead zone are introduced to prevent the estimated parameters from drifting.

[0100] In order to reduce the impact of noise, the concept of dead zone is introduced:

[0101]

[0102] Among them, ε i is a positive constant, indicating the size of the dead zone. The dead zone concept can be used when the origin is the equilibrium point. Therefore, according to the formula and The updated estimation error equation is:

[0103] At the same time, the following Lyapunov function is designed for the estimation error: Based on the new estimation error equation, the Lyapunov function is obtained:

[0104] Derivative Lyapunov function yields:

[0105]

[0106] Generally speaking, the transition process of a first-order filter can be designed to be very fast, so:

[0107] Among them, X is a very small positive constant. If the cutoff frequency of the first-order filter is designed to be large, the parameter X can be very small;

[0108] At this point, the derivative of the Lyapunov function can be written as:

[0109] In order to further enhance the robustness under interference, a projection operator is added to the invariant manifold. The mathematical expression of the projection operator is Proj(·), and it satisfies the following equation:

[0110]

[0111] The update law based on the projection operator is redesigned as follows:

[0112]

[0113] The Lyapunov function is re-derived as:

[0114]

[0115] Among them, Γ i is the update law gain, Can be obtained with the formula The same conclusion, that is, the Lyapunov function W ei is semi-negative definite, in general, is not the main part of the update law, and when the estimated parameters are not out of bounds, It does not work in the update law, so the improved update law based on the projection operator is finally expressed as:

[0116]

[0117] From the above formula, it can be seen that the update law design of the invariant manifold estimator is independent of the actuator, so the saturation of the actuator will not affect the performance of the estimator;

[0118] Compared with the existing technology, the present invention solves the "integration barrier" problem existing in the classic invariant manifold estimator by introducing a first-order filter to solve the derivative terms of the regression matrix, and then adopts an analytical method to solve the nonlinear function in the invariant manifold estimator. Its update law design is independent of the tracking error. When the actuator is saturated and the tracking error is too large, the stability of the invariant manifold estimator is not affected, and it has strong robustness under interference.

[0119] In order to better understand the technical solution of the present invention, the following describes a specific embodiment in conjunction with a hypersonic aircraft speed tracking controller. It should be understood that the specific embodiments described herein are only used to explain the present application and are not intended to limit the present application.

[0120] The structure diagram of the backstepping tracking controller of the hypersonic aircraft is shown in the figure below. Figure 2 As shown;

[0121] The longitudinal mathematical model of the winged-cone conical hypersonic vehicle considering parameter perturbations is:

[0122]

[0123]

[0124]

[0125]

[0126] The longitudinal plane model of the hypersonic aircraft includes the state variables velocity V, height h, ballistic inclination θ, angle of attack α, pitch angle rate ω z ; where the uncertainty vector θ V1 ,θ V2 ,θ θ ,θ ω1 ,θ ω2 and the regression vector Expressed as:

[0127]

[0128]

[0129]

[0130]

[0131]

[0132]

[0133] Among them, the parameterized uncertain parameter is v i ,i=1,2,...,12, for mass and moment of inertia, m=v 13 m0 and J z =v 14 J z0 ; The expression of the parameterized uncertain parameters in the aerodynamic coefficients is:

[0134] C L =v10.6203α

[0135] C D =v20.6450α 2 +v30.0043378α+v40.003772

[0136] C T =v50.02576φ

[0137] m z (α)=-v60.035α 2 +v70.036617α+v85.3261e-6

[0138]

[0139] m z (δ e )=v 12 c e (δ e -α)

[0140] Among them, v1 and v7 are both 0.8, v5 and v 12 All are 0.8, and other parameters remain nominal;

[0141] Define velocity tracking error:

[0142]

[0143] Where V d is the reference speed curve; the speed tracking error equation is:

[0144]

[0145] The designed valve opening instruction is:

[0146]

[0147] Among them, k V is a positive constant, (ξ V1 +η V1 ) is θ V1 The estimated value of (ξ V2 +η V2 ) is θ V2 estimated value of;

[0148] The design invariant manifold update law is:

[0149]

[0150] Among them, (ξ V +η V ) is θ V The estimated value of The adaptive parameters and nonlinear functions are and yes The filtered regression vector,

[0151]

[0152] Among them, ω Vf is the cutoff frequency of the first-order filter, take ω Vf =30rad / s; and The initial values ​​are equal;

[0153] Design nonlinear function η V for:

[0154]

[0155] Among them, Γ V is a positive definite update matrix, designed as Γ V =0.01diag(1,1,1,1);ε V It is a small positive dead zone;

[0156] The velocity tracking error equation is expressed as:

[0157]

[0158] Among them, k V =4, ΔV is bounded interference, ΔV max is the upper bound of the interference;

[0159]

[0160] The values ​​of the initial equilibrium state of the hypersonic vehicle are shown in Table 1:

[0161] Table 1 Initial equilibrium state of hypersonic vehicle

[0162]

[0163] Among them, Figure 3 (Simulation comparison between invariant manifold estimator and nonlinear disturbance observer: velocity tracking error, height tracking error, angular velocity and control efficiency parameter estimation response curves) Figure 3 The response results of a backstepping controller based on an invariant manifold estimator (I&I) for tracking altitude and velocity curves are presented when considering multiple system uncertainties. The response results of a backstepping controller based on an invariant manifold estimator (I&I) for tracking altitude and velocity curves are presented when considering uncertain control efficiency parameters and uncertain disturbances. The response curves are presented when considering uncertain disturbances. The response results of a backstepping controller based on a nonlinear disturbance observer (NDO) for tracking altitude and velocity curves are presented. The simulation curves show that the control performance of the backstepping controller based on an invariant manifold estimator is similar when considering the I&I and simplified I&I cases. Figure 3 Figure d shows the estimated curves of the yaw control efficiency of the invariant manifold estimator under two different conditions. The simulation curves show that the estimated curves are close to the true value of 0.6. Since the estimated variables of the invariant manifold estimator are reduced when considering both components of the system uncertainty, the required computational effort is reduced. For the backstepping controller based on a nonlinear disturbance observer, the simulation curves show that the pitch angular velocity convergence time is prolonged and the oscillation amplitude is larger. Therefore, the backstepping controller based on the invariant manifold estimator has better disturbance rejection performance than the backstepping controller based on a nonlinear disturbance observer.

[0164] Finally, it should be noted that the above preferred embodiments are only used to illustrate the technical solutions of the present invention and are not limiting. Although the present invention has been described in detail through the above preferred embodiments, those skilled in the art should understand that various changes can be made in form and details without departing from the scope defined by the claims of the present invention.

Claims

1. A disturbance estimator design method based on immersion and invariant manifold theory, characterized in that: include: Step 1: Based on the theory of immersion and invariant manifold, immerse the system dynamics into the target dynamics to complete the definition and design of the invariant manifold. This includes: Obtain an n-order nonlinear system: n-1 ； in, is the system state vector, is the control input, , for the reason The vector formed, and is the nonlinear function to be solved; Convert the n-order nonlinear system into a parametric expression: n-1 ； in, and is the parameter to be solved, and is a known regression vector The two components of and is smooth and bounded, and the function The sign of is known and does not change, so we get and The expression is: ； based on and The expression and parameterized expression of The first-order nonlinear system is simplified to obtain a unified expression: n-1 ； According to the unified form expression, the invariant manifold interference estimator is first The derived definition of the estimated parameter to be solved in the equation is: ； in, is the adaptive parameter, is the target nonlinear function, for estimated value; The invariant manifold is defined as: ; Step 2: Solve the integral barrier problem of the nonlinear function introduced in the invariant manifold definition design in step 1 using the first-order filtering method; specifically, A first-order filter is introduced for the regression vector, and the derivative of the regression vector after filtering is obtained as: ;in, is a known regression vector, is the filtered regression vector, is the first-order filter cutoff frequency; The regression vector filtering error is defined as: ; when When , the estimated error equation is: ； The pre-defined update law is obtained as: ; According to the update law, the simplified estimation error equation is obtained: ; According to the simplified estimation error equation, the nonlinear function is designed as follows: ,in, is the update law gain; Step 3: Based on the projection operator and dead zone theory, the problem of estimated parameter drift is solved; Step 4: Based on the relevant parameters calculated in steps 1 to 3, complete the design of the invariant manifold interference estimator.

2. The interference estimator design method based on immersion and invariant manifold theory according to claim 1 is characterized in that: The estimation error of the invariant manifold is: .

3. The interference estimator design method based on immersion and invariant manifold theory according to claim 1 is characterized in that: Step 3 includes: Based on the dead zone theory, the concept of dead zone is determined; Obtain the Lyapunov function, and based on the dead zone concept, transform the derivative of the Lyapunov function to obtain the objective function ; Based on the projection operator theory, the update law is recalculated; According to the update law, the Lyapunov function is re-derived to obtain a new ; When the new With the objective function The conclusion is the same as that of , and the final expression of the improved update law based on the projection operator is: 。 4. The interference estimator design method based on immersion and invariant manifold theory according to claim 3 is characterized in that: Based on the dead zone theory, the dead zone concept is determined, including: When the n-order nonlinear system When the origin is the equilibrium point, the dead zone concept is obtained: ； in, It is a positive constant, indicating the size of the dead zone.

5. A method for designing an interference estimator based on immersion and invariant manifold theory according to claim 3 or 4, characterized in that: Obtain the Lyapunov function, and based on the dead zone concept, transform the derivative of the Lyapunov function to obtain the objective function ,include: based on The origin is the equilibrium point, and , the new estimation error equation is: ； Based on the new estimation error equation, we can obtain the Lyapunov function: ; Derivative Lyapunov function yields: ； based on , is a positive constant, and the derivative of the transformed Lyapunov function is: 。 6. The method for designing an interference estimator based on immersion and invariant manifold theory according to claim 3, characterized in that: Based on the projection operator theory, the update law is recalculated, including: Get the projection operator theory, where the mathematical expression of the projection operator is , and satisfy the following equation: ； Based on the projection operator, the update law is recalculated as: ； in, is the adaptive parameter, is the target nonlinear function, is the estimation error of the invariant manifold, is the filtered regression vector, is a known regression vector, is the derivative of the regression vector after filtering.

7. A method for designing an interference estimator based on immersion and invariant manifold theory according to claim 3 or 6, characterized in that: According to the update law, the Lyapunov function is re-derived to obtain a new ,include: According to the recalculated update law, the Lyapunov function is re-derived as follows: ； in, is the update law gain, is the adaptive parameter, is the target nonlinear function, is the estimation error of the invariant manifold, is the filtered regression vector, is a known regression vector, is the derivative of the regression vector after filtering.

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