A method and system for introducing composite learning into nonlinear system output feedback adaptive control

Abstract

The application belongs to the technical field of nonlinear system control, and discloses a method and system for introducing composite learning into nonlinear system output feedback adaptive control. The application directly reuses the existing K-filter in standard output feedback backstepping control to construct an extended prediction error, and does not need to establish an additional observer or a state estimation model (such as a series-parallel estimation model, a fuzzy observer, etc.) in parallel with the K-filter. This functional reuse design eliminates the calculation overhead of the additional dynamic system, reduces the storage requirement and real-time calculation burden of the controller, and significantly reduces the hardware implementation cost and system debugging complexity. Meanwhile, through the composite driving of the tracking error and the extended prediction error, the cumulative information is continuously introduced by using the historical memory of the regression quantity, so that the adaptive and self-learning ability of the control system is significantly enhanced, and the parameter estimation convergence and the system response performance are improved.

Description

Technical Field

[0001] This invention relates to the field of nonlinear system control technology, and specifically to a method and system for introducing composite learning into the output feedback adaptive control of a nonlinear system. Background Technology

[0002] Adaptive control technology, by adjusting controller parameters online to address system uncertainties, has become an important means of handling parameter uncertainties in nonlinear systems. For practical engineering systems where the state is not fully measurable, adaptive output feedback control reconstructs the internal state using a state observer, achieving closed-loop control using only the output signal. To further improve parameter estimation performance, composite adaptive control methods introduce prediction error information (or identification error) and tracking error to jointly drive parameter updates, thereby enhancing parameter learning ability and convergence performance.

[0003] However, in the engineering practice of applying composite learning to adaptive output feedback control, the following objective technical problems still exist:

[0004] (1) The system has high implementation complexity and heavy real-time computational burden. To construct the prediction error, existing composite adaptive output feedback control methods usually require the establishment of additional state estimation models (such as serial-parallel estimation models, fuzzy logic observers, or neural network observers). These additional dynamic systems run simultaneously with the original state observers, causing the controller to need to maintain the real-time solution of multiple sets of differential equations, which significantly increases the consumption of computing resources and storage requirements. More importantly, the independent tuning of multiple sets of observer parameters places a heavy burden on engineering debugging, and the increase in system structural complexity directly drives up hardware implementation costs and reliability risks.

[0005] (2) Limited parameter learning efficiency and slow convergence speed. Existing composite adaptive methods usually only use the instantaneous prediction error at the current moment for parameter updates, failing to effectively utilize the continuous excitation information in historical data. In scenarios where system parameters change slowly or the reference signal excitation is insufficient, this update method that relies solely on instantaneous information leads to slow parameter estimation convergence speed, which in turn prolongs the tracking error convergence time and increases the overshoot, affecting the dynamic response quality of the system.

[0006] (3) Control signals are prone to singularities, resulting in insufficient system safety. In output feedback design based on backstepping control, the estimated value of the control gain parameter appears directly in the denominator of the control law. Existing technologies lack an effective mechanism to ensure that this estimated value is always far from zero. In cases of improper parameter initialization or transient processes, the estimated value may approach zero or even change sign, leading to singular jumps in the control signal. This not only seriously affects tracking performance but may also excite unmodeled dynamics of the controlled object, causing system instability or equipment damage, thus limiting the application of the method in safety-critical areas.

[0007] (4) Limited applicability to system types and insufficient versatility. Existing composite adaptive methods are mostly aimed at nonlinear systems with strict feedback, while research on composite learning for output feedback form systems is relatively insufficient. In output feedback form systems, due to the coupling between control gain and system dynamics, the direct application of existing technologies faces structural compatibility obstacles, making it difficult to apply composite learning to improve the performance of many real physical systems with only measurable outputs (such as some aircraft and mechanical systems).

[0008] In view of this, the present invention is proposed. Summary of the Invention

[0009] The present invention aims to solve at least one of the above technical problems, and provides a method and system for introducing composite learning into the output feedback adaptive control of a nonlinear system.

[0010] To achieve the above objectives, the first technical solution adopted by the present invention is as follows:

[0011] A method for introducing composite learning into the output feedback adaptive control of a nonlinear system includes the following steps:

[0012] For nonlinear controlled objects with output feedback, a K-filter is used to estimate the unmeasurable state inside the system, generating a filtered signal and a regression matrix. The K-filter directly reuses the state observer structure in the standard output feedback back-calculation control law, without the need to establish an additional observer or state estimation model in parallel with the K-filter.

[0013] Using the regression matrix, an extended prediction error is constructed through integral operation with a moving time window. The extended prediction error represents the cumulative prediction bias based on the historical memory of the regression quantity within the integration period.

[0014] The tracking error generated during the backstepping control process and the extended prediction error jointly drive the update law of the unknown parameters, constructing a composite parameter adaptive law, and using the composite parameter adaptive law to update the estimated value of the unknown parameters; wherein the control gain parameter is double-estimated using an overparameterization scheme, and its estimated value is kept far from zero and its sign remains unchanged through a projection mechanism;

[0015] Based on the updated parameter estimates, an actual control law is generated through backstepping recursion to drive the controlled object to achieve asymptotic tracking of the system output to the reference signal.

[0016] Preferably, the requirement to establish an additional observer in parallel with the K-filter specifically includes the requirement to establish a serial-parallel estimation model, a fuzzy state observer, or a neural network observer.

[0017] Preferably, the design of the K-filter includes:

[0018] The filter dynamic equations are designed to generate the filtered signal and the regression matrix, and the parameter vectors of the K-filter are selected such that the filter dynamic matrix is ​​a Herwitz matrix.

[0019] The system state is reconstructed using the filtered signal and the regression matrix to obtain a state estimation expression containing unknown parameters, wherein the state estimation error converges exponentially.

[0020] Preferably, the step of constructing the extended prediction error through integral calculation via a moving time window specifically includes:

[0021] The static regression relationship between the system output and the unknown parameters is established using the K-filter described above;

[0022] The current and historical values ​​of the regression matrix are decomposed to construct a regression matrix containing control gain components and residual parameter components.

[0023] An extended regression matrix is ​​constructed by integrating the product of the regressor matrix and its transpose over a shift time interval;

[0024] The extended prediction error is calculated based on the extended regression matrix, the integral value of the system output, and the parameter estimates.

[0025] Preferably, the dual estimation of the control gain parameter using an overparameterization scheme specifically includes:

[0026] In the first step of the backstep control, the stabilization function is calculated using the first control gain estimate;

[0027] In the subsequent backstepping step, a second control gain estimate, independent of the first control gain estimate, is used to process the tracking error coupling term;

[0028] The first control gain estimate is updated using the projection mechanism to ensure that its sign is known and its absolute value is greater than a preset lower bound. The second control gain estimate is updated independently of the first control gain estimate as part of the parameter vector.

[0029] Preferably, the method further includes a stability verification step:

[0030] We construct a Lyapunov functional that includes tracking error, parameter estimation error, and observation error, and includes a double integral term to handle the moving time window integral, so as to eliminate the influence of residual state estimation error in the extended prediction error on the time window integral.

[0031] By selecting appropriate gain parameters to make the time derivative of the Lyapunov functional negative definite, it is proven that all signals in the closed-loop system are globally uniformly bounded and the tracking error converges asymptotically.

[0032] The second technical solution adopted in this invention is:

[0033] A system that incorporates composite learning into the output feedback adaptive control of a nonlinear system includes:

[0034] The estimation module is used to estimate the unmeasurable internal state of a nonlinear controlled object with output feedback using a K-filter, generating a filtered signal and a regression matrix. The K-filter directly reuses the state observer structure in the standard output feedback back-calculation control law, without the need to establish an additional observer or state estimation model in parallel with the K-filter.

[0035] The first construction module is used to construct an extended prediction error by using the regression matrix and performing integral operations through a moving time window. The extended prediction error represents the cumulative prediction bias based on the historical memory of the regression quantity within the integration period.

[0036] The second construction module is used to drive the update law of unknown parameters together with the tracking error generated during the backstep control process and the extended prediction error, construct a composite parameter adaptive law, and use the composite parameter adaptive law to update the estimated value of the unknown parameters; wherein the control gain parameter is double-estimated using an overparameterization scheme, and its estimated value is kept far from zero and its sign remains unchanged through a projection mechanism;

[0037] The drive module is used to generate the actual control law through backstepping recursion design based on the updated parameter estimates, and drive the controlled object to achieve asymptotic tracking of the system output to the reference signal.

[0038] Compared with the prior art, the present invention has the following beneficial effects:

[0039] This invention directly reuses the existing K-filter in standard output feedback backstepping control to construct the extended prediction error, without the need to establish an additional observer or state estimation model (such as a serial-parallel estimation model, fuzzy observer, etc.) parallel to the K-filter. This functional reuse design eliminates the computational overhead of the additional dynamic system, reduces the controller's storage requirements and real-time computation burden, and significantly reduces hardware implementation costs and system debugging complexity.

[0040] This invention expands the instantaneous prediction error into a cumulative prediction error that includes the memory of historical regression values ​​by introducing integral operations based on a moving time window. This time-domain expansion mechanism provides a richer source of information for the parameter adaptation process and can significantly improve the convergence speed and accuracy of parameter estimation when the continuous excitation condition is met. Simulation results show that, compared with conventional composite adaptive methods that only use instantaneous prediction errors, the convergence time of the parameter estimation error norm of this invention is significantly shortened.

[0041] This invention employs an overparameterization scheme to perform dual estimation of the control gain parameter, and combines this with a projection mechanism to ensure that the estimated value always stays far from zero and its sign remains unchanged. This design effectively solves the technical problem of control law singularity caused by the control gain estimate approaching zero in output feedback backstepping control, and improves the robustness and engineering practicality of the closed-loop system.

[0042] This invention is particularly applicable to physical systems where only some states are measurable, such as aircraft control and robotic arm control. Taking the roll dynamics control of a delta-wing aircraft as an example, this invention can reconstruct the roll angular velocity through a K-filter by measuring only the roll angle, and quickly estimate aerodynamic parameters using composite learning to achieve high-precision attitude tracking. Moreover, it requires no additional sensors or hardware circuits; performance improvement can be achieved solely through software-level integral calculations. Attached Figure Description

[0043] Figure 1 A flowchart illustrating a method for introducing composite learning into the output feedback adaptive control of a nonlinear system, provided for an embodiment of the present invention;

[0044] Figure 2 A schematic diagram of the structure of a system that incorporates composite learning into the output feedback adaptive control of a nonlinear system, provided for an embodiment of the present invention;

[0045] Figure 3 For the roll dynamics of delta-wing aircraft, an open-loop limit cycle is provided.

[0046] Figure 4 The system outputs a tracking response to the reference instruction;

[0047] Figure 5 To control the time history of the input;

[0048] Figure 6 The convergence curve for the parameter estimation error;

[0049] Figure 7 The convergence curve for controlling the gain estimation error. Detailed Implementation

[0050] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be described in detail below with reference to specific embodiments. It should be noted that, unless otherwise specified, the embodiments and features described in these embodiments can be arbitrarily combined with each other. Those skilled in the art should understand that the following descriptions are merely exemplary means of implementing this invention and not limiting conditions; any technical means employed to achieve the same or similar technical effects as this invention should fall within the protection scope of this invention.

[0051] like Figure 1The diagram shown illustrates the overall process of this invention's method for introducing composite learning into the output feedback adaptive control of nonlinear systems. This method targets nonlinear controlled objects with output feedback. Its core lies in directly reusing the K-filters structure from standard output feedback backpropagation control, constructing an extended prediction error through time window integration, and combining this extended prediction error with the tracking error generated by backstepping control to form a composite adaptive law. This improves parameter convergence and tracking performance without requiring an additional observer.

[0052] This invention is applicable to nonlinear systems with output feedback. As an example, the method of this invention is illustrated below using a typical nonlinear single-input single-output (SISO) system with output feedback. It should be emphasized that this specific mathematical form is only used to clarify the technical principles and is not intended to limit the scope of protection of this invention. The method of this invention can also be applied to other nonlinear systems that meet the conditions of "only the output is measurable, the state can be estimated using K-filters, and the unknown parameters enter the system dynamics in a linear manner."

[0053] Consider the following nonlinear SISO system with output feedback, whose system state equation can be expressed as:

[0054] ；

[0055] ；

[0056] in, ; ; ; ; ;

[0057] x is the system state vector; Let x be the derivative of x with respect to time; u be the system control input; and y be the system output that satisfies y = x1 (i.e., the first element of the state vector). and Let R be an unknown constant parameter vector, where a1, ..., a2 are real numbers. q Let q be a specific element of vector a, and R be the dimension of vector a. q Let b represent a q-dimensional real vector space. m ..., b0 are the specific elements of vector b, m is the index of the highest-order term of vector b, m+1 is the dimension of vector b, and R... m+1 Denotes the (m+1)-dimensional real vector space, with the superscript T indicating the transpose; A is the system matrix, I n-1 It is an (n-1)×(n-1) dimensional identity matrix, where n is the dimension of the system state vector; Let be the standard basis vectors; ϕ(y) and Φ(y) be known nonlinear function matrices. Both and σ(y) are smooth nonlinear functions; F(y,u) is the regression function matrix. Let I be a zero matrix of dimension (ρ-1)×(m+1), where ρ is the relative order of the system. m+1 Let θ be an (m+1)×(m+1) dimensional identity matrix, and let θ be a parameter vector of dimensional p=q+m+1, where p represents the total number of dimensions of the parameter vector θ.

[0058] To make the above control problem solvable, the following assumptions are made regarding the state equations of the above system:

[0059] Assumption 1. Parameter b m The sign of is known, and its absolute value |b m The lower bound of | is known to be ;

[0060] Assumption 2. Given the polynomial The Hurwitz is stable, where B(s) is the transfer function polynomial and s is the Laplace variable;

[0061] Assumption 3. For For all cases, σ(y)≠0. Let R denote a real number for all 10 ...

[0062] Assumption 4. Reference signal y r Its first n derivatives are known, it is bounded and piecewise continuous.

[0063] refer to Figure 1 The first embodiment of the present invention provides a method for introducing composite learning into the output feedback adaptive control of a nonlinear system, comprising the following steps:

[0064] S101, for a nonlinear controlled object with output feedback, a K-filter is used to estimate the unmeasurable state inside the system, generating a filtered signal and a regression matrix; wherein, the K-filter directly reuses the state observer structure in the standard output feedback back-calculation control law, without the need to establish an additional observer or state estimation model in parallel with the K-filter.

[0065] For the aforementioned system where only the output is measurable but the internal state is not, a K-filter is used to estimate the system's internal state, generating a filtered signal and a regression matrix. It should be noted that the design of a K-filter is a common technique in the field of adaptive output feedback control, and those skilled in the art can refer to existing techniques in this field for design.

[0066] As an example, the design of a K-filter includes: designing the filter dynamic equation to generate the filtered signal and regression matrix, wherein the parameter vector of the K-filter is selected such that the filter dynamic matrix is ​​a Herwitz matrix; and reconstructing the system state using the filtered signal and regression matrix to obtain a state estimation expression containing unknown parameters, wherein the state estimation error converges exponentially.

[0067] Specifically, the dynamic equations of the filter are designed to generate the filtered signal. ξ And the regression matrix Ω. Select the parameter vector of the k-filter. k =[ k 1,..., k n ] T Where k1,…,k n These are elements of the parameter vector k, with the superscript T indicating the transpose. n Let be the dimension of the system state vector. This vector is chosen such that the matrix... Let A0 be the Hurwitz matrix, where A0 is the filter dynamic matrix. There are several ways to choose the parameter vector k, such as through pole placement or solving the Riccati equation, as long as A0 is a Hurwitz matrix.

[0068] Based on the Herwitz matrix, the following dynamic equations for the k-filter are constructed:

[0069] ;

[0070] ;

[0071] In the formula: the superscript T is the transpose symbol. Let ξ be the derivative of the filtered signal with respect to time. Ω T The derivative with respect to time.

[0072] The regression matrix Ω can also be constructed in a way that involves fewer dynamic orders: designing an auxiliary filter. ,in λ As an auxiliary filtering signal, e is the derivative of the auxiliary filtered signal with respect to time. n This is the nth standard basis vector; and then through... Generate input driving vector v j ， Where j = 0, ..., m (j is an integer index from 0 to m), a total of m+1 filtered signal vectors v are generated. m ..., v1, v0; simultaneously design filters Generate the output driving matrix Ξ. This is the derivative of the driving matrix with respect to time. The final regression matrix satisfies... .

[0073] Using the filtered signal ξ Given the regression matrix Ω, the state estimation expression is: In the formula: This is the state estimate.

[0074] The state estimation error ε is defined as This error satisfies the dynamic equation .in This is the derivative of ε with respect to time. Since... A 0 is the Herwitz matrix, and the error exponential is stable and converges exponentially to zero.

[0075] From the above state estimation expression, the static relationship between the system state and the unknown parameters can be obtained as follows: This static relationship forms the basis for subsequently constructing the expanded prediction error.

[0076] The key improvement in this step lies in the fact that the K-filter directly reuses the state observer structure in the standard output feedback back-calculation control law, eliminating the need to build an additional observer or state estimation model parallel to the K-filter. As an example, there is no need to build parallel dynamic systems such as serial-parallel estimation models, fuzzy state observers, or neural network observers. This direct reuse design eliminates the computational overhead of additional dynamic systems, significantly reducing system structural complexity and hardware implementation costs.

[0077] S102, using the regression matrix, an extended prediction error is constructed through integral operation with a moving time window. The extended prediction error represents the cumulative prediction bias based on the historical memory of the regression quantity within the integration period.

[0078] Prediction error is a key concept in composite adaptive control, reflecting the deviation between the predicted system output based on the current parameter estimates and the actual output. Conventional composite adaptive methods typically utilize only the instantaneous prediction error at the current moment to drive parameter updates. However, this approach, relying solely on instantaneous information, fails to fully leverage the continuous excitation information in historical data, resulting in limited parameter convergence speed in scenarios with insufficient reference signal excitation or slow changes in system parameters.

[0079] The key to this invention lies in utilizing the memory of the regressor quantity to continuously introduce more information through integral operations with a moving time window, thereby constructing an extended prediction error. This extended prediction error characterizes the cumulative prediction bias based on the historical memory of the regressor quantity within the integration period, which can significantly enhance the learning ability of the parameter adaptive process and improve the convergence of parameter estimation.

[0080] Specifically, the static regression relationship between the system output and the unknown parameters is first established using a K-filter. The static regression relationship of the output can be obtained from the state estimation expression: Where x1 is the first element of the system state vector. ξ 1 is the filtered signal ξ First element; Ω1 is the first column of the regression matrix Ω (corresponding to the column vector of the system output), and the superscript T is the transpose symbol; ε 1 is the first element of the state estimation error ε.

[0081] Based on the above static regression relationship, using ξ and v j And Ξ, further define the following regression vector for subsequent composite adaptive law and backstep control design:

[0082] ;

[0083] ;

[0084] ;

[0085] Where ξ2 is the second element of ξ, v j,2 (j=0,…,m) represents v j The second component, Φ(1) and Ξ(2), are the first row of Φ(y) and the second row of Ξ, respectively. ω 0 represents a known combination of signals. ω For the regression vector, To correct the regression vector.

[0086] Based on the above regression vector, the original system state equation can be transformed into the following equivalent measurable state form:

[0087] ;

[0088] ;

[0089] ;

[0090] Where i has no practical meaning and represents the corresponding sequence number in this invention, k i k ρ Let i and ρ be the i-th and ρ-th elements of the parameter vector k, respectively. b is the derivative of the system output with respect to time. m Let εm be the m-th element of vector b, which also represents the control gain, and ε2 be the second element of ε. For v m The i-th component v m,i The derivative with respect to time, For v mThe ρ-th component v m,ρ The derivative with respect to time, with the superscript T indicating the transpose.

[0091] This dynamic equation transforms the original system, which only has measurable outputs, into an equivalent dynamic system with measurable states composed of K-filter signals. This allows all system states to be directly measured or calculated through K-filters, laying the foundation for subsequent backstepping recursive control design.

[0092] Then, the current and historical values ​​of the regression matrix are decomposed to construct a regression matrix M containing control gain components and residual parameter components. Ω Specifically, Decomposed into , where Ω 1,1 The first element of Ω1, which corresponds to the control gain b. m Element; For the portion corresponding to the remaining parameters, the superscript T is the transpose symbol. The regressor matrix is ​​then constructed as follows: .

[0093] Then, by moving the time interval Integrating the product of the regressor matrix and its transpose, we construct an extended regression matrix; where t represents time, also indicating the current moment. d The length of the moving time window can be selected based on the system's dynamic characteristics and real-time requirements. Simultaneously, the system output y and the first element ξ1 of the filtered signal are integrated over the same interval to obtain iy and iξ1, where iy represents the weighted integral over y, and iξ1 represents the weighted integral over ξ1. The specific calculation is as follows:

[0094] Where h is the integration time variable, representing a historical moment.

[0095] Estimating values ​​using unknown parameter vectors and control gain estimate Introducing the predicted value of iy for .

[0096] Finally, based on the extended regression matrix, the integral value of the system output, and the parameter estimates, the extended prediction error is calculated. The extended prediction error ϵ is:

[0097] ;in, The estimation error of θ For b m The estimation error.

[0098] Based on the extended prediction error ϵ, design and The update law is as follows: ,in for The derivative with respect to time, for The derivative with respect to time, Γ>0 is the gain matrix, γ is the gain scalar, γ p >0 represents the composite learning gain. If M Ω Since it is bounded, the update law itself guarantees stability.

[0099] This extended prediction error expands instantaneous regression information into cumulative information that includes historical data. It effectively suppresses measurement noise through integration and makes full use of continuous excitation conditions, thereby improving the convergence of parameter estimation.

[0100] Those skilled in the art will understand that the specific implementation of the above-mentioned moving time window integration can adopt conventional numerical integration methods such as discrete accumulation, moving average, or low-pass filtering, and the length of the integration interval can be adjusted according to the sampling rate and dynamic response characteristics of the actual system.

[0101] S103, the tracking error generated during the backstepping control process and the extended prediction error are used together to drive the update law of the unknown parameters, construct the composite parameter adaptive law, and use the composite parameter adaptive law to update the estimated value of the unknown parameters; wherein the control gain parameter is double-estimated using an overparameterization scheme, and its estimated value is kept far from zero and its sign remains unchanged through a projection mechanism.

[0102] In adaptive control, the design of the parameter update law directly determines the system's learning ability and robustness. Conventional adaptive output feedback control typically updates parameters using algorithms such as gradient descent or least squares, relying solely on the tracking error generated during backstepping control. While this method ensures stability, the parameter convergence speed often depends on the continued presence of the tracking error, and inaccurate parameter estimation may occur during transient processes.

[0103] The key to this invention lies in constructing a composite parameter adaptive law, which uses the tracking error generated during the backstepping control process and the extended prediction error constructed in step two to jointly drive the update law of the unknown parameters. Furthermore, considering the characteristic that the control gain parameter directly appears in the denominator of the control law, an overparameterization scheme is used for dual estimation, and a projection mechanism is employed to ensure that its estimated value is far from zero and its sign remains unchanged, thereby avoiding singularities in the control signal.

[0104] The method of using an overparameterized scheme to perform dual estimation of the control gain parameter specifically includes: in the first step of backstepping control, using the first control gain estimate to calculate the stabilization function; in subsequent backstepping steps, using a second control gain estimate independent of the first control gain estimate to process the tracking error coupling term; wherein, the first control gain estimate is updated using the projection mechanism to ensure that its sign is known and its absolute value is greater than a preset lower bound, and the second control gain estimate is updated independently of the first control gain estimate as part of the parameter vector.

[0105] In one specific embodiment, the composite parameter adaptive law is designed as follows:

[0106] The composite adaptive law is composed of two parts: one part is the driving component of the tuning functions constructed based on the tracking error generated in each step of the backstep control process; the other part is the driving component of the extended prediction error constructed based on the aforementioned steps.

[0107] Regarding the construction of the regulation function driven by tracking error: In the recursive design process of backstepping control, after defining the initial regulation function τ1 in step 1, the regulation function τ is gradually constructed from step 2 to step ρ (where ρ is also the relative order of the system, i.e., the number of integrators required from the control input to the system output, which takes different forms in different formulas). i (i=2,…,ρ), the specific construction process is as follows: In step 1, based on the tracking error Z1 in step 1 (defined as Z1=y−y r， y r Given the reference signal, the regression vector ω, and the stabilization function (virtual control law) α1 designed in step 1, define the initial adjustment function τ1: .

[0108] In step i (i=2,…,ρ), based on the tracking error Z in step i... i (defined as) , The filtered signal vector v m The i-th element), the regression vector ω, and the stabilization function α designed in the (i-1)-th step. i-1 Partial derivative with respect to the system output y Recursively define the adjustment function τ i : , where τ i-1 This is the adjustment function for the (i-1)th step.

[0109] After the above recursive process, the adjustment function τ for parameter updating is finally obtained in the last step ρ. ρ It integrates the tracking error information from all steps and can be expressed as:

[0110] ;in, (i=2,…,ρ) reflects the sensitivity of the stabilization function in the (i-1)th step to the output, Z ρ Let ρ be the tracking error at step ρ.

[0111] Regarding the utilization of the extended prediction error: directly use the aforementioned constructed extended prediction error ϵ, and extract its first p components ϵ. 1:p (corresponding to b) m (dimensions of other uncertain parameters) and the last component ϵ p+1 (corresponding control gain b) m The prediction error component).

[0112] Based on the aforementioned adjustment function and extended prediction error, an estimate of the unknown parameter vector is constructed. and control gain estimate The composite update law. Unknown parameter estimation error. satisfy:

[0113] ;in, For the estimated value of the unknown parameter vector The derivative with respect to time.

[0114] Accordingly, in actual controller implementation, the estimated value of the unknown parameter vector Update in the opposite direction, i.e., along The direction of the update.

[0115] This step controls the gain b. m A parameterization scheme is used for dual estimation, and a projection mechanism is used to ensure that the estimated value is far from zero.

[0116] The estimation error of the first estimation satisfy:

[0117] ;in, That is, the first control gain estimate. for The time derivative, Proj(•), is the projection operator used to ensure parameter estimation of the control gain. Its absolute value is greater than or equal to at all times. , , They represent b respectively m The estimated values ​​at time t and the initial time. express The (p+1)th element.

[0118] First control gain estimate The above formula is updated independently and is specifically used for calculating the stabilization function in the first step of S104. Because b m It is the first element of θ, in the compound update law middle, The first component naturally constitutes b m The second control gain estimate is denoted as . This is used for coupling compensation in the second step of S104.

[0119] The above update laws can be uniformly expressed in matrix form:

[0120] ;

[0121] in, satisfy Z is the tracking error vector, i.e., all Z... i The set of , where the superscript T is the transpose symbol; This is the state estimation error weight vector. For the parametric regression weight matrix, and The definition of is: , .

[0122] S104, based on the updated parameter estimates, generates the actual control law through backstepping recursive design, driving the controlled object to achieve asymptotic tracking of the system output to the reference signal.

[0123] After updating the estimated values ​​of the unknown parameters using the composite parameter adaptive law, the actual control law is generated through backstepping recursion based on the updated parameter estimates, driving the controlled object to achieve asymptotic tracking of the system output to the reference signal.

[0124] Backstepping recursive design is achieved by recursively designing virtual control variables (called stabilization functions). Specifically, coordinate transformation is used to convert the original system into an error dynamic system, thereby decoupling the tracking error between the system output and the reference signal.

[0125] In some implementations, the actual control law is generated through backstepping recursive design, including: defining a tracking error coordinate transformation to decouple the tracking error of the system output from the reference signal and constructing a virtual control error; recursively designing a stabilization function, constructing a nonlinear damping term in each backstepping design step to offset the influence of the state estimation error, and designing a coupling compensation term to achieve the antisymmetric characteristics of the error system matrix; and finally generating the actual control law, which includes a feedforward compensation term based on the parameter estimation value and a feedback stabilization term based on the tracking error.

[0126] The stabilization function is designed recursively, and a nonlinear damping term is constructed in each backstep design step to offset the influence of state estimation errors. Specifically, in the first step, the stabilization function α1 is designed:

[0127] In the formula: c 1 and d 1 is a positive constant. This is the derivative of the reference signal with respect to time. We use it directly here. b m The estimated value Instead of its reciprocal estimate, this works in conjunction with the aforementioned projection mechanism to ensure that the denominator is not zero.

[0128] In the second step, design the stabilization function α2:

[0129] ;

[0130] Where c2 and d2 are positive constants, and α2 is added to it. To compensate for the coupling terms that appear in the first step of tracking error dynamics. (This item originates from the overparameterized scheme b) m The dual estimation, through which the antisymmetry of the error system matrix is ​​achieved (Φ is Φ(y), k) j It is the j-th element of the parameter vector k, λ1, λ j+1 v represents the 1st and (j+1)th components of λ, respectively. m,1 For v m The first element, the nonlinear damping term Used to offset the effects of state estimation errors.

[0131] For the i step( i =3,…, ρ Design a stabilization function α i :

[0132] ;

[0133] Among them, c i d i Z is a positive constant. j Let the tracking error be at step j. This is a coupling compensation term (constructed based on the backstep recursive design, used to eliminate the cross-coupling between the parameter update law and the stabilization function of the previous step, and to realize the antisymmetric property of the error system matrix). The coupling coefficient is... , Let a be the stabilization function at step i-1. i-1 Partial derivatives of parameter estimation, coupling terms Used to reverse the tracking error between adjacent steps. Nonlinear damping term. Used to offset the effects of state estimation errors.

[0134] In the final step, the actual control law is generated: In the formula: α ρ v is the stabilization function for the ρth step; m,ρ+1 For the filtered signal v m The (ρ+1)th element. This actual control law includes a feedforward compensation term based on parameter estimates and a feedback stabilization term based on tracking error, driving the controlled object to achieve asymptotic tracking of the reference signal by the output.

[0135] Based on the above stabilization function α i and control law The design can obtain the tracking error. The dynamic equation is:

[0136] ; Z is the derivative of Z with respect to time, representing the error system.

[0137] The tracking error system matrix It is given by the following formula:

[0138] ;

[0139] Where the superscript T denotes transpose, σ ij With σ i,j Both represent coupling coefficients, σ ij Coupling coefficient representing adjacent coupling ， σ i,j This represents the coupling coefficient of the skip coupling.

[0140] This implementation also includes a stability verification step: constructing a Lyapunov function that includes tracking error, parameter estimation error, and observation error, which includes a double integral term to handle the time delay effect introduced by the moving time window integration; by selecting an appropriate gain parameter to make the time derivative of the Lyapunov function negative definite, thereby proving that all signals in the closed-loop system are globally uniformly bounded and that the tracking error asymptotically converges.

[0141] Specifically, to prove the stability of the parameter update law based on the expanded prediction error (assuming M... Ω (Bounded), firstly, a Lyapunov functional V1 specifically for this update law is constructed. Unlike the regular Lyapunov function, this functional includes a double integral term for the state estimation error ε (which is also the state estimation residual term here), to handle the effect of the moving-time window integral of the residual term on stability.

[0142] Specifically, we first prove that based on the expanded prediction error ϵ and The stability of the update law (assuming M) Ω bounded).

[0143] Consider candidate Lyapunov functional V1: η is the integral variable ， Let P be the observation error weight, where P is a positive definite matrix, satisfying... , I is the identity matrix, and the superscript T is the transpose symbol.

[0144] The double integral term in the above functional V1 This term is specifically designed to handle the influence of the state estimation residual term ε on the moving-time window integral signal. When calculating the time derivative of V1, this term generates a strongly stable term that completely dominates the cross-integral terms appearing in the time derivative. The influence of this makes the Lyapunov functional V1 negative definite, thus proving the stability of the renewal law.

[0145] The time derivative of V1 is:

[0146] .

[0147] Note:

[0148] ;in, This represents the estimated value of θ at time t.

[0149] Substituting this result into the formula for the time derivative of V1, and noting... yes Part of, namely We can obtain:

[0150] ;

[0151] Select V1 and noticed If it is a positive semi-definite matrix, then we have Therefore, when When bounded, the update law based on the expanded prediction error is stable.

[0152] The stability of the entire closed-loop system is demonstrated below:

[0153] According to the error system The dynamic equations satisfied by the update law and error are used to select the candidate Lyapunov function of the system as follows: ;

[0154] Its time derivative is calculated as follows:

[0155] ;

[0156] Among them, A Z For A (Z,t) The abbreviation for W ε For W (ε,t) The abbreviation for W θ For W (θ,t) The abbreviation of .

[0157] Utilizing the properties of the projection operator and The defining formula is:

[0158] For the sake of notation, the equation in the second step above introduces [a certain notation]. . use The result is:

[0159] ;

[0160] Because A Z It has antisymmetry, and can be used to:

[0161] ;

[0162] Substitution The calculation formula yields:

[0163] ;ε n It is the nth component of ε;

[0164] As can be seen from the last step, all terms are either negative definite or semi-negative definite, therefore we have .according to By calculating the results and applying the Lyapunov method, we can conclude that: , , and Both are bounded.

[0165] According to the LaSalle-Yoshizawa theorem The calculation results mean Therefore, there is Where y(t) represents the output of the controlled object at time t, y r (t) represents the tracking signal at time t.

[0166] Candidate Lyapunov function V indicates , and Boundedness. Regarding the standard stability analysis of adaptive output feedback backstepping control, under assumptions 1-4, the globally uniform boundedness of the filtered signals ξ,Ξ,λ, and x can be directly derived from the exponential stability and input-state stability properties of the K-filter. The proof of this part is prior art and will not be elaborated here.

[0167] Figure 2 This is a schematic diagram of the structure of an adaptive output feedback control system 200 for nonlinear systems based on composite learning, referenced from... Figure 2 The second embodiment of the present invention provides a system 200 for introducing composite learning into the output feedback adaptive control of a nonlinear system, comprising: an estimation module 201, a first construction module 202, a second construction module 203, and a driving module 204, the functions of each module are described below.

[0168] The estimation module 201 is used to estimate the unmeasurable state inside the system for a nonlinear controlled object with output feedback by using a K-filter, and to generate a filtered signal and a regression matrix; wherein the K-filter directly reuses the state observer structure in the standard output feedback back-inference control law, without the need to establish an additional observer or state estimation model in parallel with the K-filter.

[0169] The first construction module 202 is used to construct an extended prediction error by using the regression matrix and performing an integral operation through a moving time window. The extended prediction error represents the cumulative prediction deviation based on the historical memory of the regression quantity within the integration period.

[0170] The second construction module 203 is used to drive the update law of unknown parameters together with the tracking error generated during the backstep control process and the extended prediction error, construct a composite parameter adaptive law, and use the composite parameter adaptive law to update the estimated value of the unknown parameters; wherein the control gain parameter is double-estimated using an overparameterization scheme, and its estimated value is kept far from zero and its sign remains unchanged through a projection mechanism.

[0171] The drive module 204 is used to generate an actual control law based on the updated parameter estimates through backstepping recursive design, and drive the controlled object to achieve asymptotic tracking of the system output to the reference signal.

[0172] Simulation experiment verification

[0173] The embodiments of this invention are particularly applicable to the roll dynamics control of aircraft. To verify the effectiveness of the control method of this invention, in one specific embodiment, this invention is applied to the roll dynamics control of a delta-wing aircraft. This system has an output feedback mechanism; the roll angle is measurable, but the roll angular velocity is not.

[0174] The dynamic model of the roll yaw of the delta wing aircraft is as follows:

[0175] ;

[0176] ;

[0177] In the formula: This is the roll angle (in radians, which can be measured and output). for The derivative with respect to time; The rolling angular velocity is unmeasurable. for The derivative with respect to time; δ a This represents the differential aileron deflection angle (control input, in radians). The actual values ​​of the constant parameters are: θ1=0.015, θ2=-0.0295, θ3=-0.018, θ4=0.021, θ5=0.01, b0=0.75. It can be verified that when... δ a =0 Furthermore, when θ5=0, the above delta wing aircraft roll-gyratory dynamics model has an unstable equilibrium point at the origin, and in... There exists a limit cycle near 35°, such as Figure 3 As shown, this model representation preserves the unstable wing roll phenomenon of delta-wing aircraft.

[0178] The control objective is to actively control the roll angle of a delta-wing aircraft using the method of this invention. To track the reference roll angle command φ r Refer to the roll angle command φ r By the roll command φ c Generates the input second-order filter:

[0179] ,

[0180] in φ r The second derivative with respect to time, φ r The first derivative with respect to time, ω n =1.6, ξ=0.95. The controller design will also use... and .

[0181] In this example, the parameter vector k of the k-filter is chosen to be [8, 16]. T In the formula for calculating the stabilization function α1 in the first step, c1=0.2 and d1=0.01 are chosen. In the formula for calculating the stabilization function α2 in the second step, c2=0.2 and d2=0.01 are chosen. In the formula for calculating the extended prediction error ϵ, t... dThe value is chosen as 4. In the matrix form of the unified representation of the update law, Γ=diag([10,10,10,10,10,1]), γ=10,γ p =100, , , Select 0.2.

[0182] Figures 4 to 7 The diagram illustrates the closed-loop response of the system tracking a series of step commands with an initial roll angle set to 0.2 radians. For comparison, simulation results of conventional adaptive output feedback control without composite learning are also presented in the figure, where γ is represented in matrix form of the unified update law. p Set it to 0, and the rest is the same as above.

[0183] Figure 4 The results show that both adaptive laws can make the system's roll angle φ... a Effective tracking of roll command φ c Both control methods exhibit self-learning and adaptive capabilities, and the system response improves as the closed-loop control process progresses. However, the control method employing composite learning clearly demonstrates a faster convergence speed for the tracking error. Figure 5 The time-domain response curves of the differential aileron deflection angle show that the control inputs under both control methods are smooth and without singularities, indicating that the projection mechanism effectively ensures that the estimated control gain is far from zero. At the same time, the control quantity using composite learning has a smaller amplitude and converges faster, demonstrating that the present invention improves the convergence of parameter estimation without increasing the burden on the actuator. Figure 6 and Figure 7 The parameter estimation error norm also indicates that the composite learning approach has better learning capabilities than the non-composite learning approach. When composite learning is enabled, the estimation error converges faster. This advantage stems from the fact that the method utilizes more information from the K-filter (which is used to construct the system state), thereby improving the estimation of unknown parameters in the system.

[0184] As can be seen, by using the method of this invention, the roll angular velocity is reconstructed based on the measured roll angle using a K-filter, and a filtered signal and regression matrix are generated; then, an extended prediction error is constructed by integrating through a moving time window to estimate aerodynamic parameters and control gain; based on the composite parameter update law and backstep recursive design, the actual control law is generated to achieve asymptotic tracking of the roll angle to the reference command.

[0185] The above embodiments are only used to illustrate the technical solutions of this application, and are not intended to limit them. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of this application.

Claims

1. A method for introducing composite learning into the output feedback adaptive control of a nonlinear system, characterized in that, Includes the following steps: For nonlinear controlled objects with output feedback, a K-filter is used to estimate the unmeasurable state inside the system, generating a filtered signal and a regression matrix. The K-filter directly reuses the state observer structure in the standard output feedback back-calculation control law, without the need to establish an additional observer or state estimation model in parallel with the K-filter. Using the regression matrix, an extended prediction error is constructed through integral operation with a moving time window. The extended prediction error represents the cumulative prediction bias based on the historical memory of the regression quantity within the integration period. The tracking error generated during the backstepping control process and the extended prediction error jointly drive the update law of the unknown parameters, constructing a composite parameter adaptive law, and using the composite parameter adaptive law to update the estimated value of the unknown parameters; wherein the control gain parameter is double-estimated using an overparameterization scheme, and its estimated value is kept far from zero and its sign remains unchanged through a projection mechanism; Based on the updated parameter estimates, an actual control law is generated through backstepping recursion design, which drives the controlled object to achieve asymptotic tracking of the system output to the reference signal. Specifically, the construction of the extended prediction error through the integral operation of the moving time window includes: The static regression relationship between the system output and the unknown parameters is established using the K-filter described above; The current and historical values ​​of the regression matrix are decomposed to construct a regression matrix containing control gain components and residual parameter components. An extended regression matrix is ​​constructed by integrating the product of the regressor matrix and its transpose over a shift time interval; The extended prediction error is calculated based on the extended regression matrix, the integral value of the system output, and the parameter estimate. The method of using an overparameterization scheme to perform dual estimation of the control gain parameters specifically includes: In the first step of the backstep control, the stabilization function is calculated using the first control gain estimate; In the subsequent backstepping step, a second control gain estimate, independent of the first control gain estimate, is used to process the tracking error coupling term; The first control gain estimate is updated using the projection mechanism to ensure that its sign is known and its absolute value is greater than a preset lower bound. The second control gain estimate is updated independently of the first control gain estimate as part of the parameter vector.

2. The method for introducing composite learning into the output feedback adaptive control of a nonlinear system as described in claim 1, characterized in that, The requirement to eliminate the need for additional observers to run in parallel with the K-filter specifically includes eliminating the need to build a serial-parallel estimation model, a fuzzy state observer, or a neural network observer.

3. The method for introducing composite learning into the output feedback adaptive control of a nonlinear system as described in claim 1, characterized in that, The design of the K-filter includes: The filter dynamic equations are designed to generate the filtered signal and the regression matrix, and the parameter vectors of the K-filter are selected such that the filter dynamic matrix is ​​a Herwitz matrix. The system state is reconstructed using the filtered signal and the regression matrix to obtain a state estimation expression containing unknown parameters, wherein the state estimation error converges exponentially.

4. The method for introducing composite learning into the output feedback adaptive control of a nonlinear system as described in claim 1, characterized in that, The method also includes a stability verification step: We construct a Lyapunov functional that includes tracking error, parameter estimation error, and observation error, which includes a double integral term to handle the moving time window integral in order to eliminate the influence of residual state estimation error in the extended prediction error on the time window integral. By selecting appropriate gain parameters to make the time derivative of the Lyapunov functional negative definite, it is proven that all signals in the closed-loop system are globally uniformly bounded and the tracking error asymptotically converges.

5. A system for incorporating composite learning into the output feedback adaptive control of a nonlinear system, for performing the method as described in any one of claims 1-4, characterized in that, include: The estimation module is used to estimate the unmeasurable internal state of a nonlinear controlled object with output feedback using a K-filter, generating a filtered signal and a regression matrix. The K-filter directly reuses the state observer structure in the standard output feedback back-calculation control law, without the need to establish an additional observer or state estimation model in parallel with the K-filter. The first construction module is used to construct an extended prediction error by using the regression matrix and performing integral operations through a moving time window. The extended prediction error represents the cumulative prediction bias based on the historical memory of the regression quantity within the integration period. The second construction module is used to drive the update law of unknown parameters together with the tracking error generated during the backstep control process and the extended prediction error, construct a composite parameter adaptive law, and use the composite parameter adaptive law to update the estimated value of the unknown parameters; wherein the control gain parameter is double-estimated using an overparameterization scheme, and its estimated value is kept far from zero and its sign remains unchanged through a projection mechanism; The drive module is used to generate the actual control law through backstepping recursion design based on the updated parameter estimates, and drive the controlled object to achieve asymptotic tracking of the system output to the reference signal.

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