Adaptive nonlinear time-invariant system state and parameter joint estimation method based on time scale separation

Abstract

The application discloses a kind of adaptive nonlinear time-invariant system state and parameter joint estimation method based on time scale separation, belong to adaptive state and parameter joint estimation field.It includes: according to the time scale separation gain parameter and design constant set, the high gain state observer of nonlinear time-invariant system is designed;At each sampling time, the measurement output and known external input vector of nonlinear time-invariant system are obtained;According to the error between the measurement output and the measurement output estimate value, the high gain state observer is driven to perform update, and the state vector estimate value at the current time is obtained;Auxiliary filter is constructed, and state vector estimate value is filtered to generate filter output;Based on filter output, regression vector for parameter estimation is constructed;According to regression vector, auxiliary matrix and auxiliary vector are designed;According to auxiliary matrix and auxiliary vector, parameter adaptive law is designed;At each sampling time, parameter adaptive law is integrated to obtain the parameter vector estimate value at the current time.The application effectively suppresses the common severe oscillation phenomenon in traditional coupling design.

Description

Technical Field

[0001] This invention relates to an adaptive joint estimation method for the state and parameters of a nonlinear time-invariant system based on time scale separation, belonging to the field of adaptive joint estimation of state and parameters. Background Technology

[0002] In fields such as industrial control, marine engineering, and aerospace, many nonlinear time-invariant systems (such as wave energy generation devices, UAV attitude control, and servo motor drives) can be simplified or decoupled into second-order nonlinear time-invariant system models. For these systems, it is usually necessary to estimate unmeasurable internal states (such as angular velocity) and unknown system parameters (such as moment of inertia and damping coefficient) in real time.

[0003] Existing adaptive high-gain observer methods couple high-gain state observation with adaptive parameter updates. This coupling structure suffers from a fundamental "tuning dilemma":

[0004] Parameter oscillations caused by coupling state estimation and parameter estimation: To improve the convergence speed of state estimation, it is necessary to increase the gain of the high-gain state observer. However, increasing the gain inevitably amplifies both measurement noise and unmodeled dynamics; the amplified noise then directly enters the parameter update law through the coupling channel, causing the parameter estimation to oscillate violently or even diverge.

[0005] The inability to achieve both: Engineers are forced to compromise between "state tracking speed" and "parameter estimation stability", and cannot obtain high performance in both at the same time.

[0006] Therefore, a solution is needed to achieve dynamic separation of state estimation and parameter estimation from a structural perspective. Summary of the Invention

[0007] This invention provides an adaptive method for joint estimation of state and parameters of nonlinear time-invariant systems based on time scale separation, which to some extent solves the parameter oscillation problem caused by strong coupling between state estimation and parameter estimation in the prior art.

[0008] The technical solution of this invention is:

[0009] According to a first aspect of the present invention, an adaptive method for joint estimation of state and parameters of a nonlinear time-invariant system based on time scale separation is provided, comprising:

[0010] Establish a nonlinear time-invariant system with unknown states and parameters;

[0011] Based on the set time scale, the gain parameters and design constants are separated, and a high-gain state observer is designed for the nonlinear time-invariant system. The measured output estimate is generated based on the state estimate of the previous time step. ;

[0012] At each sampling time, acquire the measurement output of the nonlinear time-invariant system. and known external input vector According to the measurement output Compared with the measured output estimated value The error between the two values ​​drives the high-gain state observer to perform an update, obtaining the estimated state vector value at the current time. ;

[0013] Construct an auxiliary filter to estimate the state vector. Filtering is performed to generate a filtered output; based on the filtered output, a regression vector for parameter estimation is constructed. ;

[0014] Based on the regression vector, design auxiliary matrices and auxiliary vectors; based on the auxiliary matrices and auxiliary vectors, design parameter adaptive laws.

[0015] At each sampling time, the parameter adaptation law is integrated to obtain the estimated value of the unknown parameter vector at the current time. .

[0016] Furthermore, the adaptive nonlinear time-invariant system state and parameter joint estimation method based on time scale separation includes:

[0017] S1. Establish a second-order nonlinear time-invariant system with unknown states and parameters:

[0018] ;

[0019] in, The state vector of a nonlinear time-invariant system; For measurement output of a nonlinear time-invariant system; Given a known external input vector; It is a regression matrix; Let be the vector of unknown parameters to be estimated; For the system matrix; This is the output matrix;

[0020] S2. Design a high-gain state observer for a nonlinear time-invariant system to generate a measurement output estimate based on the state estimate from the previous time step:

[0021] ;

[0022] in, This is the estimated value of the state vector. For the parameter vector estimate, The regression matrix is ​​calculated using the state vector estimates. The gain matrix of a high-gain state observer is designed based on the separation of gain parameters and design constants according to a set time scale. To measure and output an estimated value;

[0023] S3. At each sampling time, acquire the measurement output of the nonlinear time-invariant system. and known external input vector According to the measurement output Compared with the measured output estimated value The error between the two values ​​drives the high-gain state observer to perform an update, obtaining the estimated state vector value at the current time. ;

[0024] S4. Construct an auxiliary filter to estimate the state vector. Filtering is performed to generate a filtered output; based on the filtered output, a regression vector for parameter estimation is constructed. ;

[0025] S5. Regression vector As input, and by introducing a forgetting factor Design auxiliary matrix and auxiliary vector ;

[0026] S6. Based on the auxiliary matrix and auxiliary vector The adaptive law for design parameters is:

[0027] ;

[0028] in, This is a preset adaptive gain matrix;

[0029] S7. Simultaneously adjust both sides of the adaptive law with respect to time. Integrating yields the vector of unknown parameters. The estimated value .

[0030] Furthermore, the high-gain state observer gain matrix designed based on the set time scale separation gain parameters and design constants... as follows:

[0031] ;

[0032] in, For time-scale separation gain parameters, For scaling matrices, To presuppose solutions to the Lyapunov equations, This is a design constant.

[0033] Furthermore, the preset Lyapunov equation is:

[0034] ;

[0035] Solving for: ;

[0036] That is, the gain matrix of the high-gain state observer. for:

[0037] .

[0038] Furthermore, S4 specifically involves: introducing a set of first-order low-pass filters with a set of time constants as auxiliary filters to estimate the state vector value. The regression matrix obtained using the state vector estimates and For filtering, the auxiliary filter is:

[0039] ;

[0040] in, It is a time constant. State vector estimate The filtered output, The regression matrix calculated using the state vector estimates The filtered output, for Filtered output;

[0041] Based on the filtered output, the regression vector used for parameter estimation is defined as follows:

[0042] .

[0043] Furthermore, the auxiliary matrix and auxiliary vector The design is as follows:

[0044] ;

[0045] in, The regression matrix calculated using the state vector estimates The filtered output.

[0046] According to a second aspect of the present invention, an adaptive nonlinear time-invariant system state and parameter joint estimation system based on time scale separation is provided, comprising the module of the adaptive nonlinear time-invariant system state and parameter joint estimation method based on time scale separation described in any one of the above.

[0047] According to a third aspect of the present invention, a processor is provided for performing operations including the step of performing the adaptive nonlinear time-invariant system state and parameter joint estimation method based on time scale separation as described above.

[0048] The beneficial effects of this invention are:

[0049] Structural decoupling: This invention constructs a hierarchical observer framework for nonlinear time-invariant systems with unknown states and parameters, consisting of a high-gain state observer (fast subsystem) and a parameter adaptive law (slow subsystem). The fast subsystem is responsible for rapidly tracking state changes, while the slow subsystem is responsible for smoothing convergent parameter estimation. The fast subsystem is decoupled by introducing an adjustable high-gain parameter. To design the gain matrix of a high-gain state observer This improves the convergence speed of the state estimation error. Much faster than the convergence speed of parameter estimation errors This achieves an explicit separation between state dynamics and parameter dynamics. The update operations of both are independent. Furthermore, since the parameter adaptive law is based on... The construction of auxiliary filters is completely unaffected by the high gain parameters in the fast subsystem. Directly driven by the slow subsystem, measurement noise and unmodeled dynamics are isolated from the slow subsystem. Even under strong noise or unmodeled dynamic conditions, parameter estimation can still maintain smooth convergence, effectively suppressing the violent oscillation phenomenon commonly found in traditional coupled designs.

[0050] Reducing the difficulty of parameter tuning and computational load: Gain matrix of high-gain state observer Based on Offline pre-calculation and maintenance of constant parameters. During online runtime, the fast and slow subsystems can be independently tuned. Controlling the convergence speed of the state The convergence speed and robustness of control parameters are similar, and they do not interfere with each other. Furthermore, unlike standard adaptive high-gain observers, the parameter update process of this invention does not require real-time solution of dynamic matrix equations, resulting in low computational load and suitability for embedded real-time applications. Attached Figure Description

[0051] Figure 1 This is a schematic diagram of the present invention.

[0052] Figure 2 This is a comparison chart of the state vector estimates of the present invention and the comparison method.

[0053] Figure 3 This is a comparison chart of the parameter vector estimates of the present invention and the comparative method. Detailed Implementation

[0054] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention. It should be noted that, unless otherwise specified, the embodiments and features in the embodiments of this application can be arbitrarily combined with each other.

[0055] Example 1: As Figures 1-3 As shown, according to a first aspect of the present invention, an adaptive method for joint estimation of state and parameters of a nonlinear time-invariant system based on time-scale separation is provided, comprising: establishing a nonlinear time-invariant system containing unknown states and parameters for a controlled object; designing a high-gain state observer for the nonlinear time-invariant system based on a set time-scale separation gain parameter and design constant; and generating a measurement output estimate based on the state estimate of the previous time step. At each sampling time, acquire the measurement output of the nonlinear time-invariant system. and known external input vector According to the measurement output Compared with the measured output estimated value The error between them (error is the difference between the two) The high-gain state observer is driven to perform an update to obtain the estimated state vector value at the current time. Construct an auxiliary filter to estimate the state vector. Filtering is performed to generate a filtered output; based on the filtered output, a regression vector for parameter estimation is constructed. Based on the regression vector, design auxiliary matrices and auxiliary vectors; based on the auxiliary matrices and auxiliary vectors, design parameter adaptive laws; at each sampling time, integrate the parameter adaptive laws to obtain the estimated value of the unknown parameter vector at the current time. .

[0056] Furthermore, the adaptive nonlinear time-invariant system state and parameter joint estimation method based on time scale separation includes:

[0057] Step 1: Establish a second-order nonlinear time-invariant system with unknown states and parameters:

[0058] ;

[0059] in, The state vector of a nonlinear time-invariant system; For measurement output of a nonlinear time-invariant system; Given a known external input vector; It is a non-linear function matrix (i.e., a regression matrix); The unknown parameter vector to be estimated; the system matrix. Output matrix .

[0060] Step 2: Design a high-gain state observer for the nonlinear time-invariant system to achieve fast tracking of unmeasurable states of the system. The high-gain state observer design is as follows:

[0061] ;

[0062] in, For the system matrix, For the output matrix, This is the estimated value of the state vector. The parameter vector estimate is calculated by the slow subsystem described later, and the high-gain state observer treats it as a known value. The regression matrix is ​​calculated using the state vector estimates. Given a known external input vector, Here is the gain matrix of the high-gain state observer to be designed. To measure and output an estimated value.

[0063] Step 3: To force the state estimation error to converge quickly and to achieve time-scale separation from the slowly varying parameter dynamics, the gain parameter and design constant are separated according to the set time scale, and the high-gain state observer gain matrix is ​​designed. :

[0064] ;

[0065] in, For time-scale separation gain parameters, For scaling matrices, To presuppose solutions to the Lyapunov equations, For design constants, The output matrix is ​​T, which represents the transpose.

[0066] As can be seen from the above, the gain matrix L of this high-gain state observer explicitly depends on This makes the convergence speed of the high-gain state observer proportional to... .

[0067] Furthermore, the preset Lyapunov equation is:

[0068]

[0069] Solving for: ;

[0070] That is, the gain matrix of the high-gain state observer. for:

[0071] .

[0072] Step 4: At each sampling time, acquire the measurement output of the nonlinear time-invariant system. and known external input vector According to the measurement output Compared with the measured output estimated value The error between the two values ​​drives the high-gain state observer to perform an update, obtaining the estimated state vector value at the current time. ;

[0073] Step 5: Construct an auxiliary filter to estimate the state vector. Filtering is performed to generate a filtered output; based on the filtered output, a regression vector for parameter estimation is constructed. ;

[0074] To eliminate the impact of parameter updates on the unmeasurable true state vector Due to the dependence on [the specific parameters], this step designs a set of auxiliary filters, utilizing measurable state vector estimates. To construct a computable regression vector.

[0075] We first consider an ideal filter based on real-world conditions to provide a theoretical benchmark for subsequent practically implementable filters. A time constant is introduced. A first-order low-pass filter. Define the ideal filter output based on the real-state conditions. The following conditions must be met:

[0076] ;

[0077] in, Right now , , The ideal filtered output.

[0078] Transforming the above equation, we obtain the linear regression equation under ideal conditions: But the real situation It is unpredictable and cannot be directly realized.

[0079] Therefore, the real-time output from step 2 is used. Replace the real state To construct a practically computable filtered output, a set of first-order low-pass filters with time constants is introduced as auxiliary filters to estimate the state vector. The regression matrix obtained using the state vector estimates and For filtering, the auxiliary filter is:

[0080] ;

[0081] in, It is a time constant. State vector estimate The filtered output, The regression matrix calculated using the state vector estimates The filtered output, for The filtered output.

[0082] Based on the filtered output, the regression vector used for parameter estimation is defined as follows:

[0083] .

[0084] Step 6: To estimate parameters online and effectively suppress the effects of noise and unmodeled dynamics, the regression vector constructed in Step 5 is... As input, a forgetting factor is introduced. And design the following auxiliary matrix. and auxiliary vector :

[0085] ;

[0086] in, Let be a symmetric positive semi-definite auxiliary matrix, with initial values... ; For auxiliary vectors, initial values ; The dimension of the unknown parameter vector; forgetting factor Its physical function is to gradually "forget" outdated data during system operation, enabling the algorithm to effectively track parameters while ensuring... It will not grow indefinitely, thus maintaining numerical stability.

[0087] Step 7: Based on the auxiliary matrix and auxiliary vector The adaptive law for design parameters is:

[0088] ;

[0089] in, The preset adaptive gain matrix, For the parameter vector estimate, for The first derivative of the parameter. The adaptive law of this parameter is mainly determined by the filter time constant. and forgetting factor The convergence rate of the corresponding parameter estimation error is approximately [missing information]. Therefore, by selecting a sufficiently large The convergence speed of the state estimation error can be significantly faster than that of the parameter estimation error, thus achieving an explicit time scale separation between the two: the state error dynamically constitutes a fast subsystem, while the parameter error dynamically constitutes a slow subsystem.

[0090] Step 8: Simultaneously adjust both sides of the adaptive law of parameters in Step 7 with respect to time. Integrating yields the vector of unknown parameters. The estimated value Furthermore, the obtained state vector estimate and the unknown parameter vector estimate are synchronously input to an external actuator with a controller; the actuator can adjust the control command based on the above joint estimation results, thereby applying closed-loop control to the controlled object.

[0091] The method of this invention enables real-time acquisition of the measurement output of the controlled object. and the known external input vector Then, these are simultaneously input into the high-gain state observer of step 2, the auxiliary filter of step 5, and the parameter adaptive law of step 7. The high-gain state observer uses... The fast convergence and output of the state vector estimate. The parameter adaptive law is relatively slow. Rate convergence and output parameter vector estimates The two updates independently, achieving "explicit time-scale separation." Ultimately, a real-time joint estimate of the system's unknown state and unknown parameters is obtained. The online operation logic is as follows:

[0092] Initialization: Offline preset parameters , , , , Pre-calculate the constant gain matrix. Set the initial state vector estimate. Initial parameter vector estimate Initial values ​​of auxiliary variables and and the initial value of the filtered output .

[0093] Fast subsystem: Reading sensor output and input Only the high-gain state observer equations from step 2 are executed. This process only involves... Multiplication and addition operations on matrices and vectors have extremely low computational cost.

[0094] Slow subsystem: Run filter bank update first Then calculate the regression vector. Next, update the auxiliary matrix. and auxiliary vector Finally, perform adaptive parameter updates. .

[0095] Two-way data interaction: The fast subsystem needs to use the latest calculations from the slow subsystem. (Because the parameters change slowly, this unidirectional dependency does not compromise stability); the slow subsystem needs to use the latest output of the fast subsystem. .

[0096] For illustrative purposes, the convergence rates of state estimation error and parameter estimation error are described below:

[0097] Define state estimation error Its dynamic equation can be written as:

[0098] ;

[0099] In the formula, For parameter estimation error, This is a bounded perturbation term determined by the nonlinear function and the true values ​​of its parameters. The Lipschitz continuity has a constant. Make:

[0100] ;

[0101] The error dynamic matrix of the fast subsystem is:

[0102] ;

[0103] Its eigenvalues Satisfy the characteristic equation .because and The system matrix is ​​Hurwitz, and the real parts of its dominant eigenvalues ​​satisfy... (in To and (Irrelevant constants). Therefore, the state estimation error homogeneous system The solution satisfies:

[0104]

[0105] in, Let be the condition number. The above equation shows that the state estimation error... The convergence process is exponential, and the convergence rate is 1 / 2. .

[0106] The adaptive law of parameters is mainly determined by the filter time constant. and forgetting factor The convergence rate of the corresponding parameter estimation error is approximately [missing information]. Therefore, by selecting a sufficiently large The convergence speed of the state estimation error can be significantly faster than that of the parameter estimation error, thus achieving an explicit time scale separation between the two: the state error dynamically constitutes a fast subsystem, while the parameter error dynamically constitutes a slow subsystem.

[0107] The following stability analysis is conducted from the perspectives of fast and slow subsystems:

[0108] This invention employs singular perturbation theory as the core tool for stability analysis. The core idea of ​​this theory is that if a system consists of two subsystems with significantly different time scales (one fast and one slow), they can be analyzed separately first, and then the results can be combined. As long as each subsystem is stable when separated, the whole system will also be stable when combined.

[0109] The tachy subsystem of this invention is a high-gain state observer that operates at a rate of High-speed convergence; the slow subsystem uses a parameter adaptive law, which converges at a certain rate. The convergence is relatively slow. Since the rates of convergence differ by at least an order of magnitude, the prerequisites for applying singular perturbation theory are met. The stability of the two subsystems is then mathematically proven separately.

[0110] 1. Proof of the exponential stability of the fast subsystem (state estimation error)

[0111] The dynamic equation for the state estimation error described above is: On a rapidly changing timescale The state error is dynamically converted into:

[0112] ;

[0113] The boundary layer system matrix is:

[0114] ;

[0115] when When the root of the characteristic equation has a negative real part, i.e., the boundary layer system matrix... It's Hurwitz's work. Therefore, boundary layer systems on rapidly changing timescales... The following is exponentially stable. According to singular perturbation theory, this means that on the original timescale... Below, state estimation error will be The rate exponential convergence of the rate to its quasi-steady state.

[0116] Quasi-steady state of state estimation error Defined as the equilibrium point of the boundary layer system:

[0117] ;

[0118] The analytical expression for quasi-steady state is:

[0119] ;

[0120] in, ;

[0121] Regression Matrix and its input vector The parameter vector estimates of a globally bounded, nonlinear time-invariant system. It is also bounded. Therefore, there exists a constant. , making That is, there exists a constant. , making There exists a constant. This allows for the combination of Lipschitz continuity, resulting in .

[0122] The quasi-steady-state expression is:

[0123] ;

[0124] Therefore, a constant exists. This ensures that the quasi-steady state satisfies:

[0125] ;

[0126] Right now ;

[0127] in, This result shows that the state estimation error is limited to a value relative to the observer gain. Within the relevant small neighborhood, i.e., the proof The boundedness converges exponentially to Magnitude.

[0128] 2. Proof of Globally Consistent Final Boundedness for Slow Subsystems (Parameter Estimation)

[0129] The error between the actual filter output and the ideal filter output is defined as:

[0130] The difference between the actual state filter output value and the ideal state filter output value;

[0131] : The difference between the actual regression matrix filter output and the ideal regression matrix filter output;

[0132] : Actual filter output and The difference between the ideal filter output and the output.

[0133] Subtracting the ideal filter equation from the actual filter equation yields the filter error dynamics:

[0134] ;

[0135] First, the regression equation under ideal conditions (based on the real conditions) is:

[0136] ;

[0137] The actual regression vector Subtract the above formula and use the defined filter error. Substituting, we get:

[0138] ;

[0139] Will Substituting into the above equation and rearranging, we get:

[0140] ;

[0141] Among them, the bounded reconstruction error The definition of is:

[0142] ;

[0143] Since it has been proven The boundedness converges exponentially to The magnitude of the error, therefore the filter error It also converges exponentially, and its steady-state limit is... Therefore, there exists a constant. , so that:

[0144] ;

[0145] This invention introduces an auxiliary matrix and auxiliary vector And the parameter estimation error is defined. To prove that the parameter estimation error converges, an auxiliary variable residual is constructed. Then, its dynamic equation is derived.

[0146] right Differentiate and combine auxiliary variables and Substitute:

[0147] ;

[0148] The adaptive law of parameters is Convert it to... and The error dynamic equation of the slow subsystem is expressed as:

[0149] ;

[0150] This is a pair of state variables ( and A linear system that is coupled together, whose input is a bounded perturbation. The next task is to prove that this system is stable. The basic idea of ​​the Lyapunov method is: if we can find an "energy function"... It is always positive, and its derivative is... If the parameter error is always negative (or controllably negative), then the system tends towards an equilibrium point (i.e., the parameter error tends towards 0). Below, we construct such a function.

[0151] Constructing Lyapunov functions:

[0152] ;

[0153] in: , is an adjustable constant used to balance the two state variables in Weights in;

[0154] , is a constant guaranteed by the continuous excitation condition, satisfying ; It is the identity matrix;

[0155] , which is the upper bound of the regression matrix filter output, satisfying .

[0156] right Differentiate:

[0157] Substituting the dynamic equation of the slow subsystem error into the equation and differentiating it over time:

[0158] ;

[0159] Using Young's inequality, the form can be bounded:

[0160] ;

[0161] ;

[0162] Substituting these two inequalities into In the expression:

[0163] ;

[0164] make: ;

[0165] The above equation simplifies to:

[0166] ;

[0167] Depend on Definition ( From this, we can know that:

[0168] ;

[0169] in, express The largest eigenvalue.

[0170] Therefore, there exists a constant:

[0171] ;

[0172] Make:

[0173] ;

[0174] Right now:

[0175] ;

[0176] make Substitute Combination :

[0177] ;

[0178] make The final result is:

[0179] ;

[0180] According to the comparison lemma, the solution to the differential inequality satisfies:

[0181] ;

[0182] Take the limit :

[0183] ;

[0184] This indicates that the slow subsystem is globally consistent and eventually bounded.

[0185] 3. Overall system stability

[0186] Combining the analysis of fast and slow subsystems and applying singular perturbation theory: if the fast subsystem is exponentially stable and the slow subsystem is globally uniform and eventually bounded, then the original coupled system... It is also globally consistent and ultimately bounded.

[0187] According to a second aspect of the present invention, an adaptive nonlinear time-invariant system state and parameter joint estimation system based on time scale separation is provided, comprising the module of the adaptive nonlinear time-invariant system state and parameter joint estimation method based on time scale separation described above. Specifically, the system includes: a setup module for establishing a nonlinear time-invariant system containing unknown states and parameters; a first design module for separating gain parameters and design constants according to a set time scale, designing a high-gain state observer for the nonlinear time-invariant system, and generating a measurement output estimate based on the state estimate from the previous time step; a first acquisition module for acquiring the measurement output and a known external input vector of the nonlinear time-invariant system at each sampling time step; driving the high-gain state observer to perform an update based on the error between the measurement output and the measurement output estimate, thereby obtaining the state vector estimate at the current time step; a construction module for constructing an auxiliary filter to filter the state vector estimate and generate a filtered output; and constructing a regression vector for parameter estimation based on the filtered output; a second design module for designing an auxiliary matrix and auxiliary vector based on the regression vector; and designing a parameter adaptive law based on the auxiliary matrix and auxiliary vector; and a second acquisition module for integrating the parameter adaptive law at each sampling time step to obtain the parameter vector estimate at the current time step. For details not described in the above modules, please refer to the relevant descriptions in this embodiment.

[0188] According to a third aspect of the present invention, a processor is provided for performing operations, the operations including the step of performing the adaptive nonlinear time-invariant system state and parameter joint estimation method based on time scale separation as described above.

[0189] Example 2: To verify the effectiveness and robustness of the joint estimation method described in this invention, an optional implementation of this invention will be described in detail below.

[0190] This embodiment uses the pitching degree-of-freedom dynamic model of a hybrid wind and wave energy system as the specific controlled physical object (controlled object) for testing. In practice, the system can be built based on a co-simulation platform of ANSYS / AQWA solver and MATLAB / Simulink. External inputs to the system (such as wave environment excitation forces, relative angular velocities, etc.) are calculated in real time by the dynamic link library (DLL) of the AQWA module and input into the main control program. Because the AQWA model contains complex hydrodynamic characteristics, this introduces real structural model mismatch and unmodeled dynamic disturbances into the testing of the algorithm of this invention.

[0191] Step 1: Establish a dynamic model of the pitching degree of freedom of the hybrid wind and wave energy system, and rewrite the dynamic model as a nonlinear time-invariant system with unknown states and parameters;

[0192] The pitching degree-of-freedom dynamic model of the hybrid wind and wave energy system can be described as a second-order mass-damped-stiffness system:

[0193] ;

[0194] in, The pitch angle (rad) of the hybrid wind and wave energy system. The pitch angular velocity (rad / s) represents the unmeasurable internal state. It is the pitch acceleration; The moment of inertia of the hybrid wind and wave energy system (kg·m²) is given. The damping coefficient (N·m·s / rad) of the hybrid wind and wave energy system. The stiffness coefficient (N·m / rad) of the hybrid wind and wave energy system. , The damping coefficients are for the two power outputs PTO1 and PTO3. All of the above parameters are unknown and need to be estimated. Given the known external wave environment load (N·m), it can be calculated in real time and input by an external wave solver (such as ANSYS / AQWA); , The control torque applied to the two power outputs of the hybrid wind and wave energy system. The relative angular velocity input between the power output PTO1 and the hybrid wind and wave energy system is... The relative angular velocity input between the power output PTO3 and the hybrid wind and wave energy system can be calculated in real time by the AQWA module dynamic link library (DLL).

[0195] Based on the pitching degree-of-freedom dynamic model of the hybrid wind and wave energy system, the state vector is defined. (in, (Unknown state), measurement output Given the external input vector unknown parameter vector ;

[0196] Therefore, the second-order mass-damped-stiffness system can be rewritten as a second-order nonlinear time-invariant system with unknown states and parameters as follows:

[0197] ;

[0198] in: ; ; For ease of description, functions relating to time variables can be written in a simplified form, such as... Written as , Written as .

[0199] In this embodiment, the true value of the corresponding parameter vector is:

[0200] .

[0201] Step 2: Design a high-gain state observer for the nonlinear time-invariant system as a fast subsystem;

[0202] The high-gain state observer is designed as follows:

[0203] ;

[0204] in, For the system matrix, For the output matrix, This is the estimated value of the state vector. For the parameter vector estimate, The regression matrix is ​​calculated using the state vector estimates. Given a known external input vector, Here is the gain matrix of the high-gain state observer to be designed. To measure and output an estimated value.

[0205] Step 3: Separate the gain parameters according to the set time scale. and design constants Design the gain matrix of a high-gain state observer. :

[0206] ;

[0207] in, For high-gain parameters on the time scale, For scaling matrices, To presuppose the Lyapunov equation The solution, , This is a design constant.

[0208] Step 4: At each sampling time, acquire the measurement output of the nonlinear time-invariant system. and known external input vector According to the measurement output Compared with the measured output estimated value The error between the two values ​​drives the high-gain state observer to perform an update, obtaining the estimated state vector value at the current time. ;

[0209] Step 5: Construct an auxiliary filter and generate the filtered output; based on the filtered output, construct a regression vector for parameter estimation. ;

[0210] Introduce a set of time constants A first-order low-pass filter is used as an auxiliary filter to estimate the state vector. The regression matrix obtained using the state vector estimates and For filtering, the auxiliary filter is:

[0211] ;

[0212] in, It is a time constant. State vector estimate The filtered output, for The first derivative, The regression matrix calculated using the state vector estimates The filtered output, for The filtered output.

[0213] Define the regression vector based on the filtered output. :

[0214] ;

[0215] Step 6: Design auxiliary matrices and auxiliary vectors

[0216] Introducing the forgetting factor Design auxiliary matrix and auxiliary vector :

[0217] ;

[0218] in, for 3D auxiliary matrix, for 3D auxiliary vector, Forgetting factor, For the regression matrix The transpose of the filtered output.

[0219] Step 7: Based on the auxiliary matrix and auxiliary vector Design the following adaptive law with parameters:

[0220] ;

[0221] Wherein, the adaptive gain matrix , for 3D auxiliary matrix, for 3D auxiliary vector.

[0222] Step 8: Simultaneously adjust both sides of the adaptive law of parameters in Step 7 with respect to time. Integrating yields the vector of unknown parameters. The estimated value .

[0223] Step 9: Online Joint Estimation Implementation Process

[0224] (1) Offline initialization: default , , , , Pre-calculate the gain matrix of the high-gain state observer. Set initial values ​​for the state vector estimate. Initial values ​​for parameter vector estimation initial value of auxiliary matrix initial value of auxiliary vector Filter output initial value , , .

[0225] (2) At each sampling time, obtain the measured value output by the AQWA solver. and external input vector .

[0226] (3) Perform fast subsystem update: Calculate according to the following equation Update the state vector estimate .

[0227]

[0228] (4) Perform slow subsystem update: Update the filter output sequentially according to the following equations. Calculate the regression vector Update the auxiliary matrix and auxiliary vector Finally, update the parameter estimates. .

[0229]

[0230]

[0231]

[0232]

[0233] (5) Data interaction exists between the fast subsystem and the slow subsystem: the estimated state vector of the fast subsystem The parameters are passed to the slow subsystem for filter updates, thus updating the filter output; the parameter vector of the slow subsystem is estimated. Passed to the fast subsystem for regression matrix The calculation.

[0234] Step 10: Simulation Verification and Comparative Analysis

[0235] To verify the effectiveness of this invention, a co-simulation platform based on ANSYS / AQWA and MATLAB / Simulink can be built. The system measurement output comes directly from the high-fidelity AQWA simulation without adding artificial sensor noise. The AQWA model contains complex fluid dynamics (such as frequency-dependent radiation and diffraction forces). These unmodeled dynamics are not captured by the simplified second-order observer model, thus naturally introducing structural model mismatch and providing strict conditions for testing robustness.

[0236] The Standard Adaptive High-Gain Observer (SAHGO) was selected as the baseline for comparison. This observer employs a dynamic gain coupling design based on the Riccati equation. Both methods are configured with the same gain parameters. Use the same initial conditions to ensure a fair comparison.

[0237] The simulation results are as follows: the black solid line True represents the true value, the blue dashed line SAHGO represents the estimated value of the standard adaptive high-gain observer method, and the red dashed line Proposed represents the estimated value of the proposed time-separated scale joint state parameter estimation method. Figure 2The figures represent the state estimates from two methods (True represents the true value, SAHGO represents the standard adaptive high-gain observer, and Proposed represents the present invention; the estimation results of both methods are in high agreement with the true values, and the curves largely overlap in the figure). Figure 2 It can be seen that both methods can accurately track the actual pitch state changes of the system. The method described in this invention achieves convergence speed and tracking accuracy comparable to the standard adaptive high-gain observer method. Figure 3 The figure shows the parameter estimates of the two methods. It can be seen from the figure that the parameter estimation method described in this invention has a better convergence speed and tracking accuracy than the standard adaptive high-gain observer method, indicating high stability of parameter estimation. As can be seen from the above, this invention, while possessing comparable state estimation convergence speed and tracking accuracy, achieves even faster parameter estimation convergence speed and better tracking accuracy without compromising performance between the two results, thus overcoming the bottleneck of the standard method where it is difficult to simultaneously achieve both performance advantages.

[0238] The specific embodiments of the present invention have been described in detail above with reference to the accompanying drawings. However, the present invention is not limited to the above embodiments. Within the scope of knowledge possessed by those skilled in the art, various changes can be made without departing from the spirit of the present invention.

Claims

1. An adaptive method for joint estimation of state and parameters of a nonlinear time-invariant system based on time scale separation, characterized in that, include: Establish a nonlinear time-invariant system with unknown states and parameters; Based on the set time scale, separate the gain parameters and design constants, design a high-gain state observer for the nonlinear time-invariant system, and generate a measurement output estimate based on the state estimate of the previous time step. At each sampling time, the measurement output of the nonlinear time-invariant system and the known external input vector are acquired; based on the error between the measurement output and the estimated measurement output, the high-gain state observer is driven to perform an update to obtain the estimated state vector at the current time. Construct an auxiliary filter to filter the state vector estimate and generate a filtered output. Based on the filtered output, a regression vector for parameter estimation is constructed; Based on the regression vector, design auxiliary matrices and auxiliary vectors; based on the auxiliary matrices and auxiliary vectors, design parameter adaptive laws. At each sampling time, the parameter adaptive law is integrated to obtain the parameter vector estimate for the current time.

2. The adaptive nonlinear time-invariant system state and parameter joint estimation method based on time scale separation according to claim 1, characterized in that, The method specifically includes: S1. Establish a second-order nonlinear time-invariant system with unknown states and parameters: ； in, The state vector of a nonlinear time-invariant system; For measurement output of a nonlinear time-invariant system; Given a known external input vector; It is a regression matrix; Let be the vector of unknown parameters to be estimated; For the system matrix; This is the output matrix; S2. Design a high-gain state observer for a nonlinear time-invariant system to generate a measurement output estimate based on the state estimate from the previous time step: ； in, This is the estimated value of the state vector. For the parameter vector estimate, The regression matrix is ​​calculated using the state vector estimates. The gain matrix of a high-gain state observer is designed based on the separation of gain parameters and design constants according to a set time scale. To measure and output an estimated value; S3. At each sampling time, acquire the measurement output of the nonlinear time-invariant system. and known external input vector According to the measurement output Compared with the measured output estimated value The error between the two values ​​drives the high-gain state observer to perform an update, obtaining the estimated state vector value at the current time. ; S4. Construct an auxiliary filter to estimate the state vector. Filtering is performed to generate a filtered output; based on the filtered output, a regression vector for parameter estimation is constructed. ; S5. Regression vector As input, and by introducing a forgetting factor Design auxiliary matrix and auxiliary vector ; S6. Based on the auxiliary matrix and auxiliary vector The adaptive law for design parameters is: ； in, This is a preset adaptive gain matrix; S7. Simultaneously adjust both sides of the adaptive law with respect to time. Integrating yields the vector of unknown parameters. The estimated value .

3. The adaptive nonlinear time-invariant system state and parameter joint estimation method based on time scale separation according to claim 2, characterized in that, The high-gain state observer gain matrix designed based on the set time scale, separation gain parameters, and design constants. as follows: ； in, For time-scale separation gain parameters, For scaling matrices, To presuppose solutions to the Lyapunov equations, This is a design constant.

4. The adaptive nonlinear time-invariant system state and parameter joint estimation method based on time scale separation according to claim 3, characterized in that, The pre-defined Lyapunov equation is: ； Solving for: ; That is, the gain matrix of the high-gain state observer. for: 。 5. The adaptive nonlinear time-invariant system state and parameter joint estimation method based on time scale separation according to claim 2, characterized in that, Specifically, S4 involves introducing a set of first-order low-pass filters with time constants as auxiliary filters to estimate the state vector. The regression matrix obtained using the state vector estimates and For filtering, the auxiliary filter is: ； in, It is a time constant. State vector estimate The filtered output, The regression matrix calculated using the state vector estimates The filtered output, for Filtered output; Based on the filtered output, the regression vector used for parameter estimation is defined as follows: 。 6. The adaptive nonlinear time-invariant system state and parameter joint estimation method based on time scale separation according to claim 1, characterized in that, The auxiliary matrix and auxiliary vector The design is as follows: ； in, The regression matrix calculated using the state vector estimates The filtered output, Forgetting factor, This is the regression vector.

7. An adaptive nonlinear time-invariant system state and parameter joint estimation system based on time scale separation, characterized in that, The module includes the adaptive nonlinear time-invariant system state and parameter joint estimation method based on time scale separation as described in any one of claims 1-6.

8. A processor for performing operations, characterized in that, The operation includes the step of performing the adaptive nonlinear time-invariant system state and parameter joint estimation method based on time scale separation as described in any one of claims 1-6.

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