CN118068686A - Single-connecting-rod mechanical arm periodic fault estimation method based on iterative learning - Google Patents
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Abstract
The invention relates to the technical field of periodic fault estimation, solves the technical problem that the periodic fault is difficult to estimate by the traditional fault estimation method, in particular to a single-connecting-rod mechanical arm periodic fault estimation method based on iterative learning, which comprises the following steps: introducing a nonlinear autotransformer into the structural design process of the state observer, and performing fault estimation by adopting an embedded PD type recursive autotransformer scheme to form a fault estimation observer for effectively estimating the periodic faults of the single-link mechanical arm; proving the final boundedness of fault estimation errors by using a recursive analysis method and a robust control theory; a learning gain matrix is designed that ensures that the fault estimation error converges to a preset boundary. According to the invention, the iterative learning algorithm and the observer theory are combined to construct the fault estimation observer, so that the effective estimation of the periodic faults of the single-link mechanical arm is realized, and the problem that the periodic faults are difficult to estimate when the actuator faults exist in the single-link mechanical arm system is solved.
Description
Technical Field
The invention relates to the technical field of periodic fault estimation, in particular to a single-link mechanical arm periodic fault estimation method based on iterative learning.
Background
In actual industrial production, single-link mechanical arms are widely applied to various fields such as industrial manufacturing, medical treatment, aerospace, automobile manufacturing and the like due to unique high-precision and high-speed operation capability. However, due to the design structure of the single-link mechanical arm, the probability of failure of the system after multiple operations is greatly increased, and accidents such as economic loss and personnel safety hazard are easily caused. In recent years, with the remarkable improvement of the industrial automation level and the wide application of advanced control systems, the demands for monitoring the health condition of the system in real time and ensuring high reliability and high safety of the system are increasing. In this context, studies of fault estimation are receiving a great deal of attention.
Most industrial systems operate in a batch-to-batch manner, periodically performing the same tasks, such as robotic arms that grasp objects in the process industry, conveyor belts that repeatedly run items, numerically controlled machine tools, and the like. The dynamic characteristics of the periodic system faults are complex and difficult to model, and the traditional method cannot be used for fault estimation. Currently, the research of the estimation method aiming at periodic faults is still in a preliminary stage, a good tracking estimation effect is difficult to achieve, and the fault estimation method is concentrated on a deterministic system. Meanwhile, the long-time operation of the repetitive system is affected by environmental factors such as disturbance and noise, which requires deep analysis of the characteristics of faults in the repetitive system and the influence thereof on the operation performance of the system.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides a single-connecting-rod mechanical arm periodic fault estimation method based on iterative learning, which solves the technical problem that the periodic faults are difficult to estimate by the traditional fault estimation method.
In order to solve the technical problems, the invention provides the following technical scheme: a periodic fault estimation method of a single-link mechanical arm based on iterative learning comprises the following steps:
S1, introducing a nonlinear autotransformer into the structural design process of a state observer, and performing fault estimation by adopting an embedded PD type recursive autotransformer scheme to form a fault estimation observer for effectively estimating the periodic faults of the single-link mechanical arm;
S2, proving the final bouncy of a fault estimation error by using a recursive analysis method and a robust control theory;
S3, designing a learning gain matrix for ensuring that the fault estimation error converges to a preset boundary;
S4, reasonably selecting parameters which ensure final bounded fault estimation errors in the fault estimation observer, and verifying the actual effect of the periodic fault estimation method of the single-link mechanical arm by utilizing numerical simulation.
Further, in step S1, the fault estimation observer is composed of an iterative learning observer and an embedded PD-type fault estimator.
Further, in step S1, the specific process includes the following steps:
S11, constructing a dynamic model of the single-link mechanical arm system, wherein the expression of the model is as follows:
；
；
In the above-mentioned method, the step of, Is the angular position of the mechanical arm; /(I)Is the rotational inertia of the joint; /(I)Is joint torque; /(I)Is an exogenous disturbance torque; /(I)Is the mass of the mechanical arm; /(I)The weight of the tip load of the mechanical arm; /(I)Is the length of the mechanical arm; /(I)Gravitational acceleration;
s12, carrying out discretization analysis on a dynamic model of the single-link mechanical arm system to obtain a discrete state space model, wherein the discrete state space model is as follows:
；
In the above-mentioned method, the step of, Is a discrete time; /(I)；
S13, constructing a mechanical arm nonlinear state variable power equation of a single-link mechanical arm system based on a discrete state space model to form a nonlinear time-varying system model considering bounded disturbance and measurement noise, wherein the expression of the nonlinear time-varying system model is as follows:
；
In the above-mentioned method, the step of, Is a state variable; /(I)Is a control input; /(I)For measuring output; for periodic actuator failure, i.e./> At each time interval/>The same periodic dynamic performance is achieved; /(I)Is a measurement matrix; /(I)Is a fault distribution matrix; /(I)And/>Disturbance and measurement noise, respectively; /(I)And/>Are all nonlinear functions meeting Lipschitz conditions;
s14, setting a hypothesis condition satisfied by the nonlinear time-varying system model;
s15, constructing a fault estimation observer consisting of an iterative learning observer and an embedded PD type fault estimator according to the nonlinear time-varying system model and the satisfied hypothesis conditions.
Further, in step S14, the assumption condition is satisfied that:
Suppose 1: operator Bounded, i.e. there is a constant/>So that/>；
Suppose 2: in each iteration, the initialization error is bounded;
Suppose 3: matrix array For all/>Are all non-singular.
Further, in step S15, the expressions of the nonlinear iterative learning observer and the failure estimator are:
；
；
In the above-mentioned method, the step of, Respectively estimating the system state and the output of the single-link mechanical arm system; subscript/>Learning the batch times for iteration; /(I)Virtual faults introduced for a single-link mechanical arm system; /(I)AndAre nonlinear functions; /(I)Is a fault distribution matrix; /(I)Is a measurement matrix; /(I)Is a control input; /(I)For measuring output; /(I)A gain matrix of the observer to be designed; /(I)Run-time for observer/>Outputting an estimation error for the second iteration; /(I)Is the gain matrix to be designed.
Further, in step S2, the specific process includes the following steps:
S21, based on the first Minor/>Time/>, in iterative processState error/>Definition of (2)And the nonlinear iterative learning observer derives the/>Time/>, in a secondary iterative processState error/>Is an expression of (2);
s22, simplifying subsequent operation by using Lipschitz conditions and a method for taking upper bounds of each system matrix, and obtaining the first by using a recursive analysis method Time/>, in a secondary iterative processThe norm of the state error, i.e./>Is a recursive inequality of (2);
s23 based on Norm operation, deducing the/>Time/>, in a secondary iterative processState error and fault estimation errorNorms, i.e./>And/>Inequality relation between them;
S24, based on Derived idea of (1) >, get (1) >Fault estimation error/>, in a secondary iterative processIs a recursive inequality of (2);
s25, introduction of Obtain the/>Minor iteration and/>Error estimation/>, in the secondary iteration processNorms, i.e./>And/>Relationship between them.
Further, in step S3, the learning gain matrix includes a learning gain matrix of the iterative learning observer and the embedded PD-type fault estimator。
Further, a gain matrix is learnedAnd/>The method comprises the following steps of:
；
In the above-mentioned method, the step of, And/>A diagonal matrix to be designed; matrix/>Is a full rank matrixIs the pseudo-inverse of (a); wherein the diagonal matrix/>Diagonal line elementSatisfy/>；
Diagonal matrixDiagonal element/>Satisfy the following requirements。
Based on the technical scheme, the invention provides a single-link mechanical arm periodic fault estimation method based on iterative learning, which has at least the following beneficial effects:
1. according to the invention, the iterative learning algorithm and the observer theory are combined to construct the fault estimation observer, so that the effective estimation of the periodic faults of the single-link mechanical arm is realized, and the problem that the periodic faults are difficult to estimate when the actuator faults exist in the single-link mechanical arm system is solved.
2. The invention solves the technical problem that the periodic faults of the noisy disturbance system are difficult to estimate, and the nonlinear autotransformer embedded with the PD-type recursive autotransformer scheme is adopted to successfully update the fault estimation error through the previous output estimation error and input of the system, so that the invention has the advantages of simple realization, strong expandability and the like.
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The accompanying drawings, which are included to provide a further understanding of the application and are incorporated in and constitute a part of this specification, illustrate embodiments of the application and together with the description serve to explain the application and do not constitute a limitation on the application. In the drawings:
FIG. 1 shows a fault estimation method and a robust class of the present invention A fault estimation result graph of a fault estimation algorithm changing with time;
FIG. 2 shows a fault estimation method and a robust class of the present invention Fault estimation error map of fault estimation algorithm as a function of iteration number.
Detailed Description
In order that the above-recited objects, features and advantages of the present application will become more readily apparent, a more particular description of the application will be rendered by reference to the appended drawings and appended detailed description. Therefore, the realization process of how to apply the technical means to solve the technical problems and achieve the technical effects can be fully understood and implemented.
Referring to fig. 1-2, a specific implementation manner of the present embodiment is shown, and a periodic fault estimation method of a single-link mechanical arm based on iterative learning is provided, in this embodiment, a nonlinear autotransformer is introduced in a structural design process of a state observer, an embedded PD type recursive autotransformer is adopted to perform fault estimation, and a recursive analysis method and a robust control theory are utilized to prove final boundedness of a fault estimation error.
Secondly, in order to ensure that the fault estimation error converges to a preset limit, the invention provides a design scheme of an observer gain matrix. In the specific research process, the selection condition of each gain matrix is analyzed by means of matrix theory.
Finally, reasonably selecting parameters in the iterative learning observer, verifying the actual effect of the periodic fault estimation algorithm based on iterative learning by using numerical simulation, and performing performance comparison research with a classical robust fault estimation algorithm based on an unknown input observer to further highlight the superior performance of the invention.
Based on the simulation verification result, the single-link mechanical arm periodic fault estimation method provided by the embodiment comprises the following steps:
S1, introducing a nonlinear autotransformer into the structural design process of a state observer, and performing fault estimation by adopting an embedded PD type recursive autotransformer scheme to form a fault estimation observer for effectively estimating the periodic faults of the single-link mechanical arm; the fault estimation observer consists of an iterative learning observer and an embedded PD-type fault estimator. In this step, the implementation procedure for step S1 includes the following steps:
S11, constructing a dynamic model of a single-link mechanical arm system; in this embodiment, for the construction of the dynamics model, the corresponding parameter composition is mainly obtained from a known single-link mechanical arm system, or directly obtained from known literature, which is taken as an example for illustration, the following type of single-link mechanical arm system dynamics model is adopted, and the expression of the model is as follows:
；
；
In the above-mentioned method, the step of, Is the angular position of the mechanical arm; /(I)Is the rotational inertia of the joint; /(I)Is joint torque; /(I)Is an exogenous disturbance torque; /(I)Is the mass of the mechanical arm; /(I)The weight of the tip load of the mechanical arm; /(I)Is the length of the mechanical arm; /(I)Gravitational acceleration.
S12, carrying out discretization analysis on a dynamic model of the single-link mechanical arm system to obtain a discrete state space model; by setting a suitable sampling periodAnd discretizing the dynamic model to further obtain a discrete state space model of the single-link mechanical arm system as follows:
；
In the above-mentioned method, the step of, Is a discrete time; /(I)。
S13, constructing a mechanical arm nonlinear state variable power equation of a single-link mechanical arm system based on a discrete state space model, and forming a nonlinear time-varying system model considering bounded disturbance and measurement noise; without loss of generality, the discrete state space model can be generalized to a nonlinear time-varying system model taking bounded disturbance and measurement noise into consideration, and the expression is:
；
In the above-mentioned method, the step of, Is a state variable; /(I)Is a control input; /(I)For measuring output; for periodic actuator failure, i.e./> At each time interval/>The same periodic dynamic performance is achieved; /(I)Is a measurement matrix; /(I)Is a fault distribution matrix; /(I)And/>Disturbance and measurement noise, respectively; nonlinear function/>And/>Meets the Lipschitz condition, and exists Lipschitz constant，/>Such that:
；
；
s14, setting a hypothesis condition satisfied by the nonlinear time-varying system model; namely, the following assumption condition is satisfied:
Suppose 1: operator Bounded, i.e. there is a constant/>So that/>；
Suppose 2: in each iteration, the initialization error is bounded;
Suppose 3: matrix array For all/>Are all non-singular.
S15, constructing a fault estimation observer consisting of an iterative learning observer and an embedded PD type fault estimator according to the nonlinear time-varying system model and the satisfied hypothesis conditions.
To make the output estimated valueMeasurement output/>, converged to nonlinear time-varying system modelIt is necessary to design a virtual fault (also known as an estimated fault)/>Is a rule for iterative learning. Wherein/>Equivalent to the control quantity/>, in the traditional iterative learning control scheme. The virtual fault form is equivalent to the control law designed in the traditional iterative learning control protocol. Therefore, in order to estimate the faults of the single-link mechanical arm system, a nonlinear iterative learning observer and a fault estimator can be designed, and in order to estimate the additive faults of the actuator of the single-link mechanical arm system, the nonlinear iterative learning observer and the fault estimator are introduced as follows:
；
；
In the above-mentioned method, the step of, Respectively estimating the system state and the output of the single-link mechanical arm system; subscript/>Learning the batch times for iteration; /(I)Virtual faults introduced for a single-link mechanical arm system; /(I)AndAre nonlinear functions; /(I)Is a fault distribution matrix; /(I)Is a measurement matrix; /(I)Is a control input; /(I)For measuring output; /(I)A gain matrix of the observer to be designed; /(I)Run-time for observer/>Outputting an estimation error for the second iteration; /(I)Is the gain matrix to be designed.
Taking into account causal relationships in the discrete time domain, appropriate extrapolation (static) or prediction (dynamic) schemes may be used to obtainTracking error of time of day. In practice, if the sampling time is assumed to be/>Then. Wherein/>For/>At/>A gradient of time of day. It can be seen that the iterative learning control scheme is essentially an embedded PD-type scheme.
S2, proving final boundedness of fault estimation errors by using a recursive analysis method and a robust control theory, wherein the implementation specific process of the step S2 comprises the following steps:
S21, based on the first Minor/>Time/>, in iterative processState error/>Definition of (2)And the nonlinear iterative learning observer derives the/>Time/>, in a secondary iterative processState error/>Wherein/>Is a state variable; /(I)Is the system state of a single-link mechanical arm system. According to/>Definition of/>And a nonlinear iterative learning observer, obtainable:
；
In the method, in the process of the invention, ，/>。
S22, simplifying subsequent operation by using Lipschitz conditions and a method for taking upper bounds of each system matrix, and obtaining the first by using a recursive analysis methodTime/>, in a secondary iterative processThe norm of the state error, i.e./>Is a recursive inequality of (2); the Lipschitz condition and the method of taking the upper bound of each system matrix are utilized to simplify the subsequent operation. And for the/>Taking norms from two sides of the expression can obtain:
；
After having obtained And/>Based on the recursive inequality of (2), using recursive analysis to rewrite the inequality as:
；
In the formula, due to nonlinear function At variable/>With globally consistent Lipschitz continuity thereon,Are known as Lipschitz constants. /(I)Is defined as/>The result of three additions, i.e；/>Control inputs/>, respectivelyDisturbance/>And measuring noise/>The upscaling of norms, i.e./>；/>Gain matrix/>, respectively, of the observer to be designedMeasurement matrix/>Fault distribution matrix/>The upscaling of norms, i.e.；/>For/>Is the upper bound of (i.e.)；/>Is an additive variable that is incremented from 0 to/>Wherein/>Is the accumulated termination value; In/> Representing the system pass/>Iterating the time steps; consider from the current moment/> To the past moment/>Evolution over all time steps of (1), wherein/>Indicating that iterations are performed over past time steps. Specifically, when/>And at the same time, the current time is represented; When/>When it indicates the past time step/>And so on.
S23 based onNorm operation, deducing the/>Time/>, in a secondary iterative processState error and fault estimation errorNorms, i.e./>And/>Inequality relation between them; for follow-up deduction/>And/>Time/>, in a secondary iterative processFault estimation error/>The relationship between the following conditions need to be used:
（1）、 I.e. any real number/> />The norm is equal to itself.
（2）、And/>，/>I.e. for any real number greater than 1/>And/>Function/>At the position ofThe upper bound in all non-negative real ranges is 1.
（3）、If/>Then/>. Wherein, vector/>/>Norms are defined as/>Wherein/>In the present invention, it is assumed that. At a given time interval/>In, if/>At/>Previously, vector/>/>Norms do not exceed/>/>Norms.
；
In addition, due toAnd/>Can/>Exponential/>, in inequalityScaling, i.e./>Deformation into/>. And multiply/>, on both sides of the aboveThe above formula can be obtained.
Vector quantity/>The norm is defined as: /(I)Wherein/>，/>In this embodiment, it is assumed that/>. Finally, the above formula can be passed/>The norms are calculated as:
；
In the above-mentioned method, the step of, Respectively/>The upscaling of norms, i.e.；/>Is for the variables/>The result of the summation of the series, i.e./>Wherein/>From 0 to/>And (5) taking a value. For the variables/>Sum of the series of (i) i.e./>Wherein/>From 0 to/>Take the value of/>Is the result of summing and re-taking the upper bounds. Expressed by the following expression: /(I)，/>Wherein/>。
S24, based onDerived thinking of (1) >, get/>Is a recursive inequality of (2); similar to the derivation thinking described above, it is also possible to define/>, first by fault estimationAnd virtual fault iterative learning law/>Starting, the following is obtained:
；
After item transfer finishing, obtaining: ；
In the above-mentioned method, the step of, Run-time for observer/>Outputting an estimation error for the second iteration; A gain matrix to be designed; /(I) Is a measurement matrix; /(I)Is a control input; /(I)A gain matrix of the observer to be designed; /(I)Is a disturbance; /(I)For measuring noise; Is an identity matrix.
The Lipschitz condition and the method of taking the upper bound of each system matrix are utilized to simplify the subsequent operation. And to the aboveExpression of left multiplier/>And taking norms from two sides:
；
In the above-mentioned method, the step of, For/>The upscaling of norms, i.e.；/>For/>The upscaling of norms, i.e.；/>；/>Is defined as/>The result of the three additions, i.e./>; Due to the function/>In variable quantityWith globally consistent Lipschitz continuity thereon,/>Is a known Lipschitz constant.
S25, introduction ofObtain/>And/>Relationship. By introduction/>The related inequality can be used for carrying out/>, on the two sides of the upper partThe norm operation is specifically as follows:
；
wherein, ，，/>。
Further merging and finishing can obtain:；
In the method, in the process of the invention, 。
Properly designing gain matrixMake it meet the condition/>Then it is possible to obtain:
；
By the presence of a sufficiently large Can ensure/>The condition is satisfied in whichIs a given constant. Combining the inequality to obtain the following:
；
Finally, deducing if the condition is satisfied Wherein/>For a given parameter, when the number of iterations/>When, failure estimation error/>/>The norms converge to a preset boundary. There is a large enough/>So that the final convergence boundary of the system can be expressed as:
；
In the above-mentioned method, the step of, ，/>For/>The upper bound of the matrix, i.e. >；/>For a given parameter, when the number of iterations/>Error of fault estimation/>The norm converges on a preset boundary; /(I)。
S3, designing a learning gain matrix for ensuring that the fault estimation error converges to a preset boundary, namely, iterative learning observer and embedded PD type fault estimator; To ensure failure estimation error/>The final finite nature of the norms requires the design of a suitable gain matrix/>To ensure/>. If/>The relevant conditions are satisfied, namely:
；
In the method, in the process of the invention, Is Lipschitz constant, and/>。
To ensure fault estimation errorsThe final constraint of norms is satisfied at the same time/>Conditions. Due to the matrix/>The diagonalization of the matrix can be achieved by selecting the appropriate matrix for the column full rank. Will beAre respectively set as/>. In the method, in the process of the invention,，/>And meet the following. According to the norm definition used in the present embodiment, one can applyThe conditions were rewritten as follows:
；
Due to the matrix Is of full column rank, the pseudo-inverse of the matrix can be derived as:
；
deriving a learning gain matrix And/>The following are provided:
；
In the above-mentioned method, the step of, And/>For the diagonal matrix to be designed, matrix/>Is a full rank matrixIs a pseudo-inverse of (a). Wherein the diagonal matrix/>Diagonal element/>Satisfy/>；
Diagonal matrixDiagonal element/>Satisfy the following requirements。
S4, reasonably selecting parameters which ensure final bounded fault estimation errors in the fault estimation observer, and verifying the actual effect of the periodic fault estimation method of the single-link mechanical arm by utilizing numerical simulation. In order that the implementation of the invention can be better understood by the researchers in the field, the invention uses Matlab software for simulation verification. The specific information of the simulation software is as follows:
Software name: MATLAB
Version information: 9.8.0.1380330 (R2020 a) Update 2
License number: 919961
Operating system: microsoft Windows 10 Chinese Version 10.0 (Build 19042)
Java version: java 1.8.0_202-b08 with Oracle Corporation Java HotSpot (TM) 64-Bit Server VM mixed mode
A special tool box: STATISTICS AND MACHINE LEARNING Toolbox-11.7 (R2020 a), aircraft Control Toolbox-1.0
The bounded disturbance and the measurement noise are selected as random quantities, and the boundaries are respectively 0.05 and 0.1. In the method, in the process of the invention,To introduce errors, the dynamic performance is as follows:
；
Rational selection of parameters in observer and fault estimator to ensure fault estimation errors Final constraint of norms, let/>Obtaining the parameters:
，
，
，
；
Substituting the relevant parameters, the final boundary of the fault estimation error can be calculated as follows:
；
the actual effect of the periodic fault estimation algorithm of the nonlinear time-varying system, which is influenced by bounded disturbance and measurement noise, is verified through simulation. In the simulation experiment, fault estimation is performed through system output information influenced by noise, and a fault estimation result is represented as a random signal influenced by noise. Meanwhile, the method is compared with a classical robust class based on an unknown input observer The fault estimation method is used for performance comparison research. FIG. 1 shows a fault estimation method and a robust/>, as proposed by the present embodimentThe fault estimation results of the fault estimation algorithm with time change, and fig. 2 shows the fault estimation errors of the two algorithms with iteration times. From simulation results, it can be found that the fault estimation error of the algorithm provided after 6 iterations can be reduced to a preset error boundary. The fault estimation algorithm provided after 4 iterations can obtain more accurate estimation results than the latter method.
Those of ordinary skill in the art will appreciate that all or a portion of the steps in a method of implementing an embodiment described above may be implemented by a program to instruct related hardware, and thus, the present application may take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present application may take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, and the like) having computer-usable program code embodied therein.
In this specification, each embodiment is described in a progressive manner, and each embodiment is mainly described in a different manner from other embodiments, so that the same or similar parts between the embodiments are referred to each other. For each of the above embodiments, since it is substantially similar to the method embodiment, the description is relatively simple, and reference should be made to the description of the method embodiment for relevant points.
The foregoing embodiments have been presented in a detail description of the invention, and are presented herein with a particular application to the understanding of the principles and embodiments of the invention, the foregoing embodiments being merely intended to facilitate an understanding of the method of the invention and its core concepts; meanwhile, as those skilled in the art will have variations in the specific embodiments and application scope in accordance with the ideas of the present invention, the present description should not be construed as limiting the present invention in view of the above.
Claims (8)
1. The method for estimating the periodic faults of the single-connecting-rod mechanical arm based on iterative learning is characterized by comprising the following steps of:
S1, introducing a nonlinear autotransformer into the structural design process of a state observer, and performing fault estimation by adopting an embedded PD type recursive autotransformer scheme to form a fault estimation observer for effectively estimating the periodic faults of the single-link mechanical arm;
S2, proving the final bouncy of a fault estimation error by using a recursive analysis method and a robust control theory;
S3, designing a learning gain matrix for ensuring that the fault estimation error converges to a preset boundary;
S4, reasonably selecting parameters which ensure final bounded fault estimation errors in the fault estimation observer, and verifying the actual effect of the periodic fault estimation method of the single-link mechanical arm by utilizing numerical simulation.
2. The method according to claim 1, wherein in step S1, the fault estimation observer consists of an iterative learning observer and an embedded PD type fault estimator.
3. The method for estimating periodic faults of a single link mechanical arm according to claim 1, wherein in step S1, the specific process includes the steps of:
S11, constructing a dynamic model of the single-link mechanical arm system, wherein the expression of the model is as follows:
；
；
In the above-mentioned method, the step of, Is the angular position of the mechanical arm; /(I)Is the rotational inertia of the joint; /(I)Is joint torque; /(I)Is an exogenous disturbance torque; /(I)Is the mass of the mechanical arm; /(I)The weight of the tip load of the mechanical arm; /(I)Is the length of the mechanical arm; /(I)Gravitational acceleration;
s12, carrying out discretization analysis on a dynamic model of the single-link mechanical arm system to obtain a discrete state space model, wherein the discrete state space model is as follows:
；
In the above-mentioned method, the step of, Is a discrete time; /(I)；
S13, constructing a mechanical arm nonlinear state variable power equation of a single-link mechanical arm system based on a discrete state space model to form a nonlinear time-varying system model considering bounded disturbance and measurement noise, wherein the expression of the nonlinear time-varying system model is as follows:
；
In the above-mentioned method, the step of, Is a state variable; /(I)Is a control input; /(I)For measuring output; /(I)For periodic actuator failure, i.e./>At each time interval/>The same periodic dynamic performance is achieved; /(I)Is a measurement matrix; /(I)Is a fault distribution matrix; /(I)And/>Disturbance and measurement noise, respectively; /(I)And/>Are all nonlinear functions meeting Lipschitz conditions;
s14, setting a hypothesis condition satisfied by the nonlinear time-varying system model;
s15, constructing a fault estimation observer consisting of an iterative learning observer and an embedded PD type fault estimator according to the nonlinear time-varying system model and the satisfied hypothesis conditions.
4. The method for estimating a periodic failure of a single link mechanical arm according to claim 3, wherein in step S14, the assumption condition is satisfied that:
Suppose 1: operator Bounded, i.e. there is a constant/>So that/>；
Suppose 2: in each iteration, the initialization error is bounded;
Suppose 3: matrix array For all/>Are all non-singular.
5. The method for estimating periodic faults of a single link mechanical arm according to claim 3, wherein in step S15, expressions of the nonlinear iterative learning observer and the fault estimator are:
；
；
In the above-mentioned method, the step of, Respectively estimating the system state and the output of the single-link mechanical arm system; subscript/>Learning the batch times for iteration; /(I)Virtual faults introduced for a single-link mechanical arm system; /(I)And/>Are nonlinear functions; /(I)Is a fault distribution matrix; /(I)Is a measurement matrix; /(I)Is a control input; For measuring output; /(I) A gain matrix of the observer to be designed; /(I)Run-time for observer/>Outputting an estimation error for the second iteration; /(I)Is the gain matrix to be designed.
6. The method for estimating periodic faults of a single link mechanical arm according to claim 1, wherein in step S2, the specific process includes the steps of:
S21, based on the first Minor/>Time/>, in iterative processState error/>Definition of (2)And the nonlinear iterative learning observer derives the/>Time/>, in a secondary iterative processState error/>Is an expression of (2);
s22, simplifying subsequent operation by using Lipschitz conditions and a method for taking upper bounds of each system matrix, and obtaining the first by using a recursive analysis method Time/>, in a secondary iterative processThe norm of the state error, i.e./>Is a recursive inequality of (2);
s23 based on Norm operation, deducing the/>Time/>, in a secondary iterative process/>, In-place state error and fault estimation errorNorms, i.e./>And/>Inequality relation between them;
S24, based on Derived idea of (1) >, get (1) >Fault estimation error/>, in a secondary iterative processIs a recursive inequality of (2);
s25, introduction of Obtain the/>Minor iteration and/>Error estimation/>, in the secondary iteration processNorms, i.e.And/>Relationship between them.
7. The method according to claim 1, wherein in step S3, the learning gain matrix includes a learning gain matrix of an iterative learning observer and an embedded PD type fault estimator。
8. The method for periodic fault estimation of a single link mechanical arm of claim 7, wherein a gain matrix is learnedAnd/>The method comprises the following steps of:
；
In the above-mentioned method, the step of, And/>A diagonal matrix to be designed; matrix/>Is a full rank matrixIs the pseudo-inverse of (a); wherein the diagonal matrix/>Diagonal line elementSatisfy/>；
Diagonal matrixDiagonal element/>Satisfy the following requirements。
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