Self-adaptive pseudo-inverse control method for output constraint interconnection system with hysteresis

Abstract

The invention relates to the technical field of intelligent material driving control, provides a self-adaptive pseudo-inverse control method of an output constraint interconnection system with hysteresis, and aims to realize high-precision control of an intelligent material actuator. Comprising the following steps: designing a high-gain state observer, and deducing a stability condition of a system under disturbance; thirdly, designing a hysteresis temporary controller to improve the robustness and response capability of the controller; and finally, designing a pseudo-inverse algorithm to improve the accuracy and real-time performance of system response. The control precision of the provided control scheme is improved by two times compared with the control precision of a control scheme without considering hysteresis, and is improved by three times compared with the control precision of a PID (Proportion Integration Differentiation) control scheme, so that the provided control scheme has a better control effect.

Description

Technical Field

[0001] The present invention relates to the field of intelligent material drive control technology, and proposes an adaptive pseudo-inverse control method for an output constraint interconnected system with a hysteresis loop, which realizes high-precision control of a multi-axis rigid intelligent material drive by a multi-subsystem output constraint interconnected system. Background Art

[0002] In recent years, smart material microactuators with hysteresis-induced nonlinear inputs have been widely used in micro- and nanoscale systems, metal cutting systems, and other ultra-high-precision positioning systems, such as scanning probe microscopes, optical aligners, diamond cutting machines, active vibration control systems, and bioengineering equipment. Due to their ability to meet ultra-high precision requirements and their wide applicability, they have attracted increasing research attention. Among them, giant magnetostrictive actuators, as one type of rigid smart material actuator, offer excellent performance in the robotics industry, precision positioning, and microelectronics, including high energy density, fast response speed, and high mechanical output. However, rigid smart material actuators inherently exhibit strong hysteresis, resulting in system delays and severe nonlinearity, limiting their wider application. Therefore, achieving fast and precise control of rigid smart material actuators has been a hot topic of research both domestically and internationally.

[0003] For interconnected systems with multiple subsystems, decentralized adaptive control schemes have emerged as an effective approach to solving control problems due to the unknown interactions between the different subsystem models in the control system. The key advantage of decentralized control is that it can reduce the computational burden associated with centralized control and enhance the robustness and reliability of the interconnected system. Consequently, decentralized control of uncertain nonlinear interconnected systems has made significant progress over the past few decades.

[0004] Currently, how to deal with hysteresis nonlinearity in the control process is a great challenge. In the actual motion control process of smart material actuators, the output variables are usually constrained due to physical conditions or safety reasons and cannot take infinite values. In fact, constraints are common in control system applications. For example, the restricted workspace of the mechanical system, the preset force and speed range, and the limited communication distance. In order to ensure the safety of the control system, constraints should usually be considered in advance. If the preset constraints are violated, the system control performance may be reduced or even threaten the system safety. Therefore, it is of great significance to propose effective methods to deal with control problems with output or state constraints.

[0005] It's important to note that designing control schemes becomes more challenging when time delays are involved. Long time delays can lead to system instability and complicate the learning process for neural networks. Time delays are common in real-world applications. The Lyapunov-Krasovskii function method is a common approach for dealing with time delays. However, this method requires certain assumptions about the time delay function, imposes strict parameter requirements, and cannot guarantee semiglobally uniformly bounded convergence. Summary of the Invention

[0006] To solve the above technical problems, the present invention provides an adaptive pseudo-inverse control method for an output-constrained interconnected system with hysteresis, which includes:

[0007] Construct a high-gain state observer to estimate the state of the interconnected system;

[0008] Using finite cover gravity and RBF neural network approximator to process unknown functions of the interconnected system;

[0009] Based on the high-gain state observer, a hysteresis temporary controller is designed taking into account the interconnection relationship terms of the interconnected system to determine a temporary control signal and a temporary control law;

[0010] Based on the temporary control signal, the actual controller is determined by a pseudo-inverse algorithm.

[0011] As a preferred embodiment, the high-gain state observer is expressed as:

[0012]

[0013] Among them, k i ≥1 is a constant parameter, represents the nth unit vector of the ith subsystem, and

[0014]

[0015] Where, is state x i Estimate of the observer error ∈ i for but

[0016]

[0017] where ∈ i,1 Represents ∈ i The first term of , i = 1, ..., N;

[0018] The actual state is estimated to be

[0019]

[0020] in, It is b i,0 The estimated value of yes estimated value.

[0021] As a preferred embodiment, the interconnected system is represented as a system with N n i A nonlinear time-delay system with interconnected subsystems of order 1 and output constraints and hysteresis:

[0022]

[0023] y i =x i,1 ,i=1,…,N,j=1,…,n i -1,

[0024] where d i (t), i=1...n and ι i,j are the interference term and the unknown time-lag constant respectively; is the state vector of the system, is the interconnection term between the jth subsystem and the ith subsystem; output y i ∈R is restricted to a compact set, i.e. in is a positive constant; is an unknown smooth nonlinear function; b i,0 represents a positive constant; w i ∈R indicates that the unknown hysteresis loop output is:

[0025] w i (u i )=P re (u i (t)),

[0026] Among them, P re (·) represents the output of the hysteresis model operator, u i ∈R is the input.

[0027] As a preferred embodiment, the assumptions satisfied by the interconnected system include:

[0028] Assumption 1: Interconnected Items satisfy:

[0029]

[0030] Where m = 1,…,N, j = 1,…,n i , Λ m (y m ) represents an unknown nonlinear function; represents the strength of the interaction; ||·|| represents the Euclidean norm.

[0031] Assumption 2: Reference signal y ri is a smooth function that satisfies where Y i,0 , is a positive constant, vector Bounded, and where Ω o It is a tight set.

[0032] Assumption 3: Constant ι i,j , satisfying 0≤ι i,j ≤ι M , where ι M for ι i,j The maximum value of i=1,...,N,j=1,…,n i .

[0033] Assumption 4:d i,j Represents disturbance, satisfying

[0034] Assumption 5: i,j Represents a constant, given b i,0 The symbol of b i,0 >0.

[0035] As a preferred embodiment, the unknown smooth function f in the interconnected system i,j Using RBF neural network to determine,

[0036]

[0037] For all where |ε i,j |≤ε i,e , is the basis function, where ζ i,k ∈R q and η i >0 are basis functions the center and width of is θ i,j The optimal weight vector, ε i,j >0 indicates approximate error

[0038] As a preferred embodiment, according to Substituting into the equations of the interconnected system we obtain:

[0039] Further transformed into the system equation in state space form:

[0040]

[0041] in e i,1 =[1,0,…,0], b i =[0,…,b i,0 ] T ;

[0042] make B i =ε i +δ i,0 +d i , where A i,0 is a Hurwitz matrix related to the vector q, and The system equations can be rewritten in state space form as:

[0043]

[0044] As a preferred embodiment, the steps of the hysteresis temporary controller include:

[0045] Step 1: Define S i,1 For S i,1 =y i -y ri

[0046] where y ri is the reference signal, i=1,…,N, according to the rewritten system equation formula, the derivative of S1 is

[0047]

[0048] According to the observation error ∈ i The definition of

[0049]

[0050] Among them, ∈ i,2 is the second term of the observation error ∈. i,(2) is the second term of Ξ; there exists the following formula:

[0051]

[0052] in, is a virtual control signal designed to:

[0053]

[0054] in yes The estimated value of And there are:

[0055]

[0056] where l i,1 is a positive design parameter, yes The estimated value of Is a positive constant. Define a compact set set up

[0057]

[0058] where υ i is a constant. and The adaptive law is designed as:

[0059]

[0060] Introduce a first-order low-pass filter and obtain a new variable z through the first-order filter i,2 ,as follows

[0061]

[0062] in, The input of the above low-pass filter, ι i,2 is a positive time constant;

[0063] Step 2: Define S i,2 for

[0064] S i,2 =v i(0,2) -z i,2

[0065] According to the rewritten system equation formula, S i,2 The derivative of

[0066]

[0067] in, It is a virtual control signal. The designed virtual control signal is:

[0068]

[0069] Among them, l i,2 is a designed normal number, It is b i,0 The estimated value of The parameter update law is designed as:

[0070] Introduce a first-order low-pass filter and get the variable z i,3 as follows

[0071]

[0072] in, is the input of the low-pass filter, ι i,3 is a positive time constant;

[0073] Step j(3≤j≤n i -1): Define S i,j For S i,j =v i(0,j) -z i,j

[0074] According to the rewritten system equation, we can get S i,j The derivative of is:

[0075]

[0076] in, is a virtual control signal designed to:

[0077]

[0078] Among them, l i,j >0 is the design parameter, a first-order low-pass filter is introduced to obtain the variable z i,j+1

[0079]

[0080] in, is the input of the low-pass filter, ι i,j+1 is a positive time constant;

[0081] Step n i :definition for

[0082]

[0083] According to the rewritten system equation The derivative of is:

[0084]

[0085] Design temporary control signal w i (t) is:

[0086]

[0087] and The adaptive law is designed as:

[0088]

[0089] in, is a positive design parameter.

[0090] As a preferred embodiment, the Preisach model representing the hysteresis characteristics of the interconnected system is defined as

[0091] w i (t)=∫∫ β≥α γ α,β [u i ](t)μ(α,β)dαdβ,

[0092] Among them, w i (t) is the output of the Preisach model, u i (t) is the input of the model, and μ(α,β) represents the weight function.

[0093] Using iterative search variables Finding a near-optimal signal So that the following formula holds:

[0094]

[0095] And determine the actual controller

[0096] As a preferred embodiment, an iterative search variable is used Finding a near-optimal signal include:

[0097] Set the actual input of the drive to [u i,min ,u i,max ],consider and γ α,β [u i ](t), ∫∫ β≥α γ α,β [u i ](t)μ(t,α,β)dαdβin[u i,min ,u i,max ] is monotonically increasing, and is defined as

[0098]

[0099] The following formula exists:

[0100]

[0101] Define a new variable Its input is in Let u i,0 (t) = u i,min , the following formula holds

[0102]

[0103] When w i,max (t)<w i (t), let u * (t) = u i,max ;

[0104] When w i,min (t)>w i (t), let u * (t) = u i,min ;

[0105] When w i,max (t)≥w i (t)≥w i,min (t), It can be obtained according to steps 1 to 3.

[0106] Step 1: Make Increase from 0 to w i,min (t);

[0107] Step 2: Calculation if Increase until Go to step 3;

[0108] Step 3: Make To stop the growth, we can Assigned to make

[0109] The actual controller can be obtained using the pseudo-inverse algorithm

[0110] Compared to existing technologies, this invention achieves the following benefits: By designing a high-gain state observer, it derives the stability conditions for the system under disturbances. It then designs a temporary controller with hysteresis to improve the controller's robustness and responsiveness. Finally, it employs a pseudo-inverse algorithm to enhance the accuracy and real-time performance of the system's response. The proposed control scheme achieves a two-fold improvement in accuracy compared to a control scheme without hysteresis considerations and a three-fold improvement compared to a PID control scheme, further demonstrating its superior control effectiveness. BRIEF DESCRIPTION OF THE DRAWINGS

[0111] Figure 1 Diagram of the experimental platform for the precise operation of the 3-axis giant magnetostrictive actuator provided by the present invention

[0112] Figure 2 Prediction accuracy diagram of the Preisach model provided by the present invention

[0113] Figure 3 Comparison results between the GMA reference signal and the actual output signal provided by the present invention

[0114] Figure 4 GMA tracking error diagram provided by the present invention

[0115] Figure 5 GMA control signal diagram provided by the present invention

[0116] Figure 6 The estimated value provided by the present invention Trajectory diagram

[0117] Figure 7 The tracking error diagram of the control scheme provided by the present invention without considering the hysteresis loop

[0118] Figure 8 Tracking error diagram of the PID solution provided by the present invention;

[0119] Figure 9 Flow chart of the method of the present invention. DETAILED DESCRIPTION

[0120] The present invention will be further described below in conjunction with the accompanying drawings. The following embodiments are only used to more clearly illustrate the technical solutions of the present invention and are not intended to limit the scope of protection of the present invention.

[0121] This embodiment provides an adaptive pseudo-inverse control method for an output-constrained interconnected system with hysteresis, the main steps of which are as follows:

[0122] Design of high-gain state observer: The high-gain state observer constructs a state estimator by introducing the gain matrix K To estimate the system state. Its design is based on the error system and Lyapunov function, combined with the observer error term ∈ i The convergence analysis of the system is performed to derive the stability conditions of the system under disturbances, ensuring the asymptotic convergence of the state estimation error.

[0123] Design of hysteresis temporary controller: The design of hysteresis temporary controller is based on high gain state observer, and the output constraint problem of the system is handled by selecting the barrier Lyapunov function to construct the error signal sequence S i,j , design virtual control signals and temporary control laws, introduce low-pass filters to process intermediate variables, and implement system state tracking. The control law has an adaptive update mechanism to improve the robustness and responsiveness of the controller.

[0124] Design of pseudo-inverse algorithm: The pseudo-inverse algorithm estimates the density function and combines it with the Preisach model to build the relationship between the control signal and the actual controller. Regulated input To approach the target control quantity w i (t), thereby achieving the controller output u i The effective calculation of (t) improves the accuracy and real-time performance of system response.

[0125] Through these three key steps, the solution of the present invention is dedicated to solving the hysteresis control problem of interconnected systems. For a class of interconnected nonlinear time-delay systems with input hysteresis and output constraints, this chapter designs an adaptive pseudo-inverse dynamic surface control scheme based on neural networks, providing an innovative and comprehensive solution for its application in different application scenarios. The invention content is mainly divided into several parts:

[0126] (1) Design of high-gain state observer

[0127] Since only the output can be measured in the interconnected system, a high-gain K state observer is constructed to estimate the state x i , as shown below:

[0128]

[0129] Among them, k i ≥1 is a constant parameter, represents the nth unit vector of the ith subsystem, and Then, according to formula (1), we can get:

[0130]

[0131] Where, is state x i Estimation of . Define the observer error ∈ i for but

[0132]

[0133] where ∈ i,1 Represents ∈ i The first term of , i=1,...,N.

[0134] Lemma 1: Consider the high-gain K state observer in Equation (1), assuming the quadratic function in, satisfy make

[0135]

[0136] Where, ||B imax || is||B i The maximum value of ||. Then, the following equation holds

[0137]

[0138] Among them, k i >1, because b in formula (1) i,0 and is an unknown variable, then the actual state estimate is

[0139] in, It is b i,0 The estimated value of yes The estimated value of , and its update law design process is as follows.

[0140] (2) System Description

[0141] Consider a class with N n i A nonlinear time-delay system with interconnected subsystems of order 1 and output constraints and hysteresis:

[0142]

[0143] where d i (t), i=1...n and ι i,j are the interference term and the unknown time-lag constant respectively; is the state vector of the system, is the interconnection term between the jth subsystem and the ith subsystem; output y i ∈R is restricted to a compact set, i.e. in is a positive constant; is an unknown smooth nonlinear function; b i,0 represents a positive constant; w i ∈R indicates that the unknown hysteresis loop output is:

[0144] w i (u i )=P re (u i (t)),(9)

[0145] Among them, P re (·) represents the output of the hysteresis model operator, u i ∈R is the input.

[0146] In order to design the control scheme, the following assumptions need to be made first:

[0147] Assumption 1: Interconnected Items satisfy:

[0148]

[0149] Where m = 1,…,N, j = 1,…,n i , Λ m (y m ) represents an unknown nonlinear function; represents the strength of the interaction; ||·|| represents the Euclidean norm.

[0150] Assumption 2: Reference signal y ri is a smooth function that satisfies where Y i,0 , is a positive constant, vector Bounded, and where Ω o It is a tight set.

[0151] Assumption 3: Constant ι i,j , satisfying 0≤ι i,j ≤ι M , where ι M for ι i,j The maximum value of i=1,...,N,j=1,…,n i .

[0152] Assumption 4:d i,j Represents disturbance, satisfying

[0153]

[0154] Assumption 5: i,j Represents a constant, given b i,0 The symbol of b i,0 >0.

[0155] (3) Hysteresis model

[0156] The flexibility of the Preisach model makes it an effective tool for describing the hysteresis characteristics of 3-axis giant magnetostrictive actuators. The Preisach model is defined as

[0157] w i (t)=∫∫ β≥α γ α,β [u i ](t)μ(α,β)dαdβ, (12)

[0158] where w i (t) is the output of the Preisach model, u i (t) is the input of the model, μ(α,β) represents the weight function. α,β [u i ](t) is the relay operator, defined as:

[0159]

[0160] Where α and β are thresholds,

[0161] (4) Radial Basis Function (RBF) Neural Network Approximator

[0162] RBF neural network is an efficient universal approximator of unknown nonlinear functions. In this invention, RBF neural network is introduced to approximate the set An unknown smooth function f on i,j ,and is a sufficiently large compact set. By choosing a suitable η i and ζ i,k , there exists a constant ε i,e >0, j=1,…,n i ,k=1,...,N, when N i When >0, the output of the RBF neural network can be described by the following formula

[0163]

[0164] For all where |ε i,j |≤ε i,e . is the basis function, where ζ i,k ∈R q and η i >0 are basis functions The center and width. is θ i,j The optimal weight vector is expressed as follows:

[0165]

[0166] The approximate error is

[0167] It should be noted that when i,j When it is a time delay vector, the RBF neural network cannot directly approximate the unknown smooth function f i,j (x i,j ,x i,j (t-τ i,j )), because the delay constant τ and state x i,j Therefore, the finite covering lemma and RBF neural network are combined to approximate the unknown function f i,j (x i,j ,x i,j (t-τ i,j ))

[0168] Lemma 2: Assume There exists a smooth function f oni,j (ξ i,j ),in is a compact set and where ξ i,j is uniform and continuous about t, and ξ i,j =(ξ i,j (t),ξ i,j (t-τ i,j )). Delay constant τ i,j ∈[0,τ M ]. Then, for a given δ i,0 >0,[0,τ M ] exists on a finite interval independent of t, with

[0169] 0≤t1<...<t s ≤τ M ,(17) Then for Point of Existence So that the following equation holds:

[0170]

[0171] Considering Lemma 2 and Assumption 4, there exists a point where t1,...,t s Defined in the lemma, the unknown nonlinear function is approximated by formula (15) as follows:

[0172]

[0173] Where, ε i,j >0 indicates approximate error, Estimated value It is given in formula (7).

[0174] Substituting formula (19) into formula (8) yields:

[0175]

[0176] Then, formula (20) can be transformed into the following state space form:

[0177]

[0178] in e i,1 =[1,0,...,0], b i =[0,…,b i,0 ] T .

[0179] make

[0180]

[0181] Among them, A i,0 is a Hurwitz matrix related to the vector q, and Then, formula (21) can be rewritten as:

[0182]

[0183] (5) Design of hysteresis temporary controller

[0184] Based on the high-gain state observer of formula (1), the details of designing the hysteresis temporary controller are as follows:

[0185] Step 1: Define S i,1 for

[0186] S i,1 =y i -y ri (24) where y ri is the reference signal, i=1,…,N. According to formula (23), the derivative of S1 is

[0187]

[0188] According to the observation error ∈ i The definition of

[0189]

[0190] Among them, ∈ i,2 is the second term of the observation error ∈. i,(2) is the second term of Ξ. Then, the following equation exists:

[0191]

[0192] in, is a virtual control signal designed to:

[0193]

[0194] in yes The estimated value of And there are

[0195]

[0196] where l i,1 is a positive design parameter, yes estimated value. is a positive constant.

[0197] Definition of compact set set up

[0198]

[0199] where υ i is a constant. and The adaptive law is designed as

[0200]

[0201] Introduce a first-order low-pass filter and obtain a new variable z through the first-order filter i,2 ,as follows

[0202]

[0203] in, The input of the above low-pass filter, ι i,2 is a positive time constant.

[0204] Step 2: Define S i,2 for

[0205] S i,2 =v i(0,2) -z i,2 (34) Considering formula (23), S i,2 The derivative of

[0206]

[0207] in, It is a virtual control signal. The designed virtual control signal is:

[0208]

[0209] Among them, l i,2 is a designed normal number, It is b i,0 The estimated value of The parameter update law is designed as:

[0210]

[0211] Introduce a first-order low-pass filter and get the variable z i,3 as follows

[0212]

[0213] in, is the input of the low-pass filter, ι i,3 is a positive time constant.

[0214] Step j(3≤j≤n i -1): Define S i,j for

[0215] S i,j =v i(0,j) -z i,j (39) Considering formula (23), S i,j The derivative of

[0216]

[0217] in, Is a virtual control signal. Designed as:

[0218]

[0219] Among them, l i,j >0 is the design parameter. Introduce a first-order low-pass filter and get the variable z i,j+1

[0220]

[0221] in, is the input of the low-pass filter, ι i,j+1 is a positive time constant.

[0222] Step n i :definition for

[0223]

[0224] Considering formula (23), The derivative of

[0225]

[0226] Design temporary control signal w i (t) is:

[0227]

[0228] and The adaptive law is designed as:

[0229]

[0230] in, is a positive design parameter.

[0231] (6) Design of pseudo-inverse algorithm

[0232] Actual controller u i Coupled to the hysteresis temporary controller w i (t)=∫∫β≥α γ α,β [u i ](t)μ(t,α,β)dαdβ. Therefore, from the temporary controller w i (t) find the actual controller u i The following is the design process of the pseudo-inverse algorithm for the Preisach model.

[0233] Designed estimates Instead of the unknown density function μ(t,α,β), a search mechanism will be developed to find an approximately optimal signal It makes the following equation true.

[0234]

[0235] The actual input of the driver is set to [u i,min ,u i,max ].consider and γ α,β [u i ](t), ∫∫ β≥α γ α,β [u i ](t)μ(t,α,β)dαdβin[u i,min ,u i,max ] is monotonically increasing. Then, we define

[0236] Then, the following exists:

[0237]

[0238] In addition, define a new variable Its input is in Let u i,0 (t) = u i,min , the following formula holds

[0239]

[0240] When w i,max (t)<w i (t), let u * (t) = u i,max .

[0241] When w i,min (t)>w i (t), let u * (t) = u i,min .

[0242] When w i,max(t)≥w i (t)≥w i,min (t), It can be obtained according to steps 1 to 3.

[0243] Step 1: Make Increase from 0 to w i,min (t).

[0244] Step 2: Calculation if Increase until Go to step 3.

[0245] Step 3: Make To stop the growth, we can Assigned to make Finally, the pseudo-inverse algorithm can be used to obtain the actual controller u i (t), as shown below

[0246]

[0247] (7) Tracking error and stability analysis

[0248] The stability analysis of the control system is as follows, defining

[0249]

[0250] make The following formula exists:

[0251]

[0252] Using Young's inequality, we can get

[0253]

[0254] Then, considering formula (25) and the above inequality, we can get

[0255]

[0256] Where, Similarly, according to formula (35), formula (36), formula (40), formula (41), formula (44), and formula (45), we can obtain:

[0257]

[0258] According to formula (28)-formula (32), formula (36), formula (37), formula (41), formula (45), formula (46)

[0259] You can get:

[0260]

[0261] Among them, B i,j+1 represents a continuous function. The overall barrier Lyapunov function is defined as

[0262]

[0263] in

[0264]

[0265] Quadratic function As defined in Lemma 2,

[0266] Then, the designed adaptive pseudo-inverse dynamic surface control scheme can be proved by the following theorem.

[0267] Theorem 1: For the interconnected nonlinear time-delay system (8), considering Assumption 1, the initial value satisfies By using a high-gain state observer (1), a practical controller (51), and parameter update laws (31), (32), (37), and (46), the designed decentralized adaptive pseudo-inverse control scheme can ensure that: (1) all signals in the closed-loop control system are bounded; (2) the system output remains within a preset range, i.e. (3) By properly selecting the design parameters, all signals in the stable state of the closed-loop system can converge to an arbitrarily small compact set.

[0268] Proof: Define the following compact set

[0269]

[0270] Among them G i,0 >0 is a constant, B i,j+1 is a compact set Ω i,1 ×Ω i,2 A continuous function on a has a maximum value L i,j+1 , that is, L i,j+1 ≥|B i,j+1 |, i=1,...,N, j=1,...,n i -1. From formula (63), we can get

[0271]

[0272] Where a>0 is a constant. Therefore, V in formula (64) i The derivative of

[0273]

[0274] According to Assumption 1, the coupling strength between subsystems is given by and Decision, among which is a constant, then:

[0275]

[0276] Then V i The derivative of

[0277]

[0278] in is defined in Lemma 2. Consider the inequality

[0279]

[0280] make From the definition of χ(x) in formula (24), we know that h i Bounded, then

[0281] Among them H i yes The upper bound of .

[0282]

[0283] Select the following design parameters:

[0284]

[0285] l i,j ≥C i1 ,(76)

[0286]

[0287] Among them C i,1 is a normal number, satisfying

[0288]

[0289] in,

[0290]

[0291] And C i1 satisfy

[0292]

[0293] Then, when V i =p, This means that for all t ≥ 0, if V i(0)≤p, then V i (t)≤p. Therefore, we can get V i (t)≤p. Considering formula (84), we can get

[0294]

[0295] therefore,

[0296]

[0297] Then, all variables y in the closed-loop control system i,j+1 , S i,j , ∈ i are all semi-globally uniformly eventually bounded. In addition, by choosing appropriate parameters l i,j , ι i,j+1 ,j=1,...,n i , so that all signals in the closed-loop system can be made arbitrarily small. For the initial condition From Lemma 1 and formula (84), we can get Because y i =S i,1 +y ri , and |y ri |≤Y i,0 ,but Therefore, the system output always remains within the preset range. This completes the proof.

[0298] In another specific embodiment, in order to realize the application of the proposed control scheme in the rigid smart material actuator, an attached Figure 1 The experimental platform shown in the figure verifies the control performance of the proposed control scheme by applying it to a 3-axis giant magnetostrictive actuator precision operation experimental platform. The 3-axis giant magnetostrictive actuator precision operation experimental platform includes the following components:

[0299] (1) M3-GMA70 is selected as the giant magnetostrictive actuator. Under the action of the driving voltage, the force generated is not less than 30N and the displacement is not less than 50μm.

[0300] (2) dSPACE CP-1104 data converter, with 8 DAC output ports, used for data transmission and signal conversion between the host and the amplifier.

[0301] (3) Micro-Epsilon's DL6530 high-resolution multi-channel capacitive sensor offers excellent linearity, repeatability, and resolution. It can achieve submicron accuracy in industrial environments and even subnanometer accuracy in laboratory settings. It is used to measure the displacement of a 3-axis giant magnetostrictive actuator.

[0302] (4) The 7228AE Techron power amplifier is used to amplify current and can provide 360A output, generating an RMS power output exceeding 1000W. It has a low noise and fast conversion rate design.

[0303] (5) Lenovo 510S personal computer equipped with an AMDR7-5800X motherboard, using MATLAB software to process the data received by dSPACE.

[0304] The working process of the 3-axis giant magnetostrictive drive motion control experimental platform is as follows: the host computer sends the designed control signal to the dSPACE CP-1104 data converter through Matlab software. After signal conversion, the analog signal is sent to the 7228AE power amplifier. The amplified signal is sent to the 3-axis giant magnetostrictive drive to control its precise movement. The displacement of the movement is measured by the capacitive sensor DL6530 and transmitted back to dSPACE, and then received by the host computer. This is the working process of the established 3-axis giant magnetostrictive drive experimental platform.

[0305] The dynamic model of the 3-axis giant magnetostrictive actuator precision motion control system can be expressed as:

[0306]

[0307] Where m1, m2, m3 are the masses of the three-axis giant magnetostrictive actuator, w(t) is the hysteresis output of the three-axis giant magnetostrictive actuator control system, and y i (t) is the displacement of the 3-axis giant magnetostrictive actuator; μ i,1 represents the viscous friction coefficient, μ i,2 Represents the spring stiffness, and satisfies μ i,1 <<1,μ i,2 >1.Θ i represents unknown interactions from other subsystems, c i Represents the equivalent spring of the 3-axis giant magnetostrictive actuator. τ i,j Denotes the time constant of the system. Let y i =x i,1 , Then formula (87) can be expressed as:

[0308]

[0309] Where x i,1 is the output displacement of the 3-axis giant magnetostrictive actuator, x i,2 is the motion speed of the 3-axis giant magnetostrictive actuator, μ i,1 ,μ i,2 ,m i ,c i is an unknown constant.

[0310] The accuracy of the Preisach model in predicting the hysteresis loop in a 3-axis giant magnetostrictive actuator is verified by two open-loop experiments. Figure 3 The prediction accuracy when the input is u(t) = 2sin(10πt) is shown in Figure 2. The density function is designed as:

[0311]

[0312] Among them, ρ1,ρ2∈[0,5].

[0313] In the experiment, for the RBF neural network, the Gaussian function is selected as the basis function Right now Basis function center ζ 1,k Uniformly distributed in [-1,1], η j =1,j=1,...,7 is the width of the basis function. Among them, t1 = 0.25, t4 = 0.85, t5 = 1.05, and the initial value of the density function is set to For a high-gain observer, the parameter is set to Φ -1 =diag{0,1 / k},q=[q1,q2] T =[5,3] T , k=5, υ(0)=0, ζ(0)=0, Ξ(0)=0.

[0314] Select the tracking reference signal as y r (t) = 5cos(5×2πt) + 6(μm). The output constraint of the system is set to The control system parameters are designed as l1=10.5, l2=5, γ θ =0.3,σ b =0.1, γ b =0.3,γ r =0.1,σ θ =0.8,σ r =0.6.

[0315] This chapter proposes a decentralized neural network adaptive pseudo-inverse dynamic surface control scheme for a class of nonlinear, large-scale, interconnected time-delay systems with hysteresis and limited output. First, by combining RBF neural networks with the finite covering lemma, a time delay function estimation method is proposed. This method replaces the traditional Lyapunov-Krasovskii function for processing time delays and avoids assumptions about the time delay function. Second, a hysteresis pseudo-inverse algorithm is designed to address the coupling problem in the hysteresis-induced temporary control signal without constructing an exact inverse model. Finally, a three-axis giant magnetostrictive actuator precision motion control experimental platform is constructed and the proposed control scheme is applied to the experimental platform. The tracking performance graph shows that the proposed control scheme exhibits excellent tracking performance, demonstrating that it can accurately control the motion of the three-axis giant magnetostrictive actuator. Furthermore, by comparing the tracking error of the proposed control scheme without considering hysteresis with that of a PID control scheme, the proposed control scheme achieves the lowest tracking error, achieving an MVTE of approximately 5% and a NRMSE of approximately 3% for all three axes of the giant magnetostrictive actuator.

Claims

1. An adaptive pseudo-inverse control method for an output-constrained interconnected system with hysteresis, characterized in that: include: Construct a high-gain state observer to estimate the state of the interconnected system; Using finite cover gravity and RBF neural network approximator to process unknown functions of the interconnected system; Based on the high-gain state observer, a hysteresis temporary controller is designed taking into account the interconnection relationship terms of the interconnected system to determine a temporary control signal and a temporary control law; Based on the temporary control signal, the actual controller is determined by a pseudo-inverse algorithm.

2. The adaptive pseudo-inverse control method according to claim 1, characterized in that: The high-gain state observer is expressed as: Among them, k i ≥1 is a constant parameter, represents the nth unit vector of the ith subsystem, and Where, is state x i Estimate of the observer error ∈ i for but where ∈ i,1 Represents ∈ i The first term of , i = 1, ..., N; The actual state is estimated to be in, It is b i,0 The estimated value of yes estimated value.

3. The adaptive pseudo-inverse control method according to claim 2, characterized in that: The interconnected system is represented as a system with N n i A nonlinear time-delay system with interconnected subsystems of order 1 and output constraints and hysteresis: y i =x i,1 ,i=1,…,N,j=1,…,n i -1, where d i (t), i=1...n and ι i,j are the interference term and the unknown time-lag constant respectively; is the state vector of the system, is the interconnection term between the jth subsystem and the ith subsystem; output y i ∈R is restricted to a compact set, i.e. in is a positive constant; is an unknown smooth nonlinear function; b i,0 represents a positive constant; w i ∈R indicates that the unknown hysteresis loop output is: w i (u i )=P re (u i (t)), Among them, P re (·) represents the output of the hysteresis model operator, u i ∈R is the input.

4. The adaptive pseudo-inverse control method according to claim 3, characterized in that: The assumptions satisfied by the interconnected system include: Assumption 1: Interconnected Items satisfy: Where m = 1,…,N, j = 1,…,n i , Λ m (y m ) represents an unknown nonlinear function; represents the strength of interaction; ||·|| represents the Euclidean norm; Assumption 2: Reference signal y ri is a smooth function that satisfies where Y i,0 , is a positive constant, vector Bounded, and where Ω o is a compact set; Assumption 3: Constant ι i,j , satisfying 0≤ι i,j ≤ι M , where ι M for ι i,j The maximum value of i=1,...,N,j=1,…,n i ; Assumption 4:d i,j Represents disturbance, satisfying Assumption 5: i,j Represents a constant, given b i,0 The symbol of b i,0 >0.

5. The adaptive pseudo-inverse control method according to claim 3, characterized in that: The unknown smooth function f in the interconnected system i,j Using RBF neural network to determine, For all where |ε i,j |≤ε i,e , is the basis function, where ζ i,k ∈R q and η i >0 are basis functions the center and width of is θ i,j The optimal weight vector, ε i,j >0 indicates approximate error.

6. The adaptive pseudo-inverse control method according to claim 5, characterized in that: according to Substituting into the equations of the interconnected system we obtain: Further transformed into the system equation in state space form: in e i,1 =[1,0,...,0], b i =[0,...,b i,0 ] T ; make B i =ε i +δ i,0 +d i , where A i,0 is a Hurwitz matrix related to the vector q, and The system equations can be rewritten in state space form as:

7. The adaptive pseudo-inverse control method according to claim 6, characterized in that: The steps of the hysteresis temporary controller include: Step 1: Define S i,1 For S i,1 =y i -y ri where y ri is the reference signal, i=1,...,N, according to the rewritten system equation formula, the derivative of S1 is According to the observation error ∈ i The definition of Among them, ∈ i,2 is the second term of the observation error ∈; i,(2) is the second term of Ξ; there exists the following formula: in, is a virtual control signal designed to: in yes The estimated value of And there are: where l i,1 is a positive design parameter, yes The estimated value of is a positive constant; define a compact set set up where υ i is a constant; and The adaptive law is designed as Introduce a first-order low-pass filter and obtain a new variable z through the first-order filter i,2 ,as follows in, The input of the above low-pass filter, ι i,2 is a positive time constant; Step 2: Define S i,2 for With i,2 =in i(0,2) -z i,2 According to the rewritten system equation formula, S i,2 The derivative of in, It is a virtual control signal. The designed virtual control signal is: Among them, l i,2 is a designed normal number, It is b i,0 The estimated value of The parameter update law is designed as: Introduce a first-order low-pass filter and get the variable z i,3 as follows in, is the input of the low-pass filter, ι i,3 is a positive time constant; Step j(3≤j≤n i -1): Define S i,j for With i,j =in i(0,j) -z i,j According to the rewritten system equation, we can get S i,j The derivative of is: in, is a virtual control signal designed to: Among them, l i,j >0 is the design parameter, a first-order low-pass filter is introduced to obtain the variable z i,j+1 in, is the input of the low-pass filter, ι i,j+1 is a positive time constant; Step n i :definition for According to the rewritten system equation The derivative of is: Design temporary control signal w i (t) is: and The adaptive law is designed as: in, is a positive design parameter.

8. The adaptive pseudo-inverse control method according to claim 7, characterized in that: The Preisach model representing the hysteresis characteristics of interconnected systems is defined as w i (t)=∫∫ β≥α c α,β [u i ](t)μ(α,β)dαdβ, Among them, w i (t) is the output of the Preisach model, u i (t) is the input of the model, μ(α,β) represents the weight function; Using iterative search variables Finding a near-optimal signal So that the following formula holds: And determine the actual controller 9. The adaptive pseudo-inverse control method according to claim 8, characterized in that: Using iterative search variables Finding a near-optimal signal include: Set the actual input of the driver to [u i,min ,u i,max ],consider and γ α,β [u i ](t), ∫∫ β≥α γ α,β [u i ](t)μ(t,α,β)dαdβin[u i,min ,u i,max ] is monotonically increasing, and is defined as The following formula exists: Define a new variable Its input is in Let u i,0 (t) = u i,min , the following formula holds When w i,max (t)<w i (t), let u * (t) = u i,max ; When w i,min (t)>w i (t), let u * (t) = u i,min ; When w i,max (t)≥w i (t)≥w i,min (t), According to step 1-step 3, Step 1: Make Increase from 0 to w i,min (t); Step 2: Calculation if Increase until Go to step 3; Step 3: Make To stop the growth, we can Assigned to make The actual controller can be obtained using the pseudo-inverse algorithm

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