Design method of global tracking controller based on integral barrier lyapunov function

Abstract

The application discloses a design method of a global tracking controller based on an integral barrier Lyapunov function, comprising the following steps: constructing a strict feedback system model, setting a constraint condition, dividing a global time into one or more time segments according to a preset convergence time, adopting a time scale coordinate translation mapping to map each time segment to a corresponding infinite time domain, and respectively generating improved system models, all of which form a switching system; adopting a backstepping method to construct a tracking controller of the improved system model, the controller being based on an integral barrier Lyapunov function, recursively designing a control input, and adopting a fixed dwell time method to ensure global stable tracking control of the controller, solve the convergence of the system after the preset time, and optimize the global adaptive tracking effect.

Description

Technical Field

[0001] This invention relates to the field of control, and in particular to a design method for a global tracking controller based on an integral barrier Lyapunov function. Background Technology

[0002] In robot control, vehicle control, and other physical systems, there are stringent requirements for achieving state convergence within a pre-set time. Because pre-set time control methods not only stabilize the system within a fixed time but also allow for the pre-designing of convergence times independent of initial conditions based on system requirements, pre-set time control has become a cutting-edge control theory in recent years.

[0003] In existing research, Krishnamurthy et al. employed a dynamic scaling form with a time forcing term, achieving preset time stability of the system based on dynamic high-gain scaling technology. In addition to the single systems mentioned above, there are also numerous research results on preset time stability in multi-agent systems. Wang et al. established a new scaling function to realize the preset time distributed control of network multi-agent systems. Furthermore, Ning et al. proposed a time base generator method to solve the problem of actual fixed-time consistency in leaderless and leader-follower systems in integral multi-agent systems. Chen et al. constructed an event-triggered control for multi-agent systems to achieve binary consistency of preset time.

[0004] Liu et al. defined the concept of a pre-assigned finite-time performance function for time-delayed strict feedback nonlinear systems with quantized inputs. Cao et al. constructed a finite-time performance function to guarantee the convergence time and consistency of high-order nonlinear multi-agent systems under conditions of mismatched uncertainty and external disturbances. Ning et al. introduced an auxiliary quantity based on a time-varying function to ensure that a second-order multi-agent system achieves bipartite consensus tracking within a preset time.

[0005] The existing technology has the following problems:

[0006] 1) The above results are all based on solving the tracking control problem of preset time. Moreover, most preset time control methods are used to solve the stabilization problem, so the convergence of the system after the preset time is not considered.

[0007] 2) The above-mentioned preset time tracking control methods are all implemented by controlling the angle of gain. The designed function is generally defined as a normal number after the preset time. This design method cannot meet the needs of adaptation.

[0008] 3) To address the potential jump behavior at time mapping switching points in different time periods, independent Lyapunov functions for each subsystem are generated through time scale coordinate translation mapping, and the Fixed Dwell-time Method (FDT) is used to ensure stable global tracking control.

[0009] 4) In order to overcome the traditional obstacle of the conservatism of Lyapunov functions, an integral BLF is introduced to guarantee the full state constraints, rather than transforming the state constraints into known error constraints. Summary of the Invention

[0010] To address the aforementioned problems, this invention proposes a design method for a global tracking controller based on the integral barrier Lyapunov function.

[0011] The technical solution adopted in this invention is:

[0012] The design method of a global tracking controller based on the integral barrier Lyapunov function includes:

[0013] Construct a strict feedback system model, set constraints, divide the global time into one or more time segments according to the pre-set convergence time, use a time scale coordinate mapping function to map each time segment to the corresponding infinite time domain, and generate improved system models respectively. All improved system models form a switching system.

[0014] The tracking controller of the improved system model is constructed using a back-reasoning method. This tracking controller is based on the integral barrier Lyapunov function, and the virtual control inputs α1,...,α are recursively designed. i ,...,α n-1 Time-varying gain function Adaptive control law Where i (2≤i≤n-1), n ​​is the number of vectors in the system model, u is the actual control input of the improved system model, and the time-varying gain function is... Adaptive control law

[0015] Furthermore, the pre-set convergence time is denoted as T. P The time-scale coordinate mapping function is set as follows:

[0016]

[0017] Among them, T P The convergence time is a pre-defined time, t∈[0,T] P ), where τ is the time over an infinite field, τ∈[0,+∞).

[0018] Furthermore, the function of the improved system model is set as follows:

[0019]

[0020] in, and χ=[χ1,χ2,...,χ n ] T ∈R n This represents the measurable system state vector, where u∈R and y∈R represent the system input and output, respectively, and R is the set of real numbers. Let i represent the unknown differentiable nonlinear system function, i = 1, 2, ..., n, where n is the number of vectors in the system model. The control gain function represents an unknown differentiable function. Let λ(τ) represent the unknown external disturbances and system uncertainties, and be the derivative of the function of the time-scale coordinate mapping of the first pre-defined convergence time interval with respect to time τ over the infinite domain, i.e.: All system states are constrained to a compact set. In, among them, This refers to the designed obstacle function.

[0021] Furthermore, the time segment is set as: t∈[0,T] P ], t∈[T P 2T P ), t∈[2T P 3T P ), ..., [sT P ,(s+1)T P ),..., where s = 0, 1, 2,..., and each finite time interval is mapped to an infinite time domain through a scale-invariant translation transformation. The function for time-scale coordinate mapping is set as:

[0022]

[0023] Among them, T P The convergence time is preset, s = 0, 1, 2, ..., where s is a preset time interval T. P The number of time segments.

[0024] Furthermore, the functions of the subsystem model are set as follows:

[0025]

[0026] in, and χ=[χ1,χ2,...,χ n ] T ∈R nThis represents the measurable system state vector, where u∈R and y∈R represent the system input and output, respectively, and R is the set of real numbers. The function representing an unknown differentiable nonlinear system. The control gain function represents an unknown differentiable function. λ represents unknown external disturbances and system uncertainties. σ(t) (τ) The derivative of the function of the time-scale coordinate mapping with respect to time τ over the infinite domain is: All system states are constrained to a compact set. In, among them, This refers to the designed obstacle function;

[0027] To switch signals, each time-scale coordinate translation mapping can produce an independent improved system model.

[0028] Furthermore, the fixed residence time T of the switching signal σ(t) P Meet the conditions

[0029]

[0030] Where ε∈(0,C1), μ≥1, and C1 refers to a positive constant.

[0031] Furthermore, the error model of the improved system model tracking controller is as follows:

[0032]

[0033] Where e1 is the tracking error, α i-1 This serves as the system's virtual control input.

[0034] Actual control input u, time-varying gain function Adaptive control law The design steps include:

[0035] Step 1: Tracking error e1 = χ1 - y d Differentiate over time τ over the infinite field and choose the integral Lyapunov barrier function as:

[0036]

[0037] Among them, y d Let δ be the desired trajectory, and let δ be a substitution variable. A barrier function designed for time τ.

[0038] Using the integral Lyapunov barrier function Differentiating over the infinite field with respect to time τ, we use partial integration and L'Hôpital's rule to verify that the integral-type Lyapunov barrier function is bounded in the neighborhood of e1 = 0. Based on e2 = χ² - α1, we can obtain α1 and... The derivative relationship function is defined as follows: Given an unknown linear function S1(Z1), the general form of a neural network is S(Z) = Θ. *T To approximate the unknown function S1(Z1) using ψ(Z)+ε(Z), the virtual control input α1 and the time-varying gain function are designed. Adaptive control law α1 is about The function;

[0039] Step i (2≤i≤n-1): Tracking error e i =χ i -α i-1 Differentiate over the infinite field with respect to time τ, where α i-1 It is about The function sets an integral Lyapunov barrier function:

[0040]

[0041] Where, α i-1 For virtual control laws, A barrier function designed for time τ;

[0042] The above integral Lyapunov barrier function is differentiated with respect to time τ over an infinite field. The results are verified using partial integration and L'Hopital's rule. i =0 is bounded in its neighborhood, according to e i+1 =χ i+1 -α i α can be obtained i and The derivative relationship function defines an unknown linear function S. i (Z i Using neural networks to approximate and design virtual control input α i Time-varying gain function Adaptive control law

[0043] Step n: Tracking error e n =χ n -α n-1 Differentiating over time τ over an infinite field, we choose an integral-type Lyapunov barrier function:

[0044]

[0045] Where, α n-1 For virtual control laws, A barrier function designed for time τ;

[0046] The above integral Lyapunov barrier function is differentiated with respect to time τ over an infinite field. The results are verified using partial integration and L'Hopital's rule. n =0 is bounded in its neighborhood, according to e n =χ n -α n-1 α can be obtained n-1 and The derivative relationship function defines an unknown linear function S. n (Z n Using a neural network to approximate the actual control input u, a time-varying gain function is designed. Adaptive control law

[0047] Compared with existing technologies, the present invention has the following advantages:

[0048] 1) Unlike most preset time control methods that solve stabilization problems and only consider system convergence within a preset time, this paper proposes a time-scale coordinate translation mapping method to solve the tracking problem of preset time control by dividing the entire time axis into several time periods.

[0049] 2) Compared to most studies that implement preset time control from the perspective of control gain feedback, this study adopts a time-scale coordinate mapping method from a time perspective, mapping the preset time in a finite domain to a time variable in a transformed infinite domain. This provides a new general framework for studying preset time tracking control problems, applicable to various types of nonlinear systems.

[0050] 3) To address the potential jump behavior at time mapping switching points in different time periods, independent Lyapunov functions for each subsystem are generated through time scale coordinate translation mapping, and the Fixed Dwell-time Method (FDT) is used to ensure stable global tracking control.

[0051] 4) In order to overcome the traditional obstacle of the conservatism of Lyapunov functions, an integral BLF is introduced to guarantee the full state constraints, rather than converting the state constraints into known error constraints. Attached Figure Description

[0052] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present invention and should not be regarded as a limitation on the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.

[0053] Figure 1 According to the block diagram of the present invention

[0054] Figure 2 (a) The angular position χ1 of the link is tracked by the reference signal q. d ;

[0055] (b) The angular velocity χ2 of the connecting rod;

[0056] (c) The angular acceleration χ3 state quantity of the connecting rod;

[0057] Figure 3 This is the response curve for (a) error e1;

[0058] (b) Response curve for error e2;

[0059] (c) Response curve for error e3;

[0060] Figure 4 It is an adaptive law The response curve;

[0061] Figure 5 The input signal u is bounded; Detailed Implementation

[0062] 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. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations.

[0063] Therefore, the following detailed description of the embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0064] It should be noted that similar labels and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures.

[0065] In the description of this invention, it should be noted that the terms "first," "second," "third," etc., are used only for distinguishing descriptions and should not be construed as indicating or implying relative importance. Furthermore, the terms "horizontal," "vertical," etc., do not indicate that the component is required to be absolutely horizontal or suspended, but can be slightly tilted. For example, "horizontal" simply means that its direction is more horizontal than "vertical," and does not mean that the structure must be completely horizontal, but can be slightly tilted.

[0066] In the description of this invention, it should also be noted that, unless otherwise explicitly specified and limited, the terms "set," "install," "connect," and "link" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0067] like Figure 1 As shown, a strict feedback system model is constructed, and constraints are set. The global time is divided into one or more time segments according to the pre-set convergence time. A time scale coordinate mapping function is used to map each time segment to the corresponding infinite time domain, and an improved system model is generated for each segment. All improved system models form a switching system.

[0068] (a) A type of strict feedback system model is as follows:

[0069]

[0070] in, and χ=[χ1,χ2,...,χ n ] T ∈R n This represents the measurable system state vector, where u∈R and y∈R represent the system input and output, respectively. i = 1, 2, ..., n represents an unknown differentiable nonlinear system function. Represents an unknown differentiable control gain function. This represents unknown external disturbances and system uncertainties. Specifically, all system states are constrained to a compact set. middle.

[0071] Assumption 1: The expected trajectory y d It is smooth, and its nth derivative is... and It is bounded and satisfies the condition. Where B0 is a positive constant. For any t > 0, there exists a function Y0(t) such that the desired trajectory yd satisfy And the derivatives of the expected trajectory at each order satisfy i = 1, 2, ..., n.

[0072] Assumption 2: There exist known positive constants such that g im and g iM satisfy

[0073] Assumption 3: For the disturbance term There exists an unknown positive constant d. Mi Make i = 1, 2, ..., n.

[0074] Assumption 4: There are unknown positive constants. and Make And there are

[0075] Lemma 1: Suppose there exists a continuous positive definite radially unbounded function. For scalars α, η > 0, 0 < h < 1, the following holds:

[0076]

[0077] At this point, the residual set of the system solution is stable in finite time.

[0078] Lemma 2: For i = 1, 2, ..., n, The following inequalities hold:

[0079]

[0080] Where, α i-1 For virtual control laws, A barrier function designed for time τ;

[0081] (ii) The pre-set convergence time is denoted as T. P That is, it achieves convergence within a specified time interval, i.e., t∈[0,T] P Using the following time-scale coordinate mapping method, the specified time t∈[0,T] over the finite field is mapped. P The time scale mapping function for a time segment, τ∈[0,+∞), mapped to an infinite field, is set as follows:

[0082]

[0083] Differentiating the above expression with respect to the transformed time τ, we get

[0084]

[0085] At this point, λ(τ) is a monotonically decreasing bounded function that satisfies in It is a positive number.

[0086] The design of the time-scale coordinate mapping function needs to satisfy the following properties: 1) 2) 3) The function is differentiable and monotonically increasing, and its derivative is always positive, thus ensuring that the sign of the function remains unchanged when combined with the gain function. It is worth noting that the function design is not strictly limited to this form; as long as the above conditions are met, it can be of exp, tan, or logarithmic form.

[0087] Therefore, the system model can be transformed into the following form:

[0088]

[0089] When the original system model (1) is transformed into the improved system model (6), the control problem within a preset time period is transformed into a general asymptotic convergence problem. As long as stable convergence can be achieved in the infinite time domain of the new system, stable convergence can also be achieved in the preimage of the mapping relationship, that is, within the pre-specified time period.

[0090] The adaptive controller for the nonlinear system (6) is constructed using backpropagation techniques. First, the tracking error of the improved system model is defined as follows:

[0091]

[0092] Where e1 is the tracking error, α i-1 This serves as the virtual control input for the system.

[0093] Actual control input u, time-varying gain function Adaptive control law The design steps include:

[0094] Step 1: Consider the improved system model of (6) for e1=χ1-y d The derivative is as follows:

[0095]

[0096] The integral-type Lyapunov barrier function (IBLF) is chosen to be in the following form:

[0097]

[0098] One type of barrier Lyapunov function is chosen to handle fully constrained problems. Furthermore, traditional barrier Lyapunov functions have two typical forms: logarithmic barrier Lyapunov functions and tannic barrier Lyapunov functions. The logarithmic barrier Lyapunov function is the most common form, while the tan-BLF is a barrier Lyapunov function that can handle both constrained and unconstrained systems. Compared to IBLF, the two traditional barrier Lyapunov functions mentioned above share a common drawback: in handling constrained problems, it is necessary to first convert state constraints into error constraints before proceeding to the next derivation. IBLF, however, eliminates this step, thus removing the conservative restrictions in the conversion process.

[0099] because Easy to know It is positive definite, and it is in the set The given information is continuously differentiable and satisfies the decreasing condition. Therefore, the following inequalities hold:

[0100]

[0101] The variable substitution δ = ve1 is involved.

[0102] definition:

[0103]

[0104] therefore, The time derivative is expressed as:

[0105]

[0106] Then, based on the mean value theorem for integrals and uniformly continuous functions... It can be obtained

[0107]

[0108] in

[0109] Furthermore, pay attention It's about y d (τ) and The function, then The derivative is

[0110]

[0111] The second term in formula (14) satisfies

[0112]

[0113] Using integration by parts, we obtain:

[0114]

[0115] in,

[0116]

[0117] Similarly, the third term in formula (14) satisfies:

[0118]

[0119] Similarly, using integration by parts, we obtain:

[0120]

[0121] in,

[0122]

[0123] Using L'Hôpital's rule, we can obtain

[0124]

[0125] It is easy to know that and The boundary is defined in the neighborhood of e1 = 0.

[0126] From e2 = χ2 - α1, we can obtain

[0127]

[0128] Define an unknown nonlinear function S1(Z1), where

[0129]

[0130] The general form of the neural network is S(Z)=Θ. *T Using ψ(Z)+ε(Z) to approximate the unknown function S1(Z1), we obtain...

[0131]

[0132] Among them For θ i The estimated value is given, and l1 is a positive design constant. Neural network weight estimation error. Defined as

[0133] The virtual control input α1 is designed as follows:

[0134]

[0135] In the formula, c1 and κ1 are positive design constants.

[0136] Time-varying gain function It can be represented as:

[0137]

[0138] In the formula, ο1 is a positive design constant.

[0139] design The adaptive control law is as follows:

[0140]

[0141] Where ρ1 and σ1 are both positive design constants, for When choosing It can be inferred that The conclusion.

[0142] Substituting (25) into (24), we get:

[0143]

[0144] in Using Cauchy's inequality and Young's inequality, we have,

[0145]

[0146] Using the definition in (26), we can obtain:

[0147]

[0148] According to equations (29) and (30), (28) can be further expressed as:

[0149]

[0150] Step i (2≤i≤n-1): Each step i recursively uses a similar derivation process.

[0151] Using e i =χ i -α i-1 Then e i The dynamic equations are described as follows:

[0152]

[0153] Where α i-1 It is about The function, therefore for α i-1 Differentiation yields:

[0154]

[0155] Consider the following integral barrier Lyapunov function:

[0156]

[0157] The variable substitution δ = ve i And the following inequalities hold:

[0158]

[0159] in These are positive constants, due to the virtual controllers α1,...,α n-1 It is a continuously differentiable function and satisfies

[0160] Similar to the derivation process in step one, The first-order time derivative can be further expressed as:

[0161]

[0162] The second term of formula (36) satisfies,

[0163]

[0164] in,

[0165]

[0166] The third term of formula (36) satisfies:

[0167]

[0168] in,

[0169]

[0170] It is easy to know that and In e i Bounded is defined within the neighborhood of 0.

[0171] via e i+1 =χ i+1 -α i Substituting equations (32), (37), and (40) into equation (36) yields...

[0172]

[0173] Define an unknown nonlinear function S i (Z i )

[0174]

[0175] in

[0176] The general form of the neural network is S(Z)=Θ. *T To approximate the unknown function S using ψ(Z)+ε(Z) i (Z i ),get

[0177]

[0178] Among them l i It is a positive design constant.

[0179] Design virtual control input α i as follows:

[0180]

[0181] In the formula c i and κ i It is a positive design constant.

[0182] Time-varying gain function It can be represented as:

[0183]

[0184] In the formula i It is a positive design constant.

[0185] design The adaptive control law is as follows:

[0186]

[0187] Where ρ1 and σ1 are both positive design constants. Substituting equation (44) into (43) yields...

[0188]

[0189] in Using Cauchy's inequality and Young's inequality, we have:

[0190]

[0191] Using the definition in (45), we can obtain:

[0192]

[0193] According to equations (48) and (49), (47) can be further expressed as:

[0194]

[0195] Step n: According to en =χ n -α n-1 Then e n The dynamic equations are described as follows:

[0196]

[0197] in It can be represented as:

[0198]

[0199] Consider the following integral barrier Lyapunov function:

[0200]

[0201] The variable substitution δ = ve n And the following inequalities hold:

[0202]

[0203] Similar to step one, The first-order time derivative can be expressed as:

[0204]

[0205] The second term in formula (55) satisfies:

[0206]

[0207] in:

[0208]

[0209] Similarly, the third term of formula (55) satisfies,

[0210]

[0211] in:

[0212]

[0213] Similarly, and In e n Bounded is defined within the neighborhood of 0.

[0214] via e n =χ n -α n-1 Substituting equations (56) and (58) into equation (55) yields:

[0215]

[0216] Define an unknown nonlinear function S n (Z n )as follows:

[0217]

[0218] in,

[0219]

[0220] Using neural networks to approximate the unknown function S n (Z n ),get The first time derivative is:

[0221]

[0222] Among them l n It is a positive design constant.

[0223] The actual control input u is designed as follows:

[0224]

[0225] In the formula c n and κ n It is a positive design constant.

[0226] Time-varying gain function It can be represented as:

[0227]

[0228] In the formula n It is a positive design constant.

[0229] design The adaptive control law is as follows:

[0230]

[0231] Where, ρ n and σ n All are positive design constants. Substituting equation (63) into equation (62), we get...

[0232]

[0233] in, Using Cauchy's inequality and Young's inequality, we have,

[0234]

[0235] Using the definition in (64), we have,

[0236]

[0237] Based on equations (67) and (68), equation (66) can be further expressed as:

[0238]

[0239] The design process of the adaptive tracking controller in the infinite time domain has been completed.

[0240] Considering the nonlinear system (6) under assumptions 1-3, the constructed virtual control laws are (25) and (44), and the adaptive laws are (27), (46) and (65). Based on the designed control laws, the designed actual control law is (63). When the initial conditions are satisfied... At that time, the designed control scheme can guarantee that: (1) the signal of the closed-loop system is bounded and converges to an arbitrarily small region in a finite time; (2) all state constraints are in the set middle.

[0241] λ(τ) is used in the denominator of the control law term. Due to the properties of the mapping function λ(τ), the transformed system exhibits singularities as it approaches convergence (τ→+∞). To avoid this singularity problem, a finite-time control scheme is proposed to accelerate the convergence speed. Therefore, the transformed system can achieve convergence in a finite time T. max The system converges to the compact set within the range ≤+∞, and correspondingly, the original system can also converge to the compact set in a finite time T. max (τ s (t))≤T P The internal convergence is achieved.

[0242] Preset time T P The subsequent tracking and control issues are addressed using the following methods:

[0243] The process of mapping using time-scale coordinates not only involves mapping the preset time t∈[0,T] over a finite field. P Mapping this to the infinite time domain also requires setting the preset time T. P The time interval is then divided into several segments, which are mapped one by one in the subsequent process, for example, t∈[T P 2T P ), t∈[2T P 3T P ), ..., [sT P ,(s+1)T P ),..., where s=0,1,2,…, and each finite time interval is mapped to an infinite time domain through a scale-invariant translation transformation.

[0244]

[0245] Differentiating the above expression with respect to the transformed time variable τ, we get...

[0246]

[0247] Therefore, system equation (1) can be rewritten as follows:

[0248]

[0249] in To switch signals, each time-scale coordinate translation mapping can generate an independent subsystem, and this new system can be regarded as a switching system.

[0250] Using the same design approach, the Lyapunov function L of each improved system σ(t) All can be derived into the following form:

[0251]

[0252] Since the inverse transformation of formula (70) is:

[0253]

[0254] Replacing the time variable τ in the control law with t, the method of achieving preset time control through a time mapping function from a time scale perspective is universal. Therefore, the virtual control law and actual control input are designed as follows:

[0255]

[0256]

[0257] parameter The adaptive update law is:

[0258]

[0259] Note t s The time parameter is the instant of switching when the system changes. For convenience, it is assumed that t0 = 0 < t1 < t2 < ... < t s-1 <t s <.... Given a positive constant, if the inequality t satisfies for all s = 0, 1, 2, ... s+1 -t s =T P Then the constant T is called P Let σ(t) be the fixed dwell time of the switching signal. It is worth noting that the fixed dwell time of the switching signal in this section is the system's preset convergence time T. P Let N(t) s ,T) represents the switching signal σ(t) in the time interval (t).s The total number of items on T is then...

[0260]

[0261] Consider an uncertain switching nonlinear system (72) with adaptive control laws (75) and (76) and an adaptive parameter update law (77), if the fixed residence time T of the switching signal σ(t) P Conditions met:

[0262]

[0263] Where ε∈(0,C1), μ≥1. Then all closed-loop signals of the switching system are globally bounded, and the explicit boundary of the steady-state tracking error is:

[0264]

[0265] Conditions in formula (79) That is, the fixed dwell time T of the switching signal P The larger the parameter C1, the larger its allowable range ε. As shown in (80), increasing the value of ε helps to reduce the steady-state tracking error, which means that the tracking error performance can be improved by increasing C1. In addition, decreasing Ξ and Δ also helps to reduce the tracking error.

[0266] Reference Figures 2-5 As shown:

[0267] Example 1: Trajectory tracking control of a single-link robotic arm driven by a brushed DC (BDC) motor;

[0268] The dynamic equations of the single-link robotic arm powered by a BCD motor are expressed as follows:

[0269]

[0270] Among them, q, and These represent the angular position, velocity, and acceleration of the connecting rod, respectively; I is the motor armature current; and V is the input control voltage.

[0271] The parameter values ​​are set as follows: M = 1 kg m² represents mechanical inertia, B = 1 Nm s / rad represents the viscous friction coefficient at the connection, N = 10 represents a normal coefficient related to the load mass and gravity coefficient, L = 0.05 H represents armature inductance, R = 0.5 Ω represents armature resistance, K B =10Nm A represents the back electromotive force coefficient, Δ I =sin(χ2)cos(χ1) represents the disturbance corresponding to the motor current;

[0272] The first subsystem represents the dynamic equations of the mechanical subsystem, i.e., the single-link robot, and the second subsystem represents the dynamic equations of the BDC electrical subsystem.

[0273] Let χ1 = q, If χ3 = I, then the system equation can be expressed as:

[0274]

[0275] in, g² = 1 / M, d² = Δ I / M, u=V, f3(χ)=-Rχ3 / LK B χ2 / L, g3=1 / L.

[0276] The reference signal required for tracking by the single-link robot system is set as follows: In practical applications, the angular position q(t) and velocity of the connecting rod... and acceleration All are subject to tight clustering The constraints, wherein the symmetric time-varying constraint boundary is set as

[0277] Control objective: Design an adaptive tracking controller so that the angular position χ1 of the link can follow the reference signal q within a preset time. d And all states do not exceed the agreed compact set.

[0278] The parameters in formulas (25)-(27) of step 1 are designed as follows: c2=1, κ1=0.5, l1=1, ρ1=0.1, σ1=0.5; the parameters in formulas (44)-(46) of step 2 are designed as follows: c2=1, κ2=0.5, l2=1, ρ2=0.1, σ2=0.5; the parameters in formulas (63)-(65) of step 3 are designed as follows: c3=1, κ3=0.5, l3=1, ρ3=0.5, σ3=0.2. The initial conditions are set as [χ1(0),χ2(0),χ3(0)]=[0.5,0,0]. T P = 10 seconds. The RBFNN used to approximate S1 (Z1) contains 5 nodes, the RBFNN used to approximate S2 (Z2) contains 11 nodes, and the RBFNN used to approximate S3 (Z3) contains 13 nodes. Each radial basis function has a width of 2.

[0279] Beneficial effects:

[0280] The simulation results are shown in the figure. Figure 2 As can be seen in (a), at the pre-set time T P Within the system, the angular position χ1 of the connecting rod can follow the reference signal q with good tracking performance.d The state variables, angular velocity χ2 and acceleration χ3 of the connecting rod, are shown below. Figure 2 (b) and Figure 2 As shown in (c), all system state variables are bounded signals. Figure 3 The trajectory of the tracking error signal is described, and it can be concluded that errors e1, e2, and e3 are all bounded and gradually approach zero. This demonstrates a bounded and continuous adaptive law. The response curve is as follows Figure 4 As shown. Furthermore, by Figure 5 It can be shown that the control input u is bounded and falls within the practically acceptable range. Compared with traditional controllers, this method has a faster convergence speed and higher tracking accuracy, and achieves adaptive tracking control with a preset time.

[0281] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention in any way. Any simple modifications, alterations, and equivalent changes made to the above embodiments based on the inventive essence shall still fall within the protection scope of the present invention.

Claims

1. A design method for a global tracking controller based on the integral barrier Lyapunov function, including: Construct a strict feedback system model, set constraints, divide the global time into one or more time segments according to the pre-set convergence time, use a time scale coordinate mapping function to map each time segment to the corresponding infinite time domain, and generate improved system models respectively. All improved system models form a switching system. The tracking controller of the improved system model is constructed using a back-reasoning method. This tracking controller is based on the integral barrier Lyapunov function, and the virtual control input is designed recursively. ,..., ,..., Time-varying gain function ,..., ,..., Adaptive control law ,..., ,..., ,in, (2≤ ≤ -1), Given the number of vectors in the system model, design the actual control input for the improved system model. Time-varying gain function Adaptive control law ; Set the time period as follows: , ,..., ,...,in The function for time-scale coordinate mapping is set as follows: in, The pre-set convergence time, , Preset time period The number of time segments.

2. The design method of the global tracking controller based on the integral barrier Lyapunov function according to claim 1, characterized in that, The pre-set convergence time is denoted as The time-scale coordinate mapping function is set as follows: ； in, The pre-set convergence time, , For time over an infinite field, .

3. The design method of the global tracking controller based on the integral barrier Lyapunov function according to claim 2, characterized in that, The function of the improved system model is set as follows: ； in, and Represents the measurable system state vector. and These represent the system input and output, respectively. For the set of real numbers, The function representing an unknown differentiable nonlinear system. , The number of vectors in the system model. The control gain function represents an unknown differentiable function. This represents unknown external disturbances and system uncertainties. The function that maps the timescale coordinates of the first pre-defined convergence time interval is differentiated with respect to time τ over the infinite domain, i.e.: All system states are constrained to a compact set. In, among them, This refers to the designed obstacle function.

4. The design method of the global tracking controller based on the integral barrier Lyapunov function according to claim 1, characterized in that, The function of the improved system model is set as follows: ； in, and Represents the measurable system state vector. and They represent the system input and output, respectively. For the set of real numbers, , , representing an unknown differentiable nonlinear system function, The control gain function represents an unknown differentiable function. This represents unknown external disturbances and system uncertainties. The function that maps time scale coordinates is differentiated with respect to time τ over the infinite domain, i.e.: All system states are constrained to a compact set. In, among them, This refers to the designed obstacle function; To switch signals, each time-scale coordinate translation mapping can produce an independent improved system model.

5. The design method of the global tracking controller based on the integral barrier Lyapunov function according to claim 4, characterized in that, Switching signals Fixed stay time The following conditions must be met: ； in, , ,in It refers to a positive constant.

6. The design method of the global tracking controller based on the integral barrier Lyapunov function according to claim 3 or 5, characterized in that, The error model of the improved system model tracking controller is as follows: , ； in To track errors, This serves as the virtual control input for the system. Actual control input Time-varying gain function Adaptive control law The design steps include: Step 1: Tracking Error Differentiate over time τ over the infinite field and choose the integral-type Lyapunov barrier function as: ； in, For the desired trajectory, As a substitution variable, For time The barrier function designed above; Using the integral Lyapunov barrier function Differentiating over time τ in an infinite field, using partial integration and L'Hopital's method, we can find that the integral Lyapunov barrier function... The neighborhood is bounded, according to , can obtain and The derivative relation function defines an unknown linear function. It adopts the general form of neural networks. To approximate an unknown function Design the virtual control input Time-varying gain function Adaptive control law , It is about , , , The function; step Tracking error Differentiate over the infinite field with respect to time τ, where, It is about The function sets an integral Lyapunov barrier function: ； in, For virtual control laws, For time The barrier function designed above; The above integral-type Lyapunov barrier function is differentiated with respect to time τ over an infinite field. The partial integration method and L'Hopital's rule are used to verify that the integral-type Lyapunov barrier function... The neighborhood is bounded, according to , can obtain and The derivative relation function defines an unknown linear function. Using neural networks to approximate and design virtual control inputs Time-varying gain function Adaptive control law ; Step n: Tracking error Differentiating over time τ over an infinite field, we choose an integral-type Lyapunov barrier function: ； in, For virtual control laws, For time The barrier function designed above; The above integral-type Lyapunov barrier function is differentiated with respect to time τ over an infinite field. The partial integration method and L'Hopital's rule are used to verify that the integral-type Lyapunov barrier function... The neighborhood is bounded, according to , can obtain and The derivative relation function defines an unknown linear function. By using neural networks to approximate and design the actual control input u, a time-varying gain function is obtained. Adaptive control law .

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