A biomimetic fishtail event triggering and limiting control method with unknown perturbations

Abstract

This invention discloses an event-triggered constraint control method for a bionic fish tail with unknown perturbations. The method involves the following steps: Based on Hamiltonian principles, a dynamic model of a flexible bionic fish tail with unknown boundary perturbations is constructed; utilizing an auxiliary system and considering output constraints, a tracking error model is constructed; considering event triggering schemes, a Lyapunov function is designed based on the error model; based on the Lyapunov function, an adaptive boundary controller is constructed; and event-triggered adaptive constraint control is implemented for the flexible bionic fish tail with unknown boundary perturbations. This invention effectively controls the oscillation of the flexible bionic fish tail, addresses the impact of boundary perturbations and output constraints on control performance, and allows the flexible bionic fish tail to trigger control signals based on fixed thresholds, improving communication efficiency while ensuring the stability of the bionic fish tail.

Description

Technical Field

[0001] This invention relates to the fields of event-triggered control, vibration tracking control, adaptive control, and boundary control, and specifically to a biomimetic fishtail event-triggered constraint control method with unknown disturbances. Background Technology

[0002] Oceans cover 70% of the Earth's surface. Since the 21st century, humans have widely utilized autonomous underwater vehicles (AUVs) to develop and utilize marine resources. AUVs have shown significant application value in topographic surveying, underwater environmental monitoring, deep-sea salvage, and even military fields. Traditional research has mainly focused on AUVs with propeller structures. However, as a type of AUV, the flexible bionic fish possesses characteristics such as small mass, strong stealth, and high mobility, making it applicable to tasks such as underwater confined space exploration, reconnaissance, and information transmission. The flexible bionic fish is primarily controlled by its tail. In research on the flexible bionic fish tail, a non-uniform Euler Bernoulli beam is typically used as the basic model. However, how to control the flexible bionic fish tail to track reference vibration signals using an adaptive boundary controller remains a problem that urgently needs to be solved.

[0003] In practical engineering applications, flexible bionic fish tails will inevitably encounter unknown boundary disturbances, which can affect the control performance of the actuators and even lead to system instability. Therefore, the controller of a bionic fish tail should have a certain degree of anti-interference capability.

[0004] Output constraints are a crucial factor affecting system performance; in practical engineering, excessively high system output can even compromise system stability. This is especially true for nonlinear systems, where limiting output to a certain range is essential.

[0005] Typically, control signals are transmitted by the control system via a network. However, network communication resources are limited. Therefore, how to effectively utilize these resources while ensuring the stability of the control system and allocating remaining resources to handle other tasks is crucial in the control research of flexible biomimetic fishtail systems. Summary of the Invention

[0006] The purpose of this invention is to overcome the above-mentioned defects in the prior art and provide a biomimetic fish tail event triggering and limiting control method with unknown disturbances.

[0007] The objective of this invention can be achieved by adopting the following technical solutions:

[0008] A biomimetic fish tail event triggering and limiting control method with unknown perturbation, the biomimetic fish tail event triggering and limiting control method comprising the following steps:

[0009] S1. Based on the dynamic characteristics of the flexible bionic fish tail, and considering Hamilton's principle and unknown boundary perturbations, a dynamic model of the flexible bionic fish tail is constructed.

[0010] S2. Using an auxiliary system based on variable separation and considering output constraints, construct a tracking error model for a flexible bionic fish tail.

[0011] S3. Considering the event triggering scheme, construct a Lyapunov function based on the tracking error model;

[0012] S4. Based on the Lyapunov function, construct an adaptive boundary controller for a flexible bionic fish tail;

[0013] S5. Based on the adaptive boundary controller, the event-triggered limit control of the flexible bionic fish tail with unknown boundary disturbances is achieved through the boundary actuator located at the fish tail.

[0014] Furthermore, since the model of the flexible bionic fish tail is non-uniform, it cannot be established using a conventional uniform Bernoulli beam model. Considering the dynamic characteristics of the flexible bionic fish tail in step S1, including its kinetic energy, potential energy, and the virtual work done by non-conservative forces on the tail, the kinetic energy, potential energy, and virtual work are substituted into Hamilton's principle, and the unknown boundary perturbation is considered, resulting in the following dynamic model of the flexible bionic fish tail:

[0015]

[0016] In the formula, A(s) is the variable damping coefficient of the flexible bionic fish tail, ρ(s) is the non-uniform linear density of the fish tail, and EI(s) is the bending stiffness of the fish tail. s is a spatial position variable, t is a time variable, L represents the length of the flexible bionic fish tail; r(s,t) represents the vibration offset of the flexible fish tail at position s and time t; where ρ(s) and A(s) need to satisfy the assumptions: |ρ(s)|≤m1, |A(s)|≤m2, where m1 and m2 are the upper bounds of ρ(s) and A(s), respectively. These are the first and second partial derivatives of r(s,t) with respect to time t, respectively. [EI(s)r″(s,t)]″ is the second partial derivative of [EI(s)r″(s,t)] with respect to position s, and r″(s,t) is the second partial derivative of r(s,t) with respect to position s.

[0017] The boundary conditions for the flexible bionic fish tail are:

[0018]

[0019] In the formula Let d(t) represent the control output at the flexible fishtail connection, and let d(t) represent the unknown boundary disturbance at the flexible fishtail connection, satisfying: in Let d(t) be the first-order partial derivative with respect to time t. and Let d(t) and d(t) represent respectively The upper bound of the equation is EI(0), which is the bending stiffness of the flexible fishtail at s=0; r(0,t) represents the vibration offset of the flexible fishtail at position s=0 and time t; r″(L,t) and r″′(L,t) represent the second and third position partial derivatives of the vibration offset of the fishtail at position s=L and time t, respectively. From the above model, it can be seen that the vibration of the flexible bionic fishtail can be controlled by adjusting the output u(t).

[0020] Furthermore, in step S2, the flexible bionic fish tail needs to track the reference signal to swing. However, the vibration offset r(s,t) of the flexible fish tail is a two-dimensional function coupled with the position variable s and the time variable t. Therefore, we consider using the auxiliary function Γ(s)T(t) with variable separation to construct an auxiliary system, and define the tracking error h(s,t) as follows:

[0021] h(s,t)=r(s,t)-Γ(s)T(t)

[0022] In the formula Γ(s) is a function that depends only on the position s, and T(t) represents a time-dependent oscillating signal with a fixed amplitude, defined as... Where Ξ is a matrix that can be diagonalized along the imaginary axis and has eigenvalues ​​that guarantee T(t) to maintain a constant amplitude oscillation, and the reference signal r of the flexible fishtail at s = L is defined. d (s,t)=Γ L T(t), where Γ L Let Γ(s) be the value of Γ(s) at s = L, and let Γ(s) satisfy the following equation:

[0023]

[0024] Where Γ(0) is the value of Γ(s) at s=0, and Γ″(L) and Γ″′(L) are the second and third positional partial derivatives of Γ(s) at position s=L with respect to position s, respectively;

[0025] The resulting error model is as follows:

[0026]

[0027] in, These are the first and second partial derivatives of h(s,t) with respect to time t, respectively; h″(s,t) is the second partial derivative of h(s,t) with respect to position s; [EI(s)h″(s,t)]″ is the second partial derivative of [EI(s)h″(s,t)] with respect to position s; h(0,t) represents the magnitude of the tracking error of the flexible fishtail at position s=0 and time t; h″(L,t) and h″′(L,t) represent the second and third position partial derivatives of the tracking error of the fishtail at position s=L and time t, respectively; and h″(0,t) is the second position partial derivative of h(0,t) at position s=0 and time t.

[0028] Meanwhile, considering output limitations, the tracking error |ε| is defined as |r(L,t)-r d (L,t)|, and ε must satisfy |ε|<k a The output limit k a It is a positive constant. From the above tracking error model, it can be seen that the vibration signal tracking problem has been transformed into an error signal convergence problem.

[0029] Furthermore, control signals are typically sent by the control system via a network. However, network communication resources are limited. The advantage of event-triggered control lies in its ability to effectively utilize these resources while ensuring control stability and allocating remaining resources to handle other tasks. Step S3 considers a time-triggered scheme based on a fixed threshold, defining an increasing time sequence {t0, t1, ..., t...}. i} represents the event time for sampling the controller, where t0 represents the initial sampling time, t1 represents the next sampling time, and so on, increasing sequentially up to the i-th sampling time t. i When at time t i When the triggering condition is met, the control law will be updated at time t. i to t i+1 The internal value remains constant, and the adaptive event-triggered control input u(t) is defined as follows:

[0030]

[0031] Where, v(t) i ) is a continuous control law, and the control input u(t) will be in the range [t]. i ,t i+1 The value remains unchanged within the range and is equal to v(t). i Define the control input u(t) and the continuous control law v(t). i Error between ) Define a fixed threshold strategy to specify event triggering conditions. Where t(0) = 0, p represents a fixed trigger threshold, which is a known positive constant, and it is assumed that a continuously time-varying function exists. Make Established, among which The following conditions must be met:

[0032]

[0033] Based on the aforementioned error model and event triggering scheme, the Lyapunov function is constructed as follows:

[0034]

[0035] in,

[0036]

[0037]

[0038]

[0039] In the formula, λ, These are the first and second control parameters for adaptive boundary control of the flexible bionic fish tail, respectively; k1 and k2 are the first and second control gains of the flexible fish tail, respectively. The estimation error for the flexible fishtail boundary perturbation is defined as: in θ represents the boundary perturbation estimate for the flexible fishtail, and θ is the first scaling parameter.

[0040] Furthermore, in step S4, it is very difficult to design a controller by solving conventional state equations based on the higher-order Lyapunov function. Therefore, from the perspective of energy change, the process of constructing an adaptive boundary controller using the Lyapunov direct method is as follows:

[0041] Taking the first derivative of the Lyapunov function with respect to time t, and based on Lyapunov stability theory, combined with the above event triggering scheme based on a fixed threshold, the following adaptive boundary controller with unknown perturbation is finally constructed:

[0042]

[0043] In the formula, k0 is the third control gain of the flexible bionic fish tail; φ1(t) is defined as: Defined as: Define the bounding perturbation estimate of the flexible fishtail as: Where c and ζ1 are the second and third scaling parameters, respectively.

[0044] Furthermore, after implementing event-triggered constraint control on the flexible bionic fish tail with unknown boundary perturbations, the process further includes verifying the stability of the flexible bionic fish tail with unknown boundary perturbations under adaptive event-triggered constraint control, as follows:

[0045] By constraining the first control parameter λ and the second control parameter in the Lyapunov function The first control gain k1 and the second control gain k2 of the flexible fishtail ensure the positive definiteness of the Lyapunov function;

[0046] Find the first derivative of the Lyapunov function with respect to time t, and determine that the first derivative of the Lyapunov function with respect to time t is negative semi-definite.

[0047] By applying Lyapunov's bounded stability theory and adjusting the control gain and parameters, the vibration of the flexible bionic fish tail is determined to be uniformly stable in the zero-point region, thereby verifying the stability of the bionic fish tail under event-triggered constraint control with unknown disturbances.

[0048] Furthermore, in step S5, the process of implementing event-triggered constraint control based on the adaptive boundary controller and the flexible bionic fish tail with unknown boundary perturbations is as follows:

[0049] In a real-world experimental environment, as boundary perturbations are generated, an adaptive boundary perturbation update law is calculated in the boundary controller of the flexible bionic fish tail. The adaptive boundary controller v(t) is calculated, and after completion, the boundary actuator of the flexible bionic fish tail is updated considering the event trigger threshold. As the control time increases, the boundary actuator continuously outputs control force, constantly adjusting the vibration offset r(s,t). Finally, the motion trajectory of the flexible bionic fish tail tracks the target vibration trajectory, realizing event-triggered adaptive boundary control. This process has a fast response speed, low implementation cost, and good control effect.

[0050] The present invention has the following advantages and effects compared with the prior art:

[0051] (1) The biomimetic fish tail event triggering limit control method with unknown disturbance proposed in this invention can use an adaptive estimation method to estimate and cancel the boundary disturbance of the flexible fish tail. Therefore, this invention can effectively suppress the influence of boundary disturbance on the system.

[0052] (2) Compared with the existing flexible bionic fish tail control method, the bionic fish tail with output limitation proposed in this invention can use the obstacle Lyapunov function to constrain the error output of the flexible fish tail within a certain range. Therefore, this invention can limit the output error, thereby improving the system control performance.

[0053] (3) In the control method proposed in this invention, the flexible bionic fish tail intermittently triggers the controller signal according to a fixed threshold event triggering scheme, and can effectively achieve the control objective. Therefore, this invention can improve the communication efficiency of the control system while ensuring its stability, based on the fixed threshold event triggering control method. Attached Figure Description

[0054] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this application, illustrate exemplary embodiments of the invention and, together with their description, serve to explain the invention and do not constitute an undue limitation thereof. In the drawings:

[0055] Figure 1 This is a flowchart of a biomimetic fishtail event triggering and limiting control method with unknown disturbances disclosed in this invention;

[0056] Figure 2 This is a schematic diagram of the Euler-Bernoulli beam model of the flexible bionic fish tail in Embodiment 1 of the present invention;

[0057] Figure 3 This is a schematic diagram of the flexible bionic fish tail structure in Embodiment 1 of the present invention;

[0058] Figure 4 This is a schematic diagram of the vibration offset simulation results of the bionic fish tail with unknown boundary perturbation under event-triggered limiting control in Embodiment 2 of the present invention.

[0059] Figure 5 This is a schematic diagram of the boundary perturbation simulation results of the bionic fish tail with unknown boundary perturbation under event triggering limit control in Embodiment 2 of the present invention;

[0060] Figure 6 This is a schematic diagram of the output limitation simulation results of the bionic fish tail with unknown boundary perturbation in the event triggering limitation control of Embodiment 2 of the present invention;

[0061] Figure 7 This is a schematic diagram of the simulation results of the continuous control signal v(t) and trigger time when the bionic fish tail with unknown boundary disturbance is controlled by event triggering limitation in Embodiment 2 of the present invention. 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, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0063] Example 1

[0064] This embodiment specifically discloses a biomimetic fishtail event triggering limitation control method with unknown disturbances, such as... Figure 1 As shown, Figure 1 This is a flowchart of a biomimetic fishtail event triggering and limiting control method with unknown disturbances disclosed in Embodiment 1, which specifically includes the following steps:

[0065] S1. Based on the dynamic characteristics of the flexible bionic fish tail, and considering Hamilton's principle and unknown boundary perturbations, a dynamic model of the flexible bionic fish tail is constructed.

[0066] Figure 2 The figure shows a schematic diagram of the Euler-Bernoulli beam model of the flexible bionic fish tail. As shown in the figure, the flexible bionic fish tail can be approximated as a non-uniform Euler-Bernoulli beam model, and the fish tail vibrates only in the OSR plane. Figure 3 The diagram shown is a schematic of a flexible biomimetic fish tail structure. Figure 3 The shaded rectangle on the left represents the boundary actuator of the flexible bionic fishtail, which houses a DC motor capable of generating torque to output control force. Due to its non-uniform shape, the parameters of the flexible fishtail along the S-axis, such as the variable damping coefficient A(s), linear density ρ(s), and bending stiffness EI(s), are all related to the spatial position variable s.

[0067] The dynamic model of the flexible fish tail is as follows:

[0068]

[0069] In the formula, A(s) is the variable damping coefficient of the flexible bionic fish tail, ρ(s) is the non-uniform linear density of the fish tail, and EI(s) is the bending stiffness of the fish tail. s is a spatial position variable, t is a time variable, L represents the length of the flexible bionic fish tail; r(s,t) represents the vibration offset of the flexible fish tail at position s and time t; where ρ(s) and A(s) need to satisfy the assumptions: |ρ(s)|≤m1, |A(s)|≤m2, where m1 and m2 are the upper bounds of ρ(s) and A(s), respectively. These are the first and second partial derivatives of r(s,t) with respect to time t, respectively. [EI(s)r″(s,t)]″ is the second partial derivative of [EI(s)r″(s,t)] with respect to position s, and r″(s,t) is the second partial derivative of r(s,t) with respect to position s.

[0070] The boundary conditions for the flexible bionic fish tail are:

[0071] In the formula u(t) represents the control input at the flexible fishtail connection, d(t) represents the boundary disturbance at the flexible fishtail connection, and d(t) satisfies: in Let d(t) be the first-order partial derivative with respect to time t. and Let d(t) and d(t) represent respectively The upper bound is EI(0), which is the bending stiffness of the flexible fish tail at s=0; r(0,t) represents the vibration offset of the flexible fish tail at position s=0 and time t; r″(L,t) and r″′(L,t) represent the second and third position partial derivatives of the vibration offset of the fish tail at position s=L and time t, respectively.

[0072] S2. Using an auxiliary system based on variable separation and considering output constraints, construct a tracking error model for a flexible bionic fish tail.

[0073] The control of a flexible bionic fish tail is essentially to track the target oscillation signal, thereby achieving oscillation. To establish a tracking error model for the flexible bionic fish tail, we consider the vibration offset r(s,t) of the flexible fish tail to be a two-dimensional function coupled with the position variable s and the time variable t. Using the auxiliary function Γ(s)T(t) with variable separation, we construct an auxiliary system and define the tracking error h(s,t) as follows:

[0074] h(s,t)=r(s,t)-Γ(s)T(t),

[0075] In the formula Γ(s) is a function that depends only on the position s, and T(t) represents a time-dependent oscillating signal with a fixed amplitude, defined as... The matrix Ξ can be diagonalized along the imaginary axis and has eigenvalues ​​that guarantee T(t) maintains a constant amplitude oscillation. The reference signal r for the flexible fishtail at s = L is defined. d (s,t)=Γ L T(t), where Γ L Let Γ(s) be the value of Γ(s) at s = L, and let Γ(s) satisfy the following equation:

[0076]

[0077] Where Γ(0) is the value of Γ(s) at s=0, and Γ″(L) and Γ″′(L) are the second and third positional partial derivatives of Γ(s) at position s=L with respect to position s, respectively.

[0078] The error model can then be obtained as follows:

[0079]

[0080] in, These are the first and second partial derivatives of h(s,t) with respect to time t, respectively. h″(s,t) is the second partial derivative of h(s,t) with respect to position s. [EI(s)h″(s,t)]″ is the second partial derivative of [EI(s)h″(s,t)] with respect to position s. h(0,t) represents the magnitude of the tracking error of the flexible fishtail at position s=0 and time t. h″(L,t) and h″′(L,t) represent the second and third position partial derivatives of the tracking error of the fishtail at position s=L and time t, respectively. h″(0,t) is the second position partial derivative of h(0,t) at position s=0 and time t.

[0081] Meanwhile, considering output limitations, the tracking error |ε| is defined as |r(L,t)-r d (L,t)|, and ε must satisfy |ε|<k a The output limit k a It is a normal number.

[0082] Note that the aforementioned non-uniform parameters A(s), ρ(s), and EI(s) are all assumed to be known and bounded, along with the boundary perturbation d(t) and its first-order reciprocal with respect to time. All are assumed to be unknown and bounded. The dynamic model constructed based on Hamilton's principle and the derived error model are high-order fourth-order partial differential equations that are related to both time and spatial position. The boundary conditions are a set of ordinary differential equations that are only related to time. The construction of the dynamic model and boundary conditions is to find the relationship between the state variables of the flexible bionic fish tail and the output vibration offset, thereby providing a theoretical basis for the subsequent construction of the Lyapunov function of the flexible bionic fish.

[0083] S3. Considering the event triggering scheme, construct the Lyapunov function based on the tracking error model.

[0084] Define an increasing time series {t0, t1, ..., t2} i} represents the event time for sampling the controller, where t0 represents the initial sampling time, t1 represents the next sampling time, and so on, increasing sequentially up to the i-th sampling time t. i When at time t i When the triggering condition is met, the control law will be updated at time t. i to t i+1 The value remains constant. The adaptive event-triggered control input u(t) is defined as follows:

[0085]

[0086] Where, v(t) i ) is a continuous control law, and the control input u(t) will be in the range [t]. i ,t i+1The value remains unchanged within the range and is equal to v(t). i Define the control input u(t) and the continuous control law v(t). i Error between ) Define a fixed threshold strategy to specify event triggering conditions. Where t(0) = 0, and p represents a fixed trigger threshold, which is a known positive constant. Assume the existence of a continuously time-varying function. Make Established, among which The following conditions must be met:

[0087]

[0088] When the control input u(t) and the continuous control law v(t) i If the difference q(t) between the control input u(t) and the continuous control law v(t) does not exceed the set fixed trigger threshold p, the original value will be maintained; however, if the difference between the control input u(t) and the continuous control law v(t) is less than the set fixed trigger threshold p, the original value will be maintained; i When the difference q(t) between the control input u(t) and the control input u(t) exceeds the set fixed trigger threshold p, the control input u(t) will be updated to v(t). i+1 During the time when the control input u(t) remains constant, the communication system can perform other communication tasks. By selecting an appropriate fixed trigger threshold p, the communication efficiency of the controller can be improved while ensuring the control effect.

[0089] Based on the aforementioned error model and event triggering scheme, the Lyapunov function is constructed as follows:

[0090]

[0091] in,

[0092]

[0093]

[0094]

[0095] In the formula, λ, These are the first and second control parameters for adaptive boundary control of the flexible bionic fish tail, respectively; k1 and k2 are the first and second control gains of the flexible fish tail, respectively. The estimation error for the flexible fishtail boundary perturbation is defined as: in θ represents the boundary perturbation estimate for the flexible fishtail, and θ is the first scaling parameter.

[0096] The Lyapunov function mentioned above contains the dynamic model of the flexible bionic fish tail and the state variables in the boundary conditions, as well as the error variables of parameter estimation. It can reflect the overall energy situation of the flexible fish tail system. The subsequent construction of the bionic fish tail adaptive boundary controller with unknown disturbances needs to be based on the ability to reduce the overall energy of the flexible bionic fish tail system.

[0097] S4. Based on the Lyapunov function, construct an adaptive boundary controller.

[0098] Taking the first derivative of the Lyapunov function with respect to time t, and based on Lyapunov stability theory, combined with the above event triggering scheme based on a fixed threshold, the following adaptive boundary controller is finally constructed:

[0099]

[0100] In the formula, k0 is the third control gain of the flexible bionic fish tail; φ1(t) is defined as: Defined as: Define the bounding perturbation estimate of the flexible fishtail as: Where c and ζ1 are the second and third scaling parameters, respectively.

[0101] In the aforementioned adaptive boundary controller, the output limitation is compensated by taking the partial derivative of the barrier Lyapunov function Π3(t) with respect to time, the boundary disturbance d(t) is estimated by the adaptive estimation technique, and the coupling term generated by the fixed threshold event triggering mechanism is canceled by the last term of the controller. Therefore, the present invention can handle the impact of boundary disturbance and output limitation on control performance, and can also perform event triggering control based on a fixed threshold to improve communication efficiency.

[0102] Note that the state signals in the above-mentioned bionic fish tail adaptive boundary controller with unknown disturbances are all obtained by sensor sampling or finite difference method calculation. After obtaining the state signals, they need to be calculated in the calculation unit of the flexible bionic fish tail controller before the actuator can be updated, and then adaptive boundary control can be applied to the flexible bionic fish tail.

[0103] Note that the above process of constructing a bionic fishtail adaptive boundary controller with unknown perturbations is based on the premise that the adaptive boundary controller with unknown perturbations can reduce the total energy of the flexible bionic fishtail. Therefore, it is necessary to verify the stability of the flexible bionic fishtail under the action of the adaptive boundary controller with unknown perturbations to demonstrate the rationality of the adaptive boundary controller with unknown perturbations. The specific process is as follows:

[0104] (1) By constraining the control parameter λ in the Lyapunov function, To ensure the positive definiteness of the Lyapunov function by controlling the gains k1 and k2, we first introduce the following inequality lemma:

[0105] Lemma 1: For any two functions have:

[0106]

[0107]

[0108] In the formula, ζ is the scaling factor;

[0109] Lemma 2: For a first-order continuously differentiable function have:

[0110]

[0111] ,in They are The first and second partial derivatives of s.

[0112] Lemma 3: For any >0 has:

[0113]

[0114] By applying the above inequality lemmas 1-2 to the scaling of Π4(t) in the Lyapunov function Π(t), we can obtain In the formula, The symbol min{*,...,*} represents the minimum value of the elements in the set {*,...,*}, and the symbol max{*,...,*} represents the maximum value of the elements in the set {*,...,*}; ζ2 is the second scaling factor.

[0115] Add to both sides of the inequality -χ0Π4(t)≤Π4(t)≤χ0Π4(t) We can obtain: In the formula, χ2=min{1-χ1,1}, χ3=max{1+χ1,1}, and λ is constrained. The positive definiteness of the Lyapunov function can be guaranteed by setting k1 and k2 such that 1-χ1>0.

[0116] (2) Take the first derivative of the Lyapunov function with respect to time t, as follows:

[0117] Find the first derivative of the Lyapunov function P(t) with respect to time t. By applying inequality lemmas 1-2 and expanding, we can obtain:

[0118]

[0119] In the formula, ζ1, ζ3, ζ4, and ζ5 are the first, third, fourth, and fifth scaling coefficients, respectively. The above formula simplifies to:

[0120] Where χ and θ are both positive constants.

[0121] make in

[0122]

[0123] Then we can obtain: in

[0124] (3) Applying Lyapunov's bounded stability theory, the stability of the flexible bionic fish tail under the action of an adaptive boundary controller with unknown perturbations is verified, as follows:

[0125] Regarding the above Multiply both sides of the inequality by e ηt Integrating from 0 to t and simplifying, we get:

[0126]

[0127] Based on the definition of the Lyapunov function, the final result is:

[0128]

[0129]

[0130] Based on the above results, it can be seen that the tracking error h(s,t) will eventually converge to the neighborhood of 0. Applying the Lyapunov bounded stability theory, it can be concluded that the flexible bionic fish tail is uniformly and ultimately bounded stable under the action of the bionic fish tail adaptive boundary controller with unknown perturbation.

[0131] The stability verification results above show that the overall energy of the flexible bionic fishtail system reflected by the Lyapunov function is attenuated under the action of the bionic fishtail adaptive boundary controller with unknown perturbation. Therefore, the bionic fishtail event triggering limit controller with unknown perturbation constructed in this step is reasonable.

[0132] S5. Based on the adaptive boundary controller, an actuator located at the boundary of the fish tail is used to implement event-triggered limit control for the flexible bionic fish tail with unknown boundary disturbances.

[0133] For a flexible fishtail with unknown boundary perturbations, the process of applying event-triggered constraint control specifically refers to calculating the adaptive boundary perturbation update law in the computing unit of the controller of the flexible bionic fishtail. The event-triggered boundary continuous controller v(t) maintains a constant control input signal when the difference between the control input signal and the continuous control signal is within a fixed threshold. When the difference exceeds the fixed threshold, the actuator input signal is updated. As the control time increases, the boundary actuator continuously outputs control force to adjust the vibration offset r(s,t), ultimately allowing the flexible bionic fishtail to track the target trajectory, thus achieving adaptive boundary control.

[0134] In summary, this embodiment provides event-triggered constraint control for a bionic fish tail with unknown boundary perturbations. This includes: constructing a dynamic model based on the dynamic characteristics of the flexible bionic fish tail; building an error model considering boundary perturbations and output constraints using an auxiliary system; constructing a Lyapunov function based on the error model, considering an event triggering scheme based on a fixed threshold; constructing an adaptive boundary controller based on the Lyapunov function; and implementing adaptive boundary control for the flexible fish tail with unknown boundary perturbations based on the adaptive boundary controller. This invention not only enables vibration tracking control of the flexible bionic fish tail but also constrains its output, handles the influence of boundary perturbations, and improves communication efficiency while ensuring system control performance through event triggering control based on a fixed threshold.

[0135] Example 2

[0136] Based on the bionic fishtail event triggering and limiting control method with unknown perturbations in Example 1, a flexible bionic fishtail is digitally simulated using MATLAB simulation software to further verify the effectiveness of the proposed bionic fishtail event triggering and limiting control method with unknown perturbations. Therefore, this embodiment is based on each step of the event triggering and limiting control method with unknown perturbations disclosed in Example 1, to... Figure 3 The structural model of the flexible bionic fish is used as a reference. Based on the simulation parameters in Tables 1, 2 and 3, the flexible bionic fish is digitally simulated.

[0137] Table 1. Simulation Fixed Parameter Values ​​for Flexible Bionic Fish

[0138] parameter Numerical meaning <![CDATA[A0]]> 0.046 N / (m / s) Variable damping coefficient at s=0 <![CDATA[ρ0]]> 3.2832 kg / m Non-uniform linear density at s=0 <![CDATA[EI0]]> <![CDATA[0.0395Nm 2 ]]> Bending stiffness at s=0 L 0.2m Flexible fishtail length

[0139] Table 2. Non-uniform parameter function table for flexible biomimetic fishtail system

[0140] parameter function A(s) <![CDATA[A0[1-0.42(s / L) 2 ] <!-- 11 -->]]> ρ(s) <![CDATA[ρ0[1-0.42(s / L) 2 ]]]> EI(s) <![CDATA[EI0[1-0.42(s / L) 2 ]]]>

[0141] Table 3. Control and Adjustment Parameters for Flexible Bionic Fish Tail

[0142]

[0143] In this embodiment, an error model is constructed using an auxiliary system in conjunction with a flexible biomimetic fishtail dynamic model. The initial value of the quasi-function T(t) at t=0 is... The function Γ(s) terminates at s = L with the value Γ(L) = [0.08 0.05], and the matrix...

[0144] Figure 4 , Figure 5 , Figure 6 All images shown are simulation results from this embodiment, with a total simulation time of 40 seconds. Figure 4 The figure shows a schematic diagram of the simulation results of the vibration offset r(s,t) of a flexible bionic fish tail under event-triggered constraint control with unknown boundary perturbations. This can be seen from... Figure 4 The result is that the flexible bionic fish tail exhibits periodic vibrations according to a set trajectory. Figure 5 The figure shows a schematic diagram of the simulation results of the boundary perturbation d(t) of the flexible fishtail after applying an event-triggered constraint control method to the flexible fishtail with unknown boundary perturbation. Figure 5 This indicates that, under event-triggered constraint control with unknown boundary perturbations, the boundary perturbations of the flexible fishtail are quickly addressed by the adaptive update law. The estimated and offsetted effects of boundary disturbances are thus mitigated, thereby suppressing their impact on control performance. Figure 6 The figure shows a simulation result of the error output h(L,t) of the flexible fishtail at s=L after applying an event-triggered constraint control method to the flexible fishtail with unknown boundary perturbation. Figure 6 This indicates that, under event-triggered constraint control with unknown boundary perturbations, the output error of the flexible fishtail is well constrained within ±k. a Within. Figure 7 As shown, under a fixed threshold event triggering scheme, the flexible bionic fish tail can not only ensure good control performance, but also improve the communication efficiency of the control system.

[0145] The above embodiments are preferred implementation modes of the present invention, but the implementation modes of the present invention are not limited to the above embodiments. Any other changes, modifications, substitutions, combinations, and simplifications that do not deviate from the spirit and principles of the present invention should be considered as equivalent replacement methods and are included in the scope of protection of the present invention.

Claims

1. A biomimetic fishtail event triggering and limiting control method with unknown perturbations, characterized in that, The bionic fish tail event triggering restriction control method includes the following steps: S1. Based on the dynamic characteristics of the flexible bionic fish tail, and considering Hamilton's principle and unknown boundary perturbations, a dynamic model of the flexible bionic fish tail is constructed. The dynamic characteristics of the flexible bionic fish tail in step S1 include the kinetic energy, potential energy, and virtual work done by non-conservative forces on the tail. Substituting the kinetic energy, potential energy, and virtual work into Hamilton's principle, and considering unknown boundary perturbations, the dynamic model of the flexible bionic fish tail is obtained as follows: In the formula, A(s) is the variable damping coefficient of the flexible bionic fish tail, ρ(s) is the non-uniform linear density of the fish tail, and EI(s) is the bending stiffness of the fish tail. s is a spatial position variable, t is a time variable, L represents the length of the flexible bionic fish tail; r(s,t) represents the vibration offset of the flexible fish tail at position s and time t; where ρ(s) and A(s) need to satisfy the assumptions: |ρ(s)|≤m1, |A(s)|≤m2, where m1 and m2 are the upper bounds of ρ(s) and A(s), respectively. These are the first and second partial derivatives of r(s,t) with respect to time t, respectively. [EI(s)r″(s,t)]″ is the second partial derivative of [EI(s)r″(s,t)] with respect to position s, and r″(s,t) is the second partial derivative of r(s,t) with respect to position s. The boundary conditions for the flexible bionic fish tail are: In the formula u(t) represents the control output at the flexible fishtail connection, d(t) represents the unknown boundary disturbance at the flexible fishtail connection, and d(t) satisfies: in Let d(t) be the first-order partial derivative with respect to time t. and Let d(t) and d(t) represent respectively The upper bound is EI(0), which is the bending stiffness of the flexible fish tail at s=0; r(0,t) represents the vibration offset of the flexible fish tail at position s=0 and time t; r″(L,t) and r″′(L,t) represent the second and third position partial derivatives of the vibration offset of the fish tail at position s=L and time t, respectively. S2. Using an auxiliary system based on variable separation and considering output constraints, construct a tracking error model for a flexible bionic fish tail. S3. Considering the event triggering scheme, construct a Lyapunov function based on the tracking error model; S4. Based on the Lyapunov function, construct an adaptive boundary controller for a flexible bionic fish tail; S5. Based on the adaptive boundary controller, the event-triggered limit control of the flexible bionic fish tail with unknown boundary disturbances is achieved through the boundary actuator located at the fish tail.

2. The biomimetic fishtail event triggering and limiting control method with unknown perturbation according to claim 1, characterized in that, In step S2, the flexible bionic fish tail needs to track the reference signal to swing. Since the vibration offset r(s,t) of the flexible fish tail is a two-dimensional function coupled with the position variable s and the time variable t, we consider using the auxiliary function Γ(s)T(t) with variable separation to construct an auxiliary system. The tracking error h(s,t) is defined as follows: h(s,t)=r(s,t)-Γ(s)T(t) In the formula Γ(s) is a function that depends only on the position s, and T(t) represents a time-dependent oscillating signal with a fixed amplitude, defined as... Where Ξ is a matrix that can be diagonalized along the imaginary axis and has eigenvalues ​​that guarantee T(t) to maintain a constant amplitude oscillation, and the reference signal r of the flexible fishtail at s = L is defined. d (s,t)=Γ L T(t), where Γ L Let Γ(s) be the value of Γ(s) at s = L, and let Γ(s) satisfy the following equation: Where Γ(0) is the value of Γ(s) at s=0, and Γ″(L) and Γ″′(L) are the second and third positional partial derivatives of Γ(s) at position s=L with respect to position s, respectively; The resulting error model is as follows: in, These are the first and second partial derivatives of h(s,t) with respect to time t, respectively; h″(s,t) is the second partial derivative of h(s,t) with respect to position s; [EI(s)h″(s,t)]″ is the second partial derivative of [EI(s)h″(s,t)] with respect to position s; h(0,t) represents the magnitude of the tracking error of the flexible fishtail at position s=0 and time t; h″(L,t) and h″′(L,t) represent the second and third position partial derivatives of the tracking error of the fishtail at position s=L and time t, respectively; and h″(0,t) is the second position partial derivative of h(0,t) at position s=0 and time t. Meanwhile, considering output limitations, the tracking error |ε| is defined as |r(L,t)-r d (L,t)|, and ε must satisfy |ε|<k a The output limit k a It is a normal number.

3. The biomimetic fishtail event triggering and limiting control method with unknown disturbance as described in claim 2, characterized in that, In step S3, a time-triggered scheme based on a fixed threshold is considered, defining an incremental time sequence {t0, t1, ..., t...}. i } represents the event time for sampling the controller, where t0 represents the initial sampling time, t1 represents the next sampling time, and so on, increasing sequentially up to the i-th sampling time t. i When at time t i When the triggering condition is met, the control law will be updated at time t. i to t i+1 The internal value remains constant, and the adaptive event-triggered control input u(t) is defined as follows: Where, v(t) i ) is a continuous control law, and the control input u(t) will be in the range [t]. i ,t i+1 The value remains unchanged within the range and is equal to v(t). i Define the control input u(t) and the continuous control law v(t). i Error between ) Define a fixed threshold strategy to specify event triggering conditions. Where t(0) = 0, p represents a fixed trigger threshold, which is a known positive constant, and it is assumed that a continuously time-varying function exists. Make Established, among which The following conditions must be met: Based on the aforementioned error model and event triggering scheme, the Lyapunov function is constructed as follows: in, In the formula, λ, These are the first and second control parameters for adaptive boundary control of the flexible bionic fish tail, respectively; k1 and k2 are the first and second control gains of the flexible fish tail, respectively. The estimation error for the flexible fishtail boundary perturbation is defined as: in θ represents the boundary perturbation estimate for the flexible fishtail, and θ is the first scaling parameter.

4. The biomimetic fishtail event triggering and limiting control method with unknown disturbance as described in claim 3, characterized in that, The process of constructing the adaptive boundary controller based on the Lyapunov function in step S4 is as follows: Taking the first derivative of the Lyapunov function with respect to time t, and based on Lyapunov stability theory, combined with the above event triggering scheme based on a fixed threshold, the following adaptive boundary controller with unknown perturbation is finally constructed: In the formula, k0 is the third control gain of the flexible bionic fish tail; φ1(t) is defined as: Defined as: Define the bounding perturbation estimate of the flexible fishtail as: Where c and ζ1 are the second and third scaling parameters, respectively.

5. The biomimetic fishtail event triggering and limiting control method with unknown disturbance as described in claim 4, characterized in that, After implementing event-triggered constraint control on the flexible bionic fish tail with unknown boundary perturbations, the process further includes verifying the stability of the flexible bionic fish tail with unknown boundary perturbations under adaptive event-triggered constraint control. The process is as follows: By constraining the first control parameter λ and the second control parameter in the Lyapunov function The first control gain k1 and the second control gain k2 of the flexible fishtail ensure the positive definiteness of the Lyapunov function; Find the first derivative of the Lyapunov function with respect to time t; We apply Lyapunov's bounded stability theory to verify the stability of a biomimetic fishtail under event-triggered constraint control with unknown perturbations.

6. The biomimetic fishtail event triggering and limiting control method with unknown perturbation according to claim 5, characterized in that, In step S5, the process of implementing event-triggered constraint control based on the adaptive boundary controller and the flexible bionic fish tail with unknown boundary perturbations is as follows: Calculating the adaptive boundary perturbation update law in a boundary controller for a flexible bionic fish tail The adaptive boundary controller v(t) is calculated and then the boundary actuator of the flexible bionic fish tail is updated. As the control time increases, the boundary actuator continuously outputs control force to continuously adjust the vibration offset r(s,t). Finally, the motion trajectory of the flexible bionic fish tail tracks the target vibration trajectory, thus realizing adaptive boundary control.

Images