Non-singular fast terminal sliding mode vibration active control method based on compensation function observer

Abstract

The application discloses a non-singular fast terminal sliding mode vibration active control method based on a compensation function observer, 1) according to elastic dynamics, a fourth-order modal equation of a four-side fixed support plate structure driven by an electromagnetic inertial actuator is established; 2) in view of external unknown interference and modeling uncertainty existing in the system, a compensation function observer is designed to realize real-time estimation and compensation; 3) a non-singular fast terminal sliding mode surface is constructed, and a sliding mode control law is designed in combination with the observer output, so that the singularity problem of a traditional terminal sliding mode is solved, and the time for system state to reach a balance point is shortened through a fast convergence term; 4) a comparison experiment is designed, and the composite control method designed in the application can significantly reduce the vibration amplitude of the four-side fixed support plate while guaranteeing the finite time stability of the system, and has stronger anti-interference ability and higher vibration suppression precision compared with the traditional sliding mode control.

Description

Technical Field

[0001] This invention relates to the field of active vibration control, and in particular to an active vibration control method for nonsingular fast terminal sliding mode based on a compensation function observer. Background Technology

[0002] Lightweight and highly rigid thin-plate structures are widely used in industrial production, precision manufacturing, and aerospace. However, these structures are highly susceptible to external excitation, leading to vibration. Harmful vibrations not only significantly reduce the operational accuracy of equipment but also cause structural fatigue due to cyclic stress, thus shortening service life and even causing catastrophic engineering accidents. Active vibration control algorithms are crucial for vibration suppression in thin-plate structures, and their performance directly determines the system's vibration suppression limit. Therefore, designing efficient vibration suppression schemes for typical structures such as four-sided fixed plates has significant scientific research value and engineering practical implications.

[0003] In existing active vibration control systems, the Extended State Observer (ESO), based on Active Disturbance Rejection Control (ADRC) theory, is the mainstream technique for handling internal and external disturbances. ESO can combine the system's nonlinear dynamics, parameter uncertainties, and external load fluctuations into a "total disturbance" and perform online estimation and feedforward compensation. However, the traditional Linear Extended State Observer (LESO) has inherent limitations in practical applications. First, phase lag and amplitude attenuation: LESO's disturbance estimation essentially exhibits characteristics of a Type I system. When facing high-frequency sinusoidal disturbances, ramp disturbances, or higher-order time-varying disturbances, the observation results are often accompanied by severe phase lag and amplitude deviation, leading to insufficient accuracy in feedforward compensation and difficulty in completely eliminating complex vibration signals. Secondly, the tracking capability is limited: For acceleration signals or higher-order signals of the third order or above, traditional ESOs often cannot achieve tracking without steady-state error. Their observation accuracy is highly dependent on the observer bandwidth, but blindly increasing the bandwidth will significantly amplify sensor noise and affect system stability.

[0004] Traditional sliding mode control (SMC) is widely used in vibration controller design due to its high robustness to parameter disturbances. However, in practical engineering implementation, traditional sliding mode control still faces the following challenges. First, there is a contradiction between convergence speed and nonsingularity: While traditional non-singular terminal sliding mode control (NTSMC) can guarantee system convergence within a finite time, its convergence speed is even lower than that of a linear sliding surface in regions far from the equilibrium point. Furthermore, to eliminate singularity issues in the control terms, some dynamic response performance often needs to be sacrificed. Second, there is insufficient optimization of convergence efficiency: Existing technologies often struggle to achieve fast convergence across the entire space without increasing system chattering. Although some improved algorithms attempt to introduce higher-order terms, how to achieve composite control by combining high-precision observers while ensuring nonsingularity remains a pressing problem in the field of intelligent structure vibration control.

[0005] In addition, the traditional “algorithm simulation-manual programming-chip burning” development model has problems such as relatively independent stages, long development cycle, difficulty in writing complex algorithms and difficulty in troubleshooting errors. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention provides an active control method for non-singular fast terminal sliding mode vibration based on a compensation function observer (CFO). Employing a model-based design (MBD) approach, using MATLAB / Simulink as the core and integrating an NI PCIe-6343 data acquisition card and a Desktop Real-Time environment, it achieves a complete development process from control algorithm construction to automatic real-time code generation. This hardware-in-the-loop system not only improves the verification efficiency of advanced composite control algorithms but also ensures the real-time performance and reliability of the composite control strategy of non-singular terminal sliding mode control (NFTSMC) based on the compensation function observer (CFO) on a real physical platform.

[0007] The objective of this invention is achieved as follows: a method for active control of non-singular fast terminal sliding mode vibration based on a compensation function observer, comprising the following steps:

[0008] Step 1) Based on the theory of elastic dynamics, establish the m-th modal equation of the four-sided fixed plate structure driven by an electromagnetic inertial actuator;

[0009] Step 2) Treat the external unknown disturbances and modeling uncertainties existing in the system as total disturbances and expand them into new state variables, and design a high-type compensation function observer for real-time estimation and compensation;

[0010] Step 3) Design a non-singular fast terminal sliding surface for the system state variables estimated by the compensation function observer, and design a sliding mode control law in combination with the observer output. At the same time, shorten the time for the system state to reach the equilibrium point by using a fast convergence term.

[0011] Step 4) Design a smart structure vibration suppression verification experiment. In the MATLAB / Simulink environment, build a composite controller that combines a compensation function observer and a non-singular terminal sliding mode control, and drive an electromagnetic inertial actuator in real time to generate electromagnetic force to suppress the vibration of the plate surface.

[0012] Furthermore, step 1) specifically includes: intelligent structural dynamics modeling based on inertial actuators and four-sided fixed plates:

[0013] Based on the theory of elastic dynamics, the dynamic model of a four-sided fixed plate structure is described by the following equation:

[0014] (1)

[0015] Where ρ and h are the density and thickness of the thin plate, respectively, i.e., the unit mass of the thin plate structure is ρh; D and δ dB are the bending stiffness and viscous damping coefficient of the plate structure, F(x,y,t) is the time-varying external unknown disturbance, and w(x,y,t) is the lateral offset of the thin plate. This is the double Laplace operator, and its expression is:

[0016] (2)

[0017] According to the modal superposition theory in vibration theory, the actual displacement of any point in a thin plate structure under external excitation can be expressed as a linear combination of the vibration displacements of each natural mode:

[0018] (3)

[0019] Similar to the displacement w, the external disturbance F is represented by a linear superposition of the various modes:

[0020] (4)

[0021] Among them, W m (x,y) represents the m-th order modal function of the thin plate, η m (t) and F m (t) represents the m-th order modal displacement and disturbance; according to the double operator in equation (2), W is obtained. m The expression for (x,y) is:

[0022] (5)

[0023] ω mLet be the natural angular frequency of the thin plate in the m-th mode;

[0024] By combining equations (3) and (5) and substituting them into the elastic dynamics model of the four-sided fixed plate, the forced vibration equation of the thin plate structure is obtained as follows:

[0025] (6)

[0026] When the x-axis and y-axis lengths of a homogeneous quadrangularly fixed plate are close, it is assumed that it simultaneously satisfies the boundary and orthogonality conditions. Based on the principle of orthogonality, the infinite number of modes in the summation symbol are decoupled, the m-th order mode equation is extracted, and equation (6) is simplified to:

[0027] (7)

[0028] When using an inertial actuator for active vibration control, F m Considered as the overall disturbance excluding the inherent motion characteristics of the plate structure, including the force f generated by the inertial actuator. bm and external disturbances or excitation forces d cm Rewrite equation (7) in the form of a second-order differential equation:

[0029] (8)

[0030] Among them, f bm The force generated by the inertial actuator, d cm For external disturbances or excitation forces, ξ m ω m The damping ratio and natural angular frequency of the thin plate for the m-th order mode are given by η. m Let ξ be the m-th modal displacement. m =δ / 2ρhω m b m To control the force gain.

[0031] Furthermore, step 2) specifically includes: displacing the four-sided fixed plate structure. As state variables, reconstruct equation (8):

[0032] (9)

[0033] The force f generated by the inertial actuator b Using control voltage u and base plate displacement x b This indicates that, therefore , This represents the uncertainty of the inertial actuator output related to the displacement of the base plate. When the uncertainty is ignored, the input voltage u and the applied force f b The relationship between them is linear; equation (9) can be further expressed as:

[0034] (10)

[0035] in, For modal error, For actuator error, External disturbances To control the quantity;

[0036] For equation (10), the following extended state-space equation is established:

[0037] (11)

[0038] The total disturbance b0 is the controller gain;

[0039] The compensation function observer extends the integral term of the state variable observations by introducing a disturbance compensation term.

[0040] First, construct an observer with integral form for equation (11):

[0041] (12)

[0042] in, for The observed values, , For observation error, and For observer gain;

[0043] Write the error state equation according to equation (12):

[0044] (13)

[0045] Introducing compensation function Equation (12) can be rewritten as:

[0046] (14)

[0047] in, For total disturbance The observed values; after adding the compensation function, the error state equation... Replace with compensation error Construct the compensation function :

[0048] (15)

[0049] From equation (15), we obtain the compensation function. Compared with observed values An iterative relationship is formed through a first-order filter. When is at its maximum, ; Write the time-domain form of Equation (16):

[0050] (16)

[0051] Substitute Equation (16) into Equation (12) and let the compensation function be the extended state variable :

[0052] (17)

[0053] Write the observation error state equation of the compensation function observer according to Equation (17):

[0054] (18)

[0055] where ;

[0056] Then, calculate the disturbance observation error transfer function of the compensation function observer CFO according to Equation (18): (19)

[0057] Convert Equation (19) into the form of an equivalent unit negative feedback open-loop transfer function:

[0058] (20)

[0059] Furthermore, the non-singular fast terminal sliding mode surface described in step 3) is designed as follows:

[0060] (21)

[0061] where the tracking error , r is the target displacement; k1, k2, , are the sliding mode surface parameters, 1 < c < 2, c < a. When the system state approaches the sliding mode surface, the influence of the high-order term is ignored, and the above equation is approximated to the traditional non-singular terminal sliding mode surface; when the system state is far from the sliding mode surface, the high-order term plays a dominant role. At this time, let s = 0 and obtain:

[0062] (22)

[0063] where , when the error , if there is a non-integer power , there are complex numbers in the sliding mode surface operation. Therefore, introduce the product of the sign function and the power of the absolute value to eliminate the influence of complex numbers and singularities on the sliding mode surface; update the sliding mode surface as follows:

[0064] (twenty three)

[0065] Among them 1 <c<2,c<a, ;

[0066] The approach law is chosen to employ both power-law approach law and constant-rate approach law:

[0067] (twenty four)

[0068] Where M represents the power-law approach rate gain, and N represents the constant-rate approach rate gain. It is expressed as the absolute value of the sliding surface.

[0069] Furthermore, the design of the smart structure vibration suppression verification experiment described in step 4) is a smart structure vibration suppression verification experiment designed with the NI PCIe-6343 acquisition card as the core.

[0070] Compared with the prior art, the beneficial effects of the present invention are as follows: 1) The CFO obtained by the present invention through type reconstruction is a three-type system, which has stronger observation performance compared with the traditional ESO as a one-type system, and can more accurately observe high-order complex vibration signals. The CFO makes full use of the observation errors of plate displacement and acceleration, which helps to improve the observation accuracy of disturbances. By introducing a compensation function, the influence of unknown characteristics of disturbances is reduced, which helps to improve the tracking accuracy of the observer.

[0071] 2) The composite control scheme designed in this invention utilizes CFO to perform real-time feedforward compensation for the total system disturbance, significantly reducing the dependence of sliding mode control on switching gain. Simultaneously, the NFTSM designed in this invention overcomes the global asymptotic convergence characteristic of traditional SMC, achieving finite-time convergence, and by introducing higher-order terms, enables the state variables to converge faster than traditional NTSMC at points far from the equilibrium point.

[0072] 3) This invention uses the NI PCIe-6343 data acquisition card as its core and leverages graphical programming combined with automatic code generation technology to construct a complete algorithm design and development process. This approach avoids logical errors that are easily generated when manually writing complex algorithms, and allows for real-time observation of system state changes using a host computer. It not only achieves a combination of simulation and hardware, further accelerating the system development process, but also significantly improves the anti-interference performance and steady-state performance of the four-sided fixed plate under disturbed conditions through a composite control method combining CFO and NFTSMC. Attached Figure Description

[0073] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0074] Figure 1 This is a flowchart of the development process based on model design in this invention.

[0075] Figure 2 This is a block diagram of the intelligent structure CFONFTSMC composite sliding mode vibration control system in this invention.

[0076] Figure 3 This is a diagram showing the experimental model setup of the CFO-based NFTSMC composite controller in this invention.

[0077] Figure 4 This is a diagram showing the first-order mode observation results of CFO on superimposed velocity and acceleration disturbances in this invention.

[0078] Figure 5 This is a diagram showing the multimodal observation results of CFO on superimposed constants and velocity disturbances in this invention.

[0079] Figure 6 This is a diagram illustrating the vibration suppression effect of CFOSMC single-mode in this invention.

[0080] Figure 7 This is a diagram showing the vibration suppression effect of CFONFTSMC single-mode vibration suppression in this invention.

[0081] Figure 8 This is a magnified view of the vibration suppression effect of CFOSMC in this invention.

[0082] Figure 9 This is a magnified view of the vibration suppression effect of CFONFTSMC in this invention.

[0083] Figure 10 This is a comparison diagram of the control voltages of the two controllers in this invention.

[0084] Figure 11 This is the single-mode frequency domain amplitude diagram in this invention.

[0085] Figure 12 This is a diagram illustrating the multimodal time-domain vibration suppression effect in this invention.

[0086] Figure 13 This is a diagram illustrating the multimodal frequency domain vibration suppression effect in this invention. Detailed Implementation

[0087] 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. 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.

[0088] This invention focuses on the dynamic modeling of an intelligent structure based on an inertial actuator and a four-sided fixed plate. Addressing the problems of traditional development methods, it implements a composite control algorithm based on a compensation function observer and non-singular fast terminal sliding mode using a model-based design with the NI PCIe-6343 as the core. This algorithm suppresses plate vibration and combines MATLAB / Simulink simulation with hardware, significantly improving development efficiency and control performance.

[0089] like Figure 1 The method for active control of nonsingular fast terminal sliding mode vibration based on a compensation function observer, as shown, includes the following steps:

[0090] Step 1) Based on the theory of elastic dynamics, establish the m-th modal equation of the four-sided fixed plate structure driven by an electromagnetic inertial actuator; the modeling method based on structural modes makes it unnecessary to remodel when facing complex multimodal vibration control, and has better versatility than the traditional modeling method.

[0091] Dynamic modeling of a smart structure based on an inertial actuator and a four-sided fixed plate: Based on the theory of elastic dynamics, the dynamic model of the four-sided fixed plate structure is described by the following equations:

[0092] (1)

[0093] Where ρ and h are the density and thickness of the thin plate, respectively, i.e., the unit mass of the thin plate structure is ρh; D and δ dB are the bending stiffness and viscous damping coefficient of the plate structure, F(x,y,t) is the time-varying external unknown disturbance, and w(x,y,t) is the lateral offset of the thin plate. This is the double Laplace operator, and its expression is:

[0094] (2)

[0095] According to the modal superposition theory in vibration theory, the actual displacement of any point in a thin plate structure under external excitation can be expressed as a linear combination of the vibration displacements of each natural mode:

[0096] (3)

[0097] Similar to the displacement w, the external disturbance F is represented by a linear superposition of the various modes:

[0098] (4)

[0099] Among them, W m (x,y) represents the m-th order modal function of the thin plate, η m (t) and F m (t) represents the m-th order modal displacement and disturbance; according to the double operator in equation (2), W is obtained. m The expression for (x,y) is:

[0100] (5)

[0101] ω m Let be the natural angular frequency of the thin plate in the m-th mode;

[0102] By combining equations (3) and (5) and substituting them into the elastic dynamics model of the four-sided fixed plate, the forced vibration equation of the thin plate structure is obtained as follows:

[0103] (6)

[0104] When the x-axis and y-axis lengths of a homogeneous quadrangularly fixed plate are close, it is assumed that it simultaneously satisfies the boundary and orthogonality conditions. Based on the principle of orthogonality, the infinite number of modes in the summation symbol are decoupled, the m-th order mode equation is extracted, and equation (6) is simplified to:

[0105] (7)

[0106] When using an inertial actuator for active vibration control, F m Considered as the overall disturbance excluding the inherent motion characteristics of the plate structure, including the force f generated by the inertial actuator. bm and external disturbances or excitation forces d cm Rewrite equation (7) in the form of a second-order differential equation:

[0107] (8)

[0108] Among them, f bm The force generated by the inertial actuator, d cm For external disturbances or excitation forces, ξ m ω m The damping ratio and natural angular frequency of the thin plate for the m-th order mode are given by η. m Let ξ be the m-th modal displacement. m =δ / 2ρhω m b m To control the force gain.

[0109] Step 2) Treat the external unknown disturbances and modeling uncertainties existing in the system as total disturbances and expand them into new state variables, and design a high-type compensation function observer for real-time estimation and compensation;

[0110] Displacement of the four-sided fixed plate structure As state variables, reconstruct equation (8):

[0111] (9)

[0112] The force f generated by the inertial actuator b Using control voltage u and base plate displacement x b This indicates that, therefore , This represents the uncertainty of the inertial actuator output related to the displacement of the base plate. When the uncertainty is ignored, the input voltage u and the applied force f b The relationship between them is linear; equation (9) can be further expressed as:

[0113] (10)

[0114] in, For modal error, For actuator error, External disturbances To control the quantity;

[0115] For equation (10), the following extended state-space equation is established:

[0116] (11)

[0117] The total disturbance b0 is the controller gain;

[0118] The compensation function observer extends the integral term of the state variable observation by introducing a disturbance compensation term, improves the type of the observation system, provides a new approach for accurate observation under complex vibration environments, and advances the improvement of the observer to the stage of disturbance model and system type optimization.

[0119] First, construct an observer with integral form for equation (11): (12)

[0120] in, for The observed values, , For observation error, and For observer gain;

[0121] Write the error state equation according to equation (12):

[0122] (13)

[0123] Introducing compensation function Equation (12) can be rewritten as:

[0124] (14)

[0125] in, For total disturbance The observed values; after adding the compensation function, the error state equation... Replace with compensation error That is, through compensation items To reduce the impact of unknown characteristics, a compensation function is constructed. :

[0126] (15)

[0127] From equation (15), we obtain the compensation function. Compared with observed values An iterative relationship is established between them through a first-order filter, when At its maximum, Write out the time-domain form of equation (16):

[0128] (16)

[0129] Substitute equation (16) into equation (12), and let the compensation function... For extended state variables :

[0130] (17)

[0131] Write the observation error state equation of the compensation function observer according to equation (17):

[0132] (18)

[0133] in ;

[0134] Then, according to equation (18), the perturbation observation error transfer function of the compensation function observer CFO is calculated as follows: (19)

[0135] Equation (19) is transformed into an equivalent unity negative feedback open-loop transfer function:

[0136] (20)

[0137] As can be seen from Equation (20), the compensation function observer obtained through type reconstruction is a type-III system, which has stronger observation performance compared with the traditional extended state observer as a type-I system and can observe high-order complex vibration signals more accurately.

[0138] The first-order modal frequency of the intelligent structure experimental platform is 48.5 Hz. Based on the first-order modal excitation, velocity perturbation and acceleration perturbation are added, and the observer output under the first-order mode is obtained as Figure 4 shown. The excitation is adjusted to multi-modal complex perturbation with the first-order modal frequency of 48.5 Hz superimposed on the second-order modal frequency of 120 Hz, and the constant perturbation and velocity perturbation are retained. The observer output under the condition of multi-modal and multi-signals is compared, as Figure 5 shown.

[0139] Although the CFO has the problem of observation overshoot in the multi-modal scenario, its tracking error is still significantly better than that of the CRESO and LESO. In addition, the multi-modal simulation results also intuitively show the phase lag and amplitude attenuation problems existing in the original extended state observer. The CRESO improved by TDE equivalent has a smaller phase lag compared with the original LESO. The CFO after the system type is improved has improved these problems from a higher dimension and provides more accurate observation results.

[0140] In summary, the compensation function observer has better observation performance for both high-order complex perturbations and mixed-modal periodic vibration signals.

[0141] Step 3) Design a non-singular fast terminal sliding mode surface for the system state variables estimated by the compensation function observer, and design a sliding mode control law in combination with the observer output. At the same time, shorten the time for the system state to reach the equilibrium point through a fast convergence term;

[0142] The non-singular fast terminal sliding mode surface is designed as follows:

[0143] (21)

[0144] where the tracking error , r is the target displacement; k1, k2, , are the sliding mode surface parameters, 1 < c < 2, c < a. When the system state approaches the sliding mode surface, the influence of the high-order term is ignored, and Equation (21) is approximated to the traditional non-singular terminal sliding mode surface; when the system state is far from the sliding mode surface, the high-order term plays a dominant role. At this time, let s = 0 to obtain:

[0145] (22)

[0146] where Therefore, this sliding surface has a faster convergence speed compared to traditional non-singular terminal sliding surfaces. For equation (21), when the error... When, if there exists a non-integer power Since sliding surface operations involve complex numbers, a sign function and the power product of the absolute value are introduced to eliminate the influence of complex numbers and singularities on the sliding surface; the sliding surface is updated as follows:

[0147] (twenty three)

[0148] Among them 1 <c<2,c<a, ;

[0149] In sliding mode controllers, the design of the reaching law directly determines the time it takes for the system to reach the sliding surface from the initial state and its convergence characteristics. The convergence time characteristics of commonly used reaching laws will be discussed below, and finite-time reaching laws will be designed for composite sliding mode controllers.

[0150] First, regarding the isokinetic approach law Let the initial state be... Integrating the differential equation:

[0151] (twenty four)

[0152] Solving equation (24) yields the following results. Therefore, when using a constant-rate reaching law, the convergence time is directly proportional to the initial state quantity and inversely proportional to the reaching law gain, while the convergence speed remains constant during the reaching process. This results in a slower reaching speed when the error is large, and a greater likelihood of chattering when the error is small.

[0153] For the power-approaching law Integrating the differential equation:

[0154] (25)

[0155] Solving the above equation yields... The system state can reach the sliding surface within a finite time, and the convergence time is similar to that of the initial state. Proportional, when When the value is large, the convergence time is still too long. Therefore, consider using both the power-law and the constant-rate-time convergence law simultaneously:

[0156] (26)

[0157] Where M represents the power-law approach rate gain, and N represents the constant-rate approach rate gain. It is expressed as the absolute value of the sliding surface.

[0158] For equation (26), when When the power term is large, it supplements the constant-rate reaching law, improving the convergence speed over large error intervals; while when... When the speed is small, the constant velocity term ensures a non-zero approaching velocity within a small error range, avoiding the convergence velocity decay problem when using only the power-law approaching law, and achieving a finite-time arrival at the sliding surface. The combination of the two approaching laws improves the approaching speed and dynamic performance of the controller.

[0159] Differentiating equation (23) yields:

[0160] (27)

[0161] Among them, let , ,in ,make To obtain the equivalent control law :

[0162] (28)

[0163] in Let the observed values ​​be the disturbance values. Substituting the approach law of equation (26) into the control law, the control law of the composite vibration control system is obtained as follows:

[0164] (29)

[0165] set up , To introduce the observation error, let the Lyapunov function be... We obtain the following formula:

[0166] (30)

[0167] Substituting the composite sliding mode control law of equation (29) into equation (30): (31)

[0168] Lemma: For nonlinear systems , In the origin field Internally continuous. If there exists a continuously differentiable function... The following conditions must be met:

[0169] 1. Regarding All satisfy ,and .

[0170] 2. The derivative of the function satisfies:

[0171] (32)

[0172] in , , If the initial time is given, then the system is finite-time stable, and the convergence time has an upper bound.

[0173] For equation (31), when At that time, there were:

[0174] (33)

[0175] Rewrite equation (33) in a form similar to equation (32) in the lemma:

[0176] (34)

[0177] in , Equation (33) clearly satisfies the basic conditions of the lemma. Since... The condition holds true. According to equation (34), we can obtain:

[0178] (35)

[0179] Integrating equation (35), let the convergence time be... :

[0180] (36)

[0181] For the convergence time There must be ,make Solving equation (36) yields:

[0182] (37)

[0183] Therefore, the upper bound of the system's convergence time can be obtained as:

[0184] (38)

[0185] Will , Substituting into equation (38), the convergence time boundary can be obtained as follows:

[0186] (39)

[0187] The above analysis shows that after applying the CFONFTSMC composite sliding mode controller, the Lyapunov function of the system can converge to [the desired function] within a finite time T. The convergence time has a clear upper bound, therefore the equilibrium point of the system is stable in finite time.

[0188] like Figure 2-3As shown in step 4), a smart structure vibration suppression verification experiment is designed with the NI PCIe-6343 acquisition card as the core. A composite controller combining CFO and non-NFTSMC is built in the MATLAB / Simulink environment, and the electromagnetic inertial actuator is driven in real time to generate electromagnetic force to suppress the vibration of the plate surface.

[0189] To verify its effectiveness, the following experiment was designed:

[0190] The vibration suppression effects of the CFOSMC and CFONFTSMC composite controllers were verified based on the electromagnetic inertial actuator-four-sided fixed plate intelligent structure experimental platform. To ensure the fairness of the measured data and conclusions, the following premises were established for this experiment:

[0191] 1. When uncontrolled, the amplitude of the four fixed plates is an 8V sine wave, and the disturbance applied by the exciter remains unchanged.

[0192] 2. The inertial actuator and the acceleration sensor are configured in the same position, and the sensor, exciter and actuator remain in the same position during the experiment.

[0193] 3. When performing the Fast Fourier Transform, 20,000 data points are selected to ensure the fairness of frequency domain data comparison.

[0194] 4. During the experiment, keep the channels and gains of the constant current source adapter and integrator unchanged, the sampling and output channels of the data acquisition card unchanged, and the sampling frequency of all channels is 10kHz.

[0195] 5. The experiment time was 50 seconds for all experiments, and the control variables were all intervened at 20 seconds via the switch on the host computer.

[0196] like Figure 6-7 As shown, CFONFTSMC achieves better vibration suppression without significantly increasing settling time; both controllers stabilize quickly. (Magnification at 30-30.5s) Figure 6-7 get Figure 8-9 .Depend on Figure 8 It can be seen that after the CFOSMC controller intervened, the amplitude of the four-sided fixed plate decreased from 8V to 1.85V, a decrease of 76.88%. The CFOSMC controller further reduced the amplitude to 1.44V, a further decrease of 0.41V compared to CFOSMC, a decrease of 82% compared to the open-loop amplitude, and a decrease of 77.84% compared to the CFOSMC amplitude.

[0197] While achieving the aforementioned vibration damping effect Figure 6-9 The corresponding control voltage results are as follows Figure 10 As shown, the amplitudes of the two control voltages are almost the same at this time, which ensures that the two controllers can be compared fairly under the same experimental conditions.

[0198] Performing an FFT transform on the time-domain amplitude effect diagram above yields the following result: Figure 11 The frequency domain amplitude diagram is shown. At the single octave of 48.5 Hz, the uncontrolled amplitude is 17.83 dB, and after CFOSMC, the amplitude is 4.25 dB, a decrease of 13.58 dB. The corresponding amplitude of CFOSMC is 0.42 dB, a decrease of 17.41 dB, and the first-order modal vibration of the four-sided fixed plate is effectively suppressed.

[0199] The new controller also exhibits better vibration suppression capabilities at wide-band harmonics. At the second harmonic of 97Hz, the CFOSMC reduces the amplitude from -29.86dB to -30.69dB, while the CFONFTSM reduces it to -31.09dB, a reduction of 1.23dB. At the third harmonic of 145.5Hz, the CFOSMC suppresses the amplitude from -34.31dB to -38.68dB, a suppression of 4.37dB, while the CFONFTSM further suppresses it to -41.65dB, a suppression of 7.25dB.

[0200] Finally, using a vibrator and piezoelectric elements, multimodal excitation signals of 48.5Hz and 120Hz were simultaneously applied to the four-sided fixed plate to verify the multimodal vibration suppression performance of the two controllers. Under the premise of experimental fairness, the control effects were as follows: Figure 12 As shown, to better illustrate the multimodal disturbance waveforms and control performance, the waveforms of the two controllers are overlaid and cropped at 15-25s. At this point, the time-domain vibration suppression effect of the CFOSMC controller is 51.46%, while that of the CFOSMC controller is 56.2%. Although the multimodal suppression capabilities of both controllers decline synchronously compared to the single-mode effect, they are still effective against multimodal disturbances.

[0201] To further compare the frequency domain performance, an FFT transformation was performed on the multimodal data to obtain... Figure 13 At the first mode, the amplitude of the four-sided fixed plate decreased from 5.95 dB to -2.88 dB under the action of CFOSMC, and further decreased to -4.05 dB under the action of CFONTSMC, a reduction of 10 dB. At the second mode, the CFOSMC controller reduced the amplitude from -20.51 dB to -23.3 dB, while CFONTSMC reduced it to -24.27 dB, a total reduction of 3.76 dB. Therefore, the improved composite sliding mode controller proposed in this invention can effectively suppress the multimodal complex vibration of the four-sided fixed plate structure.

[0202] In summary, both time-domain and FFT frequency-domain harmonic data in the multimodal and single-mode vibration experiments of the smart structure effectively demonstrate the excellent suppression performance of the CFONFTSMC composite controller.

[0203] The above description of the embodiments is only for the purpose of helping to understand the method and core ideas of the present invention. It should be noted that those skilled in the art can make several improvements and modifications to the present invention without departing from the principles of the present invention, and these improvements and modifications also fall within the protection scope of the claims of the present invention.

Claims

1. A method for active control of nonsingular fast terminal sliding mode vibration based on a compensation function observer, characterized in that, Includes the following steps: Step 1) Based on the theory of elastic dynamics, establish the m-th modal equation of the four-sided fixed plate structure driven by an electromagnetic inertial actuator; Step 2) Treat the external unknown disturbances and modeling uncertainties existing in the system as total disturbances and expand them into new state variables, and design a high-type compensation function observer for real-time estimation and compensation; Step 3) Design a non-singular fast terminal sliding surface for the system state variables estimated by the compensation function observer, and design a sliding mode control law in combination with the observer output. At the same time, shorten the time for the system state to reach the equilibrium point by using a fast convergence term. Step 4) Design a smart structure vibration suppression verification experiment. In the MATLAB / Simulink environment, build a composite controller that combines a compensation function observer and a non-singular terminal sliding mode control, and drive an electromagnetic inertial actuator in real time to generate electromagnetic force to suppress the vibration of the plate surface.

2. The active control method for non-singular fast terminal sliding mode vibration based on a compensation function observer according to claim 1, characterized in that, Step 1) specifically includes: intelligent structural dynamics modeling based on inertial actuators and four-sided fixed plates: Based on the theory of elastic dynamics, the dynamic model of a four-sided fixed plate structure is described by the following equation: （1） Where ρ and h are the density and thickness of the thin plate, respectively, i.e., the unit mass of the thin plate structure is ρh; D and δ dB are the bending stiffness and viscous damping coefficient of the plate structure, F(x,y,t) is the time-varying external unknown disturbance, and w(x,y,t) is the lateral offset of the thin plate. This is the double Laplace operator, and its expression is: （2） According to the modal superposition theory in vibration theory, the actual displacement of any point in a thin plate structure under external excitation can be expressed as a linear combination of the vibration displacements of each natural mode: （3） Similar to the displacement w, the external disturbance F is represented by a linear superposition of the various modes: （4） Among them, W m (x,y) represents the m-th order modal function of the thin plate, η m (t) and F m (t) represents the m-th order modal displacement and disturbance; according to the double operator in equation (2), W is obtained. m The expression for (x,y) is: （5） ω m Let be the natural angular frequency of the thin plate in the m-th mode; By combining equations (3) and (5) and substituting them into the elastic dynamics model of the four-sided fixed plate, the forced vibration equation of the thin plate structure is obtained as follows: （6） When the x-axis and y-axis lengths of a homogeneous quadrangularly fixed plate are close, it is assumed that it simultaneously satisfies the boundary and orthogonality conditions. Based on the principle of orthogonality, the infinite number of modes in the summation symbol are decoupled, the m-th order mode equation is extracted, and equation (6) is simplified to: （7） When using an inertial actuator for active vibration control, F m Considered as the overall disturbance excluding the inherent motion characteristics of the plate structure, including the force f generated by the inertial actuator. bm and external disturbances or excitation forces d cm Rewrite equation (7) in the form of a second-order differential equation: （8） Among them, f bm The force generated by the inertial actuator, d cm For external disturbances or excitation forces, ξ m ω m The damping ratio and natural angular frequency of the thin plate for the m-th order mode are given by η. m Let ξ be the m-th modal displacement. m =δ / 2ρhω m b m To control the force gain.

3. The active control method for non-singular fast terminal sliding mode vibration based on a compensation function observer according to claim 2, characterized in that, Step 2) specifically includes: displacing the four-sided fixed plate structure. As state variables, reconstruct equation (8): （9） The force f generated by the inertial actuator b Using control voltage u and base plate displacement x b This indicates that, therefore , This represents the uncertainty of the inertial actuator output related to the displacement of the base plate. When the uncertainty is ignored, the input voltage u and the applied force f b The relationship between them is linear; equation (9) can be further expressed as: （10） in, For modal error, For actuator error, External disturbances To control the quantity; For equation (10), the following extended state-space equation is established: （11） The total disturbance b0 is the controller gain; The compensation function observer extends the integral term of the state variable observations by introducing a disturbance compensation term. First, construct an observer with integral form for equation (11): （12） in, for The observed values, , For observation error, and For observer gain; Write the error state equation according to equation (12): （13） Introducing compensation function Equation (12) can be rewritten as: （14） in, For total disturbance The observed values; after adding the compensation function, the error state equation... Replace with compensation error Construct the compensation function : （15） From equation (15), we obtain the compensation function. Compared with observed values An iterative relationship is established between them through a first-order filter, when At its maximum, Write out the time-domain form of equation (16): （16） Substitute equation (16) into equation (12), and let the compensation function... For extended state variables : （17） Write the observation error state equation of the compensation function observer according to equation (17): （18） in ; Then, according to equation (18), the perturbation observation error transfer function of the compensation function observer CFO is calculated as follows: (19) Equation (19) is transformed into an equivalent unity negative feedback open-loop transfer function: （20）。 4. The active control method for non-singular fast terminal sliding mode vibration based on a compensation function observer according to claim 1, characterized in that, The non-singular fast terminal sliding surface design described in step 3) is as follows: （21） Among them, the tracking error , r is the target displacement; k1, k2, , are sliding surface parameters, 1 < c < 2, c < a. When the system state approaches the sliding surface, the influence of high-order terms is ignored, and the above formula is approximated to a traditional non-singular terminal sliding surface; when the system state is far from the sliding surface, the high-order terms play a dominant role. At this time, let s = 0 to obtain: （22） in When the error When, if there exists a non-integer power Since sliding surface operations involve complex numbers, a sign function and the power product of the absolute value are introduced to eliminate the influence of complex numbers and singularities on the sliding surface; the sliding surface is updated as follows: （23） Among them 1 <c<2,c<a, ; The approach law is chosen to employ both power-law approach law and constant-rate approach law: （24） Where M represents the power-law approach rate gain, and N represents the constant-rate approach rate gain. It is expressed as the absolute value of the sliding surface.

5. The active control method for non-singular fast terminal sliding mode vibration based on a compensation function observer according to claim 1, characterized in that, The design of the smart structure vibration suppression verification experiment described in step 4) is a smart structure vibration suppression verification experiment designed with the NI PCIe-6343 acquisition card as the core.

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