29 results about "Second order systems" patented technology

The invention provides a loading force control method for servo torque feed-forward and linear active disturbance rejection control, and belongs to the technical field of servo hydraulic equipment control. The method comprises the steps that a servo hydraulic tension pad dynamic model is constructed, and the servo torque feed-forward and linear active disturbance rejection control loading force is obtained based on the servo tension pad dynamic model and structural design parameters of a servo tension pad test bed; building a simulation model of the servo stretching pad; a simulation data set and an experiment data set are collected, the simulation data set and the experiment data set are combined, weighted and preprocessed, and a standardized feature matrix is obtained; constructing a servo feed-forward torque prediction model based on XGBoost, taking the standardized feature matrix as model input, and outputting a feed-forward torque; constructing a second-order system model of the servo stretching pad; and combining the feed-forward torque output by the servo feed-forward torque prediction model with the feedback torque calculated by the linear active disturbance rejection control to form a final torque instruction of the servo motor. According to the invention, the control precision and robustness of the loading force in the production process of the servo hydraulic stretching cushion are improved.

The invention discloses a piezoelectric micro-positioning platform adaptive fuzzy control method considering time-varying output constraint. The piezoelectric micro-positioning platform adaptive fuzzy control method comprises the steps of 1) describing a piezoelectric micro-positioning platform as a second-order system with an input hysteresis nonlinear characteristic; 2) introducing a hysteresis operator into an extended fuzzy input space, and estimating hysteresis on line by using a fuzzy dynamic hysteresis estimator; 3) taking a backstepping method as a control framework, introducing an auxiliary function related to a time-varying output constraint condition, and designing a virtual control law and a self-adaptive updating law; and 4) designing an actual control law of the piezoelectric micro-positioning platform based on the steps 1)-3) and a Lyapunov stability criterion. According to the control method, the fuzzy dynamic hysteresis estimator is used for achieving online estimation of dynamic hysteresis behaviors with the frequency dependence characteristic, output is always kept within the preset time-varying output constraint condition, the influence of hysteresis is eliminated, and high-precision positioning control over the piezoelectric micro-positioning platform is achieved.

The invention also discloses an SIDO Boost converter terminal complementary superspiral sliding mode improved active-disturbance-rejection control method. The method specifically comprises the following steps: step 1, establishing a state space averaging model by using a state space averaging method; 2, fitting the state space average model into a standard second-order system normal form; 3, designing a fractional order cascade expansion state observer according to a standard second-order system normal form; step 4, applying the observation value obtained in the step 3 to a terminal complementary sliding mode controller; 5, introducing the super-spiral sliding mode reaching law into the terminal complementary sliding mode; step 6, according to the equivalent control law and the switching control law, obtaining a total control law of the system after summation; and step 7, applying the observer and the controller to a main voltage single closed loop and a branch voltage outer loop and a current inner loop. The problems that a traditional active-disturbance-rejection control method is insufficient in external total disturbance estimation precision of the single-inductor dual-output converter and cross influence exists between two output branches are solved.

The application discloses a second-order system privacy protection average consistency method based on edge disturbance, which comprises the following steps: determining the communication topology graph and Laplace matrix of a multi-agent system, and representing the information interaction relationship between agents by the Laplace matrix; establishing a dynamic model of the multi-agent system; in the initial iteration, each agent generates a disturbance signal for each edge connected to it, and determines the type of attacker; after confirming the attack type, the agent adds a disturbance signal to the transmission of position information and speed information to the neighbor, and the disturbed information is transmitted to the neighbor; after the disturbance signal stage ends, the agent switches to a standard second-order dynamic average consistency algorithm, and updates its own state according to the neighbor feedback information. The application can guarantee privacy protection without sacrificing the convergence accuracy of the consistency algorithm, is suitable for a resource-limited environment, and can effectively resist privacy theft of internal and external attackers.

The application discloses a control method for primary frequency modulation in a power dead zone range of a hydroelectric generating set, and belongs to the technical field of hydroelectric generating set regulation. The method comprises the following steps: S1, calculating H1 and H2 of a water diversion system according to the water flow inertia time constant of the water diversion system and the mechanical inertia time constant of the generating set; S2, calculating a dynamic analytical expression of the generating set power when a frequency unit step disturbance occurs; S3, calculating a first-order differential link coefficient decomposed by a second-order correction link based on the dynamic analytical expression of the generating set power; and S4, calculating the output of a PID speed regulator in a current sampling period. The scheme calculates the oscillation link transfer function parameters in a dynamic process, and a second-order system correction transfer function of a frequency given value is designed on the basis, so that the oscillation phenomenon of the generating set power does not occur when the frequency given value of the generating set changes, the problem that the power exceeds the power dead zone in the dynamic process is prevented, and a limiting link is arranged at the frequency deviation value to ensure that the power steady-state value exceeds the power dead zone range.

The invention belongs to the technical field of chemical industry safety automatic control, and particularly relates to a control method of a two-stage chemical reactor system under spoofing attack and unknown disturbance. Comprising the following steps: establishing a state equation and a general second-order system state space expression according to operation state information of a two-stage chemical reactor system; in consideration of spoofing attack, constructing a specified performance function and error signals, introducing a first-order filter, and converting a first error signal into a specified performance error signal; according to the constructed error signals, respectively constructing a Lyapunov function for the two-stage system, and designing a corresponding virtual controller, an adaptive law and a disturbance observer under a backstepping method framework; and a dynamic event triggering mechanism is combined to reduce data transmission quantity and reduce attack risks. According to the method, the stability, the safety and the operation efficiency of the two-stage chemical reactor system in a complex environment can be effectively improved, deception attack and disturbance are reliably inhibited, and efficient and safe operation of the reaction process is guaranteed.

The invention discloses a dielectric elastomer actuator control method based on a fractional order creep hysteresis model, and relates to the field of intelligent material driving control. The method comprises the following steps: constructing a fractional order creep hysteresis model; creep inverse compensation is carried out on the fractional order creep hysteresis model through an inverse compensator; constructing a second-order system state equation of the dielectric elastomer actuator, and defining a tracking error of the dielectric elastomer actuator based on the second-order system state equation and the expected trajectory of the dielectric elastomer actuator; taking convergence of tracking errors in a preset performance boundary as a constraint condition, approaching a nonlinear function in a second-order system state equation of the dielectric elastomer driver by using a radial basis function neural network, and recursively designing a temporary control law and an adaptive law through a backstepping method; and extracting an actual control signal of the dielectric elastomer actuator through a hysteresis implicit inverse algorithm based on a temporary control law and an adaptive law of a backstepping method recursion design. The method can improve the control precision of the dielectric elastomer actuator.

The present application discloses a method, apparatus, device, and medium for controlling a second-order system with a preset convergence time, and relates to the field of automatic control technology. The method comprises: obtaining a first error state equation of the second-order system; obtaining a pre-constructed convergence function and substituting the convergence function into the first error state equation to obtain a second error state equation; the convergence function is used to make the state error of the second-order system converge from an initial state to zero along a preset convergence path and within a specified convergence time; constructing an extended state observer for the second error state equation based on a nonlinear function; constructing a control law based on the extended state observer; and controlling the second-order system using the control law. The convergence function of the present application can make the state error of the second-order system converge from an initial state to zero along a preset convergence path and within a specified convergence time, thereby realizing a control method for the second-order system with a preset convergence process and a preset convergence time, thereby improving the self-disturbance rejection performance of the second-order system.

The application discloses a high-dynamic position filtering method for a permanent magnet motor system, comprising the following steps: a second-order system model is established based on a mechanical motion equation of a permanent magnet synchronous motor; a third-order extended state observer is designed, and the position, the speed and the total disturbance are estimated simultaneously; in the position controller, the estimated rotor position output by the observer is selected as the feedback signal of the position controller, the high-frequency noise suppression capability is improved while the high dynamic performance of the system is ensured; a position-speed cascade controller is constructed, the estimated total disturbance is used for real-time feedforward compensation of a torque instruction, and a final electromagnetic torque reference instruction is generated; and the electromagnetic torque reference instruction is input into a current controller and a space vector pulse width modulation module to drive the permanent magnet synchronous motor to operate. The application solves the limitations of poor filtering performance or phase lag of the feedback signal of the existing scheme, and improves the control bandwidth and dynamic response performance of the system.

The application relates to the technical field of inverter control, and proposes an HJI inverter robust control method based on a disturbance observer. The method can effectively control a second-order system in the case of external disturbance, uncertainty and unknown disturbance. In order to eliminate the time delay influence of a remote signal on output feedback during transmission, a second-order disturbance observer is used to identify external disturbance. In addition, an SMC control law is designed according to HJI theory to improve the robustness of the system, and the stability of the system is analyzed by using the second Lyapunov method. Compared with the prior art, the application can effectively improve the control of the second-order system, and verifies the feasibility of the theory on an auxiliary power inverter.

The invention provides a new energy automobile EHB system pressure control method based on active disturbance rejection and sliding mode collaboration, which comprises the following steps: establishing a second-order system approximation model of an EHB system, and designing a nonlinear active disturbance rejection rotating speed controller as an inner ring controller based on the model; designing a disturbance feed-forward sliding mode controller as an outer ring controller based on disturbance observed by a nonlinear active-disturbance-rejection rotating speed controller in real time; based on an inner ring controller and an outer ring controller, a cascade control architecture of the main cylinder pressure is constructed, the output of the outer ring controller serves as a rotating speed instruction signal of the inner ring controller, and the output of the inner ring controller directly drives an execution motor; and based on the cascade control architecture, cooperative control is performed on the master cylinder pressure, so that the master cylinder pressure tracks the target pressure. According to the invention, the dynamic response speed and the anti-interference capability can be improved, and the robustness is enhanced.

The present application relates to the technical field of wind power, and discloses a grid-forming device control method and apparatus, a controller, a device, and a medium. The method comprises: on the basis of an impedance from a grid side converter to a power grid, a preset inertia time constant and a preset angular velocity parameter, obtaining a damping coefficient of an active power synchronous loop by using a second-order system characteristic of the active power synchronous loop; on the basis of the damping coefficient, the inertia time constant and a control parameter of the active power synchronous loop, performing active power synchronization calculation, to obtain an internal potential phase of a grid-forming device; on the basis of a control parameter of a reactive power closed loop, performing reactive closed loop control calculation, to obtain an internal potential amplitude of the grid-forming device; and by using a PWM signal generated on the basis of the internal potential phase and the internal potential amplitude, controlling the grid side converter to operate.

The invention relates to a virtual synchronous machine virtual inertia and damping coefficient adaptive control method, system and device and a medium. The method comprises the following steps: establishing a VSG typical grid-connected model; based on a VSG typical grid-connected model, the selection principle of the virtual inertia and the damping coefficient is analyzed when the load changes or the new energy output changes; establishing a mathematical relationship between the virtual inertia and the damping coefficient under the optimal damping ratio through a second-order system small signal analysis and parameter setting method, and analyzing to obtain the influence of the virtual inertia and the damping coefficient on the system stability; and according to the determined selection principle of the virtual inertia and the virtual damping, a logarithmic function-based adaptive virtual inertia and damping coefficient control strategy is adopted, and the reference values of the virtual inertia and the damping coefficient are determined according to the optimal damping ratio. The method can be widely applied to the technical field of virtual synchronous generator control.

The invention discloses a storm coupling dynamic indirect approximation method for a Spar type floating offshore wind turbine. The storm coupling dynamic indirect approximation method comprises the steps that a system equation of a Spar type floating offshore wind turbine platform is constructed; the system equation of the Spar type floating offshore wind turbine platform is converted into a second-order system; expanding the second-order system to obtain an expanded system; designing an auxiliary system according to the extended system, and obtaining a state estimation value of the auxiliary system; defining an error vector; designing an intermediate virtual control law according to the error vector; according to the error vector and the intermediate virtual control law, designing a change law of an auxiliary system control law; obtaining an auxiliary system control law according to the change law of the auxiliary system control law; and designing an indirect approximator of an unknown nonlinear function according to the state estimation value of the auxiliary system and the control law of the auxiliary system. According to the method, the dynamic influence of the wind wave coupling effect on the Spar type floating offshore wind turbine platform can be effectively approximated, and a new technical approach is provided for coupling dynamic response approximation of the Spar type floating offshore wind turbine platform.

This invention belongs to the field of motion control systems and relates to a state perception and fault-tolerant control method for motion control systems under sensor failure. The method includes: representing the dynamic characteristics of the motion control system under sensor failure using a disturbed second-order system model; constructing a state-space control model of the motion control system; correcting distorted state feedback signals using a state feedback self-corrector; estimating the lumped disturbance of the system model using an improved fuzzy approximator based on an adaptive update rate to obtain lumped disturbance observations; estimating the state variables of the motion control system using a self-correcting state observer to obtain state observations; and designing a state feedback fault-tolerant controller using the state observations and lumped disturbance observations. This invention effectively solves the key problem of feedback signal distortion in motion control systems under sensor failure scenarios, and has advantages such as accurate estimation of unmodeled disturbances, good feedback signal correction effect, and accurate system state estimation.

The present application relates to a kind of machine tool flexible ball screw system decoupling control method, first to the speed inner ring transfer function is decoupled, obtain the flexible modal item in system;Based on flexible modal item design shaping controller, the parameter of shaping controller is only determined by the anti-resonance frequency of system and anti-resonance damping ratio;Speed inner ring closed loop dynamic shaping is target second-order system, so that resonance suppression and bandwidth adjustment are realized independently by PI controller.In this way, the transfer function of position outer ring is decoupled, separates out the first order component of high-order flexible mode, and designs additional controller based on position difference feedback, the feedback gain is equal to the proportional gain size of P controller in position outer ring, and the sign is opposite, suppress the influence of the first order component of high-order flexible mode.The present application realizes the decoupling of shaping controller and P-PI controller parameter setting, simplifies parameter setting process, and improves the closed-loop bandwidth and industrial practicability of flexible ball screw feeding system.

The application discloses a spacecraft attitude preset time control method based on adjustable preset performance, first, a kinematics and dynamics model is established for spacecraft attitude control to obtain a spacecraft attitude control system model. Then, an adjustable preset performance function and a new error conversion function are designed to convert the original system into a new second-order system. Finally, a non-singular sliding mode controller is designed for the converted second-order system to realize accurate control of system error, so that the system still has the ability to keep stable and the controller can take effect under certain sudden disturbance. The method introduces an adjustable preset performance boundary, designs a new error conversion function, and combines the non-singular sliding mode control to realize accurate control of system error, improve the robustness and anti-disturbance ability of the system, and provide solid technical support for spacecraft attitude control.

The application discloses a multi-agent differential game control method based on a signal preserving matrix and belongs to the field of multi-agent cluster motion control. The application discloses a multi-agent differential game control method based on a signal preserving matrix, solves the problem that a traditional differential game system dynamic approximate solution construction method loses a feedback signal by introducing a signal preserving matrix, avoids the problem that the traditional differential game system dynamic approximate solution construction method is only applicable to a differential game system composed of 1-relative order subsystems, and obtains an approximate optimal feedback control law of a differential game system composed of 2-relative order subsystems through the signal preserving matrix. The application expands the applicable objects of the differential game system dynamic approximate solution construction method to a large range and has important application value. The application can dynamically obtain an approximate optimal solution of the differential game system and is applicable to cluster motion control of a second-order system such as an unmanned aerial vehicle or a spacecraft.

The application discloses a robot joint space variable impedance controller design method based on reinforcement learning, the method obtains a robot joint space impedance control law by analyzing a robot joint space dynamics model and combining a physical representation of a second-order system of an expected robot joint dynamic response, and designs robot joint impedance control parameters on the basis; in combination with joint position, speed, torque value and torque change speed and other information acquired by a robot sensor, a reward function is designed on demand, an approximate Q function is obtained through a linear method, a feature matrix parameter is obtained by using a time difference method, and the designed variable impedance parameter is obtained. The method can be used for impedance control of a robot system, can better realize adaptive variable impedance dynamic characteristics of the robot facing unknown environment physical interaction, and is an important, effective and low-cost means for improving flexible interaction between the robot and the environment.

The application provides a loading force control method of servo torque feedforward and linear active disturbance rejection control, and belongs to the technical field of servo hydraulic equipment control. The method comprises the following steps: constructing a servo hydraulic tensile pad dynamics model, building a simulation model of the servo tensile pad based on the servo tensile pad dynamics model and structural design parameters of a servo tensile pad test bench; collecting simulation data sets and experimental data sets, merging, weighting and preprocessing the simulation data sets and the experimental data sets to obtain a standardized feature matrix; constructing an XGBoost-based servo feedforward torque prediction model, taking the standardized feature matrix as the model input and outputting the feedforward torque; constructing a servo tensile pad second-order system model; combining the feedforward torque output by the servo feedforward torque prediction model and the feedback torque calculated by the linear active disturbance rejection control to form the final torque instruction of the servo motor. The application improves the control precision and robustness of the servo hydraulic tensile pad in the production process.

The present disclosure relates to a kind of VSG parallel control method and device based on inertia / damping parameter collaborative adaptation.The method includes: the small signal modeling of virtual synchronous generator (VSG) parallel system is carried out, and the second-order system model of the virtual synchronous generator parallel system is generated;The oscillation curve of the angular frequency of single cycle of the virtual synchronous generator parallel system is divided into four intervals, and the influence relationship of the inertia parameter J and the damping parameter D on the power vibration stability of the virtual synchronous generator parallel system is based on, and inertia parameter J adaptive control strategy and damping parameter D adaptive control strategy are respectively assigned to the four intervals;Complete VSG parallel control based on inertia / damping parameter collaborative adaptation.The present disclosure effectively makes up for the deficiency of droop control by adding inertia and damping link in the parallel control strategy of network type PCS, and realizes the collaborative adaptive control of VSG parallel system based on inertia / damping parameter.

The invention discloses a grid construction converter transient synchronous stability evaluation method considering direct voltage dynamics. The method comprises the following steps: S1, constructing a grid construction converter grid-connected system multi-scale nonlinear four-order mathematical model considering direct current bus capacitance, direct current voltage control, virtual synchronous generator control and network transient response; s2, calculating a critical mapping relation between system states under a critical stable reverse trajectory of an unstable equilibrium point of the system by adopting an offline iterative algorithm; s3, constructing a post-scaling second-order system based on a scaling method according to the real-time operation state of the grid-connected system of the network construction converter; s4, the transient stability boundary is analyzed and calculated on line for the second-order system after scaling, the transient synchronization stability of the grid-connected system is evaluated, the dynamic state of a direct current bus capacitor and the dynamic state of a direct current voltage loop are considered, the transient synchronization stability of the grid-forming converter is accurately analyzed through the offline and online combined method, and the transient synchronization stability of the grid-connected system is evaluated. And high-precision evaluation of the stability boundary of the network construction converter is realized.

The invention discloses a fullerene functionalized biosensor and a detection method thereof, relates to the technical field of biosensors, and can better describe the response process of the sensor by constructing and optimizing first-order and second-order system response models. Compared with a first-order system response model, the second-order system response model can more effectively simulate the nonlinear characteristics of the signals and the long-term stability of the system, especially when the change process of the signals is complex, the model can improve the prediction precision, and the reliability and stability of the system in practical application are ensured. In the process of adjusting the dynamic response model in combination with the response time Tx and the maximum change rate delta Imax, the system can perform real-time adjustment according to the change rate of the concentration of the target object and the response speed of the sensor. By optimizing the parameters, the system can better adapt to different experimental environments and dynamic conditions, and the adaptive capacity of the sensor in a complex environment is enhanced.

The application discloses a kind of robot compliant control method, device and electronic equipment, it is related to robot compliant control technical field, comprising: obtaining Cartesian planning position, joint angle feedback position and Cartesian feedback position;Utilize target compliant controller, Cartesian planning position and Cartesian feedback position and joint angle feedback position are handled, and joint output torque is obtained;Utilize joint output torque to control robot movement;Target compliant controller is to robot for dynamics modeling and second-order system dynamic characteristic modeling to obtain original compliant controller, and original compliant controller is simplified to obtain;According to the virtual wall numerical value of preestablished and Cartesian feedback position, and preestablished parameter, adjust the stiffness parameter and damping parameter of second-order system in target compliant controller.The influence of the accuracy of external torque on the control accuracy of robot movement can be eliminated.The influence of robot singularity on motion control can be avoided.The virtual wall function can be easily realized.

The application relates to a BP neural network-based self-disturbance control method for a maglev train suspension system. The method comprises the following steps: approximating the maglev train suspension system at an equilibrium point as a second-order system by using linearization processing technology, adopting a third-order extended state observer to estimate the gap signal and the speed signal of the system output and realizing compensation on the disturbance; obtaining the tracking signal and the differential signal of the target signal by a tracking differentiator, and forming corresponding error signals with the obtained signals; inputting the obtained error signals into a PD controller, feeding back and compensating the output of the controller by the extended state observer, and obtaining the input signal of the system; applying the input signal of the system to the suspension system, suspending the vehicle body on the track and obtaining the output signal of the system; transmitting the error signal and the output signal of the system to a BP neural network to optimize the parameters of the extended state observer; and repeating the above steps. The application realizes the control of the maglev train suspension system with strong anti-disturbance and real-time and accurate estimation of the disturbance.

The application discloses a kind of six-phase motor open-loop fault-tolerant control method suitable for low carrier ratio condition, belongs to motor fault-tolerant control field, and the present application is to solve the problem of single-phase open-loop fault-tolerant control method of existing six-phase motor in low carrier ratio condition Control performance deterioration, poor stability problem.The application includes: configuration fault-tolerant coordinate transformation matrix, the mathematical model of mutual inductance coupling in natural coordinate system is transformed into the mathematical model without mutual inductance coupling in synchronous rotating coordinate system, and the spatial voltage equation of mutual decoupling and linear independence is obtained;By coordinate positive transformation, fault-tolerant phase current is transformed into two spatial current components;Discrete second-order rotating decoupling controller is set in two spaces respectively, and voltage control quantity is generated;The motor is driven by phase voltage through coordinate inverse transformation.The application configures fault-tolerant current loop as discrete second-order system, and the distribution of closed-loop pole does not change with the reduction of carrier ratio, so that fault-tolerant control has good stability in low carrier ratio condition.

The invention relates to a decoupling control method for a flexible ball screw system of a machine tool, and the method comprises the steps: firstly carrying out the structural decoupling of a speed inner ring transfer function, and obtaining a flexible modal term in the system; designing a shaping controller based on the flexible modal item, wherein the parameters of the shaping controller are only determined by the anti-resonance frequency and the anti-resonance damping ratio of the system; a closed loop of a speed inner loop is dynamically shaped into a target second-order system, so that resonance suppression and bandwidth adjustment are independently realized through a PI controller. On the basis, structural decoupling is carried out on a position outer ring transfer function, a high-order flexible mode first-order component is separated, an additional controller based on position difference feedback is designed, the feedback gain of the additional controller and the proportional gain of a P controller in a position outer ring are equal in size and opposite in sign, and the influence of the high-order flexible mode first-order component is restrained. According to the method, decoupling of parameter setting of the shaping controller and the P-PI controller is achieved, the parameter setting process is simplified, and the closed-loop bandwidth and industrial practicability of the flexible ball screw feeding system are improved.

The invention discloses an SIDO Buck converter fractional order timing convergence sliding mode active-disturbance-rejection control method of an adaptive A-T reaching law. The method specifically comprises the following steps: step 1, establishing a state space averaging model; 2, fitting the state space average model into a standard second-order system active-disturbance-rejection normal form; 3, designing a finite time observer; 4, observing the output value by using a finite time observer; 5, designing a fractional order timing convergence sliding mode surface, and solving an equivalent control law of the system; 6, introducing an adaptive A-T reaching law, and solving a system switching control law; 7, designing a controller according to the equivalent control law and the switching control law; and step 8, applying the observer and the controller to the closed-loop system of the branch circuit and the main circuit part to realize active-disturbance-rejection control. The problems that traditional active-disturbance-rejection control has cross influence on two output branches of the single-inductor dual-output converter and traditional ESO is insufficient in external total disturbance estimation precision are solved.

This invention discloses an open-circuit fault-tolerant control method for a six-phase motor suitable for low carrier ratio conditions, belonging to the field of motor fault-tolerant control. The invention addresses the problem of degraded control performance and poor stability of existing single-phase open-circuit fault-tolerant control methods for six-phase motors under low carrier ratio conditions. The invention includes: constructing a fault-tolerant coordinate transformation matrix to transform the mutually inductively coupled mathematical model in the natural coordinate system into a mathematical model without mutual inductive coupling in the synchronous rotating coordinate system, obtaining mutually decoupled and linearly independent space and space voltage equations; transforming the fault-tolerant phase current into current components in two spaces through forward coordinate transformation; setting discrete second-order rotating decoupling controllers in both spaces to generate voltage control quantities; and obtaining the phase voltage to drive the motor through inverse coordinate transformation. This invention configures the fault-tolerant current loop as a discrete second-order system, where the closed-loop pole distribution does not change with decreasing carrier ratio, thus ensuring good stability of the fault-tolerant control under low carrier ratio conditions.