Frequency converter under-actuated system input constraint dynamic output feedback control method

Abstract

The invention relates to the technical field of automatic control and the like, and provides a frequency converter under-actuated system input constraint dynamic output feedback control method, which comprises the following steps of: establishing a non-linear dynamic model containing a driving state and an under-actuated state, setting a non-linear disturbance and speed combined observer, and estimating unknown disturbance and speed in real time by depending on position measurement. On this basis, a first coupling signal free of an integral term and a second coupling signal for strengthening swing feedback are provided, a bounded function is utilized to carry out saturation processing on an intermediate control quantity, and an adjustable gain dynamic output feedback law is synthesized by combining an observation result and the coupling signals, so that external disturbance and parameter perturbation can be resisted at the same time; the input constraint is strictly met, a speed sensor is not needed, and the driving positioning precision, the load anti-swing performance and the global asymptotic stability of the system are guaranteed.

Description

Technical Field

[0001] The present invention relates to the technical fields of automation control and the like, and in particular to an input-constrained dynamic output feedback control method for an inverter under-driven system. Background Art

[0002] Underactuated mechanical systems, where the controlled variable is less than the control input, are widely used in modern industry and manufacturing. However, in practice, underactuated mechanical systems exhibit complex nonlinearities and strong coupling, which poses challenges in effectively achieving their control objectives. Furthermore, due to their underactuated nature, underactuated state variables can only be indirectly controlled through the coupling relationship between the actuated and unactuated states, further increasing the challenge. To address these challenges and more effectively achieve control objectives, the industrial electronics and control communities have invested significant effort in the automated control of underactuated mechanical systems. In particular, bridge cranes, a classic underactuated mechanical system that plays a crucial role in industrial transportation, have been the subject of numerous attempts and a series of advanced control strategies proposed to address their primary control objectives: precise positioning of the trolley and effective anti-sway of the load. In practical applications, bridge cranes often operate in complex and harsh environments and are susceptible to unmodeled dynamics and complex, unknown external disturbances. These effects can degrade the overall control performance of these control methods and even render the system unstable. To ensure the robustness of control systems in these situations, observer-based unknown disturbance compensation techniques have been proposed. Prior art, based on a proposed continuous global sliding mode controller to ensure convergence of the closed-loop system's state variables, introduces a nonlinear disturbance observer to effectively estimate and compensate for uncertain disturbances. By combining control techniques based on interconnected and damping-distributed passivity with a superhelical algorithm, a finite-time disturbance observer is proposed to rapidly estimate and compensate for uncertain disturbances. In addition to ensuring robustness and transient performance, input constraints must also be considered in achieving the primary control objective. This is because in practical applications, the motor power of most underactuated mechanical systems (including cranes) is limited. In many cases, they may not be able to generate the required force / torque, which often leads to performance degradation and, in the long term, motor damage. However, most current control methods ignore input constraints, though a few studies have made significant efforts in this regard. A partially saturated nonlinear control scheme exists in the prior art. This scheme successfully reduces control forces, particularly initial forces, by introducing a smooth saturation function. However, when the system is subjected to disturbances, the controller may calculate input forces that exceed the constraints. To achieve strict input constraints, a saturated control scheme based on MPC has been proposed in the prior art. This control input constraint is successfully addressed by solving a constrained optimization problem using an MPC method. However, this scheme requires a linearized model and is dependent on its parameters. Control performance degrades when the system state deviates significantly from the equilibrium point or when the model parameters are uncertain. To address this, prior art has proposed a nonlinear model-free saturation control scheme based on coupling. This scheme, by appropriately selecting the tuning gain and setting a new Lyapunov function for distributable balance, produces a simple controller with boundaries.However, it suffers from a significant input conservation constraint: its calculated input is far less than the control gain required to satisfy the input constraints. This prevents the motor from fully utilizing its performance under bounded input conditions, necessitating an increase in the control gain to achieve good transient performance. However, this approach leads to the possibility of violating the input constraints. Furthermore, the coupling method introduces an integral term for the swing angle. While this helps improve the system's transient performance, it also makes the system extremely sensitive to mismatched disturbances and prone to undesirable steady-state behavior. Furthermore, the design of bridge crane controllers faces the following challenges: Closed-loop control requires a velocity signal, yet most underactuated mechanical systems (including cranes) lack velocity sensors due to increased cost, weight, or volume. Currently, numerical differentiation is a common approach, but this introduces additional noise, compromising control performance. For bridge cranes, a dual-loop controller combining a closed-loop predictive controller and an open-loop input shaper has been proposed based on optimal control theory. This controller achieves high-precision trolley tracking control and load swing suppression. The predictive controller addresses input saturation by combining a preset performance function and constraints. However, the dual-loop controller fails to account for external disturbances, parameter perturbations, and the unavailability of velocity signals in practical applications. In addition, in the existing technology, for the anti-sway problem of the bridge crane controller, a nonlinear controller is designed based on the energy method, but it does not consider external disturbances, input saturation, parameter perturbations and the unavailability of speed signals in actual applications. The anti-sway performance needs to be improved.

[0003] In summary, the control methods of underactuated mechanical systems in the existing technology do not take into account multiple practical problems that may exist simultaneously in complex underactuated systems, such as external disturbances, input saturation, system parameter perturbations, and unmeasurable speed signals. The adaptability and robustness of the control system under actual complex working conditions need to be improved. Summary of the Invention

[0004] In response to the shortcomings of the above-mentioned existing technologies, the present invention provides an input-constrained dynamic output feedback control method for an inverter under-driven system, so as to take into account multiple practical problems that may exist simultaneously in a complex under-driven system, such as external disturbances, input saturation, system parameter perturbations, and unmeasurable speed signals, and improve the adaptability and robustness of the control system under actual complex working conditions.

[0005] The present invention provides an input-constrained dynamic output feedback control method for an inverter under-driven system, comprising:

[0006] A nonlinear dynamic model is established for an underactuated mechanical system, including an actuated state and an underactuated state, to reveal the coupling relationship between the two. The underactuated mechanical system is subject to unknown external disturbances, parameter perturbations, input saturation, and unmeasurable velocity signals.

[0007] A nonlinear disturbance and velocity joint observer is set based on the nonlinear dynamic model, and the position measurement information is used to estimate the unmeasurable velocity signal and unknown external disturbance in real time. A first coupling signal without integral term is set, and the intrinsic coupling characteristics of the driven state and the underdriven state are used to inject damping and avoid positioning drift caused by mismatched disturbances. A nonlinear feedback term related to underdriven swing is introduced on the basis of the first coupling signal to form a second coupling signal to enhance the ability to suppress load swing;

[0008] An unrestricted intermediate control variable is fed into a bounded function to generate a control input with limited amplitude, ensuring that the control input does not exceed the allowable range of the driving actuator. Based on the real-time estimation results of the unmeasurable speed signal and the unknown external disturbance, as well as the first coupling signal and the second coupling signal, a dynamic output feedback control law with adjustable gain is constructed, and the amplitude-limited control input is applied to the underactuated mechanical system.

[0009] Compared with the prior art, the present invention has the following beneficial effects:

[0010] The present invention provides an input-constrained dynamic output feedback control method for an under-driven system of a frequency converter. A nonlinear dynamic model including a driven state and an under-driven state is established for the under-driven mechanical system to reveal the coupling relationship between the two. The under-driven mechanical system simultaneously has unknown external disturbances, parameter perturbations, input saturation, and unmeasurable speed signals. A nonlinear disturbance and speed joint observer is set based on the nonlinear dynamic model, and the unmeasurable speed signal and the unknown external disturbance are estimated in real time using position measurement information. A first coupling signal without integral terms is set, and damping is injected using the intrinsic coupling characteristics of the driven state and the under-driven state to avoid positioning drift caused by non-matching disturbances. Based on the first coupling signal, a phase-shifting signal related to the under-driven swing is introduced. The nonlinear feedback term of the coupling is used to form a second coupling signal to enhance the ability to suppress load swings. The unrestricted intermediate control quantity is fed into a bounded function to generate a control input with limited amplitude, ensuring that the control input does not exceed the allowable range of the driving actuator. Based on the real-time estimation results of the unmeasurable speed signal and the unknown external disturbance, as well as the first coupling signal and the second coupling signal, a dynamic output feedback control law with adjustable gain is constructed, and the amplitude-limited control input is applied to the under-actuated mechanical system. This takes into account multiple practical problems that may exist simultaneously in complex under-actuated systems, such as external disturbances, input saturation, system parameter perturbations, and unmeasurable speed signals, and improves the adaptability and robustness of the control system under actual complex working conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] The drawings described herein are used to provide a further understanding of the present invention and constitute a part of the present invention. The exemplary embodiments of the present invention and their description are used to explain the present invention and do not constitute an undue limitation of the present invention. Some specific embodiments of the present invention will be described in detail in an illustrative and non-restrictive manner with reference to the drawings. The same reference numerals in the drawings indicate the same or similar components or parts. It should be understood by those skilled in the art that these drawings are not necessarily drawn to scale. In the drawings:

[0012] Figure 1 This is a flow chart of an input-constrained dynamic output feedback control method for an inverter under-driven system according to an embodiment of the present invention;

[0013] Figure 2 This is a schematic diagram of the architecture of a bridge crane system according to an embodiment of the present invention;

[0014] Figure 3 It is a schematic diagram of a variable and parameter comparison table of a bridge crane system according to an embodiment of the present invention. DETAILED DESCRIPTION

[0015] In order to enable those skilled in the art to better understand the solutions of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the embodiments described are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts should fall within the scope of protection of the present invention.

[0016] See also Figure 1-Figure 3 This embodiment provides an input-constrained dynamic output feedback control method for an inverter under-driven system, comprising the following steps:

[0017] S101. Establishing a nonlinear dynamic model for an underactuated mechanical system including an actuated state and an underactuated state to reveal the coupling relationship between the two; the underactuated mechanical system simultaneously has unknown external disturbances, parameter perturbations, input saturation, and unmeasurable velocity signals;

[0018] S102. Setting a nonlinear disturbance and velocity joint observer based on the nonlinear dynamic model, using position measurement information to estimate unmeasurable velocity signals and unknown external disturbances in real time, and setting a first coupling signal without integral terms. Integral-term-free damping is injected using the intrinsic coupling characteristics of the driven and underdriven states to avoid positioning drift caused by mismatched disturbances. A nonlinear feedback term related to underdriven swing is introduced on the basis of the first coupling signal to form a second coupling signal, thereby enhancing the ability to suppress load swing.

[0019] S103. Feeding the unrestricted intermediate control variable into a bounded function to generate a control input with a limited amplitude, ensuring that the control input does not exceed the allowable range of the driving actuator, constructing a dynamic output feedback control law with adjustable gain based on the real-time estimation results of the unmeasurable velocity signal and the unknown external disturbance, as well as the first coupled signal and the second coupled signal, and applying the control input with a limited amplitude to the underactuated mechanical system.

[0020] It should be noted that in existing technologies, when underactuated mechanical systems simultaneously encounter unknown external disturbances, system parameter perturbations, actuator saturation, and unmeasurable velocity signals, traditional control methods often only partially address these challenges, leading to increased vehicle positioning errors, slow load swing angle decay, and even instability. In this embodiment, a nonlinear disturbance and velocity joint observer is used to estimate velocity and disturbances in real time using position signals, replacing noise-sensitive numerical differentiation and laying the foundation for subsequent compensation. At the same time, this embodiment introduces a two-level coupling signal: the first coupling signal abandons the integral term to avoid steady-state drift caused by non-matching disturbances; the second coupling signal strengthens the swing feedback on this basis and improves the load anti-sway damping, thereby ensuring rapid convergence when unknown disturbances and parameter perturbations exist simultaneously; moreover, this embodiment also places the control output within a bounded function to achieve explicit hard constraints on the motor power, which not only prevents excess instructions from damaging the actuator, but also utilizes the adjustable gain to fully tap the potential of the motor, taking into account both steady-state accuracy and transient performance, thereby achieving the synergistic technical effects of precise positioning of the drive position, rapid attenuation of the swing angle, and constant conservation of the input, significantly improving the adaptability and robustness of the under-driven system under complex working conditions.

[0021] Preferably, by constructing a Lyapunov energy function and incorporating the Lasalle invariant set theorem, under the conditions of unknown external disturbances, input saturation, and parameter perturbations, the underactuated mechanical system achieves: the drive position converges asymptotically to the desired value; the underactuated swing angle and its derivatives asymptotically approach zero; and the control input is always constrained to a set saturation value. It should be noted that after the model is perturbed, traditional equivalent linearization or error feedback cannot guarantee global convergence. In this embodiment, a Lyapunov energy function containing observation errors and coupled signals is selected to ensure that the energy decreases strictly monotonically over time. The Lasalle invariant set theorem is then used to lock onto the minimal invariant set that the system must ultimately reside in. Furthermore, actuator saturation can easily cause the closed-loop matrix to violate conventional stability criteria, resulting in slow convergence or divergence. In this embodiment, the bounded property of the saturation function is introduced into the derivative analysis of the Lyapunov function, demonstrating that the energy decay rate remains positive even when the control force is clipped. Lasalle further ensures that the clipping process does not generate new invariant tori or limit cycles.

[0022] It should be noted that the dynamic output feedback control method for a complex underactuated system can be applied to typical underactuated systems including bridge crane systems, and can be applied to complex underactuated mechanical systems that satisfy the generalized second-order autonomous form. Preferably, the bridge crane system includes system 1 and system 2;

[0023] The formula for System 1 is

[0024] The formula for System 2 is

[0025] The variables and parameters of the bridge crane system are described in Figure 3 Comparison table of variables and parameters of the middle bridge crane system.

[0026] In the formulas of System 1 and System 2, S θ and C θ Represent the functions sin(θ) and cos(θ) respectively.

[0027] Preferably, the formula of system 1 is brought into system 2 and sorted out to obtain the control input F, which is expressed as formula 3:

[0028] Among them, the control input F is used as the error dynamics.

[0029] Preferably, both sides of the equation of system 1 are integrated twice with respect to time, resulting in equation 4:

[0030]

[0031] Where c∈R is a constant determined by the initial conditions of integration; x and S in Formula 4 θ With a coupling relationship, a new composite signal ε is introduced, which is expressed as formula 5:

[0032] ε=x+k c S θ ;

[0033] where k c ∈R is any constant, and the composite error signal e is defined as Equation 6:

[0034] e=ε-x d =xx d +k c S θ

[0035] =e x +k c S θ ;

[0036] where e x =xx d is the positioning error signal of the car, xd The desired position for positioning the vehicle.

[0037] Preferably, according to Formulas 3 to 6, the control input F is converted into Formula 7:

[0038]

[0039] Formula 7 is further transformed into Formula 8:

[0040]

[0041] Preferably, Formula 8 is further converted into Formula 9:

[0042]

[0043] Represents the total disturbances to the system, including endogenous and exogenous disturbances.

[0044] Preferably, a new composite signal e is defined Λ As the control signal, it is expressed as formula 10;

[0045]

[0046] Rewrite Equation 9 into Equation 11 based on Equation 10:

[0047] Assumption 1: The payload swing angle satisfies Formula 12:

[0048]

[0049] Assumption 2, perturbations ρ, τ a and τ u It satisfies the following properties, which can be expressed as Formula 13:

[0050]

[0051] in is a positive real number, ε is less than

[0052] Preferably, the control input F is set to achieve: in the presence or absence of mismatching disturbances, the control input F can adjust the driven and underdriven state quantities x, θ and their derivatives to their desired positions, respectively, as expressed in Formula 14:

[0053]

[0054] During the entire control process, the maximum amplitude of the input force is limited to an appropriate range, which is expressed as Formula 15:

[0055] |F|≤Fmax ;

[0056] Among them F max ∈R + is the maximum allowable input force of the trolley;

[0057] The control law does not require speed feedback, but only requires the output of the trolley and pendulum angle position signals as feedback.

[0058] Furthermore, the nonlinear disturbance and velocity joint observer is expressed as Equation 16, which serves as the dynamic feedback component of the final control law. Equation 16:

[0059]

[0060] in and They represent e1(t), and (Right now )’s estimated value; k o1 ,k o2 ∈R + is the observer gain to be determined; κ1∈R + Indicates the switching rate; and represents the estimated error signal defined by the following equation 17:

[0061]

[0062] Taking the derivative with respect to time and considering the result of Equation 11 yields Equation 18:

[0063]

[0064] Based on this, The second derivative with respect to time is calculated as Equation 19:

[0065]

[0066] Formula 20:

[0067] Theorem 1: For any initial error, as t→∞, the estimated error and will converge to 0 and remain consistently bounded if the following formula 21 is satisfied:

[0068]

[0069] For any initial error, as t→∞, the estimated error sum will converge to 0 and remain consistently bounded if the following conditions are met, expressed as Equation 22:

[0070]

[0071] Furthermore, in practical applications, the motor power that provides the input force is limited. When the power required by the control system exceeds this upper limit, input saturation occurs, which often degrades system performance. However, previous research has not adequately considered the issue of input saturation. To address this issue, the following dynamic output feedback control law is constructed, expressed as Equation 23:

[0072]

[0073] where k p ,k d ,k a ∈R + are control parameters, represents the auxiliary signal, which is expressed as Formula 24:

[0074]

[0075] In Formula 23, k p is the position-related gain, used to eliminate positioning errors; k d is the differential-related gain, which is introduced to predict the overall response trend and prevent the system from acting too violently. Note that the speed signal cannot be used for feedback, and the one generated by formula 16 is used. To replace the formula 23 and formula 24 in Control gain k a By introducing more swing feedback into the control signal, the controller is given better anti-swing capability.

[0076] Furthermore, Theorem 2: For the error dynamics of the bridge crane, the dynamic output feedback bounded control method proposed in Equation 9 can drive the trolley to the desired position and suppress and eliminate the payload swing angle, which is expressed as Equation 25:

[0077]

[0078] And the control input is kept within the allowed range at the same time, expressed as Equation 26:

[0079] |F|≤F max ;

[0080] Assume that the control gain is selected as follows:

[0081]

[0082] Proof: First, construct the following non-negative Lyapunov candidate function V(t) (obviously, V(t) ≥ 0), expressed as Formula 28:

[0083]

[0084] Taking the derivative of V(t) with respect to time and inserting it into Equation 16, Equation 23, and Equation 24 yields the following result, expressed as Equation 29:

[0085]

[0086] in In order to make There are two situations.

[0087] First, when When it is always established, we can get This means that the equilibrium of Equation 9 is stable.

[0088] In addition, to make k o1 ,k d ,k a The following conditions are applied, expressed as Equation 30:

[0089]

[0090] Now define the following set, expressed as formula 31:

[0091]

[0092] in, According to conditions (30) and (31), we can get the inequality, which is expressed as formula 32:

[0093]

[0094] Therefore, by choosing If formula 32 holds, ensure that will not exceed the set Ω at the same time, and Bounded. In addition, is a uniformly bounded signal that tends to zero as time approaches infinity. At this point, the system trajectory converges to the region where the next inequality is valid, expressed as Formula 33:

[0095]

[0096] in and properties is used for derivation.

[0097] Therefore, when k o1 Satisfies formula 34:

[0098]

[0099] Can get This gives the following result, expressed as Formula 35:

[0100]

[0101] Therefore, both the observer and the crane system are stable in the Lyapunov sense. Subsequently, the asymptotic stability of the required equilibrium point is proved using the Lasalle invariant set theorem. First, the following set is defined, expressed as Equation 36:

[0102]

[0103] Further, Defined as The maximum invariant set contained in . From formula 33, we can see that when t→∞ In the formula 37,

[0104] e Λ =β1,e=Λ -1 β1,

[0105] where β i ∈R,i=1,2 represents a specific constant.

[0106] Therefore, we can get formula 38:

[0107] e=xx d +k c S θ =0

[0108]

[0109] Substituting Equation 23 into Equation 11, we obtain Equation 39:

[0110]

[0111] At this time, from Formula 22 to Formula 24, Formula 37, we can get Formula 40:

[0112]

[0113] Therefore, we can get formula 41:

[0114]

[0115] According to formula 38, we can get formula 42:

[0116]

[0117] On the other hand, Formula 43 can be derived from Formula 1:

[0118]

[0119] Then, comparing Equation 42 and Equation 43, we can obtain Equation 44:

[0120]

[0121] According to Assumption 2, combined with Formula 41, we can get Formula 45:

[0122]

[0123] Observe that, due to k c Can be any constant, when k c When k > 0, the denominator may be 0, which will cause the crane system to lose stability and fail to operate normally. In order to avoid this situation, it is stipulated that k c <0.

[0124] therefore, Right now Will changes at a constant rate. Assume Then we get formula 46:

[0125]

[0126] Obviously, this fact contradicts the conclusion of formula 35. Therefore, If the assumption does not hold, then , which satisfies Formula 47:

[0127]

[0128] where β3∈R is a constant. Performing some similar analysis will easily lead to the conclusion that , which satisfies Formula 48:

[0129] e x =β4;

[0130] Where β4∈R is a constant. Substituting this result into formula 38, we get formula 49:

[0131]

[0132] According to Assumption 1 and the properties of the cosine function, we can derive Formula 50:

[0133]

[0134] According to Formula 2 and Formula 13, as well as Formula 50, Formula 47 and Assumption 1, we can get Formula 51: S θ=0→θ=0; Substituting this condition into Formula 38 and Formula 48, we can obtain Formula 52:

[0135] e x =β4=0→x=x d ;

[0136] According to the previous analysis, we know that the largest invariant set Only include By citing the Lasalle invariance theorem, it is concluded that the system state converges asymptotically to the desired state.

[0137] Next, it is not difficult to derive formula 53 from formula 23:

[0138] |F|≤k p +k d +k a +k o2 ;

[0139] According to Formula 53 and Formula 27, the following result is obtained, which is expressed as Formula 54:

[0140] |F|≤F max ; Thus, Theorem 2 is proved.

[0141] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the above embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for controlling an input-constrained dynamic output feedback of an inverter under-driven system, characterized in that: The following steps are involved: A nonlinear dynamic model is established for an underactuated mechanical system, including an actuated state and an underactuated state, to reveal the coupling relationship between the two. The underactuated mechanical system is subject to unknown external disturbances, parameter perturbations, input saturation, and unmeasurable velocity signals. A nonlinear disturbance and velocity joint observer is set based on the nonlinear dynamic model, and the position measurement information is used to estimate the unmeasurable velocity signal and unknown external disturbance in real time. A first coupling signal without integral term is set, and the intrinsic coupling characteristics of the driven state and the underdriven state are used to inject damping and avoid positioning drift caused by mismatched disturbances. A nonlinear feedback term related to underdriven swing is introduced on the basis of the first coupling signal to form a second coupling signal to enhance the ability to suppress load swing; An unrestricted intermediate control variable is fed into a bounded function to generate a control input with limited amplitude, ensuring that the control input does not exceed the allowable range of the driving actuator. Based on the real-time estimation results of the unmeasurable speed signal and the unknown external disturbance, as well as the first coupling signal and the second coupling signal, a dynamic output feedback control law with adjustable gain is constructed, and the amplitude-limited control input is applied to the underactuated mechanical system.

2. The input-constrained dynamic output feedback control method for an inverter under-driven system according to claim 1, characterized in that: By constructing the Lyapunov energy function and combining it with the Lasalle invariant set theorem, the underactuated mechanical system can achieve the following results under the conditions of unknown external disturbances, input saturation and parameter perturbations: the drive position converges asymptotically to the expected value; the underactuated swing angle and its derivative asymptotically approach zero; and the control input is always limited to the set saturation value.

3. The input-constrained dynamic output feedback control method for an inverter under-driven system according to claim 2, characterized in that: The complex under-actuated system dynamic output feedback control method is applicable to typical under-actuated systems including bridge crane systems, and is applicable to complex under-actuated mechanical systems that satisfy the generalized second-order autonomous form.

4. The input-constrained dynamic output feedback control method for an inverter under-driven system according to claim 3, characterized in that: The bridge crane system includes system 1 and system 2; The formula for System 1 is The formula for System 2 is The variable and parameter comparison table of the bridge crane system is as follows: In the formulas of System 1 and System 2, S θ and C θ Represent the functions sin(θ) and cos(θ) respectively.

5. The input-constrained dynamic output feedback control method for an inverter underdriven system according to claim 4, wherein the formula of system 1 is substituted into system 2 and the control input F is obtained, which is expressed as formula 3:

6. The input-constrained dynamic output feedback control method for an inverter under-driven system according to claim 5, characterized in that: Integrating both sides of the equation in system 1 twice with respect to time yields equation 4: The parameter c is a real number R, which is determined by the initial conditions of the integral; x and S in formula 4 θ With a coupling relationship, a new composite signal ε is introduced, which is expressed as formula 5: ε=x+k c S θ ; Its parameter k c For real number R, define the composite error signal e, which is expressed as formula 6: e=ε-x d =x-x d +k c S θ =e x +k c S θ ; where e x =xx d is the positioning error signal of the car, x d The desired position for positioning the vehicle.

7. The input-constrained dynamic output feedback control method for an inverter under-driven system according to claim 6, characterized in that: Based on Equations 3 to 6, the control input F is converted to Equation 7: Formula 7 is further transformed into Formula 8:

8. The input-constrained dynamic output feedback control method for an inverter under-driven system according to claim 7, characterized in that: Formula 8 is further transformed into Formula 9: Represents the total disturbances to the system, including endogenous and exogenous disturbances.

9. The input-constrained dynamic output feedback control method for an inverter under-driven system according to claim 8, characterized in that: Define a new composite signal e Λ As the control signal, it is expressed as formula 10; e Λ =Λe, Rewrite Equation 9 into Equation 11 based on Equation 10: Assumption 1: The payload swing angle satisfies Equation 12: Assumption 2, perturbations ρ, τ a and τ u It satisfies the following properties, which can be expressed as Formula 13: in is a positive real number, ε is less than 10. The input-constrained dynamic output feedback control method for an inverter under-driven system according to claim 9, characterized in that: The control input F is set to achieve: in the presence or absence of mismatched disturbances, the control input F can adjust the driven and underdriven state quantities x, θ and their derivatives to their desired positions, respectively, as expressed in Equation 14: During the entire control process, the maximum amplitude of the input force is limited to an appropriate range, which is expressed as Formula 15: |F|≤F max ; Among them F max ∈R + is the maximum allowable input force of the trolley; The control law does not require speed feedback, but only requires the output of the trolley and pendulum angle position signals as feedback.

Images