A feedback control method for a nonlinear two-dimensional system

Abstract

This application discloses a feedback control method for nonlinear two-dimensional systems, belonging to the field of control theory and control engineering technology, and applicable to controlled objects with nonlinear characteristics and parameter uncertainties. The method first performs dynamic modeling of the controlled object, representing it as a two-dimensional underactuated nonlinear state system; then, it structures the nonlinear terms in the system, constructing an equivalent nonlinear mapping related to the state variables; based on this, it designs a state feedback control law to compensate for system nonlinearity and introduce stability damping; further, it introduces a continuous robust compensation term to address parameter uncertainties and external disturbances in the system; finally, it applies the feedback control law to the system to form a closed-loop control system, achieving global stability control of the system. The method of this application has a clear structure and a simple design process, and is applicable to the stability control problems of various nonlinear underactuated systems.

Description

Technical Field

[0001] This application relates to the fields of control theory and control engineering technology, and in particular to a feedback control method for nonlinear two-dimensional systems. Background Technology

[0002] In practical industrial systems, all physical systems possess some degree of nonlinearity. While the theoretical framework for linear systems is now quite comprehensive, these theories are no longer applicable to the analysis and design of most nonlinear systems. Especially with the continuous advancement of modern science and technology, the requirements for system control accuracy are gradually increasing, and research on nonlinear systems is becoming increasingly in-depth. Many existing control strategies cannot meet these requirements. Therefore, theoretical research on nonlinear systems is essential.

[0003] In recent years, nonlinear systems have become increasingly complex and large-scale, while the requirements for their accuracy have also gradually increased. This system complexity has brought unprecedented challenges to the design of control algorithms for practical industrial systems, and the control of nonlinear systems remains a critical issue. Consequently, some theories for handling nonlinear systems have been explored in depth, and many advanced control methods have emerged. Adaptive output feedback control, by combining adaptive observers and intelligent control methods, can effectively handle situations where system parameters are unknown or changing, and it has shown good adaptability for complex nonlinear systems. However, because this control method usually requires real-time adjustment of control system parameters, it may lead to high computational complexity and may exhibit insufficient robustness when facing external disturbances or system model uncertainties. Robust output feedback control, due to its ability to achieve global stability in nonlinear systems and its strong robustness against system uncertainties, external disturbances, and parameter changes, is widely used in practical engineering. However, the design of robust output feedback controllers is usually complex, and real-time calculation and adjustment of control parameters are typically required, significantly increasing the computational burden and affecting the real-time performance of the system. Sliding mode output feedback control typically exhibits fast response characteristics and strong robustness to disturbances and uncertainties. However, it is prone to chattering, which can lead to mechanical wear or damage to electronic components. Furthermore, the design of sliding mode controllers is relatively complex. Optimal output feedback control offers significant advantages in achieving system performance optimization and can be widely applied in nonlinear system control, especially in situations where the state is unmeasurable. However, its high computational complexity, strong dependence on model accuracy, and potential conservatism in certain cases require further research and improvement.

[0004] In summary, although considerable research has been achieved in feedback control of nonlinear systems, the following problems still exist: (1) the controller design of most feedback control methods is relatively complex; (2) the applicability is not strong and cannot be widely applied to linear and nonlinear systems; and (3) the computational complexity is high. Therefore, it is necessary to propose a feedback control method for designing nonlinear feedback controllers. Summary of the Invention

[0005] This invention addresses the technical problems existing in the background art by proposing a feedback control method for nonlinear two-dimensional systems.

[0006] To solve the technical problem, the technical solution of the present invention is as follows:

[0007] A feedback control method for a nonlinear two-dimensional system, the method comprising the following steps:

[0008] S1: Perform dynamic modeling on the controlled object, representing the controlled object as a two-dimensional underactuated nonlinear state system;

[0009] S2: The nonlinear terms in the state system are structured and represented as equivalent nonlinear mappings proportional to the system state variables. The equivalent nonlinear mappings are then judged to satisfy the continuity or boundedness conditions at the equilibrium point to determine whether the system meets the feedback control design requirements.

[0010] S3: Based on the equivalent nonlinear mapping, construct a state feedback control law to compensate for system nonlinearity and introduce stability damping;

[0011] S4: To address the parameter uncertainties and external disturbances present in the state system, a continuous robust compensation term based on the upper bound of uncertainty is introduced into the state feedback control law to suppress the impact of uncertainty on system stability.

[0012] S5: Apply the state feedback control law to the two-dimensional underactuated nonlinear state system in S1 to form a closed-loop control system and achieve global stable control.

[0013] Furthermore, the two-dimensional underactuated nonlinear state system includes a first state equation and a second state equation. The system control input acts directly only on the second state equation, while the first state equation is indirectly controlled by the second state equation through the internal coupling relationship of the system.

[0014] Furthermore, in step S2, the structuring of the nonlinear terms in the state system includes: reconstructing the nonlinear terms into an equivalent nonlinear mapping proportional to the second state variable of the system.

[0015] Furthermore, the ratio of the equivalent nonlinear mapping to the corresponding state variable is continuous or bounded at the system equilibrium point to avoid singularity in the feedback control law at the equilibrium point.

[0016] Furthermore, the state feedback control law constructed in step S3 includes: a nonlinear compensation term for compensating the equivalent nonlinear mapping, and a damping term related to the system state variables, for improving the stability of the closed-loop system.

[0017] Furthermore, the damping term is a linear damping term or a nonlinear damping term related to the second state variable, and its coefficient is a design parameter greater than zero.

[0018] Furthermore, the continuous robust compensation term introduced in step S4 is constructed based on the known upper bounds of system parameter uncertainties and external disturbances, and is used to suppress the impact of the uncertainties on system stability.

[0019] Furthermore, the continuous robust compensation term adopts a continuous and bounded nonlinear function form to avoid chattering caused by discontinuous control input.

[0020] Furthermore, the continuous and bounded nonlinear function includes the arctangent function, the saturation function, or a function with equivalent continuous and bounded properties.

[0021] Furthermore, the state feedback control law is configured to cause the closed-loop system state to asymptotically converge to the equilibrium point in the global domain under the conditions of system parameter uncertainty and external disturbance.

[0022] This application has the following advantages:

[0023] (1) By performing unified modeling and structuring of nonlinear two-dimensional underactuated systems, this invention transforms complex nonlinear systems into equivalent structures that facilitate feedback control design, avoiding the complex recursive design process in traditional methods and significantly simplifying the controller design steps.

[0024] (2) The feedback control law used in this invention is an explicit continuous form, which does not require the introduction of switching logic or discontinuous control structure, effectively avoiding control chattering problem and improving the actual feasibility and operational stability of the system.

[0025] (3) By introducing a robust compensation term based on the upper bound of uncertainty into the control law, the present invention can effectively suppress the influence of parameter uncertainty and external disturbance on system stability, and enhance the robustness and adaptability of the system.

[0026] (4) The method of the present invention does not depend on the precise model parameters of the system, has low requirements for system structure, and has good versatility and scalability. It can be applied to various stable control scenarios of nonlinear underactuated systems. Attached Figure Description

[0027] To more clearly illustrate the technical solutions in the embodiments of this application 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 some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0028] Figure 1 This is a flowchart illustrating a feedback control method for a nonlinear two-dimensional system provided in an embodiment of this application. Detailed Implementation

[0029] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0030] Example 1:

[0031] like Figure 1 As shown, the nonlinear two-dimensional system feedback control method proposed in this invention will be described in detail below with reference to specific embodiments. This embodiment is based on the system model and control law shown in formulas (1) to (11), and the implementation process and technical effects of each step will be explained one by one. This embodiment is only used to illustrate the technical solution of this invention and does not constitute a limitation on the scope of protection of this invention.

[0032] Step 1: Construct robust control laws for a special form of two-dimensional system, specifically including:

[0033] 1.1) In this embodiment, the controlled object is first modeled dynamically, and represented as a two-dimensional nonlinear underactuated system as follows:

[0034] (1)

[0035] in, and These are the system's state variables; It is system input; and Represents two nonlinear functions; It is the system's error term, which is a constant and In practical implementation, after determining that the system's nonlinear term satisfies the condition that (p1(s2)) / s2 is continuous at s2=0, the following state feedback control law is constructed based on this condition. :

[0036] (2)

[0037] In the formula, It is a constant greater than 0.

[0038] 1.2) Based on the above theory, a basic theory for integrated control of underactuated nonlinear systems is proposed.

[0039] Consider the following two-dimensional system:

[0040] (3)

[0041] in, , and This is the system status. Input for the system. and These are two nonlinear mappings. , , and They are , , and The uncertain part. If the system satisfies the following conditions: 1) ,2) 3) Function exist If the condition is continuous, then a feedback control law must exist. This can keep the system globally stable. A reasonable choice is:

[0042] (4)

[0043] For any and a suitable .

[0044] 1.3) In implementation scenarios with relatively low system uncertainty, to reduce the computational complexity of the controller, a simplified form of the control law described in step 1.2) can be adopted, as follows:

[0045] Consider the following two-dimensional system:

[0046] (5)

[0047] in, , and This is the system status. Input for the system. and These are two nonlinear mappings. , and They are , , The uncertain part. If the system satisfies the following conditions: 1) It is definite and fixed, 2) 3) Function exist If the condition is continuous, then a feedback control law must exist. This can keep the system globally stable. A reasonable choice is:

[0048] (6)

[0049] For any and a suitable and .

[0050] Step 2: In order to improve the applicability of the feedback control law, we extend it to more general second-order nonlinear systems and give the following theorem.

[0051] 2.1) Consider the following second-order system:

[0052] (7)

[0053] In the formula, and It is a system state variable; Input for the system; , , , and All are constants. and For about The function. For system (7), if and exist The points are continuous, and Therefore, a feedback controller must exist. It can keep the closed-loop system globally stable. A reasonable choice is:

[0054] (8)

[0055] In the formula, .

[0056] 2.2) Based on the above theory, a basic theory for control synthesis of underactuated nonlinear systems is proposed.

[0057] Consider the following two-dimensional system:

[0058] (9)

[0059] in, , and It refers to the state of the system. Input for the system. and yes A continuous function. , , , , and These are the uncertain parts of their respective parameters. If 1) ,2) , It is bounded, and If the condition is continuous, then a feedback controller must exist. It can keep the closed-loop system globally stable. A reasonable choice is:

[0060] (10)

[0061] For any and a suitable .

[0062] in:

[0063] (11)

[0064] Example 2:

[0065] For two-dimensional nonlinear systems with parameter and input uncertainties, within the design framework of step one in Embodiment 1, a two-dimensional underactuated nonlinear system containing parameter and input gain uncertainties is first considered. The dynamic equations of this system not only contain uncertainties related to the system's structural parameters, but also include unknown disturbances or gain variations superimposed on the input channels.

[0066] In this case, the feedback control law given in step 1.2) is adopted. Based on state feedback, this control law introduces a robust compensation term related to the upper bound of uncertainty. This compensation term can compensate for dynamic deviations caused by parameter perturbations, model inaccuracies, and input channel uncertainties in a continuous sense.

[0067] By introducing this control law, we can:

[0068] Effectively suppress instability caused by changes in uncertain parameters in the system;

[0069] Compensating for the uncertainty of input gain;

[0070] Maintain the global asymptotic stability of the closed-loop system;

[0071] Adaptable to system dynamics structures with strong nonlinearity and significant coupling.

[0072] Therefore, this feedback control law is particularly suitable for underactuated systems with strong nonlinear characteristics and significant uncertainties, such as the attitude subsystems of robot systems, underactuated manipulators, or unmanned systems.

[0073] Example 3:

[0074] Under the design framework of step one in embodiment 1, for simplified controller design with low uncertainty, when the uncertainty amplitude in the system is small, or the uncertainty is negligible in an engineering sense, the controller design can be based on the simplified control law given in step 1.3).

[0075] Compared to the control law in Embodiment 2, this version omits some robust compensation structures, retaining only the nonlinear compensation term and damping term, thus forming a simpler state feedback control law. This control law can still compensate for the main nonlinear terms in the system and achieve system stability by designing linear or nonlinear damping gains.

[0076] This simplified control law has the following advantages:

[0077] The computational complexity is significantly reduced;

[0078] The control law expression is simpler and easier to implement in real time;

[0079] More suitable for embedded or computing-restricted control platforms;

[0080] Even under conditions of low uncertainty, the system can still maintain stability and the desired dynamic quality.

[0081] Therefore, this control law can be applied to nonlinear underactuated systems where uncertainties can be effectively modeled or the range of variation is small.

[0082] Example 4:

[0083] Within the design framework of step one in Embodiment 1, the generalized control method proposed in step two is extended to more general second-order nonlinear systems, applicable to general second-order nonlinear systems with multiple uncertainties. This system not only contains multiple coupled linear parameter terms, but its nonlinear part is also more complex in form and may simultaneously contain multiple sources of uncertainty, including:

[0084] Uncertainty in system structural parameters;

[0085] Nonlinear term modeling error;

[0086] External disturbances;

[0087] Input channel uncertainty.

[0088] Based on this, the feedback control law in step 2.2) is introduced. This control law superimposes a continuous robust compensation function on the basic state feedback structure and constructs a compensation quantity by combining the upper and lower bound information of parameter uncertainty, thereby suppressing the influence of comprehensive uncertainty.

[0089] Through this extended design, the control method of the present invention has the following technical effects:

[0090] The controller is applicable to a wider range of system types;

[0091] It can handle various coupling uncertainties;

[0092] Improve system robustness and anti-interference capability;

[0093] Improve control precision and dynamic performance;

[0094] To ensure the stability or asymptotic stability of the closed-loop system in a global sense.

[0095] This embodiment illustrates that the control method proposed in this invention is not only applicable to specific forms of two-dimensional nonlinear systems, but can also be extended to control scenarios of general second-order nonlinear systems.

[0096] Although preferred embodiments of this application have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments as well as all changes and modifications falling within the scope of this application.

[0097] Obviously, those skilled in the art can make various modifications and variations to this application without departing from the spirit and scope of this application. Therefore, if such modifications and variations fall within the scope of the claims of this application and their equivalents, this application also intends to include such modifications and variations.

Claims

1. A feedback control method for a nonlinear two-dimensional system, characterized in that, The method includes the following steps: S1: Perform dynamic modeling on the controlled object, representing the controlled object as a two-dimensional underactuated nonlinear state system; S2: The nonlinear terms in the state system are structured and represented as equivalent nonlinear mappings proportional to the system state variables. The equivalent nonlinear mappings are then determined to satisfy continuity or boundedness conditions at the equilibrium point to determine whether the system meets the feedback control design requirements. S3: Based on the equivalent nonlinear mapping, construct a state feedback control law to compensate for system nonlinearity and introduce stability damping; S4: To address the parameter uncertainties and external disturbances present in the state system, a continuous robust compensation term based on the upper bound of uncertainty is introduced into the state feedback control law to suppress the impact of uncertainty on system stability. S5: Apply the state feedback control law to the two-dimensional underactuated nonlinear state system in S1 to form a closed-loop control system and achieve global stable control.

2. The feedback control method for a nonlinear two-dimensional system according to claim 1, characterized in that, The two-dimensional underactuated nonlinear state system includes a first state equation and a second state equation. The system control input acts directly only on the second state equation, while the first state equation is indirectly controlled by the second state equation through the internal coupling relationship of the system.

3. The feedback control method for a nonlinear two-dimensional system according to claim 1, characterized in that, In step S2, the structuring of the nonlinear terms in the state system includes: reconstructing the nonlinear terms into an equivalent nonlinear mapping proportional to the second state variable of the system.

4. The feedback control method for a nonlinear two-dimensional system according to claim 3, characterized in that, The ratio of the equivalent nonlinear mapping to the corresponding state variable is continuous or bounded at the system equilibrium point to avoid singularity in the feedback control law at the equilibrium point.

5. The feedback control method for a nonlinear two-dimensional system according to claim 1, characterized in that, The state feedback control law constructed in step S3 includes: a nonlinear compensation term for compensating the equivalent nonlinear mapping, and a damping term related to the system state variables, for improving the stability of the closed-loop system.

6. The feedback control method for a nonlinear two-dimensional system according to claim 5, characterized in that, The damping term is a linear or nonlinear damping term related to the second state variable, and its coefficient is a design parameter greater than zero.

7. The feedback control method for a nonlinear two-dimensional system according to claim 1, characterized in that, The continuous robust compensation term introduced in step S4 is constructed based on known upper bounds of system parameter uncertainties and external disturbances, and is used to suppress the impact of the uncertainties on system stability.

8. The feedback control method for a nonlinear two-dimensional system according to claim 7, characterized in that, The continuous robust compensation term adopts a continuous and bounded nonlinear function form to avoid chattering caused by discontinuous control input.

9. The feedback control method for a nonlinear two-dimensional system according to claim 8, characterized in that, The continuous and bounded nonlinear function includes the arctangent function, the saturation function, or a function with equivalent continuous and bounded properties.

10. A feedback control method for a nonlinear two-dimensional system according to claim 1, characterized in that, The state feedback control law is configured to cause the closed-loop system state to asymptotically converge to the equilibrium point in the global domain under the conditions of system parameter uncertainty and external disturbance.

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