Global smooth time-varying control method for nonholonomic systems based on dynamic gain scheduling

Abstract

The global smooth time-varying control method based on dynamic gain scheduling for nonholonomic system belongs to the technical field of nonlinear system control, and comprises the following steps: step 1, a decomposition model comprising a linear subsystem and an integral chain subsystem is constructed; step 2, a smooth time-varying dynamic gain scheduling control law and a dynamic gain updating law for the linear subsystem are designed based on the decomposition model in step 1; step 3, a linear time-varying state transformation matrix and a transformed integral chain subsystem are designed based on step 2; and step 4, a smooth time-varying dynamic gain scheduling control law and a dynamic gain updating law for the transformed linear time-varying system are designed based on step 3. The method of the application not only eliminates the need for solving complex nonlinear equations online, guarantees the global smoothness of the control signal, but also introduces non-homogeneous terms and dynamic gain variables, realizes efficient control of high-order nonholonomic systems containing dynamic characteristics, and has important engineering application value.

Description

Technical Field

[0001] This invention belongs to the field of nonlinear system control technology, and relates to a gain scheduling control method, specifically a global smooth time-varying control method for nonholonomic systems based on dynamic gain scheduling. Background Technology

[0002] Nonholonomic systems, a typical class of mechanical systems with non-integrable velocity constraints, are widely found in complex equipment such as wheeled mobile robots, underwater unmanned vehicles, and underactuated axisymmetric spacecraft. Because these systems do not satisfy the Brockett necessary condition in their physical characteristics, traditional continuous time-invariant feedback control laws struggle to directly achieve equilibrium point stability, making controller design for nonholonomic systems a challenging problem in the field of nonlinear control. While gain scheduling methods are widely used to address nonlinear control problems, gain scheduling techniques for nonholonomic systems still face significant challenges in practical engineering applications. Traditional continuous static gain scheduling methods often heavily rely on online solving of complex nonlinear algebraic equations, and their heavy computational burden significantly limits the real-time performance of the controller in computationally constrained systems. Meanwhile, discrete scheduling schemes adopted to avoid computational pressure inevitably lead to severe discontinuities in the generated control signals due to the step switching of feedback gain in time or state space. This not only causes mechanical wear and system oscillations in actuators but also easily excites unmodeled dynamics of the controlled object. Furthermore, most existing gain scheduling frameworks are limited to simplified scalar quantum systems or first-order kinematic models, neglecting the complex dynamic couplings and higher-order state constraints that are prevalent in real-world controlled objects. This results in insufficient adaptability of the algorithms when dealing with higher-order systems containing dynamic parameters such as mass and moment of inertia. Therefore, researching and designing a dynamic gain scheduling control method with low computational load, smooth control signal trajectory, and the ability to cover the dynamic level has extremely high engineering application value for improving the control quality and reliability of complex nonholonomic systems. Summary of the Invention

[0003] The purpose of this invention is to address the problems of high computational load, discontinuous control signals, and difficulty in adapting to high-order dynamic models in existing nonholonomic system control methods. This invention provides a globally smooth time-varying control method for nonholonomic systems based on dynamic gain scheduling.

[0004] This method not only eliminates the need for online solving of complex nonlinear equations and ensures the global smoothness of the control signal, but also introduces nonhomogeneous terms and dynamic gain variables, enabling efficient control of high-order nonholonomic systems containing dynamic characteristics, thus possessing significant engineering application value.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0006] A global smooth time-varying control method for nonholonomic systems based on dynamic gain scheduling, the method comprising the following steps:

[0007] Step 1: Construct a decomposition model that includes linear subsystems and integral chain subsystems;

[0008] Step 2: Based on the decomposition model, design the smooth time-varying dynamic gain scheduling control law and dynamic gain update law for the linear subsystem;

[0009] Step 3: Design the linear time-varying state transformation matrix and the transformation integral chain subsystem;

[0010] Step 4: Design the smooth time-varying dynamic gain scheduling control law and dynamic gain update law for the transformed linear time-varying system.

[0011] Furthermore, in step 1, the construction of a decomposition model comprising a linear subsystem and an integral chain subsystem is specifically expressed as follows:

[0012]

[0013]

[0014] in, Represents continuous time; the superscript indicates differentiation with respect to time. Let this be the state vector of the linear subsystem; The state vector of the integral chain subsystem; and These are the control input vectors for the linear subsystem and the integral chain subsystem, respectively. It is a controllable matrix; It is a constant matrix with a chain structure; It is a nonlinear coupling matrix; For the state dimension of the integral chain subsystem;

[0015] .

[0016] Furthermore, the specific steps of step 2 are as follows:

[0017] Step 21: Design the smooth time-varying dynamic gain scheduling control law for the linear subsystem as follows:

[0018]

[0019] in, Define constants for users; Non-homogeneous control gain; For parametric Lyapunov equations The only positive definite solution; for A column vector of all 1s; This represents the dynamic time-varying gain of the linear subsystem;

[0020] Step 22: Design the dynamic gain update law of the linear subsystem as follows:

[0021]

[0022] in, Define constants for users; ; ; for The second element is a scalar; for A column vector of all 1s; For the auxiliary coordinate transformation vector of the linear subsystem; ; Define constants for users, and ; for 3D identity matrix; This is the auxiliary coordinate transformation vector for the linear subsystem.

[0023] Furthermore, the specific steps of step 3 are as follows:

[0024] Step 31: Design the linear time-varying state transformation matrix as follows:

[0025]

[0026] in, Define constants for users; Represents continuous time; Represents the state dimension of the integral chain subsystem; Represents a diagonal matrix;

[0027] Step 32: Design the transformation integral chain subsystem:

[0028]

[0029] in, The transformed state variables are used to convert the integral chain subsystem into the following linear time-varying system: , is the state vector of the integral chain subsystem;

[0030]

[0031] in, ; for The second element; It is a constant matrix with a chain structure.

[0032] Furthermore, the specific steps of step 4 are as follows:

[0033] Step 41: Design the smooth time-varying dynamic gain scheduling control law for the transformed linear time-varying system as follows:

[0034]

[0035] in, For parametric Lyapunov equations The only positive definite solution;

[0036] Step 42: Design the dynamic gain update law for the transformed linear time-varying system:

[0037]

[0038] in, For parametric Lyapunov equations The only positive definite solution; Define constants for users, whose meanings are similar to... They are the same, both being positive design constants designed by the user. The difference is that those with a subscript of 0 are used for gain scheduling of linear subsystems, while those without a subscript of 0 represent gain scheduling of chained subsystems. Define constants for users.

[0039] The advantages of this invention over the prior art are:

[0040] 1. The method proposed in this invention generates control gain in real time by constructing a scalar dynamic differential equation. and (See steps 22 and 42) This completely avoids the problem of solving complex nonlinear algebraic equations online at each time step in the traditional static gain scheduling method, greatly reducing computational complexity and meeting the requirements of real-time control.

[0041] 2. This invention introduces a non-homogeneous decay term. This design breaks the limitation of traditional nonholonomic system control requiring the initial state of the first subsystem to be non-zero. It enables the system to achieve global exponential convergence from any initial state, even with multi-order dynamic characteristics, demonstrating strong physical applicability.

[0042] 3. The control signal designed in this invention is a globally smooth continuous signal, which theoretically completely eliminates the control signal jump phenomenon caused by discrete scheduling methods, avoids mechanical losses of actuators, and significantly improves the steady-state performance of the system and the feasibility of actual deployment.

[0043] In summary, the method of this invention can not only achieve global exponential convergence of the state and control input of a non-homogeneous chain system containing complex dynamics under any initial state, but also introduces a non-homogeneous decay term and a scalar dynamic differential equation to generate control gain in real time. Under the premise of ensuring global continuous smoothness of the control signal, it significantly reduces the online computational complexity and hardware deployment difficulty of the system, and has important engineering application value.

[0044] Among them, the non-homogeneous decay term is In , Its function is to break the restriction that nonholonomic systems require non-zero initial states.

[0045] The scalar differential equation is embodied in step 22 (for the linear subsystem), as shown in the following equation:

[0046] And in step 42 (for chain subsystems) the following equation:

[0047]

[0048] Both are used to generate smooth control gains in real time, avoiding the need to solve complex algebraic equations online. Attached Figure Description

[0049] Figure 1 This is a flowchart illustrating the global smooth time-varying control method for nonholonomic systems based on dynamic gain scheduling according to the present invention.

[0050] Figure 2 The figures below show the attitude parameter response curves of the spacecraft under different initial dynamic gain conditions according to an embodiment of the present invention; wherein:

[0051] Figure 2 (a) represents the attitude components. The corresponding curve graph;

[0052] Figure 2 (b) represents the attitude components. The corresponding curve graph;

[0053] Figure 2 (c) represents the attitude components. The corresponding curve graph.

[0054] Figure 3 The graph shows the angular velocity response curves of the spacecraft under different initial dynamic gain conditions according to an embodiment of the present invention; wherein:

[0055] Figure 3 (a) represents the angular velocity component. The corresponding curve graph;

[0056] Figure 3(b) represents the angular velocity component. The corresponding curve graph.

[0057] Figure 4 This is a time evolution curve of the dynamic gain parameter under different initial dynamic gain conditions according to an embodiment of the present invention; wherein:

[0058] Figure 4 (a) represents the dynamic gain of the linear subsystem. Evolutionary curve;

[0059] Figure 4 (b) is the dynamic gain of the chain subsystem. The evolution curve.

[0060] Figure 5 This is a graph showing the control torque input signal curves of a spacecraft under different initial dynamic gain conditions according to an embodiment of the present invention; wherein:

[0061] Figure 5 (a) is the control torque Input signal curve of the channel;

[0062] Figure 5 (b) is the control torque Input signal curve of the channel.

[0063] Figure 6 This is a comparison curve of the spacecraft's attitude parameter response with and without dynamic gain scheduling, according to an embodiment of the present invention; wherein:

[0064] Figure 6 (a) represents the attitude components. The corresponding curve graph;

[0065] Figure 6 (b) represents the attitude components. The corresponding curve graph;

[0066] Figure 6 (c) represents the attitude components. The corresponding curve graph.

[0067] Figure 7 This is a comparison curve of the spacecraft's angular velocity response with and without dynamic gain scheduling, according to an embodiment of the present invention; wherein:

[0068] Figure 7 (a) represents the angular velocity component. The corresponding curve graph;

[0069] Figure 7 (b) represents the angular velocity component. The corresponding curve graph.

[0070] Figure 8 This is a comparison curve of the time evolution of the gain parameter with and without dynamic gain scheduling in an embodiment of the present invention; wherein:

[0071] Figure 8 (a) represents the dynamic gain of the linear subsystem. Evolutionary curve;

[0072] Figure 8 (b) is the dynamic gain of the chain subsystem. The evolution curve.

[0073] Figure 9 This is a comparison curve of the spacecraft's control torque input signal with and without dynamic gain scheduling, according to an embodiment of the present invention; wherein:

[0074] Figure 9 (a) is the control torque Input signal curve of the channel;

[0075] Figure 9 (b) is the control torque Input signal curve of the channel.

[0076] Figure 10 The figures below show the attitude parameter response curves of the spacecraft under different convergence rate parameters according to an embodiment of the present invention; wherein:

[0077] Figure 10 (a) represents the attitude components. The corresponding curve graph;

[0078] Figure 10 (b) represents the attitude components. The corresponding curve graph;

[0079] Figure 10 (c) represents the attitude components. The corresponding curve graph.

[0080] Figure 11 The figures below show the angular velocity response curves of the spacecraft under different convergence rate parameters according to an embodiment of the present invention; wherein:

[0081] Figure 11 (a) represents the angular velocity component. The corresponding curve graph;

[0082] Figure 11 (b) represents the angular velocity component. The corresponding curve graph.

[0083] Figure 12 This is a time evolution curve of the dynamic gain parameter under different convergence rate parameters according to an embodiment of the present invention; wherein:

[0084] Figure 12 (a) represents the dynamic gain of the linear subsystem. Evolutionary curve;

[0085] Figure 12 (b) is the dynamic gain of the chain subsystem. The evolution curve.

[0086] Figure 13 This is a graph showing the control torque input signal curves of the spacecraft under different convergence rate parameters according to an embodiment of the present invention; wherein:

[0087] Figure 13 (a) is the control torque Input signal curve of the channel;

[0088] Figure 13 (b) is the control torque Input signal curve of the channel. Detailed Implementation

[0089] The technical solution of the present invention will be further described below with reference to the accompanying drawings, but it is not limited thereto. Any modifications or equivalent substitutions to the technical solution of the present invention that do not depart from the spirit and scope of the technical solution of the present invention should be covered within the protection scope of the present invention.

[0090] like Figure 1 As shown, this embodiment provides a globally smooth time-varying control method for a nonholonomic system based on dynamic gain scheduling. The method includes the following steps:

[0091] Step 1: Construct a decomposition model that includes linear subsystems and integral chain subsystems;

[0092] Step 2: Based on the decomposition model, design the smooth time-varying dynamic gain scheduling control law and dynamic gain update law for the linear subsystem;

[0093] Step 3: Design the linear time-varying state transformation matrix and the transformation integral chain subsystem;

[0094] Step 4: Design the smooth time-varying dynamic gain scheduling control law and dynamic gain update law for the transformed linear time-varying system.

[0095] Furthermore, in step 1, the construction of a decomposition model comprising a linear subsystem and an integral chain subsystem is specifically expressed as follows:

[0096]

[0097]

[0098] in, Represents continuous time; the superscript indicates differentiation with respect to time. Let this be the state vector of the linear subsystem; The state vector of the integral chain subsystem; and These are the control input vectors for the linear subsystem and the integral chain subsystem, respectively. It is a controllable matrix; It is a constant matrix with a chain structure; It is a nonlinear coupling matrix; For the state dimension of the integral chain subsystem;

[0099] .

[0100] Furthermore, the specific steps of step 2 are as follows:

[0101] Step 21: Design the smooth time-varying dynamic gain scheduling control law for the linear subsystem as follows:

[0102]

[0103] in, Define constants for users; Non-homogeneous control gain; For parametric Lyapunov equations The only positive definite solution; for A column vector of all 1s; This represents the dynamic time-varying gain of the linear subsystem;

[0104] Step 22: Design the dynamic gain update law of the linear subsystem as follows:

[0105]

[0106] in, Define constants for users; ; ; for The second element is a scalar; for A column vector of all 1s; For the auxiliary coordinate transformation vector of the linear subsystem; ; Define constants for users, and ; for 3D identity matrix; This is the auxiliary coordinate transformation vector for the linear subsystem.

[0107] Furthermore, the specific steps of step 3 are as follows:

[0108] Step 31: Design the linear time-varying state transformation matrix as follows:

[0109]

[0110] in, Define constants for users; Represents continuous time; Represents the state dimension of the integral chain subsystem; Represents a diagonal matrix;

[0111] Step 32: Design the transformation integral chain subsystem:

[0112]

[0113] in, The transformed state variables are used to convert the integral chain subsystem into the following linear time-varying system: , is the state vector of the integral chain subsystem;

[0114]

[0115] in, ; for The second element; It is a constant matrix with a chain structure.

[0116] Furthermore, the specific steps of step 4 are as follows:

[0117] Step 41: Design the smooth time-varying dynamic gain scheduling control law for the transformed linear time-varying system as follows:

[0118]

[0119] in, For parametric Lyapunov equations The only positive definite solution;

[0120] Step 42: Design the dynamic gain update law for the transformed linear time-varying system:

[0121]

[0122] in, For parametric Lyapunov equations The only positive definite solution; Define constants for users, whose meanings are similar to... They are the same, both being positive design constants designed by the user. The difference is that those with a subscript of 0 are used for gain scheduling of linear subsystems, while those without a subscript of 0 represent gain scheduling of chained subsystems. Define constants for users.

[0123] This invention overcomes the limitations of traditional nonholonomic system control, which heavily relies on non-zero initial states and discontinuous feedback. Through the collaborative design of smooth time-varying dynamic gains and non-homogeneous control terms, it completely overcomes the problem of direct drive under zero initial conditions, achieving global exponential stabilization without the need for pre-controllers or logic switching. At the control architecture level, it not only replaces the complex online nonlinear algebraic equations of traditional static scheduling with scalar dynamic differential equations, greatly reducing the computational load and deployment difficulty of embedded hardware, but also innovatively integrates time-varying state transformations with higher-order dynamics, significantly enhancing the algorithm's adaptability to complex physical constraints. Furthermore, the generation of globally smooth continuous control signals effectively eliminates instruction jumps caused by traditional discrete scheduling, theoretically strictly avoiding mechanical oscillations and excessive wear of actuators. This makes it highly suitable for practical engineering deployments in spacecraft and mobile robots where computational power is limited and control stability requirements are extremely high, demonstrating significant engineering application value.

[0124] Example:

[0125] This embodiment uses the attitude control of an underactuated axisymmetric spacecraft as a specific implementation method, and is designed using the method of the present invention, specifically including the following steps:

[0126] Step 1: Construct a decomposition model of the underactuated axisymmetric spacecraft, including linear and integral chain subsystems. The specific steps are as follows:

[0127] Consider a chain-like nonholonomic dynamics model for an underactuated axisymmetric spacecraft system:

[0128]

[0129]

[0130] in, These are attitude parameters; It is the angular velocity vector; and For controlling torque input; The principal moment of inertia.

[0131] The dynamic model of the chain-like nonholonomic system is decomposed into a linear subsystem and an integral chain-like subsystem, specifically represented as follows:

[0132]

[0133] ,have:

[0134]

[0135] Define the state of the linear subsystem as The state of the integral chain subsystem is ,in , The state and input matrices of the linear subsystem and the integral chain subsystem are as follows:

[0136]

[0137]

[0138] The underactuated axisymmetric spacecraft system model can be decomposed into the following linear subsystem and integral chain subsystem:

[0139]

[0140]

[0141] in, Represents continuous time; the superscript indicates differentiation with respect to time. Let this be the state vector of the linear subsystem; The state vector of the integral chain subsystem; and These are the control input vectors for the linear subsystem and the integral chain subsystem, respectively. For controllable matrix pairs; It is a constant matrix with a chain structure; It is a nonlinear coupling matrix; For the state dimension of the integral chain subsystem;

[0142] ;

[0143] Step 2: Design the smooth time-varying dynamic gain scheduling control law and dynamic gain update law for the linear subsystem. The specific steps are as follows:

[0144] Step 21: Design the smooth time-varying dynamic gain scheduling control law for the linear subsystem as follows:

[0145]

[0146] in, Define constants for users; Non-homogeneous control gain; For parametric Lyapunov equations The only positive definite solution; for A column vector of all 1s;

[0147] Step 22: Design the dynamic gain update law of the linear subsystem as follows:

[0148]

[0149] in, Define constants for users; ; ; for The second element is a scalar; for A column vector of all 1s; For the auxiliary coordinate transformation vector of the linear subsystem; ; Define constants for users, and ; for 3D identity matrix; For the auxiliary coordinate transformation vector of the linear subsystem;

[0150] Step 3: Design the linear time-varying state transformation matrix and the transformation integral chain subsystem. The specific steps are as follows:

[0151] Step 31: Design the linear time-varying state transformation matrix as follows:

[0152]

[0153] Step 32: Design the transformation integral chain subsystem:

[0154]

[0155] in, The transformed state variables are used to convert the integral chain subsystem into the following linear time-varying system:

[0156]

[0157] in, ; for The second element;

[0158] Step 4: Design the smooth time-varying dynamic gain scheduling control law and dynamic gain update law for the transformed linear time-varying system. The specific steps are as follows:

[0159] Step 41: Design the smooth time-varying dynamic gain scheduling control law for the transformed linear time-varying system as follows:

[0160]

[0161] in, For parametric Lyapunov equations The only positive definite solution;

[0162] Step 42: Design the dynamic gain update law for the transformed linear time-varying system as follows:

[0163]

[0164] in, This is the unique positive definite solution to the parametric Lyapunov equation; Define constants for users; Define constants for users;

[0165] The selected parameters are:

[0166] .

[0167] Depend on Figure 2 and Figure 3 It can be seen that the dynamic gain scheduling mechanism designed in this embodiment can effectively drive all attitude states of the underactuated spacecraft system to converge exponentially to zero with respect to the control input, while ensuring the global continuity and smoothness of the control signal. (Comparison) Figure 4 and Figure 5 The recorded gain evolution and control torque distribution show that a larger initial gain value can further accelerate the convergence speed of the system state, but in physical verification, it is also accompanied by a certain degree of state overshoot and higher control energy consumption. Based on this, Figures 6 to 9 Comparative simulation results show that the dynamic gain scheduling mechanism incorporating scalar differential equations has stronger dynamic adjustment capabilities than the traditional static gain scheduling mechanism, effectively suppressing state overshoot during transient processes and significantly improving the overall control performance of the closed-loop system. Furthermore, Figures 10 to 13 It also shows different convergence rate parameters. Simulation results related to (i.e., different convergence rate parameters defined by the user) show that engineers can adjust the convergence rate parameters according to different user-defined parameters. The design allows for flexible adjustment of the system's convergence performance, and the peak control energy consumption of the actuator will vary with... It increases accordingly as the value increases.

Claims

1. A globally smooth time-varying control method for a nonholonomic system based on dynamic gain scheduling, characterized in that: The method includes the following steps: Step 1: Construct a decomposition model that includes linear subsystems and integral chain subsystems; Step 2: Based on the decomposition model, design the smooth time-varying dynamic gain scheduling control law and dynamic gain update law for the linear subsystem; Step 3: Design the linear time-varying state transformation matrix and the transformation integral chain subsystem; Step 4: Design the smooth time-varying dynamic gain scheduling control law and dynamic gain update law for the transformed linear time-varying system.

2. The global smooth time-varying control method for nonholonomic systems based on dynamic gain scheduling according to claim 1, characterized in that: In step 1, the construction of a decomposition model comprising a linear subsystem and an integral chain subsystem is specifically represented as follows: in, Represents continuous time; the superscript indicates differentiation with respect to time. Let this be the state vector of the linear subsystem; Let be the state vector of the integral chain subsystem; and These are the control input vectors for the linear subsystem and the integral chain subsystem, respectively. It is a controllable matrix; It is a constant matrix with a chain structure; It is a nonlinear coupling matrix; For the state dimension of the integral chain subsystem; 。 3. The global smooth time-varying control method for nonholonomic systems based on dynamic gain scheduling according to claim 2, characterized in that: The specific steps of step 2 are as follows: Step 21: Design the smooth time-varying dynamic gain scheduling control law for the linear subsystem as follows: in, Define constants for users; Non-homogeneous control gain; For parametric Lyapunov equations The only positive definite solution; for A column vector of all 1s; This represents the dynamic time-varying gain of the linear subsystem; Step 22: Design the dynamic gain update law of the linear subsystem as follows: in, Define constants for users; ; ; for The second element is a scalar; for A column vector of all 1s; For the auxiliary coordinate transformation vector of the linear subsystem; ; Define constants for users, and ; for 3D identity matrix; This is the auxiliary coordinate transformation vector for the linear subsystem.

4. The global smooth time-varying control method for nonholonomic systems based on dynamic gain scheduling according to claim 3, characterized in that: The specific steps of step 3 are as follows: Step 31: Design the linear time-varying state transformation matrix as follows: in, Define constants for users; Represents continuous time; Represents the state dimension of the integral chain subsystem; Represents a diagonal matrix; Step 32: Design the transformation integral chain subsystem: in, The transformed state variables are used to convert the integral chain subsystem into the following linear time-varying system: , is the state vector of the integral chain subsystem; in, ; for The second element; It is a constant matrix with a chain structure.

5. The global smooth time-varying control method for nonholonomic systems based on dynamic gain scheduling according to claim 4, characterized in that: The specific steps of step 4 are as follows: Step 41: Design the smooth time-varying dynamic gain scheduling control law for the transformed linear time-varying system as follows: in, For parametric Lyapunov equations The only positive definite solution; Step 42: Design the dynamic gain update law for the transformed linear time-varying system: in, For parametric Lyapunov equations The only positive definite solution; Define constants for users, whose meanings are similar to... They are the same, both being positive design constants designed by the user. The difference is that those with a subscript of 0 are used for gain scheduling of linear subsystems, while those without a subscript of 0 represent gain scheduling of chained subsystems. Define constants for users.

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