Nonlinear system output constraint control method based on new barrier lyapunov function

Abstract

This invention discloses a nonlinear system output constraint control method based on a novel obstacle Lyapunov function. This invention relates to the field of trajectory tracking control, and particularly to a path tracking and output constraint control method for a nonlinear system based on a novel obstacle Lyapunov function. The method includes the following steps: Step (1): Establishing a mathematical model of a second-order nonlinear system; Step (2): Designing a novel obstacle Lyapunov function; Step (3): Obtaining a dynamic model of the tracking error of the nonlinear system; Step (4): Designing a first-order inertial filter; Step (5): Designing the Lyapunov function and virtual control input for the nonlinear output constraint system based on the novel obstacle Lyapunov function; Step (6): Designing a nonlinear system constraint controller; Step (7): Simulation verification. This invention addresses the constraint problems caused by physical limitations, system performance, and safety indicators in practical engineering applications. It constructs a novel obstacle Lyapunov function to handle system output constraints. This function can unify constrained and unconstrained (unbounded constraint) cases in the same method and has better transient and steady-state performance.

Description

Technical fields:

[0001] This invention relates to the field of nonlinear control, and in particular to a method for output constraint control of nonlinear systems based on a novel barrier Lyapunov function. Background technology:

[0002] In practical engineering applications, most real-world systems typically operate under various constraints. These constraints may include physical limitations, system performance requirements, or safety considerations. Similar to robotic arm systems, the positions and velocities of joints and end effectors in robotic arm control methods are always subject to certain limitations. Without proper adjustment, these limitations can degrade control performance, cause system instability, or even lead to accidents. Therefore, in nonlinear systems under such conditions, constraint control is crucial.

[0003] In order to meet the above control conditions and improve system performance in nonlinear control systems, this invention proposes a nonlinear system output constraint control method based on a novel obstacle Lyapunov function. It is expected that under certain special engineering application environments, it can achieve better system performance under the same constraint conditions as different obstacle Lyapunov functions. Summary of the Invention:

[0004] The present invention discloses a nonlinear system output constraint control method based on a novel barrier Lyapunov function under output constraints.

[0005] The specific implementation process of this invention is as follows:

[0006] The nonlinear system output constraint control method based on a novel barrier Lyapunov function includes the following steps:

[0007] Step (1): Establish the mathematical model of the second-order nonlinear system:

[0008] Considering the prevalence of second-order systems in practical engineering applications, a mathematical model of a single-input, single-output, strictly feedback second-order nonlinear system is established.

[0009] Step (2): Design a novel barrier Lyapunov function:

[0010] To meet the output constraints in practical engineering applications, a novel barrier Lyapunov function is proposed to solve the nonlinear output constraint problem of the system.

[0011] Step (3): Obtain the dynamic model of the tracking error of the nonlinear system:

[0012] The control system is designed using the novel obstacle Lyapunov function in step (2). Since there is an inevitable error between the expected value and the actual value of the controlled variable in the system design, i.e. the error between the expected trajectory and the tracking trajectory, a dynamic model of the system tracking error is established.

[0013] Step (4): Design a first-order inertial filter:

[0014] When designing a system controller under the backstepping framework, the main idea of ​​this control framework is to construct the Lyapunov function of the closed-loop system recursively to obtain the feedback controller. However, as the Lyapunov function becomes more complex and the system order increases, multiple derivatives will cause a "computational explosion" problem, which will make the controller design inconvenient. To solve this problem, a first-order inertial filter is now added to the system and a dynamic surface design method is adopted.

[0015] Step (5): Design the Lyapunov function and virtual control input for the nonlinear output constraint system based on the novel obstacle Lyapunov function:

[0016] Based on the novel obstacle Lyapunov function obtained in steps (2), (3), and (4), as well as the tracking error dynamic model of the nonlinear system and the first-order inertial filter, the dynamic surface design method is used to design the Lyapunov function based on the nonlinear system and the virtual control input with respect to the first-order inertial filter.

[0017] Step (6): Design the constraint controller for the nonlinear system:

[0018] Based on the first-order inertial filter and system virtual control input obtained in steps (4) and (5), design a controller with output constraints based on a nonlinear system;

[0019] Step (7): Simulation verification:

[0020] The mathematical model of the single-input single-output strictly feedback second-order nonlinear system based on the novel obstacle Lyapunov function output constraint controller designed in steps (2) to (6) and established in step (1) was simulated.

[0021] The present invention has the following beneficial effects:

[0022] The nonlinear system output constraint control method based on a novel obstacle Lyapunov function described in this invention has the following two advantages regarding the constraints designed for the desired trajectory in nonlinear systems:

[0023] (1) Compared with the constraint effect of the tangent barrier Lyapunov function, the novel barrier Lyapunov function proposed in this invention has better transient and steady-state performance;

[0024] (2) The novel obstacle Lyapunov function can handle both constrained and unconstrained cases at the same time, unifying constrained and unconstrained (unbounded) cases in the same constraint control method. Attached image description:

[0025] Figure 1 This is a flowchart of the steps described in this invention;

[0026] Figure 2 This is a graph of the novel barrier Lyapunov function described in this invention;

[0027] Figure 3 The graph shows the time-varying constraint path tracking curve of the second-order nonlinear system under the present invention.

[0028] Figure 4 The dynamic variable curve of the tracking error of the second-order nonlinear system under the present invention is shown.

[0029] Figure 5 The control force curve of the second-order nonlinear system under the present invention is shown. Detailed implementation method:

[0030] like Figure 1 The nonlinear system output constraint control method based on the novel barrier Lyapunov function includes the following steps:

[0031] Step (1): Establish a second-order mathematical model of the nonlinear system

[0032] Considering the output constraint control strategy for nonlinear systems, a mathematical model of a strict feedback second-order nonlinear system with single input and single output is established:

[0033]

[0034] In the above equation, x1 and x2 represent the system state vectors, u represents the system input, and y represents the system output. θ1, θ2, and θ3 represent the system design parameters. In this nonlinear system, considering that the system has output constraints, i.e., the system state variable x1 satisfies the conditional constraint |x1(t)|<k from the initial time T0. b (t), t≥T0;

[0035] Step (2): Design a novel barrier Lyapunov function:

[0036] Design a novel barrier Lyapunov function, in the following form:

[0037]

[0038] In the above formula, the function V BLF (e) is on an open interval D containing the origin, with respect to the nonlinear system. A function, where the independent variable e is the tracking error, k is the constraint range boundary of the variable e, and k > 0, satisfying the relation -k < e < k;

[0039] The novel barrier Lyapunov function has the following properties:

[0040] (1)V BLF (e) is smooth and positive definite;

[0041] (2) At any point on the open interval D, there exists a continuous first-order partial derivative;

[0042] (3) When e approaches the boundary of interval D, V BLF (e)→∞;

[0043] (4) When x(0)∈D, the system The solution satisfies Established;

[0044] (5) This shows that when constraint k approaches infinity, a system with output constraints can be generalized to a system without constraints.

[0045] Compared to logarithmic and integral Lyapunov functions, the novel barrier Lyapunov function can unify constrained and unconstrained (unbounded) cases in the same method. Although the tangent barrier Lyapunov function can handle both constrained and unconstrained cases simultaneously, its constraint effect is limited. The novel barrier Lyapunov function has a better constraint effect than the tangent barrier Lyapunov function.

[0046] Step (3): Obtain the dynamic model of the tracking error of the nonlinear system:

[0047] When designing a system controller using the dynamic surface design method, the tracking error is defined as: e1 = x1 - y d And e2 = x2 - ω2, therefore the dynamic model for the system tracking error is:

[0048]

[0049] In the above formula, e1 and e2 are the tracking errors, and y d Let ω2 be the desired trajectory of the system, and let ω2 be the output of the first-order filter. It is the derivative of ω2 with respect to time, and u is the system controller.

[0050] Step (4): Design a first-order inertial filter:

[0051] ω1(0)=α1(0)

[0052] In the filter defined in the above formula, ω1(0) is the initial value of the first-order filter, α1(0) is the initial value of the virtual control input, τ2 is the design parameter of the first-order filter, α1 is the input of the first-order filter, and the filter error is y2=ω2-α1.

[0053] Step (5): Design the Lyapunov function and virtual control input for the nonlinear output constraint system based on the novel obstacle Lyapunov function:

[0054] From the system tracking error and the filter error, we get:

[0055]

[0056] Therefore, x² = e² + y² + α¹

[0057] The Lyapunov function is defined as follows:

[0058]

[0059] In the above formula, k b It's an output constraint related to time;

[0060] Differentiating with respect to V1, we get:

[0061]

[0062] In the above formula, It is k b The derivative with respect to time, It is the derivative of y2 with respect to time. For ease of calculation, let... Simplified to ψ1, Simplified to ψ2;

[0063] Therefore, the above equation have to:

[0064]

[0065] Substituting x2 into the dynamic model of the system tracking error, we get:

[0066]

[0067] Therefore, substituting the above equation into... have to:

[0068]

[0069] By Young's Inequality We can obtain:

[0070]

[0071]

[0072] Therefore, the original expression get:

[0073]

[0074] To achieve system convergence and improve performance, the virtual control input α1 is designed as follows:

[0075]

[0076] Substitute the virtual control input α1 into have to:

[0077]

[0078] Because in a first-order filter ω2(0)=α1(0), and from the filter error, we can obtain:

[0079]

[0080] Therefore, the derivative of the filter error y2 with respect to time is:

[0081]

[0082] From the above formula, we get:

[0083]

[0084] In the above formula, It is the maximum value of |-α11|, so the above inequality holds. To simplify the notation, let... Let it be ζ2;

[0085] Therefore, according to Young's inequality, we can obtain:

[0086]

[0087] Substituting the above equation into... have to:

[0088]

[0089] Step (6): Design the constraint controller for the nonlinear system:

[0090] The Lyapunov function is defined as follows:

[0091]

[0092] Taking the derivative of the tracking error e², we get:

[0093]

[0094] In the above formula, u is the system controller;

[0095] Differentiating V2 and substituting it into the above equation, we get:

[0096]

[0097] The system input u is designed as follows:

[0098]

[0099] Therefore, substitution have to:

[0100]

[0101] From the above design, we can conclude that:

[0102]

[0103] Multiply both sides of the above inequality by e ρt Then, by integrating both sides of the inequality over the interval [0, t], we can obtain:

[0104]

[0105] In the above formula, ρ is the positive design parameter.

[0106] Because V(t)≥0 is a continuous function, x i (t), e i (t), i = 1, 2 and the controller output u are bounded, and |e1| ≤ k b Therefore, all signals in a closed-loop system are uniformly and eventually bounded.

[0107] Step (7): Simulation verification

[0108] The simulation environment for the output constraint controller based on the novel barrier Lyapunov function, designed based on the above steps, is set up as follows:

[0109] θ1, θ2, and θ3 are equal to 0.1, 0.2, and -0.2 respectively; output constraint boundary k b (t) = 0.65 + 0.1sin(2.5t + 10); the designed desired trajectory is y d(t)=0.2+0.3sin(t); The initial state of the system is x1(0)=0.8, x2(0)=1.5; The positive design parameters are selected as k1=3, k2=3, τ2=0.05.

[0110] A dynamic surface control method is adopted, which uses a novel barrier Lyapunov function to constrain the output of a nonlinear system. A second-order nonlinear system with strict feedback and single input and single output is used as the simulation object to verify the output constraint control method of the nonlinear system based on the novel barrier Lyapunov function of the present invention.

[0111] Figure 2 The conditional properties of the novel barrier Lyapunov function are shown, indicating that the constructed novel barrier Lyapunov function satisfies the constraint conditions and converges quickly to the constraint boundary. Figure 3 The diagram shows the constraint path tracking curve of the nonlinear system under the present invention. By comparing the expected trajectory before constraint processing, the path tracking constraint control method based on the novel obstacle Lyapunov function of the present invention has good tracking performance and short response time. Figure 4 The graph shows the dynamic variable curve of the tracking error of the nonlinear system under the present invention. The curve shows that the tracking error of the system converges asymptotically. Figure 5 The control force curve of the nonlinear system under the present invention is shown, indicating that the designed controller output is relatively smooth and conforms to practical engineering applications.

[0112] This invention constructs a novel obstacle Lyapunov function to process system output constraints, and designs a nonlinear system output constraint control method based on the novel obstacle Lyapunov function, which solves the problem of limited system output under environmental, safety, physical and other constraints in some practical engineering applications.

[0113] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A nonlinear system output constraint control method based on a novel barrier Lyapunov function, characterized by: It includes the following steps: Step (1): Establish the mathematical model of the second-order nonlinear system: Establish a mathematical model for a single-input, single-output, strictly feedback second-order nonlinear system; Step (2): Design a novel barrier Lyapunov function: A novel barrier Lyapunov function, in the following form: In the above formula, the function On a certain open interval D containing the origin, with respect to this nonlinear system A function, where the independent variable is... It is tracking error. It is a variable The constraint range boundary, and Satisfying the relationship ; The novel barrier Lyapunov function has the following properties: (1) It is smooth and positive definite; (2) At any point on the open interval D, there exists a continuous first-order partial derivative; (3) When As it approaches the boundary of interval D, ; (4) When ,system The solution satisfies Established; (5) This indicates that when constraints As the system approaches infinity, a system with output constraints can be generalized to a system without constraints. Step (3): Obtain the dynamic model of the tracking error of the nonlinear system: The control system is designed using the novel obstacle Lyapunov function in step (2). Since there is an inevitable error between the expected value and the actual value of the controlled variable in the system design, i.e. the error between the expected trajectory and the tracking trajectory, a dynamic model of the system tracking error is established. Step (4): Design a first-order inertial filter: When designing a system controller within the backstepping framework, a first-order inertial filter is added to the system using a dynamic surface design method. Step (5): Design the Lyapunov function and virtual control input for the nonlinear output constraint system based on the novel obstacle Lyapunov function: Based on the novel obstacle Lyapunov function obtained in steps (2), (3), and (4), as well as the tracking error dynamic model of the nonlinear system and the first-order inertial filter, the dynamic surface design method is used to design the Lyapunov function based on the nonlinear system and the virtual control input with respect to the first-order inertial filter. Step (6): Design the constraint controller for the nonlinear system: Based on the first-order inertial filter and system virtual control input obtained in steps (4) and (5), design a controller with output constraints based on a nonlinear system; Step (7): Simulation verification: The mathematical model of the single-input single-output strictly feedback second-order nonlinear system based on the novel obstacle Lyapunov function output constraint controller designed in steps (2) to (6) and established in step (1) was simulated.

2. The nonlinear system output constraint control method based on a novel obstacle Lyapunov function according to claim 1, characterized in that: The dynamic model of the tracking error of the nonlinear system obtained in step (3) is as follows: When designing a system controller using the dynamic surface design method, the tracking error is defined as follows: and Therefore, the dynamic model for the system tracking error is as follows: In the above formula, , Represents system state variables. , , Indicates system design parameters, , To track errors, For the system's desired trajectory, It is the output of a first-order filter. yes The derivative with respect to time, It is the system controller.

3. The nonlinear system output constraint control method based on a novel obstacle Lyapunov function according to claim 1, characterized in that: The design of the first-order inertial filter described in step (4) is as follows: The first-order inertial filter is designed within the dynamic surface design framework as follows: ， In the filter defined by the above formula, These are the initial values ​​for the first-order filter. The initial value for the virtual control input. These are the design parameters for a first-order filter. It is the input of a first-order filter, and the filter error is... .

4. The nonlinear system output constraint control method based on a novel obstacle Lyapunov function according to claim 1, characterized in that: The design of the Lyapunov function and virtual control input for the nonlinear output constraint system based on the novel obstacle Lyapunov function described in step (5) is as follows: Design of novel barrier Lyapunov functions for nonlinear systems : Differentiating the above equation and substituting it into the controller, we get: In the above formula, It concerns the output constraint boundary regarding time. yes abbreviation, , These are system design parameters. It is the virtual control rate The maximum value; Virtual control input for nonlinear systems based on novel barrier Lyapunov functions : In the above formula, This represents the derivative of the system's desired trajectory with respect to time. The derivative representing the time-varying constraint, yes The abbreviation of .

5. The nonlinear system output constraint control method based on a novel obstacle Lyapunov function according to claim 1, characterized in that: The design of the constraint controller for the nonlinear system described in step (6) is as follows: Nonlinear System Output Constraint Controller Based on Novel Barrier Lyapunov Function The design is as follows: 。

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