Preset performance control method based on normalized error boundary function

Abstract

The invention discloses a preset performance control method based on a normalized error boundary function, and relates to the field of automatic control. The method comprises the following steps: establishing a dynamic system model; selecting a preset performance function based on the kinetic model, carrying out normalized error boundary function design based on the selected preset performance function, and solving error conversion based on the normalized error boundary function and the conversion function; and introducing a conversion error, designing a nonsingular terminal sliding mode surface and a reaching law, and completing the design of the preset performance controller based on the normalized error boundary function. Compared with the prior art, the control method has the advantages that the system state can track the state instruction more efficiently, the control method is prevented from losing efficacy when the error exceeds the preset performance function boundary, and the transient performance and the steady-state performance of the control system are improved.

Description

Technical Field

[0001] The present invention relates to the field of automatic control, and specifically to a preset performance control method based on a normalized error boundary function. Background Technique

[0002] In the field of modern engineering, as a core technology, control systems have deeply penetrated into key fields such as aerospace equipment, underwater submarines, intelligent driving systems, and industrial automation equipment. These complex engineering systems generally exhibit strong nonlinear and multi-variable coupling dynamic characteristics. Typical representatives include high-dimensional robotic arm systems, multi-degree-of-freedom aircraft platforms, etc. The research on their dynamic characteristics has universal theoretical value, and thus has become an important test carrier for verifying the effectiveness of advanced control theories. With the strict requirements for control accuracy and dynamic performance, traditional control methods face huge challenges.

[0003] To address this technical bottleneck, the sliding mode control theory has received extensive attention in the field of dynamic control due to its unique structural advantages. This control method not only has the characteristics of simple algorithm and fast response, but also exhibits strong robustness to parameter perturbation and external disturbance. However, the inherent high-frequency chattering phenomenon of classical sliding mode control severely restricts its application in precision control systems. It is worth noting that the non-singular terminal sliding mode control method effectively eliminates the singularity problem of traditional methods near the equilibrium point while ensuring the finite-time convergence characteristic. Simulation experiments show that this improved algorithm can make the system state trajectory smoothly approach the sliding mode surface, reducing the amplitude of the control input chattering by more than 60%, and significantly improving the dynamic quality of high-precision servo systems.

[0004] The preset performance control method makes full use of means such as performance functions and error transformation to constrain the system error within the specified performance boundary, and can take into account both the steady-state performance and transient performance of the system. However, when traditional preset performance control methods are applied to some special occasions, such as when the initial error of the system is large or the system disturbance is large and the performance function boundary needs to be narrowed, it is easy to cause the system error to exceed the performance boundary, resulting in the distortion of the control method. If the boundary is relaxed, the degree of constraint on the system tracking error will also decrease accordingly. Summary of the Invention

[0005] In order to solve the above problems, the purpose of the present invention is to provide a preset performance control method based on a normalized error boundary function, which improves the steady-state performance and transient performance of the dynamic system by improving the normalized error in the preset performance function to obtain a new conversion error.

[0006] The technical solution of the present invention is: a preset performance control method based on a normalized error boundary function, and the normalized error of the dynamic system is:

[0007]

[0008] In the formula, \(t\) is time, \(\rho_1(t)\) represents the improved performance function with respect to time, \(\lambda_1\) is the normalized error, \(e_1\) is the state tracking error of the dynamic system, and \(\psi(t)\) is the normalized error boundary function.

[0009] The specific steps of a preset performance control method based on the normalized error boundary function are as follows:

[0010] S1: Establish a dynamic system model;

[0011] S2: Based on the established dynamic system model, establish a preset performance function;

[0012] S3: Design a normalized error boundary function based on the preset performance function, thereby obtaining the normalized error;

[0013] S4: Obtain the conversion error based on the normalized error and the error conversion function;

[0014] S5: Design a controller based on the conversion error;

[0015] S6: Verify the rationality of the system based on the stability analysis of the system.

[0016] Further, it is characterized in that the preset performance function in S2 is:

[0017]

[0018] where \(t\) is time, \(\rho_1(t)\) represents the improved performance function with respect to time, \(u_0 \gt 0\), \(u\) ∞ \(\gt 0\), \(k_0 \gt 0\) are positive constant parameters, and \(e\) is the natural constant.

[0019] Further, the specific steps of S4 are:

[0020] S401: Select an error conversion function \(T(\varepsilon_1)\), where \(\varepsilon_1\) is the conversion error;

[0021] S402: Obtain the inverse function of \(T(\varepsilon_1)\) to get the conversion error:

[0022]

[0023] Further, S5 includes:

[0024] S501: Select a control sliding mode surface, introduce the conversion error, and calculate the control sliding mode surface;

[0025] S502: Determine the exponential reaching law;

[0026] S503: Design a sliding mode controller based on the conversion error.

[0027] Furthermore, the sliding mode surface calculation formula in S501 is as follows:

[0028] S = ε1 + η|e2| γ sgn(e2);

[0029] where S is a non-singular terminal sliding mode surface function, ε1 is the conversion error, η > 0, 0 < γ < 1 are sliding mode surface parameters, sgn(·) is the sign function, and e2 is the system state tracking error;

[0030] Taking the derivative of S gives:

[0031]

[0032] where: x1, x2 represent the state variables of the system, represents the derivatives of the system state variables x1, x2 with respect to time, f1(x1,x2) and g1(x1,x2) are smooth functions, is the derivative of the tracking command x for the dynamic system state variable 2c with respect to time, is the derivative of the dynamic system state variable x2 with respect to time, is the derivative of the sliding mode surface S with respect to time, and u is the non-singular terminal sliding mode controller function.

[0033] Furthermore, the exponential reaching law in S502 is:

[0034]

[0035] where k1, μ > 0 are sliding mode control gains, is the reaching law function, and sgn(·) is the sign function.

[0036] Furthermore, the non-singular terminal sliding mode controller designed in S503 is:

[0037] Furthermore, S6 includes: analyzing the stability of the non-singular terminal sliding mode controller;

[0038] Selecting the Lyapunov function:

[0039]

[0040] where V is the Lyapunov function;

[0041] Taking its derivative with respect to time gives:

[0042]

[0043] where, is the derivative of the Lyapunov function V;

[0044] Furthermore, we obtain:

[0045]

[0046] Therefore, by selecting appropriate normal control gain constants k and μ such that the system is stable at this time.

[0047] Compared with the prior art, the advantages of the present invention are as follows:

[0048] By improving the normalized error, the present invention obtains a new conversion error, so that the preset performance control method will not fail due to the system error exceeding the preset performance function boundary. Furthermore, the control system has a faster response speed, can reach the system steady state more quickly, and has a stronger anti-interference ability. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 is a flowchart for designing the dynamic control method

[0050] Figure 2 are the system state and time curves. Among them, (a) is the system state q1 of joint 1 of the two-axis robotic arm and the time curve; (b) is the system state q2 of joint 2 of the two-axis robotic arm and the time curve, (c) is the system state tracking error e1 of joint 1 of the two-axis robotic arm and the time curve; (d) is the system state tracking error e2 of joint 2 of the two-axis robotic arm and the time curve. DETAILED DESCRIPTION OF THE INVENTION

[0051] The purpose of the present invention is to provide a preset performance control method based on the normalized error boundary function and applied to the field of automatic control, so as to improve the steady-state performance and transient performance of the dynamic system, solve the problems existing in the above-mentioned traditional preset performance control, and further improve the control performance of the dynamic system.

[0052] First of all, the present invention provides a preset performance control method based on the normalized error boundary function, and its steps include:

[0053] S1: Establish a general dynamic system model;

[0054] Specifically, the state equation of the dynamic system is:

[0055]

[0056] where x1 and x2 represent the state variables of the system, represents the derivatives of the state variables x1 and x2 of the system with respect to time, f1(x1, x2) and g1(x1, x2) are smooth functions, u is the system control input, and d is the external input disturbance.

[0057] S2: Based on the established dynamic system model, select a preset performance function;

[0058] S201: Calculate the tracking error of the dynamic system

[0059]

[0060] where e1 and e2 are the system state tracking errors, and x 1c , x 2c is the system state quantity tracking instruction.

[0061] S202: Select the preset performance function as:

[0062]

[0063] where t is time, ρ1(t) represents the specified performance function, u0 > 0, u ∞ > 0, k0 > 0 are positive constant parameters.

[0064] Take the derivative of ρ1(t)

[0065]

[0066] where, represents the derivative of the specified performance function ρ1(t) with respect to time, u0 > 0, u ∞ > 0, k0 > 0 are positive constant parameters.

[0067] S3: Design a normalized error boundary function based on the preset performance function, thereby obtaining the normalized error:

[0068] Design the normalized error of the dynamic system as:

[0069]

[0070] In the formula, e1 is the dynamic system state tracking error, λ1 is the normalized error, sgn(·) is the sign function, and ψ(t) is the normalized error boundary function and 0 < ψ(t) < ρ1(t).

[0071] Compared with the traditional normalized error λ1 = e1 / ρ1, the present invention first changes the normalized error to a piecewise function form, thereby effectively avoiding the conversion error from becoming complex when the system state tracking error exceeds the preset performance function ρ1(t), which may lead to the failure of the control method. Specifically: when the dynamic system state tracking error e1 approaches the preset performance function ρ1(t), the normalized error is defined as λ1 = ψ(t) / ρ1(t), ensuring that the conversion error ε1 is always a real number and avoiding the failure of the control method.

[0072] Take is:

[0073]

[0074] In the formula, is the derivative of the normalized error λ1 with respect to time, is the derivative of the dynamic system state tracking error e1 with respect to time, is the derivative of the performance function ρ1 with respect to time, is the derivative of the normalized error boundary function ψ(t) with respect to time;

[0075] Take

[0076]

[0077] S4: Obtain the conversion error based on the normalized error and the error conversion function;

[0078] The designed error conversion function T(ε1) is:

[0079]

[0080] In the formula, T(ε1) is the error conversion function, and ε1 is the conversion error.

[0081] Finding the inverse function of T(ε1) gives the conversion error, and at this time the conversion error ε1 is:

[0082]

[0083] Taking the derivative of the conversion error ε1 gives:

[0084]

[0085] Among them, is the derivative of the conversion error ε1 with respect to time, is the derivative of the normalized error λ1 with respect to time.

[0086] S5: Design a controller based on the conversion error using a non-singular terminal sliding mode control method;

[0087] S501: Introduce the conversion error and calculate the non-singular terminal sliding mode surface;

[0088] The calculation formula for the non-singular terminal sliding mode surface is:

[0089] S = ε1 + η|e2| γ sgn(e2) (11)

[0090] Among them, S is the designed non-singular terminal sliding mode surface function, η > 0, 0 < γ < 1 are the sliding mode surface parameters, and sgn(·) is the sign function;

[0091] Deriving with respect to S gives:

[0092]

[0093] Where: is the derivative of the tracking instruction x of the dynamic system state quantity with respect to time, 2c the derivative with respect to time, is the derivative of the dynamic system state quantity x2 with respect to time, is the derivative of the sliding mode surface S with respect to time, and u is the non-singular terminal sliding mode controller function.

[0094] S502: Determine the exponential reaching law;

[0095] The selected exponential reaching law is:

[0096]

[0097] where k1, μ > 0 are the sliding mode control gains, is the reaching law function, and

[0098] S503: Design a non-singular terminal sliding mode controller based on the conversion error;

[0099] Combining equations (12) and (13), the non-singular terminal sliding mode controller is:

[0100]

[0101] When the system error |e1| < ψ(t), that is, |e1| is less than the normalized error boundary function ψ1, the controller normally controls the system; when the system error |e1| ≥ ψ(t), that is, |e1| is greater than or equal to the normalized error boundary function ψ1, the system error e1 in the controller is replaced with the normalized error boundary function ψ1, which can effectively avoid the problem of controller failure caused by the system error exceeding the preset performance function.

[0102] S6: Based on the stability analysis of the system, verify the rationality of the system.

[0103] Analyze the stability of the non-singular terminal sliding mode controller;

[0104] Select the Lyapunov function:

[0105]

[0106] where V is the Lyapunov function;

[0107] Taking the derivative of it with respect to time gives:

[0108]

[0109] Among them, is the derivative of the Lyapunov function V.

[0110] Combining equations (13) and (14), we can obtain:

[0111]

[0112] Therefore, by choosing appropriate control gain positive constants k and μ such that the system is stable at this time;

[0113] Example:

[0114] Use MATLAB software to conduct a case simulation on the dynamic system of a two-axis robotic arm to prove the effectiveness of this method.

[0115] Specifically:

[0116] q instruction: External disturbance

[0117] (1) Establish the dynamic system of a two-axis robotic arm

[0118] The dynamic system of a two-axis robotic arm is expressed as

[0119]

[0120] In the formula: are the joint position, joint angular velocity, and joint angular acceleration respectively; M(q) ∈ R 6×6 is a positive definite inertial mass matrix; represents the centrifugal force and Coriolis force matrix; G(q) ∈ R 6×1 represents the gravity; τ ∈ R 6×1 is the control input joint torque; τ d ∈ R 6×1 is the external unknown disturbance.

[0121]

[0122] (2) Design the controller

[0123] The non-singular terminal sliding mode surface and control law expressions of the two-axis robotic arm:

[0124]

[0125] (3) Provide a set of non-singular terminal sliding mode control parameters, specifically: where ε is the conversion error, α, r are control parameters, and α = 0.5, r = 1.2, and the reaching law parameters are k1 = 5, k2 = 0.001;

[0126] (4) Provide a set of traditional preset performance function parameters, and use the exponentially convergent preset performance function for comparison. Specifically, where ρ is the hyperbolic cotangent type preset performance function, ρ0, ρ ∞ are the performance function parameters, and ρ0 = 1, ρ ∞ = 0.1, κ0 = 2;

[0127] (5) Provide a set of improved preset performance function parameters, ψ(t) = 0.98ρ(t), ρ0 = 1, ρ ∞ = 0.1, κ0 = 2;

[0128] (6) Use MATLAB software to perform the above steps for the dynamic system model of the two-axis manipulator, and conduct simulation and comparative analysis on the proposed method, the traditional non-singular terminal sliding mode control method, and the non-singular terminal sliding mode control method based on the traditional preset performance function respectively to verify the effectiveness of the present invention;

[0129] From Figure 2 it can be seen that for the traditional preset performance function in the initial stage of the system, if the initial error of the system is large, it is easy to cause the tracking error to exceed the boundary at this time, resulting in a slow convergence speed or even no convergence; if the system disturbance is large and causes the tracking error to exceed the boundary, it is easy to cause a significant reduction in control efficiency at this time. The preset performance control method based on the normalized error boundary function provided by the present invention can well solve the above problems. In the initial stage of the system, even if the initial error is large and exceeds the boundary function or the system disturbance is large and causes the tracking error to exceed the boundary, the system can still maintain good control performance through the improved normalized error, and the system state tracking performance has been significantly improved, further improving the steady-state performance and transient performance of the dynamic system.

Claims

1. A preset performance control method based on a normalized error boundary function, characterized in that The normalized error of the dynamic system is as follows: In the formula, t is time, ρ1(t) represents the performance function with respect to time, λ1 is the improved normalized error, e1 is the state tracking error of the dynamic system, and ψ(t) is the normalized error boundary function.

2. The preset performance control method based on the normalized error boundary function according to claim 1, wherein The specific steps are as follows: S1: Establish a dynamic system model; S2: Based on the established dynamic system model, establish a preset performance function; S3: Design a normalized error boundary function based on the preset performance function to obtain the normalized error; S4: Obtain the conversion error based on the normalized error and the error conversion function; S5: Design a controller based on the conversion error; S6: Verify the rationality of the system based on the stability analysis of the system.

3. A preset performance control method based on a normalized error boundary function according to claim 1, characterized in that, The preset performance function in S2 is as follows: where t is time, ρ1(t) represents the improved performance function with respect to time, u0 > 0, u ∞ > 0, and k0 > 0 are positive constant parameters, and e is the natural constant.

4. The preset performance control method based on the normalized error boundary function according to claim 1, wherein, The specific steps of S4 are as follows: S401: Select the error conversion function T(ε1), where ε1 is the conversion error; S402: Obtain the inverse function of T(ε1) to get the conversion error:

5. A preset performance control method based on a normalized error boundary function according to claim 1, characterized in that S5 includes: S501: Select the control sliding mode surface, introduce the conversion error, and calculate the control sliding mode surface; S502: Determine the exponential reaching law; S503: Design a sliding mode controller based on the conversion error.

6. A preset performance control method based on a normalized error boundary function according to claim 5, characterized in that, The calculation formula for the sliding mode surface in S501 is as follows: S = ε1 + η|e2| γ sgn(e2); Where S is the non-singular terminal sliding mode surface function, ε1 is the conversion error, η > 0, 0 < γ < 1 are the sliding mode surface parameters, sgn(·) is the sign function, and e2 is the state tracking error of the system; Take the derivative of S to get: where: x1, x2 represent the state variables of the system, represent the derivatives of the state variables x1, x2 of the system with respect to time, f1(x1,x2) and g1(x1,x2) are smooth functions, is the derivative of the tracking command x for the state variable of the dynamic system 2c with respect to time, x&2 is the derivative of the state variable x2 of the dynamic system with respect to time, is the derivative of the sliding surface S with respect to time, and u is the non-singular terminal sliding mode controller function.

7. A preset performance control method based on a normalized error boundary function according to claim 5, characterized in that, The exponential reaching law in S502 is: where \(k_1,\mu>0\) are sliding mode control gains, is the reaching law function, and sgn(·) is the sign function.

8. A preset performance control method based on a normalized error boundary function according to claim 5, characterized in that The non-singular terminal sliding mode controller designed in S503 is:

9. The preset performance control method based on the normalized error boundary function according to claim 1, characterized in that The specific steps of S6 are as follows: Analyze the stability of the non-singular terminal sliding mode controller; Select the Lyapunov function: Where V is the Lyapunov function; Take the derivative of it with respect to time to get: wherein, is the derivative of the Lyapunov function V; Furthermore, get: Therefore, select appropriate control gain positive constants \(k\) and \(\mu\) such that the system is stable at this time.

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