Track control method based on high-order finite time observer

Abstract

The invention relates to a trajectory control method based on a high-order finite time observer. The method comprises the steps that a system dynamics model is established, the high-order finite time homogeneous observer is designed, parameter drift is limited through a projection operator, and a fractional order time-varying gain adaptive law is introduced to cope with sudden change disturbance. An asymmetric obstacle function is adopted to reconstruct an observation error, and the convergence speed under an asymmetric dynamic condition is improved; injecting a time-varying scaling item into the observer, and balancing initial fast response and steady-state noise suppression; a backstepping sliding mode differential controller is constructed, a fractional order sliding mode surface is designed, multi-scale weighted gain and a nonlinear continuous operator are combined, control parameters are dynamically adjusted, high-frequency buffeting is restrained, and steady-state phase lag is eliminated. According to the invention, through finite time observation, adaptive disturbance compensation and fractional order sliding mode control, the trajectory tracking precision, the anti-disturbance capability and the system robustness are significantly improved.

Description

Technical Field

[0001] The present invention belongs to the field of motion control, and in particular relates to a trajectory control method based on a high-order finite-time observer. Background Art

[0002] Among the existing robot trajectory control methods, traditional sliding mode control has high-frequency jitter problems, and fixed-time observers are difficult to accurately estimate the state and disturbance within a limited time. Fixed gain design cannot effectively deal with sudden disturbances (such as impact loads), resulting in compensation lag or overcompensation; the convergence speed of symmetric observers decreases under unilateral disturbances (such as unidirectional friction); the proportional-integral characteristics of traditional sliding surface are prone to introduce phase lag, affecting steady-state accuracy. In the prior art, although fractional-order differentials are used to adjust dynamic response, they are not combined with multi-scale gains, and cannot take into account both high- and low-frequency error suppression; adaptive laws mostly use integer-order designs, with a high risk of parameter drift, and lack a time-varying gain mechanism to balance dynamic response and steady-state performance. In addition, the fixed convergence mechanism of the observer is susceptible to high-frequency noise interference in the initial stage, and residual oscillations cannot be effectively suppressed in the steady-state stage. Summary of the invention

[0003] The present invention provides a trajectory control method based on a high-order finite-time observer, which comprises the following steps: Step 1, establish a multi-degree-of-freedom robot dynamics model and define an extended state vector to describe position, velocity, acceleration and lumped disturbance; Step 2, design a high-order finite-time homogeneous observer, which consists of a recursive form of the observation error equation, limits the drift of adaptive parameters through a projection operator, or uses a fractional-order time-varying gain adaptive law to compensate for disturbances in real time; Step 3: Design a fractional step sliding surface and control law, where the sliding surface dynamically adjusts the error weighting through a fractional order differential operator, and the control law contains an integral term and a time-varying smooth nonlinear damping term.

[0004] Furthermore, the observer gain coefficients of the high-order finite-time homogeneous observer are configured through Hurwitz polynomials, and the gain parameters are designed in combination with the upper bound of the Lipschitz constant of the unmodeled dynamics of the system.

[0005] Furthermore, the lumped disturbance estimation uses a projection operator to constrain the parameter range, and the projection operator dynamically cuts off or adjusts the parameter update direction according to a preset disturbance amplitude upper limit.

[0006] Furthermore, the disturbance compensation term is updated through a fractional-order time-varying gain adaptive law, and its gain parameter is dynamically adjusted according to the sliding surface error.

[0007] Furthermore, an asymmetric barrier function is introduced into the observer to set the convergence index independently by the positive and negative error directions.

[0008] Furthermore, a time-varying scaling term is injected into the observer equation, which retains the fast response characteristics in the initial stage and gradually suppresses the high-frequency noise components over time.

[0009] Furthermore, the sliding film surface and the control law are: , where D is the fractional differential, ζ is the fractional differential order, m is the order of the sliding surface, Λ k is the gain matrix, M is the inertia matrix, K ω is the integral gain matrix, ω is the integral term power parameter, H(s) is the nonlinear continuous operator, and ρ is the robust gain.

[0010] Furthermore, the nonlinear continuous operator is: , δ is the smoothing factor, δ0 is the initial value of the smoothing factor, η is the power parameter, δ ∞ is the steady-state value.

[0011] The present invention also provides a trajectory control system based on a high-order finite-time observer, which is used to execute the above method.

[0012] Beneficial technical effects: By constraining the parameter range through the projection operator, the robustness of disturbance compensation is enhanced by combining the fractional-order time-varying gain adaptive law; the asymmetric barrier function is designed to optimize the convergence characteristics of the observation error; the time-varying scaling term is introduced to realize the dynamic attenuation of the observer; the fractional-order step sliding surface and multi-scale gain are constructed, and the control law is dynamically adjusted to eliminate chattering and improve tracking accuracy within a limited time. This method can significantly suppress sudden disturbances and improve the smoothness of control. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] Attached Figure 1 A control principle diagram of a trajectory control method based on a high-order finite-time observer according to the present invention; Attached Figure 2 Schematic diagram of tracking error of joint angle according to the present invention. DETAILED DESCRIPTION

[0014] like Figure 1 As shown, this embodiment provides a trajectory control method based on a high-order finite-time observer, which includes the following steps: Step 1: System dynamics modeling, define the extended state vector x=[x1 T x2 T x3 T ] T , where x1, x2, and x3 are position, velocity, and acceleration, respectively, and a third-order chain differential equation is established:

[0015] Among them, M, C, G are the inertia matrix, Coriolis force matrix, and gravity term, respectively, τ d is the lumped disturbance, including friction, external load, etc., τ is the control input, and u is the generalized control input, specifically the first-order derivative of τ.

[0016] Step 2, high-order finite-time observer design: Construct an r-order finite-time homogeneous observer:

[0017] Among them, Φ is the nonlinear dynamic term, k i , α i is the gain coefficient, which can be designed as: , where λ i , L is the observer parameter, λ i Determined by the Hurwitz polynomial root configuration, specifically: construct the observer error dynamic equation, linearize it to obtain the characteristic polynomial, and select λ according to Hurwitz stability i The roots of the polynomial are all located in the left half plane of the complex plane (i.e., the real part is negative), and L is the upper bound of the Lipschitz constant of the unmodeled dynamic and disturbance terms in the system dynamics model, which can be determined by experiments or theoretical analysis.

[0018] Among them, the lumped disturbance estimation can use the projection operator to constrain the parameter range. The projection operator dynamically truncates or adjusts the parameter update direction according to the preset upper limit of the disturbance amplitude to prevent parameter drift. The lumped disturbance estimation error can be designed as , where σ is the integral variable, representing the historical time point in the integral process, and t is the continuous time variable. , γ is the disturbance control coefficient, and proj( ) is the projection operator. In disturbance estimation, the real lumped disturbance τ d Usually bounded, but the integral term in the adaptive law may cause the adaptive parameters to exceed the physically allowed range, so the projection operator proj() is introduced to limit the value range of the adaptive parameters by forcibly truncating or adjusting the parameter update direction to prevent them from growing infinitely due to integral accumulation, that is, to prevent parameter drift. The calculation of the projection operator proj() is as follows:

[0019] Among them, τ max It is set according to the robot system characteristics, such as the maximum torque of the motor and the load limit.

[0020] Alternatively, fixed gain disturbance compensation may not be able to cope with sudden disturbances, so a fractional-order time-varying gain adaptive law can be designed. The disturbance compensation term is updated by the fractional-order time-varying gain adaptive law, and its gain parameter is dynamically adjusted according to the sliding surface error to balance the sudden disturbance compensation intensity and steady-state noise suppression. The improved adaptive disturbance compensation law is:

[0021] D is a fractional-order differential operator used to balance dynamic response and suppress high-frequency noise, which can be: , Γ(ν) is the Gamma function, f (1) is the first-order derivative of f. ν is the fractional differential order, ν∈(0,1), which reflects the weight of the historical error on the current compensation and is used to adjust the historical error weight. By adjusting ν, the adaptive compensator can simultaneously cope with slow changes (such as temperature drift) and sudden disturbances (such as impact loads). γ(t) is the time-varying gain, μ is the gain adjustment index, μ∈(0,1), which is used to determine the balance between gain growth and attenuation; κ is the gain growth rate, κ >0, reflecting the adjustment intensity according to the sliding surface error, θ is the gain attenuation rate, θ>0, to prevent the gain from growing unbounded. When the sliding surface error is large, γ(t) increases rapidly to improve the disturbance compensation intensity. When the sliding surface error approaches the steady state, γ(t) converges to the equilibrium point to avoid overcompensation.

[0022] The improved control system can resist sudden disturbances, has enhanced steady-state accuracy, and enhanced parameter adjustment robustness. When the system is suddenly loaded (such as a robotic arm grabbing an object of unknown mass), the sliding surface error increases rapidly, triggering the adjustment mechanism, causing γ(t) to jump within milliseconds. The compensation speed is several times faster than that of a fixed gain, and when in steady state, high-frequency jitter caused by excessive gain can be avoided.

[0023] In the formula, Where sign is the sign function.

[0024] Alternatively, in order to solve the problem of decreased convergence speed when there is a unilateral disturbance (such as unidirectional friction) in the observer symmetric gain design, an asymmetric barrier function can be introduced to reconstruct the observation error, that is, the observer introduces an asymmetric barrier function, and independently sets the convergence index by the positive and negative error directions to optimize the observation accuracy and convergence speed under unilateral disturbance. The asymmetric barrier function is specifically:

[0025] Thus the observer is changed to an asymmetric r-order observer:

[0026] Among them, α i ,β i∈(0,1) are the convergence indexes in the positive / negative error directions, respectively, which control the convergence speed in the positive / negative error directions, and λ max (M), λ min (M) are the maximum / minimum eigenvalues ​​of the inertia matrix, respectively.

[0027] In the presence of unilateral disturbances, the observation accuracy can be significantly improved through asymmetric design, making the observer adaptive to asymmetric dynamics.

[0028] Furthermore, in order to improve the convergence, a time-varying scaling fixed-time convergence mechanism can be introduced into the observer. The time-varying scaling term is injected into the observer equation. The fast response characteristics are retained in the initial stage, and the high-frequency noise components are gradually suppressed over time to achieve a balance between dynamic convergence speed and steady-state smoothness. Specifically, Among them, β is the scaling parameter and ε is the power parameter, which is used to balance the convergence speed and smoothness. The larger β and ε are, the faster the decay.

[0029] By injecting a time-varying scaling term into the observer equation, in the initial stage of control, the scaling factor is 1, retaining the rapid response characteristics of the original observer, and as the control progresses, the scaling factor gradually decreases and tends to 0 as time t increases, suppressing the high-frequency dynamic components of the observer. In the rapid convergence stage, the time-varying scaling term plays almost no role, the exponential decay term maintains a high power, and the observer quickly tracks the true state. In the steady-state stage, the scaling term suppresses high-frequency noise, and the exponential decay term reduces the gain to achieve smooth control.

[0030] Step 3, back-stepping synovial differential control: In order to reduce the high-frequency chattering in traditional sliding film control, the sliding film surface can be designed as a fractional-order step sliding surface, where the sliding surface dynamically adjusts the error weighting through a fractional-order differential operator, and the control law contains an integral term and a time-varying smooth nonlinear damping term to suppress high-frequency chattering and improve tracking accuracy. The sliding film surface and control law are specifically:

[0031] Where D is the fractional differential, through which the sliding surface characteristics are dynamically adjusted to reduce the sensitivity of the controller to high-frequency noise while maintaining a fast response to low-frequency errors. ζ is the fractional differential order, and m is the order of the sliding surface, that is, the number of backstepping recursions. When m is an odd number, the suppression of high-frequency error components is enhanced, and when m is an even number, the integrity of the low-frequency tracking information is retained. k is the gain matrix, Λ kis a diagonal matrix; the multi-scale weighting of the error is realized by the index k / m, that is, the gain of the higher level k is more sensitive to the high-frequency error; M is the inertia matrix, which can be directly quoted from the inertia matrix in the dynamic model, and its estimated value can be used in actual control; K ω is the integral gain matrix, which is used to adjust the weight of the integral term (the subscript ω is only a mark and has nothing to do with the integral term power parameter); ω is the integral term power parameter, which can be ω=ζ / (ζ-1). ω controls the cumulative effect of the integral term. When ω tends to 1, the integral degenerates into pure accumulation. When ω tends to infinity, it is approximate proportional control. Through the coupling design of ω and ζ, the steady-state phase lag in the traditional sliding mode control is eliminated; H(s) is a nonlinear continuous operator, specifically: , when |s| is large, it is approximate proportional control to enhance robustness, and when |s| is small, it is approximate linear damping control to eliminate high-frequency chattering, where δ is the smoothing factor, the smoothing factor δ is a time-varying parameter, δ>0, δ0 is the initial value of the smoothing factor, δ ∞ is the steady-state value; η is the power parameter. ρ is the robust gain, which can be adjusted by the adaptive regulation law , ρ0 is the initial value of the robust gain, Κρ is the robust parameter, Κρ>0, the gain is automatically increased in the initial stage to accelerate convergence, and the gain is reduced in the steady state to reduce energy consumption.

[0032] The sliding film control of this embodiment can suppress high-frequency chattering, maintain convergence within a limited time, and dynamically design the control parameters so that the controller can automatically adjust under different working conditions. Figure 2 As shown, as the control proceeds, the tracking error of the joint angle decreases rapidly.

[0033] In the several embodiments provided by the present invention, it should be understood that the disclosed methods can be implemented in other ways. For example, the above-described embodiments of the invention are merely schematic. For example, the division of method modules and steps is only a logical function division, and there may be other division methods in actual implementation. The modules and steps described as separate components or separate steps may or may not be physically separated, and the components displayed as modules may or may not be physical modules, which may be located in one place or distributed on multiple network modules. Some or all of the modules may be selected according to actual needs to achieve the purpose of the scheme of this embodiment.

[0034] In addition, each functional module and step in each embodiment of the present invention may be integrated into a processing module or method, or each module or step may exist physically separately, or two or more modules or steps may be integrated into one module or method. The above-mentioned integrated module may be implemented in the form of hardware or in the form of hardware plus software functional modules.

[0035] It is obvious to a person skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the present invention can be implemented in other specific forms without departing from the essential characteristics of the present invention.

[0036] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that the technical solution of the present invention can be modified or replaced by equivalents without departing from the spirit and scope of the technical solution of the present invention.

Claims

1. A trajectory control method based on a high-order finite-time observer, characterized in that: The following steps are involved: Step 1, establish a multi-degree-of-freedom robot dynamics model and define an extended state vector to describe position, velocity, acceleration and lumped disturbance; Step 2, design a high-order finite-time homogeneous observer, which consists of a recursive form of the observation error equation, limits the drift of adaptive parameters through a projection operator, or uses a fractional-order time-varying gain adaptive law to compensate for disturbances in real time; Step 3: Design a fractional step sliding surface and control law, where the sliding surface dynamically adjusts the error weighting through a fractional order differential operator, and the control law contains an integral term and a time-varying smooth nonlinear damping term.

2. The method according to claim 1, characterized in that The observer gain coefficients of the high-order finite-time homogeneous observer are configured through Hurwitz polynomials, and the gain parameters are designed by combining the upper bound of the Lipschitz constant of the unmodeled dynamics of the system.

3. The method according to claim 1, characterized in that The lumped disturbance estimation uses a projection operator to constrain the parameter range, and the projection operator dynamically cuts off or adjusts the parameter update direction according to a preset disturbance amplitude upper limit.

4. The method according to claim 1, characterized in that The disturbance compensation term is updated through a fractional-order time-varying gain adaptive law, and its gain parameter is dynamically adjusted according to the sliding surface error.

5. The method according to claim 1, characterized in that The observer introduces an asymmetric barrier function and sets the convergence index independently through the positive and negative error directions.

6. The method according to claim 1, characterized in that A time-varying scaling term is injected into the observer equation, which retains the fast response characteristics in the initial stage and gradually suppresses the high-frequency noise components over time.

7. The method according to claim 1, characterized in that The sliding film surface and control law are: , where D is the fractional differential, ζ is the fractional differential order, m is the order of the sliding surface, Λ k is the gain matrix, M is the inertia matrix, K ω is the integral gain matrix, ω is the integral term power parameter, H(s) is the nonlinear continuous operator, and ρ is the robust gain.

8. The method according to claim 7, characterized in that The nonlinear continuous operator is: , δ is the smoothing factor, δ0 is the initial value of the smoothing factor, η is the power parameter, δ ∞ is the steady-state value.

9. A trajectory control system based on a high-order finite-time observer, used to implement the method described in any one of claims 1 to 8.

Images