Trolley inverted pendulum control method based on differentiator

Abstract

The invention relates to a differentiator-based trolley inverted pendulum control method. The method comprises the following steps of: establishing a trolley inverted pendulum mathematical model taking trolley acceleration as control input; introducing an auxiliary variable to convert the obtained state equation into a non-strict feedback system; constructing a differentiator according to a non-strict feedback system and a backstepping method; a trolley inverted pendulum control law is designed by using the output of a differentiator and an event driving method, so that the trolley is stable at an expected position, and a swing rod is kept upright. According to the designed control method, the swing rod does not need to be located at the vertically upward position at the initial moment, and the application range of the control method is widened. In addition, the designed control method is simple in structure, and data transmission between the controller and the execution mechanism can be effectively reduced.

Description

Technical Field

[0001] This invention relates to the field of inverted pendulum control, and more particularly to a method for controlling an inverted pendulum on a trolley based on a differentiator. Background Technology

[0002] Since the advent of inverted pendulum control systems, their nonlinear, strongly coupled, and unstable characteristics have attracted considerable attention from scholars, becoming a research hotspot in the field of control. As a typical nonlinear system, the stability and balance control of the inverted pendulum are crucial to the stability of robots. Research on inverted pendulum control helps robots maintain balance in complex environments, thereby improving their performance in various tasks, such as moving and transporting objects. Furthermore, research on inverted pendulum control can drive the optimization and improvement of control algorithms. These optimizations and improvements can not only be applied to the inverted pendulum system itself but also extended to robot control systems, thereby improving the performance and efficiency of the entire robot system.

[0003] Existing control methods require linearization of the inverted pendulum system and demand that the pendulum rod be vertically upward during the initial control startup, severely limiting their applicability. Backstepping, a typical method of nonlinear control, avoids linearization, but existing backstepping methods struggle to ensure the car remains stable at the desired position while simultaneously keeping the pendulum rod upright. Traditional backstepping methods require taking the time derivative of the virtual control law, leading to complex control structures that are unsuitable for practical applications. Furthermore, traditional time-driven control methods waste limited bandwidth resources. To conserve communication resources and avoid bandwidth waste, introducing an event-driven mechanism into the inverted pendulum control system is necessary. In conclusion, designing a differentiator-based control method for an event-driven inverted pendulum system without utilizing linearization is a significant challenge in the field of inverted pendulum control. Summary of the Invention

[0004] This invention provides a control method for an inverted pendulum cart based on a differentiator. Its purpose is to address the problems of existing control methods for event-driven inverted pendulum cart systems, which, without utilizing linearization operations, cannot ensure the cart remains stable at the desired position while simultaneously keeping the pendulum upright; the control methods also suffer from high computational complexity and significant waste of communication resources in the control system.

[0005] To achieve the above objectives, the present invention adopts the following technical solution, including:

[0006] The control method for the inverted pendulum of a cart based on a differentiator consists of the following steps:

[0007] Step 1. Establish a mathematical model of the inverted pendulum with the cart's acceleration as the control input, and convert the mathematical model of the inverted pendulum into a state equation;

[0008] Step 2. Introduce auxiliary variables to transform the obtained state equations into a non-strict feedback system;

[0009] Step 3. Construct the differentiator based on the non-strict feedback system and the backstepping method;

[0010] Step 4. Using the output of the differentiator and the event-driven method, design the control law for the inverted pendulum of the trolley to make the trolley stable in the desired position and keep the pendulum upright.

[0011] Furthermore, the mathematical model of the inverted pendulum with the cart's acceleration as the control input, as described in step 1, is as follows:

[0012]

[0013] Where, χ c θ represents the position of the car. p M represents the angle between the pendulum and the vertically downward direction. p For the mass of the pendulum, L p Let I be the length of the pendulum. p Let τ be the moment of inertia of the pendulum. a This represents the control input (car acceleration), where g is the acceleration due to gravity.

[0014] Furthermore, the state equation described in step 1 is:

[0015]

[0016] in, ζ1=χ c -χ d , ζ3=θ p -π, π represents the mathematical constant pi, and χ represents the mathematical constant pi. d This indicates the desired position of the vehicle.

[0017] Furthermore, the non-strict feedback system in step 2 is as follows:

[0018]

[0019] Among them, auxiliary variables Auxiliary variables Auxiliary variable r3 = -gtanζ3, auxiliary variable

[0020] Furthermore, the differentiator in step 3 is:

[0021]

[0022] Where, εw1 ε w2 ε w3 、b 11 、b 12 、b 21 、b 22 、b 31 and b 32 ω is a design parameter and is a positive number. 11 ω 21 and ω 31 ω represents the state of the differentiator. 12 ω 22 and ω 32 Let α1, α2, and α3 represent the output of the differentiator, and let α1, α2, and α3 represent the virtual control laws. The expressions for α1, α2, and α3 are:

[0023]

[0024] Where k1, k2 and k3 are design parameters and are positive numbers, e1 = r1, e2 = r2 - α1, e3 = r3 - α2.

[0025] Furthermore, the control law for the inverted pendulum of the trolley in step 3 is as follows:

[0026]

[0027] Where, γ ma γ mi , k4 are design parameters and are positive numbers, t represents time, and q represents a positive integer. q ψ represents the update time of the control law. m (t)=τ a -α4(t), e4=r4-α3.

[0028] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0029] 1. This invention does not require linearization of the inverted pendulum system of the trolley, and does not require the pendulum to be in a vertically upward position at the initial stage of the control method, thus expanding the applicability of the control method;

[0030] 2. This invention avoids taking the time derivative of the virtual control law, making the control method simple in structure and easy to apply in practice;

[0031] 3. This invention utilizes an event-driven approach, which can reduce data transmission and save communication resources of the control system.

[0032] Based on the above reasons, this invention can be widely applied in the field of inverted pendulum control. Attached Figure Description

[0033] Figure 1 This is a flowchart of the control method of the present invention;

[0034] Figure 2 This is a schematic diagram of the inverted pendulum model of the trolley of the present invention;

[0035] Figure 3 Let χ represent the position of the trolley under different control methods in case 1. c and angle θ p Comparison chart;

[0036] Figure 4 For case 1, the control input τ is the control input for different control methods. a Comparison chart;

[0037] Figure 5 Let χ represent the position of the trolley under different control methods in case 2. c and angle θ p Comparison chart;

[0038] Figure 6 For case 2, the control input τ is different from the control input τ for different control methods. a Comparison chart. Detailed Implementation

[0039] The present invention will be described in more detail below with reference to the accompanying drawings.

[0040] This invention addresses the inverted pendulum system of a cart considering event-driven operation. It introduces auxiliary variables to transform the obtained state equation into a non-strict feedback system. Based on the non-strict feedback system and the backstepping method, a differentiator is constructed. Using the output of the differentiator and the event-driven method, a control law for the inverted pendulum of the cart is designed to achieve stability of the cart at the desired position and keep the pendulum rod upright.

[0041] like Figure 1 As shown, the present invention provides a method for controlling an inverted pendulum cart based on a differentiator, comprising the following steps:

[0042] Step 1. Establish a mathematical model of the inverted pendulum with the cart's acceleration as the control input, and convert the mathematical model of the inverted pendulum into a state equation;

[0043] The established mathematical model of the inverted pendulum cart with the cart's acceleration as the control input is as follows:

[0044]

[0045] Where, χ c θ represents the position of the car. p M represents the angle between the pendulum and the vertically downward direction. p For the mass of the pendulum, L p Let I be the length of the pendulum. p Let τ be the moment of inertia of the pendulum. aThis represents the control input (car acceleration), where g is the acceleration due to gravity.

[0046] The state equation is:

[0047]

[0048] in, ζ1=χ c -χ d , ζ3=θ p -π, π represents the mathematical constant pi, and χ represents the mathematical constant pi. d This indicates the desired position of the vehicle.

[0049] Step 2. Introduce auxiliary variables to transform the obtained state equations into a non-strict feedback system;

[0050] The non-strict feedback system is:

[0051]

[0052] Among them, auxiliary variables Auxiliary variables Auxiliary variable r3 = -gtanζ3, auxiliary variable

[0053] Step 3. Construct the differentiator based on the non-strict feedback system and the backstepping method;

[0054] The differentiator is:

[0055]

[0056] Where, ε w1 ε w2 ε w3 、b 11 、b 12 、b 21 、b 22 、b 31 and b 32 ω is a design parameter and is a positive number. 11 ω 21 and ω 31 ω represents the state of the differentiator. 12 ω 22 and ω 32 Let α1, α2, and α3 represent the output of the differentiator, and let α1, α2, and α3 represent the virtual control laws. The expressions for α1, α2, and α3 are:

[0057]

[0058] Where k1, k2 and k3 are design parameters and are positive numbers, e1 = r1, e2 = r2 - α1, e3 = r3 - α2.

[0059] Step 4. Using the output of the differentiator and the event-driven method, design the control law for the inverted pendulum of the trolley to make the trolley stable at the desired position and keep the pendulum upright;

[0060] The control law for the inverted pendulum cart is:

[0061]

[0062] Where, γ ma γ mi , k4 are design parameters and are positive numbers, t represents time, and q represents a positive integer. q ψ represents the update time of the control law. m (t)=τ a -α4(t), e4=r4-α3.

[0063] The proposed inverted pendulum control method for a trolley based on a differentiator was simulated in a virtual environment to verify its feasibility.

[0064] The inverted pendulum model of a small car, such as Figure 2 As shown, in the simulation experiment, the mathematical model of the inverted pendulum cart with the cart's acceleration as the control input is as follows:

[0065]

[0066] Where, χ c θ represents the position of the car. p M represents the angle between the pendulum and the vertically downward direction. p For the mass of the pendulum, L p Let I be the length of the pendulum. p Let τ be the moment of inertia of the pendulum. a M represents the control input (car acceleration), where g is the acceleration due to gravity. In the experiment, M... p =0.109kg, L p =0.25m, I p =0.0034 kg·m 2 g = 9.8 m / s 2 The simulation time was set to 6 seconds.

[0067] The desired position of the car is set as χ. d =0.2m, expected θ p The initial angle is 180° (meaning the desired pendulum position is vertically upward). The initial state of the inverted pendulum on the cart is χ. c (0) = 0m, To verify that the control method of the present invention can handle different θ values p (0), θ p (0) Set to two cases (Case 1: θ) p (0) = 160°; Case 2: θ p (0) = 130°).

[0068] The parameters of the control law are set as k1 = 1.01, k2 = 1.35, k3 = 12, k4 = 20, ε w1 =0.1, ε w2 =0.1, ε w3 =0.04, b 11 =10, b 12 =10, b 21 =10, b 22 =10, b 31 =10, b 32 =10, γ ma =1.5, γ mi =0.2, ω 11 (0)=0.082, ω 21 (0)=0.192, ω 31 (0)=-40.418,ω 12 (0)=0,ω 22 (0)=0,ω 32 (0) = 0.

[0069] To verify the superiority of the control method of this invention, an optimal control method is used for comparison. The expression of the optimal control method is τ. aO =-31.623ζ1-20.151ζ2+72.718ζ3+13.155ζ4.

[0070] Figure 3 Let χ represent the position of the trolley under different control methods in case 1. c and angle θ p Comparing the graphs, it can be seen that when θ p When (0) = 160°, both the control method of this invention and the optimal control method can make the trolley stable at the desired position and keep the pendulum upright.

[0071] Figure 4 For case 1, the control input τ is the control input for different control methods. a As can be seen from the comparison figures, the control inputs of both the control method of this invention and the optimal control method are reasonable. Furthermore, the control input generated by the control method of this invention is a piecewise constant, which effectively reduces data transmission and saves communication resources of the control system.

[0072] Figure 5Let χ represent the position of the trolley under different control methods in case 2. c and angle θ p Comparing the graphs, it can be seen that when θ p When (0) = 130°, the optimal control method determines the trolley position χ. c and angle θ p The curve is divergent and cannot achieve stable control of the inverted pendulum system. However, the control method of this invention can still stabilize the trolley at the desired position and keep the pendulum upright.

[0073] Figure 6 For case 2, the control input τ is different from the control input τ for different control methods. a As can be seen from the comparison figures, the control input of the optimal control method is divergent and not a reasonable control input. However, the control input of the control method of this invention is still bounded and reasonable.

[0074] The simulation results above demonstrate that, for an event-driven inverted pendulum system, this invention achieves stability of the vehicle at the desired position and keeps the pendulum upright by utilizing the output of the differentiator and the backstepping method, without requiring linearization. Furthermore, the differentiator of this invention simplifies the control method structure, and the event-driven approach reduces data transmission, conserving communication resources of the control system.

[0075] It is understood that the above specific description of the present invention is only for illustrating the present invention and is not limited to the technical solutions described in the embodiments of the present invention. Those skilled in the art should understand that modifications or equivalent substitutions can still be made to the present invention to achieve the same technical effect; as long as the use needs are met, they are all within the protection scope of the present invention.

Claims

1. A method for controlling an inverted pendulum cart based on a differentiator, characterized in that, Includes the following steps: Step 1. Establish a mathematical model of the inverted pendulum with the cart's acceleration as the control input, and convert the mathematical model of the inverted pendulum into a state equation; Step 2. Introduce auxiliary variables to transform the obtained state equations into a non-strict feedback system; Step 3. Construct the differentiator based on the non-strict feedback system and the backstepping method; Step 4. Using the output of the differentiator and the event-driven method, design the control law for the inverted pendulum of the trolley to make the trolley stable in the desired position and keep the pendulum upright.

2. The method for controlling the inverted pendulum of a cart based on a differentiator according to claim 1, characterized in that: The mathematical model of the inverted pendulum in step 1, with the cart's acceleration as the control input, is as follows: Where, χ c θ represents the position of the car. p M represents the angle between the pendulum and the vertically downward direction. p For the mass of the pendulum, L p Let I be the length of the pendulum. p Let τ be the moment of inertia of the pendulum. a This represents the control input (car acceleration), where g is the acceleration due to gravity.

3. The method for controlling the inverted pendulum of a cart based on a differentiator according to claim 1, characterized in that: The state equation in step 1 is: in, ζ1=χ c -χ d , ζ3=θ p -π, π represents the mathematical constant pi, and χ represents the mathematical constant pi. d This indicates the desired position of the vehicle.

4. The method for controlling the inverted pendulum of a cart based on a differentiator according to claim 1, characterized in that: The non-strict feedback system in step 2 is: Among them, auxiliary variables Auxiliary variables Auxiliary variable r3 = -gtanζ3, auxiliary variable 5. The method for controlling the inverted pendulum of a cart based on a differentiator according to claim 1, characterized in that: The differentiator in step 3 is: Where, ε w1 ε w2 ε w3 b 11 b 12 b 21 b 22 b 31 b 32 ω is a design parameter and is a positive number. 11 ω 21 ω 31 ω represents the state of the differentiator. 12 ω 22 ω 32 Let α1, α2, and α3 represent the output of the differentiator, and let α1, α2, and α3 represent the virtual control laws. The expressions for α1, α2, and α3 are as follows: Where k1, k2, and k3 are design parameters and are positive numbers, e1 = r1, e2 = r2 - α1, and e3 = r3 - α2.

6. The method for controlling the inverted pendulum of a cart based on a differentiator according to claim 1, characterized in that: The control law for the inverted pendulum of the trolley in step 4 is: Among them, γ ma γ mi , k4 is a design parameter and is a positive number, t represents time, q represents a positive integer, and ψ m (t)=τ a -α4(t), e4=r4-α3.

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