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The stabilization of a Rotary Inverted Pendulum based on Lyapunov stability theorem is investigated in this paper. The key of designing control laws by Lyapunov control method is the construction of Lyapunov function. A logarithmic function is constructed as the Lyapunov function and is compared with the usual quadratic function theoretically. The comparative results show that the constructed logarithmic function has higher numerical accuracy and faster convergence speed than the usual quadratic function. On this basis, the control law of stabilizing Rotary Inverted Pendulum is designed based on the constructed logarithmic function by Lyapunov control method. The effectiveness of the designed control law is verified by experiments and is compared with LQR controller and the control law designed based on the quadratic function. Moreover, the system robustness is analyzed when the system parameters contain uncertainties under the designed control law.

The Rotary Inverted Pendulum, which was proposed by Furuta et al. [

For the swing-up and stabilizing control of Rotary Inverted Pendulum, a variety of control methods had been applied. Jose et al. [

Except these classical control techniques, several advanced control methods are also used to design controllers to stabilize the pendulum in upright vertical position. For example, Chen and Huang proposed an adaptive controller to bring the pendulum close to the upright position regardless of the various uncertainties and disturbances [

On the other hand, the inverted system can be underactuated and nonholonomic systems. Yue et al. used indirect adaptive fuzzy and sliding mode control approaches to achieve simultaneous velocity tracking and tilt angle stabilization for a nonholonomic and underactuated wheeled inverted pendulum vehicle [

Among various control methods, Lyapunov control method is a simple method of designing control laws and can be applied to linear systems and nonlinear systems. Particularly, the analytic expression of designed control laws can be obtained, which help to analyze the control performances and system characteristics. Thus we use Lyapunov control method to achieve stabilizing control of Rotary Invert Pendulum in this paper. The key point of Lyapunov control method is the construction of the Lyapunov function. Aguilar-Ibanez et al. [

The remaining of this paper is organized as follows. Section

Rotary Inverted Pendulum is composed of a rotating arm which is driven by a motor and a pendulum mounted on arm’s rim, whose structure is shown in Figure

Schematic diagram of Rotary Inverted Pendulum.

By applying Newton method or Lagrange method [

Descriptions of physical parameters.

Parameter | Description |
---|---|

| Moment of inertia at the load |

| Mass of pendulum arm |

| Rotating arm length |

| Length to pendulums center of mass |

| Gravitational constant |

| Viscous damping coefficient |

| Motor torque constant |

| System gear ratio |

| Back-EMF constant |

| Armature resistance |

| Motor efficiency |

| Gear efficiency |

This paper considers the problem of stabilizing Rotary Inverted Pendulum; thus

Obviously, vertically upward position (

Block diagram of states feedback.

In the rest of this paper, we focus on the system model (

For linear control systems, lots of methods can be used to design control laws. In this paper, the control laws of stabilizing Rotary Inverted Pendulum are designed by Lyapunov control method, whose motivations are as follows: (i) the procedure of designing control laws is simple; (ii) the analytic expression of control laws can be obtained; (iii) the control laws designed by Lyapunov method can guarantee the system stability. Based on Lyapunov stability theorem, the control laws can be designed as follows:

Construct a function

Calculate the first time derivative of

Design control laws to make

In classical literatures [

In order to obtain higher numerical accuracy and faster convergence speed, a logarithmic function

from which one can see that

Construct a function

where

If

which means

thereby,

which means

According to the above analysis,

Moreover, the constructed

In order to design the control laws under the condition of

To ensure

Similarly, the first-order time derivative of

Comparing (

The actual system is usually affected by the perturbations from environment or other sources, so that the system parameters contain uncertainties. It is reasonable to request that the designed control laws can resist the variation of parameters and the effect of perturbations to the greatest extent, so the controlled systems need stronger robustness under the control laws. In this subsection, we will investigate the system robustness under the designed control law

If the perturbations make

If the perturbations make

From (

In this section, the values of parameters will be given and the characteristics of Rotary Inverted Pendulum will be analyzed. Then, three experiments are used to investigate the control law

The initial position of Rotary Inverted Pendulum is set to

The initial position of Rotary Inverted Pendulum is set to

In order to investigate the system robustness under the designed control law, let the system parameters contain uncertainties; that is, the elements

The values of the parameters in Table

Values of physical parameters.

| 0.0033 |

| 0.1675 |

| 0.0077 |

| 2.6 |

| 0.125 |

| 9.81 |

| 70 |

| 0.69 |

| 0.215 |

| 0.0040 |

| 0.0077 |

| 0.90 |

The key of state feedback is to calculate the gain vector

The initial state of Rotary Inverted Pendulum is set to

Results of stability control for vertically downward initial state.

Control law

Feedback control

System state

The results in Figures

The initial state of Rotary Inverted Pendulum is set to

The control laws

Results of comparative experiments.

System state

System state

Control laws

Energies of control laws

From Figures

In Figure

Curve of

If

Curves of

From Figure

Control performances are better in the cases of

For the case of

For the case of

For the case of

For the case of

Therefore, when the different system parameters contain uncertainties, the system robustness is also different under the control laws designed by Lyapunov control method. The better system robustness is in the following two cases:

The actual

The actual

The control law of stabilizing Rotary Inverted Pendulum is designed based on Lyapunov stability theorem. A logarithmic function is constructed as the Lyapunov function, based on which the control law is designed by Lyapunov control method. In particular, the relationships between the constructed logarithmic function and the usual quadratic function in numerical value and convergence speed are analyzed in theory. The results show that the logarithmic function has higher numerical accuracy and faster convergence speed than the quadratic function, which is also verified in experiments. Moreover, the control law designed can also achieve the swing-up control of Rotary Inverted Pendulum. On this basis, the system robustness when different system parameters contain uncertainties is investigated. The further works can be considered as follows: (1) Lyapunov control method is applicable to nonlinear system, so the nonlinear mathematical model of Rotary Inverted Pendulum, which is more accurate to describe system characteristics than linear model, can be considered as the controlled system, and the stabilizing control laws can be designed by Lyapunov method; (2) the constructed logarithmic function in this paper can be used in other control methods, such as the logarithmic function which was selected as the performance function in optimal control; (3) the research results for the general classes of the underactuated and nonholonomic systems, such as [

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by 333 Talent Project of North University of China and Natural Science Foundation of North University of China (no. XJJ2016032).