#### Abstract

A proportional derivative controller with inverse dead-zone is proposed for the control of pendulum systems. The proposed method has the characteristic that the inverse dead-zone is cancelled with the pendulum dead-zone. Asymptotic stability of the proposed technique is guaranteed by the Lyapunov analysis. Simulations of two pendulum systems show the effectiveness of the proposed technique.

#### 1. Introduction

Nonsmooth nonlinear characteristics such as dead-zone, backlash, and hysteresis are common in actuators, sensors such as mechanical connections, hydraulic servovalves, and electric servomotors; they also appear in biomedical systems. Dead-zone is one of the most important nonsmooth nonlinearities in many industrial processes, which can severely limit the system performance, and its study has been drawing much interest in the control community for a long time [1].

There is some research about control systems. In [2], the stabilization of the inverted-car pendulum is presented. The stabilization of the Furuta pendulum is introduced in [3]. In [4], the dissipative control problem is investigated for a class of discrete time-varying systems. The distributed filtering problem for a class of nonlinear systems is considered in [5]. The recursive finite-horizon filtering problem for a class of nonlinear time-varying systems is addressed in [6, 7]. In [8], the authors present a solution to the problem of the quadratic mini-max regulator for polynomial uncertain systems. The problem of a two-player differential game affected by matched uncertainties with only the output measurement available for each player is considered by [9]. The stability analysis and control for a class of discrete time switched linear parameter-varying systems are concerned in [10]. The sliding mode control problem for uncertain nonlinear discrete-time stochastic systems with performance constraints is designed in [11]. In [12], the sliding mode control problem is considered for discrete-time systems. In [13], the description about the modelling and control of wind turbine system is addressed. A model to describe the dynamics of a homogeneous viscous fluid in an open pipe is introduced in [14]. In [15], the authors consider the problems of robust stability and control for a class of networked control systems with long-time delays. The control issue for a class of networked control systems with packet dropouts and time-varying delays is introduced in [16]. From the above studies, in [2, 3, 8, 9, 13, 14], the authors propose proportional derivative controls; however, none considers systems with dead-zone inputs.

There is some work about the control of systems with dead-zone inputs. In [17–22], the authors proposed the control of nonlinear systems with dead-zone inputs. Nevertheless, they do not research about the pendulum systems. The pendulum dynamic models have different structures with respect to the nonlinear systems addressed in the above papers; thus, a new design may be developed.

In this paper, a proportional derivative controller with inverse dead-zone is proposed for the control of pendulum systems with dead-zone inputs. One main contribution of this study is that the pendulum dynamic model is rewritten as a robotic dynamic model to satisfy a property, and later, the property is applied to guarantee the stability of the proposed controller.

The paper is organized as follows. In Section 2, the dynamic model of the robotic arm with dead-zone inputs is presented. In Section 3, the dynamic model of the pendulum systems with dead-zone inputs is presented. In Section 4, the proportional derivative controller with inverse dead-zone is introduced. In Section 5, the proposed method is used for the regulation of two pendulum systems. Section 6 presents conclusions and suggests future research directions.

#### 2. Dynamic Model of the Robotic Arms with Dead-Zone Inputs

The main concern of this section is to understand some concepts of robot dynamics. The equation of motion for the constrained robotic manipulator with degrees of freedom, considering the contact force and the constraints, is given in the joint space as follows: where denotes the joint angles or link displacements of the manipulator, is the robot inertia matrix which is symmetric and positive definite, contains the centripetal and Coriolis terms and are the gravity terms, and denotes the dead-zone output. The nonsymmetric dead-zone can be represented by where and are the right and left constant slopes for the dead-zone characteristic and and represent the right and left breakpoints. Note that is the input of the dead-zone and the control input of the global system.

Define the following two states as follows: where , for . Then (1) can be rewritten as where , , and are described in (1), the dead-zone is [17, 19, 20, 22] the parameters , , , and are described in (2), and is the control input of the system. Figure 1 shows the dead-zone [17].

*Property 1. *The inertia matrix is symmetric and positive definite; that is, [23–25]
where , are known positive scalar constants; .

*Property 2. *The centripetal and Coriolis matrix is skew-symmetric, that is, satisfies the following relationship [23–25]:
where .

The normal proportional derivative controller is
where and and and are positive definite, symmetric, and constant matrices.

#### 3. Dynamic Model of the Pendulum Systems with Dead-Zone Inputs

The dynamic model of the pendulum systems can be rewritten as the dynamic model of the robotic arms; however Property 2 is not directly satisfied. Pendulum dynamic models are rewritten as the robotic dynamic models because in this study, if the above sentence is true, Property 2 of the robotic systems can be used to guarantee the stability of the controller applied to the pendulum systems. The following lemmas let to modify the Property 2 for its application in the pendulum systems.

Lemma 1. *A pendulum model can be rewritten as a robotic arm model (4). Nevertheless, it cannot satisfy Property 2.*

*Proof. *Consider for the pendulum systems in (4); it gives
where
and , , and are selected from the pendulum dynamic model, and and are defined in (3). Consequently,

Lemma 2. *Pendulum model (4) can be rewritten as follows:
**
where
**
the input is given by (5), , , , , , , and are defined in (3), and the following modified property is satisfied:
*

*Proof. *Consider for the pendulum systems in (4); it gives
Consequently, a change of variables is used as follows:
where the elements are given in (12). Note that the elements of and are selected such that the property (14) is satisfied.

In the following section, a stable controller for the pendulum systems will be designed.

#### 4. Proportional Derivative Control with Inverse Dead-Zone

The regulation case is considered in this study; that is, the desired velocity is . The proportional derivative control with inverse dead-zone is as follows: where the parameters , , , and are defined as in (2), and the auxiliary proportional derivative control is where is the tracking error, and are defined in (3), is the desired position, , are positive definite, is an approximation of , and are the nonlinear terms of (12). Figure 2 shows the inverse dead-zone [17, 22] and Figure 3 shows the proposed controller denoted as PDDZ. It is considered that the approximation error is bounded as

Now the convergence of the closed-loop system is discussed.

Theorem 3. *The error of the closed-loop system with the proportional derivative control (17) and (18) for the pendulum systems with dead-zone inputs (12) and (5) is asymptotically stable, and the error of the velocity parameter will converge to
**
where is the final time, , , , and . *

* Proof. *The proposed Lyapunov function is
Substituting (17) and (18) into (12) and (5) the closed-loop system is as follows:
Using the fact , the derivative of (21) is
where and . Substituting (22) into (23) gives
Using (14), (17), and , it gives
where . Thus, the error is asymptotically stable [26]. Integrating (25) from to yields
If , then ; (20) is established.

*Remark 4. *The proposed controller is used for the regulation case; that is, the desired velocity is . The general case when is not considered in this research.

#### 5. Simulations

In this section, the proportional derivative control with inverse dead-zone denoted as PDDZ will be compared with the proportional derivative control with gravity compensation of [23] denoted by PD for the control of two pendulum systems with dead-zone inputs. In this paper, the root mean square error (RMSE) [1, 26, 27] is used for the comparison results and it is given as where or .

##### 5.1. Example 1

Consider the inverted-car pendulum [2] of Figure 4.

Inverted-car pendulum is written as (1) and it is detailed as follows:
where , , , , the other parameters of are zero, , and the other parameter of is zero. is the sine function, is the cosine function, kgm^{2} is the pendulum inertia, kg is the mass of the car, kg is the pendulum mass, m is the pendulum length, is the angle with respect of the axis, is the motion force of the car, is the motion distance of the car, and m/s^{2} is the constant acceleration due to gravity. It can be proven that Property 2 of (7) is not satisfied.

Inverted-car pendulum is written as (12), and it is detailed as follows: where , , , and the other parameters of are zero, , ; therefore, it can be proven that the property of the lemma of (14) is satisfied.

PDDZ is given by (17) and (18) as with parameters , , , and ,with. Conditions given in (20) , , and are satisfied; consequently, the error of the closed-loop dynamics of the PDDZ applied for pendulum systems is guaranteed to be asymptotically stable.

PD is given by [23] as with parameters , , and .

Comparison results for the control functions are shown in Figure 5, position states are shown in Figure 6, and comparison results for the controller errors are shown in Figure 7. Comparison of the square norm of the velocity errors of (20) for the controllers is presented in Figure 8. From the theorem of (20), will converge to zero for the PDDZ. Table 1 shows the RMSE results using (27).

**(a)**

**(b)**

**(a)**

**(b)**

The most important variable to control is the pendulum angle , and this variable may reach zero even if it starts with other value as in this example. Note that the PD technique requires the bigger gains than the PDDZ method to obtain satisfactory results. From Figures 5, 6, and 7, it can be seen that the PDDZ improves the PD because the signal of the plant for the first follows better the desired signal than the second and in the first the inputs are smaller than in the second. From Figure 8, it is shown that the PDDZ improves the PD because the velocity error presented by the first is smaller than that presented by the second. From Table 1, it can be shown that the PDDZ achieves better accuracy when compared with the PD because the RMSE is smaller for the first than for the second.

##### 5.2. Example 2

Consider the Furuta pendulum [3, 27] of the Figure 9.

Furuta pendulum is written as (1) and it is detailed as follows:
where , , , , , , , and the other parameter of is zero. is the sine function, is the cosine function, Kgm^{2} is the arm inertia, kgm^{2} is the pendulum inertia, kg is the arm mass, kg is the pendulum mass, m is the arm length, m is the pendulum length, isthe arm angle, is the pendulum angle, is the motion torque of the arm, and m/s^{2} is the constant acceleration due to gravity. It can be proven that Property 2 of (7) is not satisfied.

Furuta pendulum is written as (12), and it is detailed as follows: where , , , , , , and the other parameter of is zero, , ; therefore, it can be proven that the property of the lemma of (14) is satisfied.

PDDZ is given by (17) and (18) as with parameters , , , ,with. Conditions given in (20) , , and are satisfied; consequently, the error of the closed-loop dynamics of the PDDZ applied for pendulum systems is guaranteed to be asymptotically stable.

PD is given by [23] as with parameters , , and .

Comparison results for the control functions are shown in Figure 10, position states are shown in Figure 11, and comparison results for the controller errors are shown in Figure 12. Comparison of the square norm of the velocity errors of (20) for the controllers is presented in Figure 13. From the theorem of (20), will converge to zero for the PDDZ. Table 2 shows the RMSE results using (27).

**(a)**

**(b)**

**(a)**

**(b)**

The most important variable to control is the pendulum angle , and this variable may reach zero even if it starts with other value as in this example. Note that the PD technique requires the bigger gains than the PDDZ method to obtain satisfactory results. From Figures 10, 11, and 12, it can be seen that the PDDZ improves the PD because the signal of the plant for the first follows better the desired signal than the second and in the first the inputs are smaller than in the second. From Figure 13, it is shown that the PDDZ improves the PD because the velocity error presented by the first is smaller than that presented by the second. From Table 2, it can be shown that the PDDZ achieves better accuracy when compared with the PD because the RMSE is smaller for the first than for the second.

#### 6. Conclusion

In this research, a proportional derivative control with inverse dead-zone for pendulum systems with dead-zone inputs is presented. The simulations showed that the proposed technique achieves better performance when compared with the proportional derivative control with gravity compensation for the regulation of two pendulum systems, and the results illustrate the viability, efficiency, and the potential of the approach especially important in pendulum systems. As a future research, the proposed study will be improved considering that some parameters of the controller are unknown [28–32], or it will consider the communication delays and packet dropout.

#### Conflict of Interests

The authors declare no conflict of interests about all the aspects related to this paper.

#### Acknowledgments

The authors are grateful to the editors and the reviewers for their valuable comments and insightful suggestions, which helped to improve this research significantly. The authors thank the Secretaría de Investigación y Posgrado, the Comisión de Operación y Fomento de Actividades Académicas del IPN, and Consejo Nacional de Ciencia y Tecnología for their help in this research.