Proportional Derivative Control with Inverse Dead-Zone for Pendulum Systems

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 𝐻_∞ 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 𝐻_∞ 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 𝐻₂ 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 𝐻_∞ control for a class of networked control systems with long-time delays. The 𝐻_∞ 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 𝑛 degrees of freedom, considering the contact force and the constraints, is given in the joint space as follows: 𝑀𝑞𝑞̈+𝐶𝑞,𝑞̇𝑞̇+𝐺𝑞=𝜏, (1) where 𝑞∈ℜ^𝑛×1 denotes the joint angles or link displacements of the manipulator, 𝑀(𝑞)∈ℜ^𝑛×𝑛 is the robot inertia matrix which is symmetric and positive definite, 𝐶(𝑞,𝑞)̇∈ℜ^𝑛×𝑛 contains the centripetal and Coriolis terms and 𝐺(𝑞) are the gravity terms, and 𝜏 denotes the dead-zone output. The nonsymmetric dead-zone can be represented by (𝑣)=𝑚𝜏=DZ^𝑟𝑣−𝑏_𝑟𝑣≥𝑏_𝑟0𝑏_𝑙<𝑣<𝑏_𝑟𝑚_𝑙𝑣−𝑏_𝑙𝑣≤𝑏_𝑙, (2) where 𝑚_𝑟 and 𝑚_𝑙 are the right and left constant slopes for the dead-zone characteristic and 𝑏_𝑟 and 𝑏_𝑙 represent the right and left breakpoints. Note that 𝑣 is the input of the dead-zone and the control input of the global system.

Define the following two states as follows: 𝑥₁=𝑞∈ℜ^𝑛×1,𝑥₂=𝑞̇∈ℜ^𝑛×1,𝑢=𝜏∈ℜ^𝑛×1, (3) where 𝑥₁=[𝑥₁₁𝑥₁₂]^𝑇=[𝑞₁𝑞₂]^𝑇, 𝑥₂=[𝑥₂₁𝑥₂₂]^𝑇=[𝑞̇₁𝑞̇₂]^𝑇 for 𝑛=2. Then (1) can be rewritten as 𝑥̇₁=𝑥₂,𝑀𝑥₁𝑥̇₂+𝐶𝑥₁,𝑥₂𝑥₂+𝐺𝑥₁=𝑢, (4) where 𝑀(𝑥₁), 𝐶(𝑥₁,𝑥₂), and 𝐺(𝑥₁) are described in (1), the dead-zone 𝑢 is [17, 19, 20, 22] (𝑣)=𝑚𝑢=DZ^𝑟𝑣−𝑏_𝑟𝑣≥𝑏_𝑟0𝑏_𝑙<𝑣<𝑏_𝑟𝑚_𝑙𝑣−𝑏_𝑙𝑣≤𝑏_𝑙, (5) the parameters 𝑚_𝑟, 𝑚_𝑙, 𝑏_𝑟, and 𝑏_𝑙 are described in (2), and 𝑣 is the control input of the system. Figure 1 shows the dead-zone [17].

Figure 1 

The dead-zone.

Property 1. The inertia matrix is symmetric and positive definite; that is, [23–25] 𝑚₁|𝑥|²≤𝑥^𝑇𝑀𝑥₁𝑥≤𝑚₂|𝑥|², (6) where 𝑚₁, 𝑚₂ are known positive scalar constants; 𝑥=[𝑥₁,𝑥₂]^𝑇.

Property 2. The centripetal and Coriolis matrix is skew-symmetric, that is, satisfies the following relationship [23–25]: 𝑥^𝑇𝑀̇𝑥₁−2𝐶𝑥₁,𝑥₂𝑥=0, (7) where 𝑥=[𝑥₁,𝑥₂]^𝑇.
The normal proportional derivative controller is 𝑢=−𝐾_𝑝𝑥₁−𝐾_𝑑𝑥₂, (8) where 𝑥₁=𝑥₁−𝑥^𝑑₁ and 𝑥₂=𝑥₂−𝑥^𝑑₂ and 𝐾_𝑝 and 𝐾_𝑑 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 𝑛=2 for the pendulum systems in (4); it gives 𝑥̇₁=𝑥₂,𝑀𝑥₁𝑥̇₂+𝐶𝑥₁,𝑥₂𝑥₂+𝐺𝑥₁=𝑢, (9) where 𝑀𝑥₁=𝑚₁₁𝑚₁₂𝑚₂₁𝑚₂₂,𝐶𝑥₁,𝑥₂=𝑐₁₁𝑐₁₂𝑐₂₁𝑐₂₂,𝐺𝑥₁=𝑔₁𝑔₂ (10) and 𝑀(𝑥₁), 𝐶(𝑥₁,𝑥₂), and 𝐺(𝑥₁) are selected from the pendulum dynamic model, and 𝑥₁ and 𝑥₂ are defined in (3). Consequently, 𝑥^𝑇𝑀̇𝑥₁−2𝐶𝑥₁,𝑥₂𝑥≠0. (11)

Lemma 2. Pendulum model (4) can be rewritten as follows: 𝑥̇₁=𝑥₂,𝑀𝑥₁𝑥̇₂=−𝐶𝑥₁,𝑥₂𝑥₂−𝑁𝑥₁,𝑥₂+𝑢, (12) where 𝑀𝑥₁=𝑚₁₁𝑚₁₂𝑚₂₁𝑚₂₂,𝐶𝑥₁,𝑥₂=𝑐̂₁₁𝑐̂₁₂𝑐̂₂₁𝑐̂₂₂,𝑁𝑥₁,𝑥₂=𝑛₁𝑛₂ (13) the input 𝑢 is given by (5), 𝑐₁₁=𝑐̂₁₁+𝑐₁₁, 𝑐₁₂=𝑐̂₁₂+𝑐₁₂, 𝑐₂₁=𝑐̂₂₁+𝑐₂₁, 𝑐₂₂=𝑐̂₂₂+𝑐₂₂, 𝑛₁=𝑔₁+𝑐₁₁𝑥₂₁+𝑐₁₂𝑥₂₂, 𝑛₂=𝑔₂+𝑐₂₁𝑥₂₁+𝑐₂₂𝑥₂₂, 𝑥₁ and 𝑥₂ are defined in (3), and the following modified property is satisfied: 𝑥^𝑇𝑀̇𝑥₁−2𝐶𝑥₁,𝑥₂𝑥=0. (14)

Proof. Consider 𝑛=2 for the pendulum systems in (4); it gives 𝑀𝑥₁𝑥̇₂+𝐶𝑥₁,𝑥₂𝑥₂+𝐺𝑥₁𝑀𝑥=𝑢,₁𝑥̇₂=−𝐶𝑥₁,𝑥₂𝑥₂−𝐺𝑥₁+𝑢. (15) Consequently, a change of variables is used as follows: 𝑀𝑥₁𝑥̇₂=−𝐶𝑥₁,𝑥₂𝑥₂−𝑁𝑥₁,𝑥₂𝑥+𝑢,_𝑇𝑀̇𝑥₁−2𝐶𝑥₁,𝑥₂𝑥=0, (16) where the elements are given in (12). Note that the elements of 𝐶(𝑥₁,𝑥₂) and 𝑁(𝑥₁,𝑥₂) 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 𝑥^𝑑₂=0. The proportional derivative control with inverse dead-zone 𝑣 is as follows: 𝑣=DZ⁻¹𝑢_pd=1𝑚_𝑟𝑢_pd+𝑏_𝑟𝑢_pd>00𝑢_pd1=0𝑚_𝑙𝑢_pd+𝑏_𝑙𝑢_pd<0, (17) where the parameters 𝑚_𝑟, 𝑚_𝑙, 𝑏_𝑟, and 𝑏_𝑙 are defined as in (2), and the auxiliary proportional derivative control is 𝑢_pd=−𝐾_𝑝𝑥₁−𝐾_𝑑𝑥₂+𝑁−𝐾sign𝑥₂, (18) where 𝑥₁=𝑥₁−𝑥^𝑑₁ is the tracking error, 𝑥₁ and 𝑥₂=𝑥₂ are defined in (3), 𝑥^𝑑₁=[𝑥^𝑑₁₁𝑥^𝑑₁₂]∈ℜ^𝑛×1 is the desired position, 𝐾_𝑝, 𝐾_𝑑 are positive definite, 𝑁∈ℜ^𝑛×1 is an approximation of 𝑁(𝑥₁,𝑥₂), and 𝑁(𝑥₁,𝑥₂)∈ℜ^𝑛×1 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 𝑁=𝑁−𝑁(𝑥₁,𝑥₂) is bounded as 𝑁≤𝑁. (19)

Figure 2 

The inverse dead-zone.

Figure 3 

The proposed controller.

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 𝑥₂ will converge to limsup_𝑇→∞𝑥₂²=0, (20) where 𝑇 is the final time, 𝑥₂=𝑥₂, 𝑁≤𝐾, 𝐾_𝑝>0, and 𝐾_𝑑>0.

Proof. The proposed Lyapunov function is 𝑉₁=12𝑥^𝑇₂𝑀𝑥₁𝑥₂+12𝑥^𝑇₁𝐾_𝑝𝑥₁. (21) Substituting (17) and (18) into (12) and (5) the closed-loop system is as follows: 𝑀𝑥₁𝑥̇₂=−𝐾_𝑝𝑥₁−𝐾_𝑑𝑥₂+𝑁−𝐾sign𝑥₂−𝐶𝑥₁,𝑥₂𝑥₂. (22) Using the fact 𝑥₂=𝑥₂, the derivative of (21) is 𝑉̇₁=𝑥^𝑇₂𝑀𝑥₁𝑥̇₂+12𝑥^𝑇₂𝑀̇𝑥₁𝑥₂+𝑥^𝑇₂𝐾_𝑝𝑥₁, (23) where ̇𝑥₁=𝑥̇₁−𝑥̇^𝑑₁=𝑥₂−𝑥^𝑑₂=𝑥₂=𝑥₂ and ̇𝑥₂=𝑥̇₂. Substituting (22) into (23) gives 𝑉̇₁=−𝑥^𝑇₂𝐾_𝑑𝑥₂+𝑥^𝑇₂𝑁−𝐾sign𝑥₂+12𝑥^𝑇₂𝑀̇𝑥₁−2𝐶𝑥₁,𝑥₂𝑥^𝑇₂. (24) Using (14), (17), and |𝑥₂|^𝑇=𝑥^𝑇₂sign(𝑥₂), it gives 𝑉̇₁≤−𝑥^𝑇₂𝐾_𝑑𝑥₂+𝑥₂^𝑇𝑁−𝑥₂^𝑇𝑉̇𝐾,₁≤−𝑥^𝑇₂𝐾_𝑑𝑥₂, (25) where 𝑁≤𝐾. Thus, the error is asymptotically stable [26]. Integrating (25) from 0 to 𝑇 yields ^𝑇₀𝑥^𝑇₂𝐾_𝑑𝑥₂𝑑𝑡≤𝑉_1,0−𝑉_1,𝑇≤𝑉_1,0,1𝑇^𝑇₀𝑥^𝑇₂𝐾_𝑑𝑥₂1𝑑𝑡≤𝑇𝑉_1,0,limsup_𝑇→∞1𝑇^𝑇₀𝑥^𝑇₂𝐾_𝑑𝑥₂𝑑𝑡≤𝑉_1,0limsup_𝑇→∞1𝑇=0. (26) If 𝑇→∞, then ‖𝑥₂‖²=0; (20) is established.

Remark 4. The proposed controller is used for the regulation case; that is, the desired velocity is 𝑥^𝑑₂=0. The general case when 𝑥^𝑑₂≠0 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 1RMSE=𝑇^𝑇₀𝑥²𝑑𝑡^1/2, (27) where 𝑥²=𝑥²₁₂ or 𝑥²=𝑢²₁.

5.1. Example  1

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

Figure 4 

Inverted-car pendulum.

Inverted-car pendulum is written as (1) and it is detailed as follows: 𝑀𝑥₁=𝑚₁₁𝑚₁₂𝑚₂₁𝑚₂₂,𝐶𝑥₁,𝑥₂=𝑐₁₁𝑐₁₂𝑐₂₁𝑐₂₂,𝐺𝑥₁=𝑔₁𝑔₂, (28) where 𝑚₁₁=𝑀+𝑚, 𝑚₁₂=𝑚₂₁=𝑚𝑙cos(𝑥₁₂), 𝑚₃₃=𝐼+𝑚𝑙², 𝑐₁₂=−𝑚𝑙sin(𝑥₁₂)𝑥₂₂, the other parameters of 𝐶(𝑥₁,𝑥₂) are zero, 𝑔₂=−𝑚𝑔𝑙sin(𝑥₁₂), and the other parameter of 𝐺(𝑥₁) is zero. sin(⋅) is the sine function, cos(⋅)is the cosine function,𝐼=0.5 kgm² is the pendulum inertia,𝑀=0.136729 kg is the mass of the car, 𝑚=0.040691 kg is the pendulum mass, 𝑙=0.15 m is the pendulum length, 𝜃₁₂ is the angle with respect of the 𝑦 axis, 𝑢=𝐹 is the motion force of the car, 𝑥₁₁=𝑥 is the motion distance of the car, and 𝑔=9.81 m/s² 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: 𝑀𝑥₁=𝑚₁₁𝑚₁₂𝑚₂₁𝑚₂₂,𝐶𝑥₁,𝑥₂=𝑐̂₁₁𝑐̂₁₂𝑐̂₂₁𝑐̂₂₂,𝑁𝑥₁,𝑥₂=𝑛₁𝑛₂, (29) where 𝑚₁₁=𝑀+𝑚, 𝑚₁₂=𝑚₂₁=𝑚𝑙cos(𝑥₁₂), 𝑚₃₃=𝐼+𝑚𝑙², 𝑐̂₁₂=𝑐̂₂₁=−(1/2)𝑚𝑙sin(𝑥₁₂)𝑥₂₂ and the other parameters of 𝐶(𝑥₁,𝑥₂) are zero, 𝑛₁=−(1/2)𝑚𝑙sin(𝑥₁₂)𝑥₂₂, 𝑛₂=−𝑚𝑔𝑙sin(𝑥₁₂)+(1/2)𝑚𝑙sin(𝑥₁₂)𝑥₂₂; therefore, it can be proven that the property of the lemma of (14) is satisfied.

PDDZ is given by (17) and (18) as 𝑢_pd=−𝐾_𝑝𝑥₁₂−𝐾_𝑑𝑥₂₂+𝑛₁−𝐾sign(𝑥₂₂) with parameters 𝐾_𝑝=200, 𝐾_𝑑=20, 𝑁=𝑛₁, and 𝑛₁=0.01,with𝐾=0.01. Conditions given in (20) 𝑁≤𝐾, 𝐾_𝑝>0, and 𝐾_𝑑>0 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 𝑢_pd=−𝐾_𝑝𝑥₁₂−𝐾_𝑑𝑥₂₂+𝑔₂ with parameters 𝐾_𝑝=400, 𝐾_𝑑=20, and 𝐺(𝑥₁)=𝑔₁.

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 ‖𝑥₂‖² of (20) for the controllers is presented in Figure 8. From the theorem of (20), ‖𝑥₂‖² will converge to zero for the PDDZ. Table 1 shows the RMSE results using (27).

Table 1 

Error results for Example 1.

Figure 5 

Input for Example  1.

(a)

(b)

Figure 6 

Position states for Example  1.

Figure 7 

Position state error for Example  1.

(a)

(b)

Figure 8 

Velocity errors for Example  1.

The most important variable to control is the pendulum angle 𝑥₁₂=𝜃, 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 ‖𝑥₂‖² 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.

Figure 9 

Furuta pendulum.

Furuta pendulum is written as (1) and it is detailed as follows: 𝑀𝑥₁=𝑚₁₁𝑚₁₂𝑚₂₁𝑚₂₂,𝐶𝑥₁,𝑥₂=𝑐₁₁𝑐₁₂𝑐₂₁𝑐₂₂,𝐺𝑥₁=𝑔₁𝑔₂, (30) where 𝑚₁₁=𝐼₀+𝑚₁(𝐿²₀+𝑙²₁sin²(𝑥₁₂)), 𝑚₁₂=𝑚₂₁=𝑚₁𝑙₁𝐿₀cos(𝑥₁₂), 𝑚₃₃=𝐽₁+𝑚₁𝑙²₁, 𝑐₁₁=𝑚₁𝑙²₁sin(2𝑥₁₂)𝑥₂₂, 𝑐₁₂=−𝑚₁𝑙₁𝐿₀sin(𝑥₁₂)𝑥₂₂, 𝑐₂₁=−𝑚₁𝑙²₁sin(𝑥₁₂)cos(𝑥₁₂)𝑥₂₁, 𝑔₂=−𝑚₁𝑔𝑙₁sin(𝑥₁₂), and the other parameter of 𝐺(𝑥₁) is zero. sin(⋅) is the sine function, cos(⋅) is the cosine function,𝐼₀=0.5 Kgm² is the arm inertia, 𝐽₁=0.5 kgm² is the pendulum inertia, 𝑚₂=0.34 kg is the arm mass, 𝑚₁=0.24 kg is the pendulum mass, 𝐿₀=0.293 m is the arm length, 𝑙₁=0.28 m is the pendulum length, 𝑥₁₁=𝜃₀isthe arm angle,𝑥₁₂=𝜃₁ is the pendulum angle, 𝑢=𝐹 is the motion torque of the arm, and 𝑔=9.81 m/s² 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: 𝑀𝑥₁=𝑚₁₁𝑚₁₂𝑚₂₁𝑚₂₂,𝐶𝑥₁,𝑥₂=𝑐̂₁₁𝑐̂₁₂𝑐̂₂₁𝑐̂₂₂,𝑁𝑥₁,𝑥₂=𝑛₁𝑛₂, (31) where 𝑚₁₁=𝑀+𝑚, 𝑚₁₂=𝑚₂₁=𝑚𝑙cos(𝑥₁₂), 𝑚₃₃=𝐼+𝑚𝑙², 𝑐̂₁₁=𝑚₁𝑙²₁sin(𝑥₁₂)cos(𝑥₁₂)𝑥₂₂, 𝑐̂₁₂=−(1/2)𝑚₁𝑙₁𝐿₀sin(𝑥₁₂)𝑥₂₂, 𝑐̂₂₁=−(1/2)𝑚₁𝑙²₁sin(𝑥₁₂)cos(𝑥₁₂)𝑥₂₂, and the other parameter of 𝐶(𝑥₁,𝑥₂) is zero, 𝑛₁=𝑚₁𝑙²₁sin(𝑥₁₂)cos(𝑥₁₂)𝑥₂₂−(1/2)𝑚₁𝑙₁𝐿₀sin(𝑥₁₂)𝑥₂₂, 𝑛₂=−𝑚₁𝑔𝑙₁sin(𝑥₁₂)−(1/2)𝑚₁𝑙²₁sin(𝑥₁₂)cos(𝑥₁₂)(𝑥₂₁−𝑥₂₂); therefore, it can be proven that the property of the lemma of (14) is satisfied.

PDDZ is given by (17) and (18) as 𝑢_pd=−𝐾_𝑝𝑥₁₂−𝐾_𝑑𝑥₂₂+𝑛₁−𝐾sign(𝑥₂₂) with parameters 𝐾_𝑝=200, 𝐾_𝑑=20, 𝑁=𝑛₁, 𝑛₁=0.01,with𝐾=0.01. Conditions given in (20) 𝑁≤𝐾, 𝐾_𝑝>0, and 𝐾_𝑑>0 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 𝑢_pd=−𝐾_𝑝𝑥₁₂−𝐾_𝑑𝑥₂₂+𝑔₂ with parameters 𝐾_𝑝=400, 𝐾_𝑑=20, and 𝐺(𝑥₁)=𝑔₂.

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 ‖𝑥₂‖² of (20) for the controllers is presented in Figure 13. From the theorem of (20), ‖𝑥₂‖² will converge to zero for the PDDZ. Table 2 shows the RMSE results using (27).

Table 2 

Error results for Example 2.

Figure 10 

Input for Example  2.

(a)

(b)

Figure 11 

Position states for Example  2.

Figure 12 

Position state error for Example  2.

(a)

(b)

Figure 13 

Velocity errors for Example  2.

The most important variable to control is the pendulum angle 𝑥₁₂=𝜃₁, 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 ‖𝑥₂‖² 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.