Shock and Vibration

Volume 2017 (2017), Article ID 8340510, 14 pages

https://doi.org/10.1155/2017/8340510

## SDRE Control Applied to the Wheel Speed of a Compressed Air Engine with Crank-Connecting-Rod Mechanism

^{1}Sao Paulo State University, Bauru, SP, Brazil^{2}Federal University of Technology-Parana, Ponta Grossa, PR, Brazil^{3}Aeronautics Technological Institute, São José dos Campos, SP, Brazil

Correspondence should be addressed to Jose Manoel Balthazar

Received 6 March 2017; Revised 11 May 2017; Accepted 23 May 2017; Published 15 August 2017

Academic Editor: Mario Terzo

Copyright © 2017 Alexandre de Castro Alves et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Renewable energy sources for vehicles have been the motivation of many researches around the world. The reduction of fossil fuels deposits and increase of the pollution in cities bring the need of more efficient and cleaner energy sources. In this way, this work will present the application of a compressed air engine applied to a bicycle. The engine is composed of two pneumatic cylinders connected to the bicycle wheel through a crank-connecting-rod mechanism. In order to control the velocity of the bicycle, a strategy of control composed of two controls was implemented: a feedback and a feedforward control. For feedback control, the State-Dependent Riccati Equation (SDRE) control and also a proportional-derivative (PD) control are considered, considering three cases for velocity bicycle variation: 10 km/h, 20 km/h, and 30 km/h. The equations of motion of the system were obtained through the Lagrangian energy method. Numerical simulations were performed in order to analyze the dynamics of the system and the efficiency of the controllers.

#### 1. Introduction

The use of energy has become every time more intense through the society in the last decades; however, most of this energy comes from nonrenewable resources like oil, natural gas, and coils, that is, fossil fuels in general.

In order to convert these kinds of energy sources into energy of movement, especially in vehicles, the main mechanism of engines, which is responsible for this transformation is the crank-connecting-rod.

The crank-connecting-rod mechanism has been windily studied with the objective of increasing the engine performance. Some of these studies consider the movement in function of the system’s geometry, noises, and vibrations induced by this mechanism [1–3].

Due to the limited current sources of fossil fuels, because of the world demand, and the need of improving vehicular performance, the study of new energy matrices like Hybrid Renewable Energy Systems (HRES) has been of great interest to numerous researchers [4, 5].

In this context, the compressed air systems become a very interesting alternative. The compressed air systems allow the energy recovery in the form of pressure, which can be applied as an extra energy source to the combustion engines, which characterizes a hybrid engine [6].

Pneumatic motors are very interesting in these applications because of their high force in relation to their masses [7]. In this way, many authors have been interested in the research of the application of compressed air engines in small vehicles, for example, motorcycles [8–10].

Therefore, this work proposes the application of a pneumatic motor to a bicycle as a main force generator. The pneumatic engine is composed of two pneumatic cylinders connected through a crank-connecting-rod mechanism to the bicycle wheel. Hence, the force generated by the compressed air is converted into angular movement of the wheel and into linear movement of the bicycle.

The crank-connecting-rod mechanism converts the linear force of the pneumatic cylinders into torque applied to the wheel considered as a single-degree-of-freedom system. In addition, the dynamic model is nonlinear because of the complexity of variables. Taking into account the fact that the velocity of the bicycle will be controlled, this work presents the application of the SDRE control and a PD control.

The SDRE control technique is a suboptimal control, which searches for local stabilities of a system [11]. The advantage of this control technique is that it does not cancel possible benefits provided by nonlinearities of the system, due to the fact that it is not necessary to linearize the system when applying this technique [12–16]. Among successful techniques implemented in real applications, there is the classical proportional-derivative (PD) controllers [17–19].

The next sections will show the mathematical modelling of the system composed of the wheel and the pneumatic engine composed of the crank-connecting-rod and the pneumatic cylinders. The SDRE and PD controllers will be presented and numerical simulations will be performed in order to analyze the system dynamics.

The remainder of this paper is organized as follows. Section 2 presents the literature related to the engineering problem and its mathematical modelling through subsections with the equations of motion and the control designs for SDRE and PD. In Section 3 the parameters and numerical results of the simulations with the results of applied controls SDRE and PD are presented with the discussions of the errors. Section 4 presents the final conclusions for the study presented in this article. In addition, posttextual elements that make up the structure of the work are described. Acknowledgments and references for literature are presented in sequence. Finally, the technical terms are presented through a glossary section with a large volume of presented terms.

#### 2. The Engineering Problem Design and Mathematical Modelling

By applying compressed air force () to the crank-connecting-rod mechanism, it changes the position of the connecting-rod and generates the angular displacement of the crank. It is considered that the links of the system have their mass distribution proportional to a mass concentered in the Center of Gravity (CG) of each link.

Figure 1 presents a schematic draw of the crank-connecting-rod mechanism. This mechanism has restrictions to move in the vertical direction because of the cylinder but can translate free in direction. This vertical displacement restriction enables the system to move in relation to its length (), which generates the angular movement because of its connection to the connecting-rod.