Journal of Control Science and Engineering

Volume 2018, Article ID 6259049, 11 pages

https://doi.org/10.1155/2018/6259049

## PID Control for Electric Vehicles Subject to Control and Speed Signal Constraints

^{1}Department of Automation and Computing, CT, Federal University of Rio Grande do Norte, 59078-970 Natal, RN, Brazil^{2}Master of Engineering in Oil and Gas, Potiguar University, 59054-180 Natal, Brazil

Correspondence should be addressed to André Felipe O. de A. Dantas; rb.pnu@satnad.erdna

Received 1 June 2018; Accepted 19 July 2018; Published 1 August 2018

Academic Editor: Darong Huang

Copyright © 2018 Amanda Danielle O. da S. Dantas 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

A PID control for electric vehicles subject to input armature voltage and angular velocity signal constraints is proposed. A PID controller for a vehicle DC motor with a separately excited field winding considering the field current constant was tuned using controlled invariant set and multiparametric programming concepts to consider the physical motor constraints as angular velocity and input armature voltage. Additionally, the integral of the error, derivative of the error constraints, and were considered in the proposed algorithm as tuning parameters to analyze the DC motor dynamic behaviors. The results showed that the proposed algorithm can be used to generate control actions taking into account the armature voltage and angular velocity limits. Also, results demonstrate that a controller subject to constraints can improve the electric vehicle DC motor dynamic; and at the same time it protects the motor from overvoltage.

#### 1. Introduction

Some researchers state that electric vehicles can be one of the renewable solutions to energy and environmental problems caused by oil based vehicles due to the various advantages associated with the use of electric energy, such as low cost [1–5]. In this scenario, direct current (DC) motors are one of the most used actuators in the construction of electric vehicles [6]. This type of actuator has numerous advantages, such as low cost, high reliability, easy maintenance, and simple control for both speed and position variables with PID being one of the main used controllers [7, 8].

The Proportional Integral Derivative Controller (PID) has been widely used for most industrial process, due to its simplicity and effectiveness in control [9, 10]. This type of controller is commonly used in level, flow, temperature, and vehicular systems, as well as electric motors [10–12]. In addition, the design of the PID controller is considered easy to implement, since it is only necessary to tune three parameters , , and and tuning methods can be performed automatically [13]. Some of most used PID tuning methods in control engineering literature are Ziegler and Nichols, Cohen and Conn, Relay method, and Relatus Apparatus. These methods are effective and achieve excellent results when controlling unconstrained monovariable systems although some of these ones are also applicable for multivariate systems [9].

Despite all advantages of PID controllers, most of tuning methods do not consider the process constraints. Thus, many researches tried to consider these conditions in the control loop using antireset windup, control signal saturation, and integrator constraints. These techniques aim to limit the control action to suit the controller to constrained processes [14, 15]. However, these methods still do not take into account the constraints while tuning the controller and, therefore, such methods are not totally appropriate; i.e., they do not lead to an optimal control signal for the constrained system.

In order to solve the optimal constrained problem many controllers are being proposed. One solution consists of maintaining the system trajectory within -contractive controlled invariant polyhedron set defined in the state space. This set contains all states for which there is a state feedback control law that maintains the trajectory of the dynamic system within [16, 17]. The state feedback control law can be calculated online, from the solution of a linear programming (LP) problem, or offline, by solving a multiparametric linear programming problem (mp-LP) [17]. This optimal solution represents an explicit PWA (PieceWise Affine) state feedback control law defined under a set of polyhedral regions in state space [14]. In a complementary way to feedback control recent research has shown that there is a state space form that allows the tuning of PID controllers using the Linear Quadratic Regulator (LQR) [13, 18], and this may allow us to combine both strategies making a new tuning method that considers constraints.

Within this context, in this paper a design of a new type of gain-scheduling PID controller to control angular velocity of electric vehicle DC motors subject to constraints in angular velocity and input voltage and PID states is proposed. To this end, the formulations in the state space of the PID controller are used, as well as the concept of controlled invariant sets together with the solution of a multiparametric programming problem [6, 9, 19, 20]. In this case, we use the same techniques applied to obtain explicit controllers (which take into account system constraints) to tune similar PID controllers (mp_PID) to constrained systems.

This work is organized as follows: At first, we will approach the concept of the -contractive controlled invariant set. In sequence, the problem of linear multiparametric programming will be described. After that, we will introduce how to tune PID controllers from multiparametric linear programming technique. An overview of electrical vehicle DC motors will be discussed later. Finally, a set of simulations will be carried out with the objective of proving the functionality of the proposed algorithm and the concept of mp_PID in the control of electrical vehicle DC motors, i.e., specified to work with electric cars.

#### 2. Controlled Invariant Sets

The concept of controlled invariant sets has become important in the design of controllers for linear discrete-time systems subject to constraints since it represents a fundamental condition to maintain system stability ensuring that the constraints are not violated [21].

Consider the linear time-invariant discrete-time system described bywhere is the sample time, is the state of the system with (where and ), and is the control input subject to the constraints .

A nonempty closed set is controlled invariant with respect to the system described in (1), if there exists a control signal such that remains inside it for every belonging to the closed set. Moreover, if a given contraction rate is considered, a set is said to be -contractive controlled invariant set with respect to system (1) if there exists a control signal such that belongs to the set , for every belonging to the closed set [16, 21]. In general, the set of constraints defined in state space is not a controlled invariant set; i.e., there is not necessarily a control law () which maintains the trajectory of the state vector completely contained in the set of constraints. However, it is possible to compute a controlled invariant set , to be as large as possible, contained within the set of constraints [22]. Therefore, before starting the controller synthesis process, it is necessary to define a controlled invariant set and then to compute a suitable control law that is able to restrict the state vector to a controlled invariant set .

By defining the maximal contractive controlled invariant set () [16, 22], a state feedback control law (), capable of maintaining the system dynamics (1), contained in , can be computed online by solving the linear programming problem (LP) as described in [16] or offline from the solution of the following multiparametric programming problem (mp-LP) [23]:

where where is the contraction rate to be minimized at each time step, is the control action to be computed, and is set of states contained inside . The expression represents a convex polyhedron in the space , and is a convex polytope that represents the constraints in the control variable.

In the design of controllers under constraints, the solution of the mp-LP (problem (2)) results in a PWA state feedback control law over the polyhedral regions in the space of parameters as follows [24]:(1)The set (controlled invariant polyhedral) is partitioned into different polyhedral regions: (2)The optimal solution is a PWA function over :

As the system is in the state space form it is possible to find the largest -contractive invariant set and, in sequence, the parameters of the control law are computed which maintain the dynamics of the states within the -contractive invariant set. In order to associate with the PID controller, we will call this “the tuning step”, because we find the controller’s parameters that guarantee positive invariance and -contractivity. That is, by using this process we will be able to find a PID control law, PWA, that allows the controller to synthesize control actions capable of controlling the process under constraints.

##### 2.1. Tuning of Gain-Scheduling PID Control Design (mp_PID)

Based on formulation that allows the reorganization of a second-order systems in state space form, described in [20], whose states are the tracking error, integral of the error, and derivative of the error, we propose the tuning of a type of gain-scheduling PID controller by using the PWA state feedback control law computed by multiparametric linear programming, described in (2).

###### 2.1.1. PID Controllers

Consider now the system presented in (1) is described by

Because the external setpoint does not affect the controller design, we assume for the system of Figure 1 [20].