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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

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 [

The Proportional Integral Derivative Controller (PID) has been widely used for most industrial process, due to its simplicity and effectiveness in control [

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 [

In order to solve the optimal constrained problem many controllers are being proposed. One solution consists of maintaining the system trajectory within

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 [

This work is organized as follows: At first, we will approach the concept of the

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 [

Consider the linear time-invariant discrete-time system described by

A nonempty closed set

By defining the maximal contractive controlled invariant set (

where

In the design of controllers under constraints, the solution of the mp-LP (problem (

The set

The optimal solution

As the system is in the state space form it is possible to find the largest

Based on formulation that allows the reorganization of a second-order systems in state space form, described in [

Consider now the system presented in (

Because the external setpoint does not affect the controller design, we assume

System controlled by a PID.

The relation

Use the following definitions:

Equation (

Considering the state space, the formulation becomes [

In this case, we assume the reference signal equal to zero and because the system is regulatory it is organized in a way that the states tend to zero and tend to eliminate the disturbance. When applied in electrical vehicle motor control, we intend to make changes in the reference, so some considerations must be realized:

The external reference does not affect the controller design.

It is possible to work step-type references in two ways, by using model illustrated in Figure

Concerning the direct change of reference, it is possible to verify that as the system is stable in closed loop, it tends to converge to the reference. However, in this case, only the error and derivative of the error states will converge to zero and the integral of the error will become a value that maintains the necessary control action to force the output to zero, in the same way conventional PID does.

The second way to control the system is using the linearization idea and it can be observed in Figure

System controlled by a PID.

Figure

Given the main considerations about the state space system, it is necessary to tune and use the controller. In this case the concept of invariant sets will be used to find an optimum tuning for a PWA PID control law, which we call mp_PID. Thus, the

1: Convert the continuous system into a state space system, where the states are the errors and the input is the

2: Compute the maximal

3: Solve the mp-LP problem described in equation (

4: Compute the integral of error, error, and derivative of error;

5: Identify which polyhedral region the computed state

6: Use the affine control law to control the system with

7: If the control routine is not interrupted, return to step 4.

As described in Algorithm

After the definition of the PID tuning method the application must be studied in order to have mp_PID applied to it, so in this section we present an overview of electric vehicle DC motors.

Recently, electric vehicles are gaining popularity among the population resulting in more demand for these types of car. Electric vehicles are efficient and need less maintenance than fuel-based cars and they do not pollute the environment [

The consuming of the electric vehicles depends heavily on the used motor type and adopted control strategy. Actually, DC motors and induction motors are being proposed to be used in the electric vehicle industry, with DC motor type being a good candidate to be used in electric vehicle applications [

DC motors have the advantage of being easy to control resulting in several control techniques for velocity control using PID type control. Techniques using metaheuristic PID [

DC motors have different configurations as compound, shunt, series, permanent magnetic DC and separated field winding where each of them has advantages and disadvantages. DC motors in series configuration are the type of motors that can be used in electric cars due to their instantaneous torque and smooth acceleration. However, these types of motors need a minimal load not to be damaged, which is a condition that occurs in electric car applications. One way to limit the damage in the DC motor series control is to limit the maximum velocity in the adopted control strategy [

In sequence, we present the DC motor modelling with excited separately field and afterwards we present the numerical examples controlling the motor and testing the performance of the tuned controllers.

A DC motor controlled by current armature with independent field is depicted in Figure

DC motor with separated winding field with constant current (

The torque induced (

The armature induced voltage (

As the DC motor has separated field, the voltage applied (

The following second-order transfer function resulted:

The parameters of a motor suitable for use in electric cars are determined.

The mp_PID controllers are tuned.

The proposed tuning algorithm is performed using different parameters and, then, the results are compared.

The influence of each parameter on the final process performance is analyzed.

At first, in order to control an electric vehicle DC motor we specify its parameters (the motor) in such a way that it is able to move the vehicle. The voltage armature, resistance and inductance armatures, moment of inertia, velocity constant, torque constant, and friction coefficient motor parameters presented in Table

Motor parameters.

| 48 V |

R | 0.1 |

L | 0.005 H |

| 0.004 |

| 0.0036 |

| 0.1 |

| 0.05 |

Replacing the motor parameters in (

then, the simplified transfer function of the used motor becomes

In the model of (

This section presents the general tuning process of the mp_PID controllers. Some important aspects to emphasize are described as follows:

The output constraints can be transformed into constraints on the error, especially when assuming the operating point at zero. The output constraints for the electric vehicle DC motor case will be the maximum speed allowed at a voltage of 48V.

The control signal constraints must be limited to

The integral and derivative of the error values will be appropriately chosen; i.e., they will be used as a tuning parameter and the influence of these constraints will be evaluated in order to emphasize how the final performance of the system is changed, in aspects such as overshooting and stabilization time.

The main tuning parameter of the mp_PID controller is the value of

In a specific case we observe the difference between varying the operating point and varying the reference.

Based on these considerations, tests are performed varying the constraints on the integral of the error, derivative of the error, and the

In this first test the objective is to observe the influence, in system’s performance, of integral of the error constraints change. Despite the direct relationship between error and the output of the motor (angular velocity), the integral of the error is an internal state of the system in state space and influences the motor dynamic performance but not necessarily forces physically the system outside its limits. Therefore, by hypothesis we assume that we can freely modify these parameters. In this way, the test conditions presented in Table

Table with the 3 tests, varying integral of error.

Simulations | | Integral of the Error Limits | Derivative of the Error Limits |
---|---|---|---|

1 | 0.999 | | |

2 | 0.999 | | |

3 | 0.999 | | |

The system’s outputs and the control actions for test condition 1 are depicted in Figures

Polyhedron test 1 for test condition 1.

System’s outputs for test condition 1.

Control actions for test condition 1.

As seen, Figure

Figure

Figure

Similarly to the first test, in this second test our objective is to observe the influence in the performance of the system, when the derivative of the error constraints is changed. This also happened to the integral of the error constraint, the derivative of the error is an internal state of the system in state space and influences the performance but not necessarily physically forces the system outside its limits (we are not considering variation effects in this case, although in some cases they are included in problems with restrictions). Therefore, by hypothesis we assume that we can freely modify these parameters. In this way, the test conditions presented in Table

Table with the 3 tests, varying derivative of error.

Simulations | | Integral of the Error Limits | Derivative of the Error Limits |
---|---|---|---|

1 | 0.999 | | |

2 | 0.999 | | |

3 | 0.999 | | |

The system’s outputs and the control signals for test condition 2 are depicted in Figures

System’s outputs for test condition 2.

Control actions for test condition 2.

Figure

In these tests the quantity of sets generated within the controlled invariant set is 8, 8, and 12. Similarly to the previous test we emphasize that it will be very inexpensive to compute the control action.

The figure of this second test (Figure

Unlike the first 2 tests, this one will not present the effect in constraints, but in the tuning parameter (

Table with the 3 tests, varying

Simulations | | Integral of Error Limits | Derivative of Error Limits |
---|---|---|---|

1 | 0.99 | | |

2 | 0.90 | | |

3 | 0.60 | | |

The system’s outputs and the control actions for the test condition 3 are depicted in Figures

System’s output for test condition 3.

Control actions for test condition 3.

This third test resulted as the quantities of sets within the invariant sets 8, 14, and 12. Also, Figure

The last figure of this third test (Figure

The first three tests aimed to show the results related to the controllers tuning when the systems are subject to reference variations (step). This means that the controlled invariant set remained unchanged as well as the operating point of the system. In this fourth test, the objective is to present the changes related to the operating point. Note that there is a possibility of differences in behaviors, because although it has the same amount of variation, this test directly affects the activated regions in the control process, which consequently changes the response of the system. Thus, the test condition is to change the setpoint by 20 units and compare it with the changed operating point (setpoint 1) by 20.

The system’s outputs and the control actions for the test condition 4 are depicted in Figures

System’s output.

Control action.

Figure

Figure

This paper presented a PID controller for electric vehicle DC motors based on proving the functionality of the proposed algorithm from a set of simulations. The algorithm proved that it is possible to generate control actions that consider the 48V limits of the motor, as well as forcing the output speed of the motors to be limited to their specified values. In addition, it was verified that the integral and derivative of the error constraints make tuning parameters “constrain” the performance of the process output. In other words, we found a direct relationship between error constraints and the DC motor dynamics. Another important point to note is that changes in

In conclusion, control of electric vehicle DC motors under constraints and tuning controllers adjusting the behavior of the system's output is possible by using the proposed algorithm. In this sense, important improvements can be proposed. The first one would be to use a methodology to automatically tune parameters, since there are no tuning rules for these constraints (in the integral of the error and in the derivative of the error). A second improvement would be to extend the tests by verifying that the behavior presented to the DC motors, i.e., verified, can be extended to other processes.

Although DC motor with excited separately field can be used in an electric vehicle, new types of motors as permanent magnet synchronous motor (PMSM) are being used recently [

The [algorithms] data used to support the findings of this study have not been made available because scientific research is still being carried out using the presented algorithms.

The authors declare that they have no conflicts of interest.

The authors would like to mention the institutions that contributed to the development of this project: UFRN (Federal University of Rio Grande do Norte), DCA (Department of Computation and Automation), CNPQ (National Council for Scientific and Technological Development), UnP (Potiguar University), and LAUT (Laboratory of Automation in Petroleum).