## Abstract

Drones, specifically quadcopters, have increased in importance during the last years due to their wide range of applications, from civil applications to military employment. One of the most important issues in quadcopters is the efficient control system. While many researchers have dealt with building control systems for symmetric quadcopters, this work presents an efficient control system for asymmetric quadcopters using evolutionary computations. The problem is well-defined throughout the paper, and the methodology is explained in detail in the respective sections. A genetic algorithm is used to tune the weighting matrix of the control system after formulating the control system as an optimization problem. The genetic algorithm was fast and active to increase the performance of the proposed system.

## 1 Introduction

The quadcopter is one of the autonomous robot types that do not need a supporting surface; it contains four symmetric rotors. The symmetry of the quadcopter is crucial because the control of the quadcopter depends on rotor speed and lift force variation [1, 2]. The symmetric structure of the quadcopter has many advantages for different applications like aerospace fields [3], military security systems [4], critical monitoring [5], and mineral exploration [6]. The key point in all quadcopter applications is the sensitive nature of the flight control of the quadcopter, which is expressed by many control techniques like Lyapunov-based control for indoor micro-quadcopter [7] and combinations of PID-based control [8]. Also, feedback linearization-based control is presented [9, 10], as well as model predictive control [11]. The control system looks like a brain that tells a dynamic system how to behave and react to responses. Adaptive backstepping control was designed [12] for the ball and beam system, considering parameter uncertainties. The backstepping controller [13] is a well-known technique in control systems that has been used in many applications. Optimal control was formulated for a quadcopter as a constrained optimization problem [14]. The PID controller is the most famous controller for the speed of motors [15]. It is worth mentioning that the PID controller is convenient for SISO systems because these systems have a single input and single output plant, which is easier to do using a PID controller. The quadcopter is an underactuated system, i.e. only longitudinal, lateral, and altitude axis and yaw angle can be directly controlled. Roll and pitch angles are systemically calculated, and that is why it is hard to control an asymmetric quadcopter.

In this paper, a unique control system based on evolutionary computing is developed for asymmetric quadcopters, which are hard to be controlled using traditional methods. The proposed methodology is simple and fast and employs the genetic algorithm for tuning the control parameters. The genetic algorithm used in this paper is one of the metaheuristic algorithms, i.e., other optimization techniques can be used for this controller like swarm intelligence methods. The proposed structure in this work is an asymmetric quadcopter that can be used for aesthetic reasons or any other purposes. The unbalanced structure is a challenging issue for developing a control system.

## 2 Controlling a quadcopter

The control system of a quadcopter, which is shown in Fig. 1, can be modeled as a rotating mass in a frictionless environment; the dominating states vector is

## 3 State space representation

*u*being added or moved over time:

In state space representation, we reform a high-order differential equation into a set of first-order differential equations. This reforming or repackaging is useful because it makes the system easy to analyze because we can notice the interconnected system's underlying behavior and how the system can be affected by an external input or multiple inputs. There are many control techniques that are built based on state space, such as:

Kalman filter [17].

Linear–quadratic regulator (LQR) control [12].

Rubost control [18].

Model predictive control (MPC) [19].

PID controller [15].

### 3.1 Pole placement

Pole placement or full-state feedback is a method used to develop a feedback controller from a model that is expressed by state space equations [20]. This method is not used extensively in the control industry like LQR; for pole placement control, the feedback occurs on the state vector instead of the output, as shown in Fig. 2. By finding the proper value of the gain K and subtracting the result from the multiplication of the reference signal and scaling term Kr we can get a new input value that can be fed to the plant. The Kr on the reference is a scalable value and is used to affect the steady-state error by controlling the input signal.

Typical feedback control system

Citation: International Review of Applied Sciences and Engineering 14, 3; 10.1556/1848.2022.00584

Typical feedback control system

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Typical feedback control system

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*A*and

*B*are matrices while

### 3.2 Linear quadratic regulator

*f*as follows [14]:

*Q*and

*R*are penalties on the performance and effort, respectively, and the following equation can estimate the gain matrix for a given dynamics of the system:

*R*and

*Q*are both positive definite matrices where

*R*acts on the input vector while

*Q*acts on the state vector. The cost function can be written as follows:

By solving the LQR problem, we can get the gain matrix that produces a low cost for a given dynamic system. The best cost or best performance can be represented by the first term of equation (4) as the area under the curve in the state-time space, as shown in Fig. 3. In other words, Fig. 3 shows the first part of the cost function that has to be minimized to the lowest value as much as possible.

Performance of the dynamic system

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Performance of the dynamic system

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Performance of the dynamic system

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*u*and

*x*in equation (6) is often penalized by adding penalty factors to the off-diagonal of the weighting matrix as follows:

## 4 Evolutionary computation

### 4.1 Evolutionary family

Evolutionary computation is focused on using natural evolution mechanisms to modify a population of individuals representing different points in the solutions space associated with a given problem to reach an optimal situation using a “survival of the fittest” procedure. The evolutionary computation family of algorithms classification divides the family into four main groups [21], as revealed in Fig. 4. Most of these categories share a common basic algorithm. Variations come from the way some operators are applied to the algorithm process and what is the basic representation of the individuals in the population.

Family of the evolutionary computation

Citation: International Review of Applied Sciences and Engineering 14, 3; 10.1556/1848.2022.00584

Family of the evolutionary computation

Citation: International Review of Applied Sciences and Engineering 14, 3; 10.1556/1848.2022.00584

Family of the evolutionary computation

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### 4.2 Genetic algorithm

Genetic algorithm GA [22] is a heuristic optimization algorithm where the evolution and natural genetics form the algorithm's base. In the 1970s, Holland [23] first presented the concept of genetic algorithms. The inspiration came from the principle of survival of the fittest after the principle had been introduced by Charles Robert Darwin [24]. Most of the search spaces have complex shapes and multimodality containing multi-local optimum solutions. However, the probabilistic mechanism of the GA leads a specific solution optimally in a search space toward a point of the minimum cost or maximum benefits. In real life, individuals of species can survive and reproduce if they can adapt to the change in environment and best fit the competition in the foraging process. Each individual has its own characteristics that make it unique.

On the other hand, the genetic content of the individual determines these characteristics. Each feature is represented by a chromosome consisting of a set of units called genes. Thus, the order or form of the set of genes is responsible for a specific feature in the individual. Hence, genes are the key points behind the survival of the individual. Evolution is successive changes in the characteristic of an individual during successive generations over time. However, we can define evolution as successive changes in the genetic content for an individual during successive generations. During reproduction, half of the genetic material comes from a male, and the other half comes from a female of a specific species. Thus, the offspring is a recombination of the genetic pool, and the role of the natural selection is to decide whether that offspring fits or does not fit to the competitive environment. This mechanism enables the individuals with best features to dominate at the expense of the individuals with weak features. Only the fittest will survive and couple to reproduce the next generation, which is another recombination of the genetic material. So, we can say that the best set of genes will survive in the existence battle. Specifically, the recombination of the genetic pool from parents to reproduce offspring is called crossover. For some reason, the arrangement of the genes in a chromosome may be changed in a probabilistic process called mutation, which leads to a change in some features of the individual. Repeated crossover and mutation may lead to the right possible individual that can be more fit for the competitive environment. In a genetic algorithm, sequences of generations are created using selection, crossover, and mutation; the crossover and mutation represent a search mechanism within the search space. Selection provides the process of choosing the best solution to survive, and many selection mechanisms were discussed in the literature. Each solution should be associated with a fitness value; the best solution is the one with the best fitness value.

## 5 Tuning weighting matrix

*Q*and

*R*can be formulated as follows:

Table 1 illustrates how tuning weighting parameters can affect the performance of the dynamic system, which is described by equations (11) to (14). The performance was recorded as the area under the curve of the response to initial conditions, as described in Fig. 3. The performance of the system can be considered as the inverse value of the area under the state-time curve. In other words, less area under a curve means the best performance. Thus, we search for a combination of

Effect of the weighting matrix on the performance of the system

No. | Parameter | Area under state-time curve | Performance | ||

1 | 1 | 1 | 1 | 17.43 | 0.05737 |

2 | 7 | 4 | 3 | 13.851 | 0.0721 |

3 | 2 | 10 | 5 | 28.595 | 0.0349 |

4 | 10 | 5 | 2 | 11.868 | 0.0842 |

5 | 8 | 8 | 8 | 17.434 | 0.0573 |

The numerical example in Table 1 shows that tuning

The minimization problem was solved using genetic algorithm, as shown in the convergence curve in Fig. 5, where the reduction of the cost function represents an amplification of the efficiency of the control system. The better efficiency can be described by the minimum time required to settle a signal to the steady state, as shown in Figs 6 and 7.

Performance of the control system

Citation: International Review of Applied Sciences and Engineering 14, 3; 10.1556/1848.2022.00584

Performance of the control system

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Performance of the control system

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Time response

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Time response

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Time response

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Input development of the control system

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Input development of the control system

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Input development of the control system

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The advantages and disadvantages of employing an optimization algorithm for a control system are competitive depending on the nature of the algorithm itself [25, 26]. However, the population size used for the statistical results in Table 1 is 50, with one independent run with three variable size of integer values. The cost function search space is convex, similar to Fig. 8, which decreases monotonically towards the global point, which is zero.

Tuning parameters search space

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Tuning parameters search space

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Tuning parameters search space

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Figure 5 shows that the genetic algorithm finds the optimal values of

## 6 Quadcopter flight dynamics

Figure 9 shows the body frame

Quadcopter frames assignment

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Quadcopter frames assignment

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Quadcopter frames assignment

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Transformation between the vehicle frame and body frame

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Transformation between the vehicle frame and body frame

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Transformation between the vehicle frame and body frame

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*C*and

*S*denote cosine and sine functions. According to Newton's second law, the acceleration in the inertial frame will be:

## 7 Optimization problem formulation

The quadcopter can be controlled efficiently only by four inputs which are position

Schematic diagram of a quadcopter control system

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Schematic diagram of a quadcopter control system

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Schematic diagram of a quadcopter control system

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The zero values in the right-hand side of equation (30) come from the assumption that the quadcopter structure is symmetric, which is necessary to solve equation (30) analytically. The problem is getting worse when the quadcopter has an asymmetric structure. In this case, the right-hand side of equation (30) components will be all nonzero values, and that makes it hard to solve for

This optimization problem can be minimized by many methods like swarm intelligence [30], biologically inspired algorithms [31], and so on. In other words, this research proposal opens the door for many possible future works that can employ metaheuristic algorithms [32] for control systems.

## 8 Conclusion

The evolutionary control system of a quadcopter introduces an efficient way to control quadcopters that have unsymmetrical structures, which are hard to solve their equations. Thus, the proposed methodology presents a direct way to control both symmetric and asymmetric structures. Building a control system with evolutionary computing is a kind of optimal control of systems. The analytical solution of the dynamic equations of an asymmetric quadcopter cannot be found during feedback linearization-based control. Thus, the evolutionary control system has the advantage of estimating the solutions for the control system. The genetic algorithm is used in this paper to find the optimal values for the weighting matrix. This algorithm has proven that tuning the weighting matrix of a control system is not a time expensive optimization problem. During a fraction of a second, the algorithm can find the optimal values of the weighting matrix that can drive the system to the best performance.

## Acknowledgments

The research was supported by the Hungarian National Research, Development and Innovation Office – NKFIH under project number K 134358.

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