Proportional-Integral-Derivative (PID) controller is widely used for quadrotors due to its usability and robustness over many years. The PID controller is easy to understand and implement, but the quality of its parameters directly determines the control e ect. Manual adjustment of three PID parameters using trial-and-error method is a timeconsuming process that can hardly reach a good result. In recent years, numerical optimization methods have been developed and expanded. In this paper, a proposed optimization by the gradient descent (GD) method is employed to tune PID controller for a quadrotor. First, the initial parameters of the GD-based PID controller are chosen, including learning rate, number of epochs and values of the parameter vector. Then the gradient-based algorithm is employed to minimize the cost function by adjusting the parameters. Both the classical and the selfadjusting PID controller were implemented to track de ned trajectory with quadrotor. Unlike conventional PID design, this method is capable of reaching the desired control performance, such as minimum overshoot. Another unique feature is that the optimized PID controller is applicable to a wide variety of time-varying and nonlinear systems. The analysis of the experiment results demonstrates that algorithm can quickly and accurately nd the optimal parameters solutions. Comparison of simulations shows that this method is robust and adaptable and can also improve control precision.
In recent years, interest in research on small unmanned aerial vehicles (UAVs)
has increased dramatically. These UAVs have become one of the major and
critical areas [
One of the most preferable unmanned aerial vehicles is four{rotor drones, also known as quadcopters or quadrotors. At a small size, quadcopters are cheaper and more durable than conventional helicopters due to their mechanical simplicity. This aircraft has the ability to take o and land vertically (VTOL) and hover in a stable air condition. However, it has a number of limitations, such as complex and di cult control and limited capacity of energy and load.
There are many controller designs and applications in the literature to control
unmanned aerial vehicles, e.g., fuzzy controller [
A proportional{integral{derivative controller is a control loop mechanism that is widely used in industrial control systems and a variety of other applications requiring continuously modulated control. A PID controller continuously calculates an error value e(t) as the di erence between a desired setpoint and a measured process variable and applies a correction based on proportional, integral, and derivative terms.
The PID controller is easy to understand and implement, but the quality of its parameters directly determines the control e ect. Manual adjustment of three PID parameters using trial-and-error method is a time-consuming process that can hardly reach a good result. In recent years, numerical optimization methods have been developed and expanded.
Many approaches have been proposed for improving the setting performance
of PID parameters using heuristic optimization methods such as Particle Swarm
Optimization (PSO) [
Gradient Descent (GD) { a rst-order iterative optimization algorithm for
nding the local minimum of a di erentiable function. To nd a local minimum
of a function using gradient descent, one takes steps proportional to the negative
of the gradient (or approximate gradient) of the function at the current point.
If, instead, one takes steps proportional to the positive of the gradient, one
approaches a local maximum of that function. Several works have been proposed
for tuning the PID parameters using the gradient descent methods such as [
In the given work the control system of the quadcopter, realizing the ight on control points in an automatic mode, is considered. Tuning of coe cients is performed during the ight using the method of numerical optimization { the gradient descent.
The paper organized as follows. In next section, a brief description about the AR.Drone 2.0 as a controlled system has given. After that the gradient descent algorithm for control parameters optimization is described. Then quadrotor control results has been demonstrated. In the last section, the main outcomes of this work are summarized. 2
The AR.Drone 2.0 quadcopter from the French company Parrot was chosen as the control object. A short description of the quadrotor is presented in the following section of this report. 2.1
The AR.Drone 2.0 is an electrically powered quadcopter intended for augmented
reality games [
The AR.Drone 2.0 is a not open source UAV introduced by parrot company as a toy in 2010. However this quadcopter has increased its popularity in the academic and developmental applications, a numerous of application has been developed for manipulate the drone from a smart phone, tablet or a computer with wi{ connection, there are also numerous publications, softwares and videos with useful information for the users.
Takeo and land functions are controlled for the embedded electronic system. Landing sequence is automatically activated in case of low battery situation. AR.Drone have also an emergency function which cut o the power on the motors and can be activated during a ight if any object comes in contact with the propellers or the emergency function is sent by code from the pilot device.
In work project, the AR.Drone 2.0 was controlled from a computer using
Ubuntu 16.04 with Robot Operating System (ROS) Kinetic Kame [
AR.Drone is mainly controlled considering four degree of freedom (4DoF) and
the control parameters to the on board controller can be given as oat values in
between [
In this section we describe control system for quadcopter.
Despite technological progress, the PID controller remains the most popular control loop mechanism and is used in many areas. This is due to the fact that the PID controller is easy to understand and implement. The controller forms a control signal, which is the sum of three parts: proportional, integral and derivative components. The success of the PID controller depends on the quality of the gain choice. The PID controller in the literature can mainly be divided into two categories.
In the rst category, the PID controller parameters remain the same during
the control. At this point the best known method is to set the PID controller
using the Ziegler-Nichols formula [
A gradient is usually counted as the sum of the gradients caused by each learning element. The vector of parameters changes in the direction of the anti{ gradient with a speci ed step. Therefore, a standard (batch) gradient descent requires a single pass through all learning data before it can change parameters.
In the case of the stochastic gradient descent, instead of cycling through each example of learning, we use only one randomly selected learning element. Thus, the model parameters change after each learning object. For large data arrays, a stochastic gradient descent can give a signi cant speed advantage over a standard gradient descent.
The compromise between the two methods is mini-batch. In this case, a small set of data is processed instead of calculating the gradient from 1 sample (SGD) or all n training samples (GD), we calculate the gradient from k training samples. 3.4
The learning rate is the most important hyper-parameter for gradient-descent
based methods [
Stochastic Gradient Descent (SGD) is a widely used optimization algorithm for machine learning. SGD usually uses a xed learning rate. This is because the SGD gradient estimator introduces a source of noise (the random sampling of m training examples), and that noise does not vanish even when the loss arrives at a minimum. A hyper-parameter 2 (0; 1) is the step of the method (or learning rate). Stochastic gradient descent updates the parameters for each observation, so tuning happens online while ying.
The parameter update rule of SGD from time (i.e., iteration) t to time t + 1 is given by: t+1 = t SGD Momentum is designed to accelerate learning, especially in the case of small and consistent gradients. The momentum algorithm accumulates an exponentially decayed moving average of past gradients and continues to move in the consistent direction. The name momentum derives from a physical analogy, in which the negative gradient is a force moving a particle through parameter space. A hyper-parameter 2 (0; 1) determines how much the past gradients to the current update of the weights. The momentum term is usually set to 0.9 or a similar value.
Its parameter update rule: (
Vt+1 =
Vt Nesterov's Momentum is a simple change to normal momentum. Here the gradient term is not computed from the current position t in parameter space but instead from a position (t + 1) = (t) + Vt. This helps because while the gradient term always points in the right direction, the momentum term may not. If the momentum term points in the wrong direction or overshoots, the gradient can still "go back" and correct it in the same update step.
The Nesterov's Momentum update rule is given by: (
Vt+1 =
Vt t+1 = t + Vt+1 3.5
Let's suppose, gradient-descent method will adjust the PID controller parameters by minimizing the squared controller error, which de ned: (3) (4) (5) (6) (7) (8) where r - the set point and y - the control output.
Let's de ne the objective function, which should be minimized: e = r
y L = 1 e2 =) 2
When used to minimize the above function, the main challenge remains behind the calculation of gradient @Lt=@ t in 2. In this case we must apply a chain rule to composites of more than two functions: where Kp=i=d - one of the PID controller parameters.
To measure the rate of change of L in any direction that is represented by a unit vector u, in multivariate calculus, we de ne the directional derivative of L at in the direction of u as: where yt - the actual output of the plant and ut - the control signal (output of the controller)
Therefore, the update rules for PID control parameters for the gradientdescent are expressed as: 8Kp = Kp > > > > > > <
Ki = Ki > > > > > >:Kd = Kd
The GD-based PID controller is shown in Figure 2. The inputs of algorithm are created by proportion, integration and derivation of error e. (9) (10) The main objective of this study is to enable the aircraft to follow a de ned trajectory with algorithm of an improved controller. When the aircraft takes o , the initial position and orientation of the aircraft is set to zero. Then the aircraft is informed about the reference waypoints. The waypoints information which are necessary for trajectory tracking includes four data. These are Xref , Yref , href and ref . The relative position between vehicle and target is calculated in PC. And the controllers provide to track desired parameters. The aircraft's position and velocity is measured by its inertial sensors. The simulation has done using Gazebo. The results of the performance of SGDMomentum method is given (see Figure 3). In the following simulations, the proposed algorithm SGD-Momentum used a set of parameters as: the learning rate = 0.0001, momentum parameter = 0.9, the number of iterations n = 100. In Figure 5, the simulation gains has been shown using GD-based autotuning. 4.2
In this section, the experimental results has shown. In Figure 4, the real experimental results has plotted. From this gure it is clear that, AR.Drone successfully had gone to the point with coordinates (2.0; 2.0; 2.0).
We can see that the parameters are adjusted faster than in the approach with classical gradient descent. The optimal coe cients have been selected, thus we reaching the desired control performance, such as minimum overshoot (see Figure 5). In this paper, the control system of the quadcopter based on PID controller with optimization of parameters using the gradient descent method was considered.
The results verify that the proposed algorithm is an e ective method for optimal tuning of PID parameters in term of no overshoot, zero steady state error and extremely short setting time and rise time. The proposed approach is highly robust and e ective.
This proposed tuning method may create some bene t in real-life and industrial applications since it saves time for none expert control engineering to nd optimal controller parameters to enhance the performance quality. For future work, gradient based method will be applied to tune PID controller of PX4 quadrotor. 6
This work was nancially supported by ITMO University: grant num. 419331 "Development of a multifunctional autonomous landing station for multi{copter with an autopilot system".