Model predictive control of quadrotor UAV linear model

Model predictive control (MPC) has gained interest in recent years due to its ability to deal with the both input and state constraints. First version of the model predictive control was used in mainly with the system that has slow dynamic behaviours. With the help of recent advancement in the technology the computational power has increased which makes the computational work eligible to be used in the embedded systems. In terms of the advantages of the MPC it can be pointed out that it can deal with the dynamics of the system while it is computing the inputs to the system. It can be understood that the basic working principle of to control algorithm is to use the mathematical model of the system itself. The model is referred is mainly used as the linear state-space model which is obtained from the non-linear equations of the system. The prediction has been carried out by using the state-space matrices and putting the results of the previous calculation to the next one. In terms of the solution of algorithm, MPC relies on the solution of the optimization problem specific to the problems having quadratic cost functions. This quadratic functions are basically shaped on input and the error between the current states and the reference point to find optimum solution. Then required matrix and vector for the general quadratic function equation are found and fed into solver. The main reason of beeing used of the MPC is to account the constraints of both the input and states. Moreover, by expanding the constraint inequality matrix and vectors through the horizon in both control and prediction, additional arguments for the quadratic programming solver are obtained. Quadrotor vehicles are very popular in recent years. This popularity drives the researchers to work on these systems to improve both their mechanical and computational capabilities. Therefore, a mathematical model is needed to be used for the interested controller algorithm. The well know 6 Degrees of Freedom (DOF) non-linear equations describing the dynamical behaviour of the vehicle is used to obtain the state-space representation used by MPC. These non-linear equations are linearised around an equilibrium point which in this case the hovering condition of the vehicle. Linearising the vehicle around hover condition makes it very sensitive to angular movements where linearisation is valid under approximately 30 degrees in both roll and pitch angles where importance of constraint inclusion of the algorithm becomes crucial. Likewise, since the attitude dynamics of the vehicle is rapid, optimal inputs makes the movement better. These optimal controls can be obtained from the solution of the quadratic programming algorithm used by MPC by considering their limits and integrating them into the solution process. Since the main control objective of the thesis is to track a desired trajectory, this should be done with zero-steady state error in order to be sure that the vehicle is reached to the desired state. This is accomplished by using the MPC with an augmentation. This augmentation is applied on state and input vectors where the augmented states are represented as the concatenation of the difference of previous and current state and the output vector and the input vector is the increment of the input needs to be added onto previous input applied. This augmentation induces integrator effect to the controller allowing it to have zero-steady state error. Having described both the system and the control scheme, few different simulations has been carried out. In the first three simulations, step inputs are applied as reference position or yaw angle. Since the X and Y axes are symmetric for the defined vehicle, only one lateral axis simulation has been done. Because of the fast dynamic behaviour in roll and pitch movement of the vehicle shown during lateral movement, change in length of the horizons have considerable amount of effect in the final output. Unlike the lateral movement, in the vertical Z-axis position or yaw rotation behaviour, change of the length of horizons are less effective. Finally, in the final simulation a trajectory tracking scenario has been simulated where it can be seen that in both cases, where the horizons are slightly different, vehicle can track the trajectory. All of the simulations and algorithms are implemented in MATLAB environment with object-oriented programming approach and it is tried to obtain flexibility like using different objects in different applications. %As a conclusion, it can be said that essentially choosing the prediction horizon carefully response of the system can be adjusted to desired one. Increasing the length of horizon allows the controller to predict farther points with the cost of increased computational burden. At this point other types of MPC are tried to be used where the optimization problem solution has taken out from the online algorithm in which solutions are obtained offline for different operational regions and specific points from table like data is chosen as the solution and used to obtain system inputs. This type of MPC is called explicit-MPC which is aimed to be studied as a next objective.