# 3D trajectory tracking control of quadrotor UAV with on-line disturbance compensation

TL;DR: Using the proposed formulation, it is shown that the performance of the control and its robustness level can be significantly improved and an in-depth discussion with respect to the control performance is highlighted by considering several scenarios and using several metrics.

Abstract: In this paper, we propose a revisited form of the so-called Model-Free Control (MFC). Herein, the MFC principle is employed to deal with the unknown part of the plant only (i.e. unmodeled dynamics, disturbances, etc.) and occurs beside an Interconnection and Damping Assignment-Passivity Based Control (IDA-PBC) strategy that is used instead of the PID structure as done in the classical MFC form. Using the proposed formulation, it is shown that we can significantly improve the performance of the control and its robustness level. This problem is studied in the case of Multi-Inputs Multi-Outputs (MIMO) system with an application to a small Vertical Take-Off and Landing (VTOL) vehicle where a stability analysis is also provided. The numerical simulations have shown satisfactory results where an in-depth discussion with respect to the control performance is highlighted by considering several scenarios and using several metrics.

## Summary (2 min read)

### INTRODUCTION

- The quadrotors are considered as a good case study to design, to analyze and to implement flight control strategies.
- Thus, its use as the basis of control allows the compensation of the uncertainties as well as other disturbances.
- Therefore, the authors propose a Revisited Model-Free Control (R-MFC) strategy to simultaneously accommodate the unmodeled and neglected dynamics and external disturbances.
- Passivity-Based Control (PBC) is well known especially in mechanical applications for controlling nonlinear systems.

### III. R-MFC FLIGHT CONTROLLER DESIGN

- The classic MFC approach proceeds by considering an ultralocal model, valid in short time that approximates the nonlinear model via input-output behavior using the experimental available data without any modeling step.
- This online numerical differentiator and estimation may fail with some highly nonlinear and/or time-varying dynamics that need to be treated carefully.
- This class of systems is widely adopted in the robotics and mechanical fields.
- Usually, model ( 6) is quite simplified with neglected and unmodeled dynamics.
- An additional effort is requested to deal with this new term 𝛿ℱ.

### Remark 1: For the existing MFC technique, 𝑣 is fixed by the user and may equal to 1 or 2.

- This estimation is valid for a short period 𝑇 only and it should be continuously updated at every iteration of the closed loop controller.
- Many significant advances on the numerical differentiation of noisy signals are elaborated in the literature [10] .
- This updated term 𝛿ℱ ̂ captures the unknown dynamics of the system as well as the disturbances during each period 𝑇 and then brings the required changes in the control input by compensation.
- Herein, the authors proceed with a different way by employing a sophisticated tool rather than the PID structure where the main goal is to ensure the asymptotic convergence, towards the origin, of the tracking errors of closed-loop of model (10) .
- To achieve the desired specifications, by using another control procedure, is more challenging.

### IV. IDA-PBC BASED AUXILIARY INPUT

- In the following, trajectory tracking control is achieved using the IDA-PBC approach.
- The authors modify the total energy function of (10) to assign the desired equilibrium and damping injection matrix to meet the asymptotic stability.
- To preserve the energy interpretation, the closed-loop system is presented in a Port-Controlled Hamiltonian (PCH) representation.

### A. System energy and PCH model

- This system has a natural stable equilibrium configuration.
- From (12), the dynamic of the quadrotor can be written as We take the reference trajectories as desired equilibrium configuration.the authors.the authors.

### C. Energy shaping & damping injection

- The controller is obtained by substituting (23) in (15) and making the resulting equations equal to (20).
- 𝐶 deals with the unknown parts and allows to maintain a certain level of robustness, 𝑢 𝐸𝑆 allows to meet the desired specification through the target model components and finally the damping injection term 𝑢 𝐷𝐼 in order to guarantee a damped response.

### D. Control design for quadrotor and stability analysis

- Closed-loop of system ( 12) written under PCH form, using control law ( 31) is asymptotically stable, also known as EQUATION Theorem 1.
- Following the IDA-PBC approach described above, closed loop system ( 12 So, closed loop of system (12) is asymptotically stable, also known as Proof.

### V. RESULTS AND DISCUSSION

- The authors test the effectiveness of the proposed controller not only in the ideal case but also in the presence of different disturbances.
- For the sake of further comparison, the authors follow the same protocol and fit the same conditions.
- The control parameters are tuned, using Genetic Algorithms (GA), in the ideal case then kept for the entire proposed scenarios and for which the objective is to reduce the steady state errors.

### Target model Eq (20)

- ℱ, ℬ promising results (for details one may refer to [5] ) whilst the second one is traditionally applied for quadrotors i.e.
- The authors add the sensor noise on the states of the system.
- These accelerations are considered as perturbations added to the equations related to the forces in the quadrotor model.
- 3 , the three controllers exhibit an acceptable behavior with moderate consumed energy regardless the external effect.
- The accuracy is almost the same using the MFC or R-MFC, which demonstrate the efficiency of the online estimation of the disturbance.

### VI. CONCLUSION

- It uses an auxiliary input and by bringing some changes (see Section III-IV), it operates in closed loop form.
- It improves the performance with respect to structured and unstructured uncertainties.
- Numerical simulations have been performed using the non-linear dynamic model of the quadrotor in order to test the effectiveness of the designed controller.
- The good efficiency of their approach is demonstrated in multiple test scenarios.
- The settling time is shown to be quite fast with good accuracy and a high level of robustness is ensured with respect to parameters uncertainties or external disturbances.

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##### References

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### "3D trajectory tracking control of q..." refers background in this paper

...Commonly, the control input of IDA-PBC is decomposed into two terms [11]....

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### "3D trajectory tracking control of q..." refers methods in this paper

...In a certain point of view, the control of a system with a model free has already been used, since many decades, on the basis of fuzzy logic control or the more popular one for linear systems through Ziegler-Nichols method [6]....

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### "3D trajectory tracking control of q..." refers background in this paper

...Many significant advances on the numerical differentiation of noisy signals are elaborated in the literature [10]....

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180 citations

### "3D trajectory tracking control of q..." refers background in this paper

...For this reason, many studies have led to the development of sophisticated and robust nonlinear control laws (as for instance [1-3])....

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### "3D trajectory tracking control of q..." refers background in this paper

...For this reason, many studies have led to the development of sophisticated and robust nonlinear control laws (as for instance [1-3])....

[...]