Simulation test 3

In this simulation test, the settings are the same as before except for the added disturbance, it is set as a slowly time-varying signal:

dz(t) = 2 + 8 sin(0.1t)

The result of altitude and its error is shown in Figure 3.12.

time(s) 0 50 100 150 200 altitude(cm) 0 50 100 150 200 z_(t) ˆ z(t + τnom) z_set time(s) 0 50 100 150 200 altitude error(cm) 0 50 100 150 200 error of altitude 100 150 200 -1 -0.5 0 0.5

Figure 3.12 – Simulation Test3-Altitude and error of altitude result It is shown that under time-varying disturbance dz, the designed controller

successfully stabilizes robot at desired altitude, and the error of altitude oscillates only in a small region near the origin, which is acceptable in our application scenario.

As for the switching signal and disturbance term estimation results, they are shown in Figure 3.13.

It can be seen that the switching signal is well estimated, and the disturbance estimation result follows the time-varying signal after it has converged.

In general, the simulation results prove that the designed predictor-based controller with disturbance compensation can successfully stabilize the NON-A blimp prototype robot at desired altitude in the presence of various perturba- tions (system parameter identification inaccuracy, time-varying delay, external constant or slowly varying disturbance). Later it will be implemented on both the NON-A blimp prototype and NON-A blimp V2 (by setting τnom= 0) and the

3.6. Conclusion 95 time(s) 0 50 100 150 200 switching signal 0 1 2 3 4 5 ˆ σ σ time(s) 0 50 100 150 200 disturbance -10 0 10 20 30 40 50 60 dz(t) ˆ dz(t) ˆ dz(t + τnom)

Figure 3.13 – Simulation Test3-Switching signal and disturbance estimation result

experiment result will be presented in Chapter 5.

3.6

Conclusion

In this chapter, we concentrate on one of the blimp decoupled dynamics, which is the altitude stabilization control.

First, the blimp altitude control nominal model is complemented with a disturbance term and the system is expressed in state space form. Based on the parameter identification results, the altitude control system for blimp NON-A prototype is considered as aswitched system with a constant time-delay comple- mented with uncertain bounded disturbances.

Then for the purpose of designing an output feedback controller for the system, we first design an observer for state and switching signal estimation. To this end, HG, HOSM and HOMD differentiators are presented briefly and compared. HOMD differentiator is finally chosen for the estimation task.

Next, for the controller design, based on the error between filter output and altitude measurement, a real-time disturbance estimator is conceived. With the aim of compensating the time-delay of NON-A prototype robot, the state is predicted by Smith predictor, then in the controller the predicted state is used together with the disturbance compensation term. The controller gain is determined by a common Lyapunov function approach for the switched

96 CHAPTER 3. Altitude Control parameters.

Finally, simulations are carried out by MATLAB Simulink to validate the designed controller and verify the performances of disturbance estimation and compensation method. The results demonstrate that designed controller is ready to be applied on the robot for real tests, which will be presented in Chapter 5.

Chapter

4

Horizontal Plane Movement Control

4.1

Introduction

From the analysis presented in Chapter 2, under reasonable assumptions, the blimp motion is decoupled into two independent parts. In Chapter 3, the blimp altitude motion controller is conceived. In this chapter, we focus on the controller design for the blimp motion in the horizontal plane.

First, a complete description for the blimp planar movement model is given (Section 4.2), moreover, two approaches are discussed to transform the under- actuated system to a simpler form for the ease of controller design. Then, for the purpose of compensating the disturbance in controller, a method to estimate perturbations in real-time is proposed (Section 4.3). Next, a disturbance compensation based robust controller is designed for tracking a predefined trajectory (Section 4.4). Finally, simulations are made to verify the designed controller performance and disturbance estimation result (Section 4.5).

4.2

System description

Similar to the altitude control system, for the NON-A blimp V2 robot, the parameter identified (with the help of OptiTrack camera system) in Section 2.6.2 can be used to establish a nominal model for the blimp planar movement control system. Then in order to ensure the accuracy of control, disturbance

98 CHAPTER 4. Horizontal Plane Movement Control terms are added to the planar movement nominal model, which represent the errors between nominal model and real one, they include the errors caused by:

• Nominal model parameter identification inaccuracy; • Airflow perturbation to the balloon;

• Temperature change in testing environment (which influences the buoy- ancy force of the balloon);

• Ignored motor dynamics during modeling process; • Disturbance caused by the altitude movement;

• Other environmental disturbances which are impossible to be accurately modeled.

Thus the planar movement nominal model (2.29) complemented with distur- bance terms becomes:

           ¨ x = cψbu + κ1(ax, ay, ψ) ˙x + κ3(ax, ay, ψ) ˙y + dx ¨

y = sψbu + κ2(ax, ay, ψ) ˙y + κ3(ax, ay, ψ) ˙x + dy

¨

ψ = bψv + aψψ + d˙ ψ

(4.1)

where dx, dyand dψ are the disturbance terms which are estimated on-line, they

are assumed to be small, bounded and smooth. Moreover, a dimensional analysis shows that dxand dyhas the same unit as ¨x (or ¨y), while dψ has the same unit as

¨

ψ.

Recall that u and v are the two control inputs, with u = uleft+ uright and

v = uleft−uright; uright and uleft are respectively the value of command signal for

right and left motors, which are dimensionless quantities (see also (2.29) on page 53); b and bψ are the coefficients related to the control inputs; coefficients

κ1(ax, ay, ψ) = axcψ2 + aysψ2, κ2(ax, ay, ψ) = ayc2ψ+ axs2ψ and κ3(ax, ay, ψ) = axcψsψ−

aycψsψ. As we have mentioned before, the two damping coefficients in lateral

and longitudinal direction are assumed to be approximately equal, i.e. ax= ay,

thus the terms κ1(ax, ay, ψ) = κ2(ax, ay, ψ) = ax = ay, and κ3(ax, ay, ψ) = 0. The

inaccuracy caused by this assumption is also included in disturbance terms, and can be compensated once dx, dy and dψ are estimated.

4.2. System description 99