# Finite-Time Height Control of Quadrotor UAVs
## 5 Simulation and Experiment Results  
Due to the working characteristics of the quadrotor UAV, it usually includes two working modes of fixed height and lifting in actual flight tasks. Therefore, the simulation in this section consists of two parts, i.e., tracking a fixed signal \(z_{d}(t)=5[\mathrm{m}]\) and a periodic signal \(z_{d}(t)=2+0.5 sin(0.01t)[\mathrm{m}]\), so as to verify the tracking effect of the simulation on different desired signals. In this simulation, the external disturbance \(\xi(t)\) is simulated as a periodic function given as \(\xi(t)=5(sin(0.01t)+cos(0.01t))\). The control period of the system is 0.002 s and the initial value of each state is zero. Figure 2: Dual closed-loop height control strategy structure diagram. In this paper, the parameters include observer parameters \(\beta_{1}\), \(\beta_{2}\), and \(\beta_{3}\), and controller parameters \(\alpha_{1}\), \(\alpha_{2}\), \(k_{1}\), \(k_{2}\), and \(k_{3}\). For the observer parameters, \(\beta_{i}\) with \(i=1,2,3\) are required to be positive. In general, we first choose parameter \(\beta_{1}\) and then solve suitable \(\beta_{2}\) and \(\beta_{3}\) according to the conditions in Theorem 1. Our experience shows that a satisfactory estimation result is usually obtained by setting \(\beta_{1}=30\), \(\beta_{2}=5\) and \(\beta_{3}=800\). For the controller parameters, positive values \(\alpha_{1}\), \(\alpha_{2}\),\(k_{1}\), \(k_{2}\) and \(k_{3}\) represent the gains of the inner- and outer-loop controllers. They are selected based on the stability conditions in Theorems 2 and 3. Increasing these gains will speed up convergence but may result in a larger oscillation in the initial stage. When tracking \(z_{d}(t)=5\)m, the parametric gains of the finite-time ESO are \(\beta_{1}=30\), \(\beta_{2}=5\), \(\beta_{3}=800\) and \(n=0.5\). The parameters of the super-twisting controller are \(\alpha_{1}=100\), \(\alpha_{2}=2\) and \(b_{0}=1\). The parameters of the integral sliding mode controller are \(k_{1}=8\), \(k_{2}=0.01\) and \(k_{3}=0.5\). Figure 3 shows the simulation results of tracking fixed signals. As can be seen from Figure 3, when tracking the fixed signal \(z_{d}(t)=5[\text{m}]\), the actual height \(z(t)\) converges to the expected height \(z_{d}(t)\) quickly and accurately. Figure 4 shows the tracking error for the fixed signal. As can be seen from Figure 4, the tracking error \(e(t)\) quickly converges to zero and remains stable with the error not exceeding 0.003 m. Figure 5 shows the curve of the inner-loop control system when the fixed signal is tracked. Figure 4: Tracking error for the fixed signal. Figure 3: Simulation result of tracking the fixed signal. As can be seen from Figure 5, \(\tau(t)\) is the expected value of the inner-loop control system. Under the control scheme designed for the inner loop, \(x_{1}(t)\) tracks \(\tau(t)\) stably, and the error converges to a small enough size within a finite time. \(z_{1}(t)\) is the estimation signal of the finite-time ESO designed in this paper. It can be seen that state \(x_{1}(t)\) can be estimated by \(z_{1}(t)\) accurately and in real time, which proves the effectiveness of the finite-time ESO. Figure 6 shows the disturbance observation curve of the finite-time ESO. Figure 6 shows the tracking effect of the external disturbance \(\xi(t)\) by the finite-time ESO. It can be seen that after a period of convergence time, the estimated value \(z_{2}(t)\) is simulated for the external disturbance \(\xi(t)\), and the peak value is almost the same as the trend. Therefore, the inner-loop controller can effectively reduce the influence of external disturbance \(\xi(t)\) by compensating the estimated value of \(z_{2}(t)\). When tracking \(z_{d}(t)=2+0.5 sin(0.01t)[\text{m}]\), the parametric gains of the finite-time ESO used in simulation are \(\beta_{1}=30\), \(\beta_{2}=5\), \(\beta_{3}=700\) and \(n=0.5\). The parameters of the super-twisting controller are \(\alpha_{1}=600\), \(\alpha_{2}=2\) and \(b_{0}=1\).