Discrete-Time Incremental Backstepping Control with Extended Kalman Filter for UAVs

Abstract

In this study, a discrete-time incremental backstepping (DTIBS) controller with an extended Kalman filter (EKF) is proposed for unmanned aerial vehicles (UAVs) with unknown actuator dynamics. The Taylor series and an approximate discrete method are employed, transforming the second-order continuous-time nonlinear system into a discrete-time nonlinear plant with an incremental input form. The incremental control laws are designed using the incremental nonlinear dynamic inversion (INDI) method and the time-delay control (TDC) method. The TDC is introduced to design the control law, eliminating the need for prior knowledge of the control effectiveness matrix involving some unknown aerodynamic coefficients. In addition, the airflow angle and body rotation rate are selected as key system states, and the EKF is used to design a state estimator to estimate the local state of the small unmanned aerial vehicle closed-loop flight control system under strong noise conditions. The effectiveness of the DTIBS control method with EKF is verified through numerical simulation. The results show that the proposed method can effectively estimate the state under the typical noise characteristics of low-cost sensors, and the closed-loop control systems has good tracking performance and can quickly and effectively track sudden commands.

1. Introduction

In recent years, with the development of unmanned aerial vehicle (UAV) control technology, UAVs have been widely used in military, civilian, and scientific research fields, sparking a global trend in developing and applying UAVs. As the heart of UAVs, the flight control system plays a vital role in sustaining their stable flight and ensuring flight safety throughout their operation. Currently, research on the flight control of UAVs mainly focuses on nonlinear control methods, including dynamic inversion [1], sliding mode control [2,3], fuzzy control [4], backstepping (BS) [5], etc. However, these advanced control methods focus more on the issues of system uncertainty or disturbance, based on the assumption that the system state can be accurately obtained. In fact, due to sensor settings and the nature of their measurements, the state vector of unmanned aerial systems is often incomplete or imprecise. Therefore, researching nonlinear robust control methods for UAVs with unknown actuator dynamics, improving the adaptability of the flight control system, and performing state estimation on the closed-loop control system are hot topics in the field of flight control.

Backstepping (BS) control method is one of the most widely applied nonlinear methodologies for the research of UAV control algorithms, which can be used to ensure the stability and improve the control performance of the control system. The BS method was first proposed by Kokovic et al. [6] in 1991. It decomposes complex nonlinear systems into subsystems that do not exceed the system order and recursively constructs the Lyapunov function of the closed-loop system to obtain feedback controllers. The BS control method is remarkably versatile and can be used in combination with other approaches. The traditional BS has certain limitations [7]: (1) in the process of controller design, the virtual control law needs to be differentiated repeatedly, which will lead to the “explosion of complexity”. (2) It is sensitive to model uncertainty, and a precise mathematical model is required. Dynamic Surface Control (DSC) [8] and Command Filter [9] avoid repeated differentiation of virtual control laws by using filters to obtain the derivatives of command virtual signals. The constant change of parameters during the UAV flight and the uncertainty of disturbances lead to the poor tracking performance of the backstepping control. Several studies have addressed the problem, including the disturbance observers [10,11,12] developed to estimate the uncertainties and disturbances and intelligent methods [13,14], such as neural network (NN) and fuzzy logic systems, so the effect of aerodynamic modeling error was compensated for through the application of these methods.

A type of sensor-based backstepping is proposed by a researcher at the Delft University of Technology (TU Delft), namely incremental backstepping (IBS) [15,16], that is robust to model uncertainty. It is relatively easy to implement, does not require a large number of model parameters, and does not generate a huge amount of computation. IBS began to be widely applied to aircraft controller design including fixed-wing aircraft [17,18,19], hypersonic vehicles [20], and UAVs [7,21,22]. In [17], IBS has been successfully used in fixed-wing flight experiments. The experimental results show that IBS can achieve robust control of easily rectifiable nonlinear control systems and only requires very little knowledge of system dynamics parameters. This also proves that it is reasonable to use IBS in attitude control. In [22], an advanced disturbance observer is used to actively observe and compensate for the comprehensive uncertainty of the reconstructed incremental model, further improving the robustness and fault adaptation of the standard IBS controller. In [7], a discrete incremental backstepping (DTIBS) controller is designed to solve the unknown actuator dynamic problem of the aircraft, and simulation results show that the method can significantly improve the tracking performance of the aircraft. In [23], a modified incremental backstepping controller, capable of stabilizing all state variables simultaneously, is proposed, and the effectiveness of the proposed generic controller is verified through simulation.

Since IBS lies between the sensor approach and the model approach, it is crucial to understand the impact of measurement defects such as bias, noise, and delay as well as model uncertainty on closed-loop systems. The extended Kalman filter (EKF) is able to estimate the state variables of a nonlinear system with minimal estimation error variance, thus appropriately estimating state variables and parameters in a noisy environment. Currently, EKF is widely used in aircraft integrated navigation [24] and state estimation [25,26]. Ref. [25] used the EKF to estimate the state of a fixed-wing UAV in a random wind field. In [26], an EKF is introduced to estimate the state of the actuator for a tilt-rotor unmanned aerial vehicle. In [27], an EKF is used for multi-sensor information fusion. The random drift error of the gyroscope is rectified using various measurement data, resulting in enhanced accuracy of the attitude angle measurement. Among the available research results, most of them are designed for the open-loop model of UAV, and relatively few studies consider the estimation methods under the closed-loop structure of the flight control system.

Time-delayed control (TDC) [28] based on Time Delay estimation (TDE), proposed in 1989, is a model-free method that does not use a control effectiveness matrix. Youcef-Toumi [29] and Hsia [30] gave stability conditions $‖1-{\overline{M}}^{-1}M‖<1$ that must be satisfied by the diagonal matrix $\overline{M}$, where $M$ is the control validity matrix. Cho et al. [31] used a modified Nussbaum function to automatically adjust. However, the improved Nussbaum function method is difficult to implement in practical applications because it is too sensitive to measurement noise. The adaptive scheme developed by Lee et al. [32,33] achieves automatic adjustment of the diagonal matrix of the TDC controller. Wang et al. [34] used TDE to design an IBS controller to reduce dependence on the mathematical model.

In this paper, a DTIBS control method with EKF is proposed, mainly considering the problem of closed-loop flight state estimation of small UAVs under strong noise conditions. After linearization and discretization, the proposed method uses INDI to design the first subsystem and TDC technology to design the virtual control law of the second subsystem. A robust signal is designed to approximate the linearization and discretization of the first subsystem to improve tracking performance. A state estimator based on EKF is designed to estimate the local state of the closed-loop control system for UAVs and use it as a state feedback input to the flight controller once, eliminating the impact of noise on the closed-loop system and achieving stable control of the small UAV under noise conditions.

The organization of this paper is as follows: In Section 2, the problem is proposed and prepared, and the UAV system model is established. In Section 3, a local flight state estimator based on EKF is designed. In Section 4, the controller is designed. Finally, in Section 5, flight simulation is performed.

2. Problem Formulation and Preliminaries

The research plant of this paper is a small UAV. The parameters of UAV are shown in the Appendix A. The model of controlled plant is as follows:

$$\left\{\begin{array}{l}{\dot{\mathit{x}}}_1={\mathit{f}}_1\left({\mathit{x}}_1\right)+\Delta {\mathit{f}}_1\left({\mathit{x}}_1\right)+\left[{\mathit{g}}_1\left({\mathit{x}}_1\right)+\Delta {\mathit{g}}_1\left({\mathit{x}}_1\right)\right]{\mathit{x}}_2+\left[{\mathit{h}}_1\left({\mathit{x}}_1\right)+\Delta {\mathit{h}}_1\left({\mathit{x}}_1\right)\right]\mathit{u}+{\mathit{d}}_1\\ {\dot{\mathit{x}}}_2={\mathit{f}}_2\left({\mathit{x}}_1,{\mathit{x}}_2\right)+\Delta {\mathit{f}}_2\left({\mathit{x}}_1,{\mathit{x}}_2\right)+\left[{\mathit{g}}_2\left({\mathit{x}}_1,{\mathit{x}}_2\right)+\Delta {\mathit{g}}_2\left({\mathit{x}}_1,{\mathit{x}}_2\right)\right]\mathit{u}+{\mathit{d}}_2\\ \mathit{y}={\mathit{x}}_1\end{array}\right.$$

(1)

where the state vectors are ${\mathit{x}}_1={\left[\varphi \hspace{0.33em}\alpha \hspace{0.33em}\beta \right]}^T\in {\Re }^3$ and ${\mathit{x}}_2={\left[p\hspace{0.33em}q\hspace{0.33em}r\right]}^T\in {\Re }^3$, $\mathit{y}$ is the output of the controlled plant, $\mathit{u}={\left[{\delta }_a\hspace{0.33em}{\delta }_e\hspace{0.33em}{\delta }_r\right]}^T\in {\Re }^3$ is the input vector of the vehicle, ${\mathit{d}}_1$ and ${\mathit{d}}_2$ are the external disturbances. The functions ${\mathit{f}}_i\left(·\right)$, ${\mathit{g}}_i\left(·\right)$, and ${\mathit{h}}_1\left(·\right)$ are expressed as follows:

$${\mathit{f}}_1\left({\mathit{x}}_1\right)=\frac{\overline{q}S}{m{V}_T}\left[\begin{array}{c}0\\ -\frac{C_{L,0}+C_{L,\alpha }\alpha +C_{L,\beta }\left|\beta \right|+T\sin \alpha -mg\left(\sin \alpha \sin \theta +\cos \alpha \cos \varphi \cos \theta \right)}{\cos \beta }\\ \begin{array}{l}{C}_{Y,\beta }\beta -T\sin \beta \cos \alpha +mg\left(\cos \alpha \sin \beta \sin \theta +\cos \beta \sin \varphi \cos \theta -\right.\\ \left.\sin \alpha \sin \beta \cos \varphi \cos \theta \right)\end{array}\end{array}\right]$$

$${\mathit{g}}_1\left({\mathit{x}}_1\right)=\left[\begin{array}{ccc}1& \sin \varphi \tan \theta & \cos \varphi \tan \theta \\ -\cos \alpha \tan \beta & 1& -\sin \alpha \tan \beta \\ \sin \alpha & 0& -\cos \alpha \end{array}\right]$$

$${\mathit{h}}_1\left({\mathit{x}}_1\right)=\frac{\overline{q}S}{m{V}_T\cos \beta }\left[\begin{array}{ccc}0& 0& 0\\ -C_{L,{\delta }_a}& -C_{L,{\delta }_e}& -C_{L,{\delta }_r}\\ C_{Y,{\delta }_a}\cos \beta & C_{Y,{\delta }_e}\cos \beta & C_{Y,{\delta }_r}\cos \beta \end{array}\right]$$

$${\mathit{f}}_2\left({\mathit{x}}_1,{\mathit{x}}_2\right)=\left[\begin{array}{c}\begin{array}{cc}\hfill & c_{1}pq+c_{2}qr+\overline{q}Sb\left(c_3{C}_{l,\beta }\beta +c_4{C}_{n,\beta }\beta \right)+\frac{\overline{q}S{b}^2}{2V_T}\left[c_3\left(C_{l,p}p+C_{l,r}r\right)\right.+\hfill \\ \hfill & \left.c_4\left(C_{n,p}+C_{n,r}r\right)\right]\hfill \end{array}\\ c_{5}pr-c_6\left(p^2-r^2\right)+\overline{q}S\overline{c}\left(C_{m,0}+C_{m,\alpha }\alpha +C_{m,\beta }\beta \right)+\frac{\overline{q}S{\overline{c}}^2}{2V_T}{c}_7{C}_{m,q}\\ \begin{array}{cc}\hfill & c_{8}pq-c_{1}qr+\overline{q}Sb\left(c_4{C}_{l,\beta }\beta +c_9{C}_{n,\beta }\beta \right)+\frac{\overline{q}S{b}^2}{2V_T}\left[c_4\left(C_{l,p}p+C_{l,r}r\right)+\right.\hfill \\ \hfill & \left.c_9\left(C_{n,p}p+C_{n,r}r\right)\right]\hfill \end{array}\end{array}\right]$$

$${\mathit{g}}_2\left({\mathit{x}}_1,{\mathit{x}}_2\right)=\overline{q}S\left[\begin{array}{ccc}b{c}_3{C}_{l,{\delta }_a}+b{c}_4{C}_{n,{\delta }_a}& b{c}_3{C}_{l,{\delta }_e}+b{c}_4{C}_{n,{\delta }_a}& b{c}_3{C}_{l,{\delta }_r}+b{c}_4{C}_{n,{\delta }_r}\\ \overline{c}{c}_7{C}_{m,{\delta }_a}& \overline{c}{c}_7{C}_{m,{\delta }_e}& \overline{c}{c}_7{C}_{m,{\delta }_r}\\ b{c}_4{C}_{l,{\delta }_a}+b{c}_9{C}_{n,{\delta }_a}& b{c}_4{C}_{l,{\delta }_e}+b{c}_9{C}_{n,{\delta }_e}& b{c}_4{C}_{l,{\delta }_r}+b{c}_9{C}_{n,{\delta }_r}\end{array}\right]$$

where $b$ is the wingspan, $S$ is the wing area, $\overline{q}$ is the dynamic pressure, $\overline{c}$ is the average aerodynamic chord of the wing, and $T$ is the thrust of engine; $C_{L,•}$, $C_{D,•}$ and $C_{Y,•}$ are the lift derivative, drag derivative, and side force derivative, respectively; and $C_{l,•}$, $C_{m,•}$, and $C_{n,•}$ are the roll moment derivative, pitch moment coefficient, and yaw moment coefficient, respectively; $V_T$ represents the true airspeed; $\alpha$ denotes the angle of attack, and $\beta$ denotes the sideslip angle; ${\left[\varphi \hspace{0.33em}\theta \hspace{0.33em}\psi \right]}^T$ are attitude angles, which includes roll angle, pitch angle, and yaw angle; ${\left[p\hspace{0.33em}q\hspace{0.33em}r\right]}^T$ represent angular rates around body axis; $m$ denotes the mass of the UAV. Additionally, the coefficient $c_i\left(i=1,2,\cdots ,9\right)$ is defined as $c_1=\frac{(I_x-I_y+I_z)I_{xz}}{c_N}$, $c_2=\frac{(I_y-I_z)I_z-I_{xz}^2}{c_N}$, $c_3=\frac{I_z}{c_N}$, $c_4=\frac{I_{xz}}{c_N}$, $c_5=\frac{I_z-I_x}{I_y}$, $c_6=\frac{I_{xz}}{I_y}$, $c_7=\frac{1}{I_y}$, $c_8=\frac{(I_x-I_y)I_x+I_{xz}^2}{c_N}$, $c_9=\frac{I_x}{c_N}$, $c_N=I_x{I}_z-I_{xz}^2$, where $I_x$, $I_y$, and $I_z$ are roll, pitch, and yaw moments of inertia, respectively; $I_{xz}$ is a product moment of inertia.

Assumption 1 ([7]). The control surface deflection has negligible effects on the aerodynamic force component; i.e., ${\mathit{h}}_1\left({\mathit{x}}_1\right)\mathit{u}\approx \mathbf{0}$.

Lee [35] et al. proved that ${\mathit{h}}_1$ can be ignored, so we made Assumption 1 based on Ref. [35].

According to Assumption 1, the controlled system can be rewritten as follows:

$$\left\{\begin{array}{l}{\dot{\mathit{x}}}_1={\mathit{f}}_1\left({\mathit{x}}_1\right)+{\mathit{g}}_1\left({\mathit{x}}_1\right){\mathit{x}}_2+{\overline{\mathit{d}}}_1\\ {\dot{\mathit{x}}}_2={\mathit{f}}_2\left({\mathit{x}}_1,{\mathit{x}}_2\right)+{\mathit{g}}_2\left({\mathit{x}}_1,{\mathit{x}}_2\right)\mathit{u}+{\overline{\mathit{d}}}_2\\ \mathit{y}={\mathit{x}}_1\end{array}\right.$$

(2)

where ${\overline{\mathit{d}}}_1=\Delta {\mathit{f}}_1\left({\mathit{x}}_1\right)+\Delta {\mathit{g}}_1\left({\mathit{x}}_1\right){\mathit{x}}_2+\left[{\mathit{h}}_1\left({\mathit{x}}_1\right)+\Delta {\mathit{h}}_1\left({\mathit{x}}_1\right)\right]\mathit{u}+{\mathit{d}}_1$ and ${\overline{\mathit{d}}}_2=\mathsf{\Delta }{\mathit{f}}_2\left({\mathit{x}}_1,{\mathit{x}}_2\right)+\Delta {\mathit{g}}_2\left({\mathit{x}}_1,{\mathit{x}}_2\right)\mathit{u}+{\mathit{d}}_2$.

Assumption 2 ([7]). The composite disturbances ${\overline{\mathit{d}}}_i\left(i=1,2\right)$ are bounded. That is, $\forall {\phi }_i\in {\Re }^{+}$ ; then, ${\overline{\mathit{d}}}_i\left(i=1,2\right)$ satisfy $‖{\overline{\mathit{d}}}_i‖\le {\phi }_i$.

3. DTIBS Controller with EKF

In this section, the design of the UAV flight controller is introduced. For the nonlinear UAV system with uncertainty, a discrete time incremental step-back controller is proposed. Since the signals of the main flight states of the small UAV system are critical for flight attitude control and are significantly affected by sensor noise, a state estimator based on an extended Kalman filter (EKF) is therefore designed to estimate the local state of the closed-loop flight control system. To this end, the aircraft incremental model is reformulated, and the novel DTIBS controller with EKF is developed.

3.1. Design of a Local Flight State Estimator Based on EKF

In this section, the state estimator based on EKF is developed. Some system states ${\mathit{x}}^F$, measurement outputs ${\mathit{y}}^m$, and control inputs $\mathit{u}$, which are severely affected by sensor noise in the Six Degrees of Freedom(6-DOF) nonlinear system of the UAV (1), can be selected for focused analysis as shown in Equation (3):

$$\left\{\begin{array}{l}{\mathit{x}}^F=[\begin{array}{ccccc}\alpha & \beta & p& q& r\end{array}{]}^{\mathrm{T}}\\ \mathit{u}=[\begin{array}{ccc}{\delta }_a& {\delta }_e& {\delta }_r\end{array}{]}^{\mathrm{T}}\\ {\mathit{y}}^m=[\begin{array}{ccccc}\alpha & \beta & p& q& r\end{array}{]}^{\mathrm{T}}\end{array}\right.$$

(3)

where ${\left[{\delta }_a\hspace{0.33em}{\delta }_e\hspace{0.33em}{\delta }_r\right]}^T$ are aileron deflection, elevator deflection, and heading deflection, respectively.

Then, the nonlinear system model and observation model of the discussed UAV system can be written as Equations (4) and (5), respectively:

$$\dot{\mathit{x}}=\mathit{f}(\mathit{x},t)+\mathit{g}(\mathit{x},t)\mathit{u}$$

(4)

$$\mathit{y}=\mathit{h}(\mathit{x},t)$$

(5)

For a small UAV system, there is unavoidable system noise $\mathit{w}(t)$ and measurement noise $\mathit{v}(t)$ in the system equation and measurement equation, respectively. The UAV system model with noise can be rewritten as Equations (6) and (7):

$$\dot{\mathit{x}}=\mathit{f}(\mathit{x},t)+\mathit{g}(\mathit{x},t)\mathit{u}+\mathit{w}(t)$$

(6)

$$\mathit{y}=\mathit{h}(\mathit{x},t)+\mathit{v}(t)$$

(7)

According to the nonlinear continuous system model (6) and (7) of the UAV, the linear discrete differential Equations (8) and (9) with noise can be derived:

$${\mathit{x}}_{k+1}^F={\mathit{\Phi }}_k{\mathit{x}}_k^F+{\mathit{G}}_k{\mathit{u}}_k+{\mathit{w}}_k$$

(8)

$${\mathit{y}}_k^F={\mathit{H}}_k+{\mathit{v}}_k$$

(9)

where $\hspace{0.33em}{\mathit{x}}_k^F$, ${\mathit{u}}_k$, and ${\mathit{y}}_k^F$ are the state vector, input vector, and measurement vector of the state estimator at time $k$, respectively; ${\mathit{\Phi }}_k$, ${\mathit{G}}_k$, and ${\mathit{H}}_k$ are the state transition matrix, control input matrix, and measurement matrix at time k, respectively; ${\mathit{w}}_k$ and ${\mathit{v}}_k$ represent process noise and measurement noise, respectively, and are usually assumed to be Gaussian white noise. Then, it has the following statistical characteristics:

$$\left\{\begin{array}{l}E({\mathit{w}}_k)=0,E({\mathit{v}}_k)=0\\ {\mathit{R}}_k=E({\mathit{v}}_k,{\mathit{v}}_k{}^{\mathrm{T}}),{\mathit{Q}}_k=E({\mathit{w}}_k,{\mathit{w}}_k{}^{\mathrm{T}})\end{array}\right.$$

(10)

The transition matrix at time $k$ can be approximated by first-order approximation using Equation (11):

$${\mathit{\Phi }}_k\approx \mathit{I}+{\mathit{F}}_k{T}_s$$

(11)

where $T_s$ is the discrete time; ${\mathit{F}}_k$ is the continuous system dynamic matrix. According to the state estimate of the system at the nearest time, the continuous nonlinear Equation (6) is linearized to obtain ${\mathit{F}}_k$:

$${\mathit{F}}_k={\left.\frac{\partial \mathit{f}\left(\mathit{x},\mathit{u}\right)}{\partial {\mathit{x}}^F}\right|}_{{\mathit{x}}^F={\widehat{\mathit{x}}}_{{}_{k/k}}^F,\mathit{u}={\mathit{u}}_k}=\left[\begin{array}{cccc}{F}_{11}& F_{12}& \dots & F_{15}\\ F_{21}& F_{22}& \cdots & F_{25}\\ ⋮& ⋮& \ddots & ⋮\\ F_{51}& F_{52}& \cdots & F_{55}\end{array}\right]$$

(12)

According to the nonlinear dynamic model of the UAV, the elements of the continuous system dynamic matrix ${\mathit{F}}_k$ in Equation (12) are as follows:

$$\begin{array}{c}{F}_{11}=\frac{\overline{q}S{b}^2{C}_{Lp}{I}_z}{2c_N}+\frac{I_x{I}_{xz}-I_{xz}{I}_y+I_{zx}{I}_z}{2c_N{V}_T}q, F_{12}=\frac{I_x{I}_{xz}-I_{xz}{I}_y+I_{zx}{I}_z}{c_N}p+\frac{I_y{I}_z-I_{xz}{}^2-I_z{}^2}{c_N}r,\hfill \\ F_{13}=\frac{\overline{q}S{b}^2(C_{Nr}{I}_{xz}+C_{Lr}{I}_z)}{2c_N{V}_T}+\frac{(I_y{I}_z-I_{xz}{}^2-I_z{}^2)}{c_N}q, F_{14}=0, F_{15}=\frac{\overline{q}Sb(I_z{C}_{L\beta }+I_{xz}{C}_{N\beta })}{c_N},\hfill \\ F_{21}=-\frac{2I_{zx}}{I_y}p+\frac{I_z-I_x}{I_y}rs, F_{22}=-\frac{\overline{q}S{c}^2{C}_{Mq}}{2I_y{V}_T}, F_{23}=\frac{I_z-I_x}{I_y}p+\frac{2I_{xz}}{I_y}r, F_{24}=\frac{\overline{q}Sc{C}_{M\alpha }}{I_y},\hfill \\ F_{25}=0, F_{31}=\frac{\overline{q}S{b}^2{C}_{Lp}{I}_{zx}}{2c_N{V}_T}+\frac{I_x^2-I_x{I}_y+I_{zx}^2}{c_N}q,\hfill \\ F_{32}=\frac{I_x^2-I_x{I}_y+I_{zx}^2}{c_N}p-\frac{I_x{I}_{xz}-I_{zx}{I}_y+I_{zx}{I}_z}{c_N}r,\hfill \\ F_{33}=\frac{\overline{q}S{b}^2(C_{Lr}{I}_{zx}+C_{Nr}{I}_x)}{2c_N{V}_T}-\frac{I_x{I}_{xz}-I_{zx}{I}_y+I_{zx}{I}_z}{c_N}q, F_{34}=0, F_{35}=\frac{\overline{q}Sb(C_{L\beta }{I}_{xz}+C_{N\beta }{I}_x)}{c_N},\hfill \\ F_{41}=0, F_{42}=1, F_{43}=0, F_{44}=\frac{\overline{q}S(C_{Z\alpha }+C_{X1})}{m{V}_T}, F_{45}=0, \hfill \\ F_{51}=0, F_{52}=0, F_{53}=-1, F_{54}=0, F_{55}=\frac{\overline{q}S{C}_{Y\beta }}{m{V}_T}.\hfill \end{array}$$

Meanwhile, the discrete control input matrix in system (8) is

$${\mathit{G}}_k={\mathit{G}}_c(k)\cdot T_s$$

(13)

where ${\mathit{G}}_c(k)$ is the continuous control input matrix, which can be calculated by Equation (14):

$${\mathit{G}}_c(k)=\left[\begin{array}{c}{\mathit{M}}_{CA}\\ {\mathit{0}}_{2\times 3}\end{array}\right]$$

(14)

where ${\mathit{M}}_{CA}$ is usually called the control efficiency matrix, which mainly reflects the efficiency of the UAV control surfaces. There is

$${\mathit{M}}_{CA}=\left[\begin{array}{ccc}b& 0& 0\\ 0& \overline{c}& 0\\ 0& 0& b\end{array}\right]\left[\begin{array}{ccc}{c}_3{C}_{L\delta a}& 0& c_4{C}_{N\delta r}\\ 0& c_7{C}_{M\delta e}& 0\\ c_4{C}_{L\delta a}& 0& c_9{C}_{N\delta r}\end{array}\right]$$

(15)

Similarly, the measurement matrix ${\mathit{H}}_k$ can also be obtained by linearizing the nonlinear measurement Equation (16) based on the estimated value of the system state ${\widehat{\mathit{x}}}_{k/k}^F$ at the nearest moment:

$${\mathit{H}}_k={\left.\frac{\partial \mathit{h}(\mathit{x})}{\partial {\mathit{x}}^F}\right|}_{{\mathit{x}}^F={\widehat{\mathit{x}}}_{{}_{k/k}}^F}=\mathit{I}$$

(16)

Then, the prediction and update equations of the EKF design process can be given by Equations (17)–(22).

Equation (17) is the state prediction. Considering that the flight control method used in this study is the DTIBS method, $\mathit{f}(\mathit{x},t)$ and $\mathit{g}(\mathit{x},\mathit{u},t)$ will be calculated at each moment during the design process of the controller. The estimated value ${\dot{\mathit{x}}}_k^F$ of the state differential ${\dot{\mathit{x}}}_k^F$ at time $k$ can be directly obtained. Moreover, the sampling time $T_s$ is small in this study, so the prediction of the system state can be calculated using Equation (18).

$${\widehat{\mathit{x}}}_{k+1/k}^F={\mathit{\Phi }}_k{\widehat{\mathit{x}}}_k^F+{\mathit{G}}_k{\mathit{u}}_k$$

(17)

(1)

State prediction equation:

$${\mathit{P}}_{k+1/k}={\mathit{\Phi }}_k{\mathit{P}}_{k/k}{\mathit{\Phi }}_k^{\mathrm{T}}+{\mathit{G}}_k{\mathit{Q}}_k{\mathit{G}}_k^{\mathrm{T}}$$

(18)

(2)

Variance matrix prediction equation:

$${\mathit{P}}_{k+1/k}={\mathit{\Phi }}_k{\mathit{P}}_{k/k}{\mathit{\Phi }}_k^{\mathrm{T}}+{\mathit{G}}_k{\mathit{Q}}_k{\mathit{G}}_k^{\mathrm{T}}$$

(19)

(3)

Kalman gain matrix equation:

$${\mathit{K}}_{k+1}=\hspace{0.33em}{\mathit{P}}_{k+1/k}{\mathit{H}}_k^{\mathrm{T}}{[{\mathit{H}}_k^{}{\mathit{P}}_{k+1/k}{\mathit{H}}_k^{\mathrm{T}}+{\mathit{R}}_k]}^{-1}$$

(20)

(4)

Covariance matrix estimating equation:

$${\mathit{P}}_{k+1/k+1}=[\mathit{I}-{\mathit{K}}_{k+1}{\mathit{H}}_k]{\mathit{P}}_{k+1/k}$$

(21)

(5)

State estimation equation:

$${\widehat{\mathit{x}}}_{k+1}^F={\widehat{\mathit{x}}}_{k+1/k}^F+{\mathit{K}}_{k+1}[{\mathit{y}}_{k+1}^F-\mathit{h}({\widehat{\mathit{x}}}_{k+1/k}^F)]$$

(22)

In the above equation, ${\widehat{\mathit{x}}}_{k+1/k}^F$ is the predicted state variable; ${\mathit{P}}_{k+1/k}$ is the predicted variance matrix; ${\mathit{K}}_{k+1}$ is the gain of Kalman filter; ${\widehat{\mathit{x}}}_{k+1/k+1}^F$ is the state vector estimated by the filter; ${\mathit{P}}_{k+1/k+1}$ is the estimated variance matrix.

3.2. Controller Design and Stability Analysis

Considering the aircraft model (2), a DTIBS controller is designed. The robust terms are introduced in the process of controller design, and the updating laws of the robust terms are obtained using a standard gradient-based adaption method and chain rule.

The state variable after EKF filtering is input to the controller, and the recursive control method backstepping is adopted. The incremental control law is designed as follows:

Step 1: For the first subsystem,

$${\dot{\mathit{x}}}_1={\mathit{f}}_1\left({\mathit{x}}_1\right)+{\mathit{g}}_1\left({\mathit{x}}_1\right){\mathit{x}}_2+{\overline{\mathit{d}}}_1$$

(23)

The composite disturbance is taken as part of function ${\mathit{f}}_1\left({\mathit{x}}_1\right)$, and Taylor series is used to expand the nonlinear dynamics around the operating point $\left[{\mathit{x}}_{1,0},{\mathit{x}}_{2,0}\right]$, rewrite the nonlinear dynamics as an incremental form, and derive the incremental nonlinear control law as follows:

$${\dot{\mathit{x}}}_1\cong {\dot{\mathit{x}}}_{1,0}+\frac{\partial }{\partial {\mathit{x}}_1}\left[{\mathit{f}}_1\left({\mathit{x}}_1\right)+{\overline{\mathit{d}}}_1+{\mathit{g}}_1\left({\mathit{x}}_1\right){\mathit{x}}_2\right]\left|{}_{\begin{array}{l}{\mathit{x}}_1={\mathit{x}}_{1,0}\\ {\mathit{x}}_2={\mathit{x}}_{2,0}\end{array}}\right.\left({\mathit{x}}_1-{\mathit{x}}_{1,0}\right)+\frac{\partial }{\partial {\mathit{x}}_2}\left[{\mathit{f}}_1\left({\mathit{x}}_1\right)+{\overline{\mathit{d}}}_1+{\mathit{g}}_1\left({\mathit{x}}_1\right){\mathit{x}}_2\right]\left|{}_{\begin{array}{l}{\mathit{x}}_1={\mathit{x}}_{1,0}\\ {\mathit{x}}_2={\mathit{x}}_{2,0}\end{array}}\right.\left({\mathit{x}}_2-{\mathit{x}}_{2,0}\right)+{\mathit{\varphi }}_{\hspace{0.33em}1}$$

(24)

where $\partial$ denotes a deviation score operator, and ${\mathit{\varphi }}_{\hspace{0.33em}1}$ is the linearization error, which generated by the expansion process.

Under the assumptions that the sampling frequency is sufficiently high, the linearization error ${\mathit{\varphi }}_{\hspace{0.33em}1}$ is small. In contrast to increment $\Delta {\mathit{x}}_2$, according to the time-scale separation (TSS) principle, $\Delta {\mathit{x}}_1$ is negligible for the system. Let $\mathsf{\Delta }t$ be processed with a constant sampling time, and approximate discretization is used to obtain

$$\left({\mathit{x}}_1-{\mathit{x}}_{1,0}\right)/\Delta \mathit{t}={\dot{\mathit{x}}}_{1,0}+{\mathit{g}}_1\left({\mathit{x}}_{1,0}\right)\Delta {\mathit{x}}_2+{\overline{\varphi }}_{\hspace{0.33em}1}+{\varsigma }_1$$

(25)

where the linearization error is constituted by ${\overline{\mathit{\varphi }}}_{\hspace{0.33em}1}={\mathit{A}}_1\Delta {\mathit{x}}_1+{\mathit{\varphi }}_{\hspace{0.33em}1}$, and ${\mathit{\varsigma }}_1$ is the discretization error. The following equation can be obtained:

$${\mathit{x}}_1\left(k+1\right)={\mathit{x}}_1\left(k\right)+{\dot{\mathit{x}}}_1\left(k\right)\Delta t+\left({\mathit{g}}_1\Delta t\right)\Delta {\mathit{x}}_2\left(k\right)+{\mathit{\zeta }}_1$$

(26)

where ${\mathit{x}}_1\left(k\right)={\mathit{x}}_{1,0}$, and ${\mathit{\zeta }}_1=\left({\overline{\mathit{\varphi }}}_{\hspace{0.33em}1}+{\mathit{\varsigma }}_1\right)\mathsf{\Delta }t$.

Compared with the standard backstepping approach, the command filtered approach proposed by Farrell [9] does not increase and may decrease the effects of measurement noise. Therefore, based on the command filter backstepping method, a discrete time incremental intermediate controller is designed as follows:

$$\Delta {\mathit{x}}_{2d}\left(k\right)=-{\left({\mathit{g}}_1\Delta t\right)}^{-1}\left[-c_1{\tilde{\mathit{x}}}_1\left(k\right)+\left({\dot{\mathit{x}}}_1\left(k\right)-{\dot{\mathit{y}}}_c\left(k\right)\right)\Delta t-{\mathit{y}}_c\left(k\right)\right]$$

(27)

where $c_1>0$ is the controller design parameter, ${\mathit{y}}_c$ is the smooth signal obtained by passing the reference signal ${\mathit{y}}_r$ through a filter, and ${\tilde{\mathit{x}}}_1\left(k\right)={\mathit{x}}_1\left(k\right)-{\mathit{y}}_c\left(k\right)$ is the tracking error.

Then, a robust term ${\widehat{\zeta }}_1\left(k\right)$ is introduced to design a virtual control law for the first subsystem, and Equation (27) can be rewriten as follows:

$$\Delta {\mathit{x}}_{2d}\left(k\right)=-{\left({\mathit{g}}_1\Delta t\right)}^{-1}\left[-c_1{\tilde{\mathit{x}}}_1\left(k\right)+\left({\dot{\mathit{x}}}_1\left(k\right)-{\dot{\mathit{y}}}_c\left(k\right)\right)\Delta t-{\mathit{y}}_c\left(k\right)+{\widehat{\zeta }}_1\left(k\right)\right]$$

(28)

The control algorithm of Equation (28) is replaced, and we have

$${\mathit{z}}_1\left(k+1\right)=c_1{\mathit{z}}_1\left(k\right)+\left({\mathit{g}}_1\Delta t\right){\mathit{z}}_2\left(k+1\right)-{\tilde{\mathit{\zeta }}}_1$$

(29)

where ${\tilde{\mathit{\zeta }}}_1={\widehat{\mathit{\zeta }}}_1-{\mathit{\zeta }}_1$ is the functional estimation error. Thus, the robust term is tuned to minimize the following Lyapunov candidate:

$$E_1\left(k\right)=0.5{\tilde{\mathit{\zeta }}}_1^T\left(k\right){\tilde{\mathit{\zeta }}}_1\left(k\right)$$

(30)

Standard gradient-based adaptive methods and chain rules are used to design the updated law of the robust term:

$$\begin{array}{ll}\mathsf{\Delta }{\widehat{\mathit{\zeta }}}_1\left(k\right)& =-{\lambda }_{1a}\left(\frac{\partial E_{1a}\left(k\right)}{\partial {\widehat{\mathit{\zeta }}}_1\left(k\right)}\right)\\ & =-{\lambda }_{1a}\left(\frac{\partial E_{1a}\left(k\right)}{\partial {\tilde{\mathit{\zeta }}}_1\left(k\right)}\right)\left(\frac{\partial {\tilde{\mathit{\zeta }}}_1\left(k\right)}{\partial {\widehat{\mathit{\zeta }}}_1\left(k\right)}\right)\\ & ={\lambda }_{1a}\left[{\mathit{z}}_1\left(k+1\right)-c_1{\mathit{z}}_1\left(k\right)-\left({\mathit{g}}_1\Delta t\right){\mathit{z}}_2\left(k+1\right)\right]\end{array}$$

(31)

Thus, the updated laws for the robust term ${\widehat{\zeta }}_1\left(k\right)$ can obtain

$${\widehat{\mathit{\zeta }}}_1\left(k+1\right)={\widehat{\mathit{\zeta }}}_1\left(k\right)+\Delta {\widehat{\mathit{\zeta }}}_1\left(k\right)={\widehat{\mathit{\zeta }}}_1\left(k\right)+{\lambda }_{1a}\left[{\mathit{z}}_1\left(k+1\right)-c_1{\mathit{z}}_1\left(k\right)-\left({\mathit{g}}_1\Delta t\right){\mathit{z}}_2\left(k+1\right)\right]$$

(32)

The proof of stability is described below. Consider the following Lyapunov function: $V_1:\hspace{0.33em}{\mathit{D}}_{{\mathit{z}}_1}\times {\mathit{D}}_{{\tilde{\mathit{\zeta }}}_1}\to \Re$

$$V_1\left(k+1\right)=V_{11}\left(k+1\right)+V_{12}\left(k+1\right)$$

(33)

where $V_{11}\left(k+1\right)={\tau }_{11}{\mathit{z}}_1^T\left(k+1\right){\mathit{z}}_1\left(k+1\right)$, $V_{12}\left(k+1\right)={\tau }_{12}{\tilde{\mathit{\zeta }}}_1^T\left(k+1\right){\tilde{\mathit{\zeta }}}_1\left(k+1\right)$.

From Equation (29), taking derivative of $V_{11}$ yields to

$$\begin{array}{ll}\mathsf{\Delta }{V}_{11}\left(k+1\right)& ={\tau }_{11}{‖c_1{\mathit{z}}_1\left(k\right)+\left({\mathit{g}}_1\Delta t\right){\mathit{z}}_2\left(k+1\right)-{\tilde{\mathit{\zeta }}}_1‖}^2-{\tau }_{11}{‖{\mathit{z}}_1\left(k\right)‖}^2\\ & \le -\left(1-c_1^2-1/{\kappa }^2-{\tau }_{11}\right){\tau }_{11}{‖{\mathit{z}}_1\left(k\right)‖}^2+3{\tau }_{11}{‖\left({\mathit{g}}_1\Delta t\right)‖}^2{‖{\mathit{z}}_2\left(k+1\right)‖}^2+\left(2+{\kappa }^2\right){\tau }_{11}{‖{\tilde{\mathit{\zeta }}}_1‖}^2\end{array}$$

(34)

Based on Equations (29) and (32), the difference of $V_{12}$ can obtain

$$\begin{array}{ll}\mathsf{\Delta }{V}_{12}\left(k+1\right)& ={\tau }_{12}{\tilde{\mathit{\zeta }}}_1^T\left(k+1\right){\tilde{\mathit{\zeta }}}_1\left(k+1\right)-{\tau }_{12}{\tilde{\mathit{\zeta }}}_1^T\left(k\right){\tilde{\mathit{\zeta }}}_1\left(k\right)\\ & ={\tau }_{12}{‖\left(1-{\lambda }_{1a}\right){\tilde{\mathit{\zeta }}}_1\left(k\right)‖}^2-{\tau }_{12}{\tilde{\mathit{\zeta }}}_1^T\left(k\right){\tilde{\mathit{\zeta }}}_1\left(k\right)\\ & \le -\left[1-{\left(1-{\lambda }_{1a}\right)}^2\right]{\tau }_{12}{‖{\tilde{\mathit{\zeta }}}_1\left(k\right)‖}^2\end{array}$$

(35)

Combining Equations (34) and (35) gives the following equations:

$$\begin{array}{ll}\mathsf{\Delta }{V}_1\left(k+1\right)\le & -\left(1-c_1^2-1/{\kappa }^2-{\tau }_{11}\right){\tau }_{11}{‖{\mathit{z}}_1\left(k\right)‖}^2\\ & -\left\{\left[1-{\left(1-{\lambda }_{1a}\right)}^2\right]{\tau }_{12}\right.\left.-\left(2+{\kappa }^2\right){\tau }_{11}\right\}\hspace{0.33em}{‖{\tilde{\mathit{\zeta }}}_1\left(k\right)‖}^2\\ & +3{\tau }_{11}{‖\left({\mathit{g}}_1\Delta t\right)‖}^2{‖{\mathit{z}}_2\left(k+1\right)‖}^2\end{array}$$

(36)

According to Equation (36), if the tracking error converges to a small neighborhood of the origin, it is stable under the effect of the virtual control laws.

Therefore, in step 2, we will give the control laws of the controlled plant.

Step 2: With respect to the second subsystem

$${\dot{\mathit{x}}}_2={\mathit{f}}_2\left(\mathit{x}\right)+{\mathit{g}}_2\left(\mathit{x}\right)\mathit{u}$$

(37)

where $\mathit{x}={\left[{\mathit{x}}_1^T,{\mathit{x}}_2^T\right]}^T$.

The dynamics of angular rates can be rewritten as an incremental form by taking the first-order Taylor series expansion around the point $\left[{\mathit{x}}_0,{\mathit{u}}_0\right]$.

$${\dot{\mathit{x}}}_2\cong {\dot{\mathit{x}}}_{2,0}+\frac{\partial }{\partial \mathit{x}}\left[{\mathit{f}}_2\left(\mathit{x}\right)+{\mathit{g}}_2\left(\mathit{x}\right)\mathit{u}\right]\left|{}_{\begin{array}{l}\mathit{x}={\mathit{x}}_0\\ \mathit{u}={\mathit{u}}_0\end{array}}\right.\left(\mathit{x}-{\mathit{x}}_0\right)+\frac{\partial }{\partial \mathit{u}}\left[{\mathit{f}}_2\left(\mathit{x}\right)+{\mathit{g}}_2\left(\mathit{x}\right)\mathit{u}\right]\left|{}_{\begin{array}{l}\mathit{x}={\mathit{x}}_0\\ \mathit{u}={\mathit{u}}_0\end{array}}\right.\left(\mathit{u}-{\mathit{u}}_0\right)$$

(38)

Let ${\mathit{\varphi }}_{\hspace{0.33em}2}$ be the error arising from the process of linearization. Therefore, Equation (35) can be rewritten as

$${\dot{\mathit{x}}}_2={\dot{\mathit{x}}}_{2,0}+{\mathit{A}}_2\Delta \mathit{x}+{\mathit{g}}_2\left({\mathit{x}}_0\right)\Delta \mathit{u}+{\mathit{\varphi }}_2$$

(39)

where $\Delta \mathit{x}=\mathit{x}-{\mathit{x}}_0$, $\Delta \mathit{u}=\mathit{u}-{\mathit{u}}_0$ and ${\mathit{A}}_2=\frac{\partial }{\partial \mathit{x}}\left[{\mathit{f}}_2\left(\mathit{x}\right)+{\mathit{g}}_2\left(\mathit{x}\right)\mathit{u}\right]\left|{}_{\begin{array}{l}\mathit{x}={\mathit{x}}_0\\ \mathit{u}={\mathit{u}}_0\end{array}}\right.$.

According to the assumption proposed in the previous section, due to the TSS principle, compared with the incremental quality $\Delta u$, $\Delta x$ can be neglected. Therefore, Equation (39) is rewritten as

$${\dot{\mathit{x}}}_2={\dot{\mathit{x}}}_{2,0}+{\mathit{g}}_2\left({\mathit{x}}_0\right)\Delta \mathit{u}+{\overline{\mathit{\varphi }}}_2$$

(40)

where ${\overline{\mathit{\varphi }}}_2={\mathit{A}}_2\Delta \mathit{x}+{\mathit{\varphi }}_2$ is the combination error.

Taking actions similar to step 1 and converting Equation (40) into discrete time form yields to:

$${\mathit{x}}_2\left(k+2\right)={\mathit{x}}_2\left(k+1\right)+{\dot{\mathit{x}}}_2\left(k+1\right)\Delta t+\left({\mathit{g}}_2\Delta t\right)\Delta \mathit{u}\left(k+1\right)+{\zeta }_2{}^{\prime }$$

(41)

where ${\zeta }_2{}^{\prime }=\left({\overline{\mathit{\varphi }}}_2+{\varsigma }_2\right)\mathsf{\Delta }t$ is the discretization error.

In actual flight, ${\mathit{g}}_2\left(x\right)$ may not be available or may interfere with aerodynamic parameters. Therefore, in this paper, we assume that the function ${\mathit{g}}_2\left(x\right)$ is unknown.

The time-delay control (TDC) is a good choice for designing incremental control laws when ${\mathit{g}}_2\left(x\right)$ is not available. In addition, Ref. [36] also confirms that TDC and INDI are essentially the same.

We use the TDC method to design an incremental controller $\Delta {\mathit{u}}_d$ as follows:

$$\Delta {\mathit{u}}_d\left(k+1\right)=-{\left({\overline{\mathit{g}}}_2\Delta t\right)}^{-1}\left[-c_2{\tilde{\mathit{x}}}_2\left(k\right)+\left({\dot{\mathit{x}}}_2\left(k+1\right)-{\dot{\mathit{x}}}_{2c}\left(k+1\right)\right)\Delta t\right.\left.+{\mathit{g}}_1^T{\mathit{z}}_1\left(k\right)-{\mathit{x}}_{2c}\left(k+1\right)\right]+\left(\mathrm{Fil}\left({\mathit{u}}_d\left(k+1\right)\right)-{\mathit{u}}_m\left(k+1\right)\right)$$

(42)

where ${\overline{\mathit{g}}}_2$ is a diagonal matrix. $c_2$ is the designed parameter, and $c_2>0$. ${\mathit{u}}_d$ is the sum of the actuator angle ${\mathit{u}}_m\left(k+1\right)$ and the incremental controller $\Delta {\mathit{u}}_d\left(k+1\right)$ measurement, and ${\mathit{u}}_d$ is the input signal of the actuator system. ${\mathit{u}}_m\left(k+1\right)$ represents the deflection angle of the control surface measured by the actuator sensor. And ${\mathit{u}}_m\left(k+1\right)={\mathit{u}}_c\left(k+1-\tau \right)+{\mathit{u}}_n\left(k+1\right)$, where ${\mathit{u}}_c\left(k+1-\tau \right)$ is the deflection angle of the control surface at the time $\left(k+1-\tau \right)$; $\tau$ is a time-delay constant; ${\mathit{u}}_n\left(k+1\right)$ is the measurement errors and noises. Therefore, ${\mathit{u}}_d\left(k+2\right)={\mathit{u}}_m\left(k+1\right)+\Delta {\mathit{u}}_d\left(k+1\right)$, and it may cause an increase in external disturbance to the controller. Therefore, in order to compensate for external disturbances based on the Lyapunov function, we introduce the term $\left(\mathrm{Fil}\left({\mathit{u}}_d\left(k+1\right)\right)-{\mathit{u}}_m\left(k+1\right)\right)$.

According to the incremental controller in Equation (42), the control law is modified as follows:

$$\Delta {\mathit{u}}_d\left(k+1\right)=-{\left({\overline{\mathit{g}}}_2\Delta t\right)}^{-1}\left[-c_2{\tilde{\mathit{x}}}_2\left(k\right)+\left({\dot{\mathit{x}}}_2\left(k+1\right)-{\dot{\mathit{x}}}_{2c}\left(k+1\right)\right)\Delta t\right.+{\mathit{g}}_1^T{\mathit{z}}_1\left(k\right)-{\mathit{x}}_{2c}\left(k+1\right)\left.+{\widehat{\zeta }}_2\right]+\left(\mathrm{Fil}\left({\mathit{u}}_d\left(k+1\right)\right)-{\mathit{u}}_m\left(k+1\right)\right)$$

(43)

where ${\widehat{\zeta }}_2$ is the robust term, and the expression and update rate are described in the following procedure.

Converting the control law Equation (43), we have

$${\mathit{z}}_2\left(k+2\right)=c_2{\mathit{z}}_2\left(k+1\right)-{\left({\mathit{g}}_1\Delta t\right)}^T{\mathit{z}}_1\left(k\right)+\left({\mathit{g}}_2\Delta t\right)\left[\left({\mathit{u}}_c\left(k+2\right)\right.\right.\left.-\left(\mathrm{Fil}\left({\mathit{u}}_d\left(k+2\right)\right)-\mathrm{Fil}\left({\mathit{u}}_d\left(k+1\right)\right)\right)\right]-{\tilde{\zeta }}_2$$

(44)

where ${\tilde{\zeta }}_2={\widehat{\zeta }}_2-{\zeta }_2$, ${\zeta }_2$ is the ideal value.

Based on the standard gradient-based adaptive method and the chain rule, the expression and updated law of the robust term are processed as follows:

$$\Delta {\widehat{\zeta }}_2\left(k+1\right)={\lambda }_{2a}\left[{\mathit{z}}_2\left(k+2\right)-c_2{\mathit{z}}_2\left(k+1\right)+{\left({\mathit{g}}_1\Delta t\right)}^T{\mathit{z}}_1\left(k\right)\right]$$

(45)

$${\widehat{\zeta }}_2\left(k+2\right)={\widehat{\zeta }}_2\left(k+1\right)+\Delta {\widehat{\zeta }}_2\left(k+1\right)={\widehat{\zeta }}_2\left(k+1\right)+{\lambda }_{2a}\left[{\mathit{z}}_2\left(k+2\right)-c_2{\mathit{z}}_2\left(k+1\right)+{\left({\mathit{g}}_1\Delta t\right)}^T{\mathit{z}}_1\left(k\right)\right]$$

(46)

The proof of stability of the controller is described below. Lyapunov function is established as $V_2:\hspace{0.33em}{\mathit{D}}_{z_2}\times {\mathit{D}}_{{\tilde{\zeta }}_2}\to \Re$

$$V_2\left(k+1\right)=V_{21}\left(k+1\right)+V_{22}\left(k+1\right)$$

(47)

where $V_{21}\left(k+1\right)={\tau }_{21}{\mathit{z}}_2^T\left(k+2\right){\mathit{z}}_2\left(k+2\right)$, $V_{22}\left(k+1\right)={\tau }_{22}{\tilde{\zeta }}_2^T\left(k+1\right){\tilde{\zeta }}_2\left(k+1\right)$.

Based on Equation (44), we obtain the difference of $V_{21}\left(k+1\right)$:

$$\begin{array}{ll}\mathsf{\Delta }{V}_{21}\left(k+1\right)\hspace{0.33em}& ={\tau }_{21}{\mathit{z}}_2^T\left(k+2\right){\mathit{z}}_2\left(k+2\right)-{\tau }_{21}{\mathit{z}}_2^T\left(k+1\right){\mathit{z}}_2\left(k+1\right)\\ & \le -{\tau }_{21}\left(1-c_2^2-3/{\kappa }^2\right){‖{\mathit{z}}_2\left(k+1\right)‖}^2+{\tau }_{21}\left(1+{\kappa }^2+2/{\kappa }^2\right){‖{\mathit{g}}_1\Delta t‖}^2{‖{\mathit{z}}_1\left(k\right)‖}^2\\ & +{\tau }_{21}\left(1+{\kappa }^2+2/{\kappa }^2\right){‖{\mathit{g}}_1\Delta t‖}^2{\psi }^2+{\tau }_{21}\left(1+3{\kappa }^2\right){‖{\tilde{\zeta }}_2‖}^2\end{array}$$

(48)

From Equations (32) and (44), we derive $\Delta V_{22}\left(k+1\right)$ as follows:

$$\mathsf{\Delta }{V}_{22}\left(k+1\right)={\tau }_{22}{‖\left(1-{\lambda }_{2a}\right){\tilde{\zeta }}_2\left(k\right)‖}^2-{\tau }_{22}{\tilde{\zeta }}_2^T\left(k\right){\tilde{\zeta }}_2\left(k\right)\le -\left[1-{\left(1-{\lambda }_{2a}\right)}^2\right]{\tau }_{22}{‖{\tilde{\zeta }}_2\left(k\right)‖}^2$$

(49)

Combining Equations (47) and (49), the difference $\Delta V_2\left(k+1\right)$ in the Lyapunov function is obtained as follows:

$$\begin{array}{ll}\mathsf{\Delta }{V}_2\left(k+1\right)\le & -{\tau }_{21}\left(1-c_2^2-3/{\kappa }^2\right){‖{\mathit{z}}_2\left(k+1\right)‖}^2+{\tau }_{21}\left(1+{\kappa }^2+2/{\kappa }^2\right){‖{\mathit{g}}_1\Delta t‖}^2{‖{\mathit{z}}_1\left(k\right)‖}^2\\ & -\left\{\left[1-{\left(1-{\lambda }_{2a}\right)}^2\right]{\tau }_{22}\right.\left.-\hspace{0.33em}{\tau }_{21}\left(1+3{\kappa }^2\right)\right\}{‖{\tilde{\zeta }}_2\left(k\right)‖}^2+{\tau }_{21}\left(1+{\kappa }^2+2/{\kappa }^2\right){‖{\mathit{g}}_1\Delta t‖}^2{\psi }^2\end{array}$$

(50)

For the entire closed-loop system, we choose the following Lyapunov function $L\left(k+1\right)$:

$$L\left(k+1\right)=V_1\left(k+1\right)+V_2\left(k+1\right)$$

(51)

where

$A_1=\left(1-c_1^2-1/{\kappa }^2-{\tau }_{11}-{\tau }_{21}\left(1+{\kappa }^2+2/{\kappa }^2\right){‖{\mathit{g}}_1\Delta t‖}^2\right){\tau }_{11}$,

$A_2={\tau }_{21}\left(1-c_2^2-3/{\kappa }^2-3{\tau }_{11}{‖\left({\mathit{g}}_1\Delta t\right)‖}^2\right)$,

$A_3=\left\{\left[1-{\left(1-{\lambda }_{1a}\right)}^2\right]{\tau }_{12}-\left(2+{\kappa }^2\right){\tau }_{11}\right\}$,

$A_4=\left\{\left[1-{\left(1-{\lambda }_{2a}\right)}^2\right]{\tau }_{22}\right.\left.-\hspace{0.33em}{\tau }_{21}\left(1+3{\kappa }^2\right)\right\}$, $D=\hspace{0.33em}{\tau }_{21}\left(1+{\kappa }^2+2/{\kappa }^2\right){‖{\mathit{g}}_1\Delta t‖}^2{\psi }^2$.

Therefore, a similar analysis method as in Ref. [37] is used. If the following conditions are satisfied, then $\mathsf{\Delta }L\left(k+1\right)<0$:

$$‖{\mathit{z}}_1\left(k\right)‖>\sqrt{\frac{D}{A_1}}=\sqrt{\frac{{\tau }_{21}\left(1+{\kappa }^2+2/{\kappa }^2\right){‖{\mathit{g}}_1\Delta t‖}^2{\psi }^2}{\left(1-c_1^2-1/{\kappa }^2-{\tau }_{11}-{\tau }_{21}\left(1+{\kappa }^2+2/{\kappa }^2\right){‖{\mathit{g}}_1\Delta t‖}^2\right){\tau }_{11}}}$$

$$‖{\mathit{z}}_2\left(k+1\right)‖>\sqrt{\frac{D}{A_2}}=\sqrt{\frac{{\tau }_{21}\left(1+{\kappa }^2+2/{\kappa }^2\right){‖{\mathit{g}}_1\Delta t‖}^2{\psi }^2}{\left(1-c_2^2-3/{\kappa }^2-3{\tau }_{11}{‖\left({\mathit{g}}_1\Delta t\right)‖}^2\right){\tau }_{21}}}$$

Cao et al. [38] compared the experiment that included the EKF in the control closed-loop system, meaning using the estimated value of EKF as the state information of the controller, with the experiment without the EKF in the control closed-loop system, meaning only using the estimated value of EKF for state monitoring and not as the state information for the controller design. The result shows that the controller, based on the EKF, was affected slightly, and the output was much smoother than before. Therefore, in the process of controller design, we included the EKF in the control closed-loop system. In the previous section, after obtaining the effective estimate ${\widehat{\mathit{x}}}_{k/k}^F$ of the system state at time $k$, the system state estimate was introduced into the control design, and a closed-loop controller was designed for the UAV system. The overall block diagram of the controller is shown in Figure 1.

4. Simulation Study

In order to verify the effectiveness of the proposed method, a digital simulation experiment is provided on the Links-Box real-time simulation system based on a small unmanned aerial vehicle system model. The nonlinear 6-DOF model is primarily designed in [39], so we refer to Ref. [39] to obtain details on the aerodynamic data. The control surfaces consist of an elevator (${\delta }_e$), aileron (${\delta }_a$), and rudder (${\delta }_r$). The flight controller is designed to stabilize and track command $\left[\begin{array}{ccc}{\varphi }_c& {\theta }_c& {\beta }_{\mathrm{c}}\end{array}\right]$ signals for roll angle, pitch angle, and yaw angle simultaneously. Second-order nonlinear links are used in the simulation model to simulate the actuator characteristics. According to the characteristics of the UAV actuator, details on the actuator dynamics are presented in

Table 1. Actuator dynamic characteristics.

Parameter	Unit	Value
Surface deflection limit	deg	−25 to 25
Surface rate limit	deg/s	−120 to 120
Damping ratio  $\xi$	---	0.75
$\mathrm{Natural} \mathrm{frequency} {\omega }_n$	rad/s	100

.

A fixed-step fourth-order Runge–Kutta algorithm (ODE4) is adopted to the integral algorithm, with a discrete sampling period of 5 ms and a state variable delay as one discrete sampling period.

The dynamics of the controlled plant are required to satisfy the TSS property to ensure the accuracy of the linearization and apply the first-order Taylor series expansion. In the UAV system, the actuator system is considered as a subsystem cascaded with the angular rate dynamic system. Since the dynamics of the actuator are faster than the dynamics of the angular rate, the angular velocity dynamics can also be regarded as a cascade subsystem of attitude dynamics. Specifically, a change in the aircraft control input of an aircraft causes a change in the torque. The change in torque directly affects the angular acceleration. In contrast, the angular rates are altered solely by integrating the angular accelerations [40]. As a result, the TSS property is assured [5].

In order to verify the control performance of the control law developed in this paper, the control command of the UAV’s sideslip angle is set to zero, and the UAV is ordered to perform doublets in both roll angle and attack angle. Due to coupling between all three channels and modeling errors, this set of control commands is somewhat challenging for the UAV. During the simulation process, a typical uncertainty is considered, namely amplitude scaling. In this case, the actual coefficients are obtained by scaling the magnitude of the nominal coefficients. Their relationship can be described as the following equation: $C_{*act}\left(*\right)=\left(1+F_{*nom}\right)C_{*nom\left(*\right)}$,where $C_{*act}\left(*\right)$ and $C_{*nom\left(*\right)}$ are the actual coefficients and nominal coefficients, respectively. $F_{mag}=-30\%$ is the scaling factor. In addition, external disturbances to the UAV are added. The disturbance of the first subsystem is a constant with a value of $diag\left(30,30,30\right)$.

The design parameters are $c_1=0.98$, $c_2=0.98$, updating gain ${\lambda }_{1a}=0.005I_{3\times 3}$, ${\lambda }_{2a}=diag\left(10,20,0.01\right)$, and ${\overline{g}}_2=diag\left(60,30,30\right)$, which is an important parameter in TDC. The sampling frequency is maintained at 100 Hz in all simulations.

According to the actual measurement of the UAV state, at the 12th second, Gaussian white noise is added to the state variables to simulate the noisy state measurement signals that are obtained from typical low-cost sensors used by the UAV. The typical noise characteristics of the sensors [38] are as follows: the noise distribution of the angular rate signal has a standard deviation of ${\sigma }_{p,q,r}=5\left[\mathrm{deg}/\mathrm{s}\right]=0.0873\left[\mathrm{rad}/\mathrm{s}\right]$ and a noise variance of ${\displaystyle \sum {}_{p,q,r}=0.0076\times I_3\left[{\mathrm{rad}}^2{/\mathrm{s}}^2\right]}$, and the noise distribution of the angle of attack signal has a standard deviation of ${\sigma }_{\alpha ,\beta }=2\left[\mathrm{deg}/\mathrm{s}\right]=0.0349\left[\mathrm{rad}/\mathrm{s}\right]$ and a noise variance of ${\displaystyle \sum {}_{\alpha ,\beta }=0.0012\times I_2\left[{\mathrm{rad}}^2{/\mathrm{s}}^2\right]}$.

Actuator d biases and measurement noise, biases, and time delay are taken into account in all simulation experiments. All actuators are assumed to have biases with a value of $+{0.3}^{\circ }$. To verify the performance of the state estimation and control effect of the closed-loop control system after adopting EKF, two sets of simulation experiments are carried out as a comparison in the simulation experiment: one set includes EKF in the control closed-loop system, and the other set does not include EKF in the closed-loop control system. The remaining conditions of the two sets of simulation experiments are exactly the same. The control parameters used are $k_1=6$, $k_2=4$.

The simulation results are shown in Figure 2 and Figure 3. The angle command tracking response is shown in Figure 2. The DTIBS controller designed in this paper can track the command signal well. It is worth mentioning that before 12 s, there was no noise added to the system. The angle command tracking response curve was relatively smooth and had good tracking performance. At the 12th second, independent Gaussian noise with a non-zero mean (0.2°) was added to the measurement value with a standard deviation of +0.1°. Although the system tracking response curve showed obvious fluctuations, it could still track the control command well. Compared to the EKF not be included in the closed-loop system, the EKF be included in the closed-loop system results with better control performance and smaller tracking errors. In Figure 2a, at the 5th and 40th seconds, the roll angle command underwent a sudden change similar to a step signal. The system was able to quickly and accurately track the sudden change in the command. In Figure 2a, the closed-loop control system with EKF state estimation has a tracking stability time of less than 0.2 s for roll angle commands and a tuning time of no more than 0.6 s. In Figure 2b, the response time for tracking the pitch angle command signal is less than 0.3 s, and the adjustment time is not more than 0.6 s.

The results of the EKF estimation of the system state under noise conditions are shown in Figure 3a–e. At the 12th second, noise simulating a low-cost drone sensor was added. From the figure, it can be seen that the true state measurement value of the UAV system was drowned out by the noise, and the true value of the system state could not be obtained. After adding EKF for state estimation in the system, it can effectively eliminate the measurement noise of the sensor. Compared to the EKF not being included in the closed-loop system, the EKF being included in the closed-loop system results in more accurate state estimation results and more obvious noise elimination effects, especially in Figure 3b,c,e. This is mainly because when the EKF is not in a closed-loop system, the system state used by the controller is the system state value containing noise, rather than the EKF estimated value. When the EKF is included in the closed-loop, the EKF estimated value is used as the input to the controller. The closed-loop system response is less affected by noise, and the output is smoother. When there is a sudden change in the signal in the system, the EKF can quickly estimate the state value of the system, such as at the 40th second in Figure 3c and at the 30th second in Figure 3d.

In this paper, the control command is set to keep the yaw angle at zero. To further verify the performance of EKF, the error curves of the yaw angle with or without EKF are shown in Figure 4. From Figure 4, it can be seen that when the closed-loop system is without EKF, the tracking error of the yaw angle is very large. After the EKF is added in the closed-loop system, the tracking error of the yaw angle indicates that the EKF can effectively eliminate noise and reduce tracking error. To demonstrate the effectiveness of EKF, the mean and root mean square are introduced to evaluate tracking errors. The quantities of the tracking errors are presented in

Table 2. Quantity table of tracking error of Yaw angle.

	Mean Value	Root Mean Square
Without EKF	1.2182	3.0513
With EKF	0.1833	0.0197

, and the mean and root mean square of tracking errors with EKF are both much smaller than without EKF.

The curve of control surface deflection is shown in Figure 5. As can be seen from Figure 5, when the control commands undergo a sudden change, the control surface can quickly recover to a stable state after deflection. The comparison of EKF in a closed-loop system and EKF not in a closed-loop system shows that EKF can effectively reduce noise in a closed-loop system.

5. Conclusions

In this study, a DTIBS with EKF is proposed to deal with uncertainty and unknown actuator dynamics, eliminate the impact of sensor noise, and minimize tracking errors. The proposed new incremental backstepping method uses standard gradient-based adaptive methods and the chain rule to design robust signals for each subsystem. Moreover, an EKF state estimator is designed to estimate the state of a UAV closed-loop system under strong noise conditions. The stability of the closed-loop system is verified using standard Lyapunov methods. Simulation results show that the designed DTIBS controller can track control commands well. When Gaussian white noise is added to the state variables, the DTIBS controller still has good tracking performance and can quickly and effectively track sudden commands. Compared to not including the EKF in the closed-loop control system, the response curve of including the EKF in the closed-loop control system is smoother. After using the EKF to estimate the state of the UAV closed-loop system, the impact of sensor noise can be effectively eliminated, improving the efficiency of control command execution. The response time is less than 0.6 s, and the tracking error is no more than 1.16%.