3.1. INDI Derivation
The nonlinear control–affine dynamic system can be expressed as
where
is the state vector,
is the control input vector,
is the mapping function
, and
g is the mapping function
, whose columns are smooth and nonsingular vector fields [
13]. In practice,
f is a nonlinear dynamic function related mainly to the state variables, and
g is the control distribution function. It is assumed that
and the output vector
. A first-order Taylor series expansion for the system in Equation (5) in the neighborhood of the point
can be expressed as
where
where
and
are the current state and control variables, respectively.
and
, indicated by the subscript ‘0’, are the current state matrix and control matrix, respectively;
denotes higher-order terms (HOT). Note that the control matrix
needs to be inverted. The sampling rate is assumed small enough that the state variable
x changes significantly more slowly than the control variable
u. Thus,
can be assumed to be comparable to the increment of
u in one sampling time step. Without considering the higher-order part and the increment of the state, Equation (6) can be simplified as
where
, which represents the control increment in one sampling time step
. For the inner-loop angular rate control,
can be estimated using a virtual angular accelerometer [
16], calculated from an onboard model (OBM) with the derivative of the angular rates [
22] or measured directly by an angular accelerometer [
23]. In addition, the input–output relative control degree of the inner-loop angular rate control is equal to one, which means there are no internal dynamics [
24]. The model information
in Equation (8) is replaced with sensor measurements; however, in a regular explicit nonlinear dynamic inversion (NDI) controller, the nonlinear feedback term
is needed [
25]. Thus, the INDI control approach is less sensitive to model mismatch [
13] and improves the robustness of NDI to model uncertainties [
26]. The virtual control input
ν is introduced and replaces
in Equation (8), and the increment control output can be calculated as
The control command to the actuator of the INDI controller is equal to the sum of the previous control input
and the increment control output, as
The requirements of the INDI controller are composed of the control derivatives , the state derivative , the virtual control input ν, and the control position , as shown in Equations (9) and (10). In addition, Equation (10) can effectively prevent the control jumping problem due to the incremental control form.
3.2. Angular Acceleration Estimation
In terms of angular rate control, the state derivative serves as an estimate of the angular acceleration, encompassing both the inner nonlinear dynamics and outer disturbances encountered by the aircraft. The accuracy of this estimation is critical for achieving optimal control performance.
CF is commonly used in attitude estimation due to its high computational efficiency and simple structure. As depicted in
Figure 2, a proportional integral (PI) compensator is used to correct the estimated error, and the state dynamics can be expressed in the Laplace transform domain as follows:
where
and
are PI compensator gains;
corresponds to gyroscope measurements, while
denotes the results obtained from the mathematical model calculations. Specially,
is equivalent to
. Additionally, the variable
signifies the estimation result for angular rate, which is equal to
. Equation (11) can be rearranged as
where
The PI compensator gains can be chosen as
and
. The transfer function of
and
are complementary, and their sum equals 1.
Figure 3 presents the Bode plot of the transfer function
and
with a cut-off frequency of
and the damping coefficient
. A higher value of
introduces more measurement information to estimate the angular acceleration, leading to a better response to external disturbances and model uncertainties and a faster response speed [
27]. However, it also amplifies the measurement noise. Moreover, the high-pass filter
introduces model information to improve the phase margin [
28]. The choice of
is crucial in CF, as it determines the weighting factor for different input signals. The optimal cut-off frequency can be obtained by analyzing the noise properties of the measurements and the estimation error using power spectral density in the frequency domain [
29]. Alternatively, least square estimation in the time domain can be used to minimize the estimation error and avoid manual selection of filter parameters [
30]. Moreover, the cut-off frequency value can be adjusted on the basis of different flight conditions to achieve better estimation results [
31].
To address the issue of measurement noise not being attenuated in the high-frequency region in the transfer function
given by Equation (13) and the resulting interference of noise with the estimated angular acceleration signal, an extended state observer (ESO) [
19] can be used to estimate the angular acceleration. By taking the mathematically calculated information
as control inputs, the following equation can be obtained as
where
;
is the angular acceleration residuals outside the model information; and
is the derivative of
, which is unknown but bounded. Then, a second-order ESO is designed to estimate the residuals, which can be formulated as
where
y is the output of the gyroscope and
;
and
are the estimation of
x and
, respectively;
and
are the observer gains. The error dynamics of the ESO can be calculated as
where
and
are the estimation error. The error equation can be rewritten as
where
The observer gains are determined using pole placement [
19], where the choices of
and
are made. Here,
represents the observer bandwidth. The characteristic polynomial of
can be expressed as
Under the assumption that the parameter
is bounded, it can be concluded that the ESO achieves bounded-input–bounded-output (BIBO) stability. When the system reaches a steady state, referring to Equation (16), we have
The absolute value of the steady-state error is
Finally, the estimated angular acceleration can be expressed as
Rearranging Equation (22) using Laplace transform, we obtain
where
A comparison between Equations (13) and (24) reveals that they share many similarities in their forms, with the main difference being the location of the part
. In the case of the CF, the part
is present in the numerator of the angular rate transfer function. However, for ESO, the part
is in the numerator of the model information transfer function. By setting the observer bandwidth to
and choosing
and
, the Bode diagrams of the CF and ESO are compared in
Figure 4.
Figure 4a clearly illustrates that the magnitude of the ESO transfer function
decreases beyond the observer bandwidth, while the magnitude of the CF transfer function
remains constant. Furthermore, it is observed that
exhibits a high degree of high-frequency attenuation with a loop roll-off slope of approximately 20 dB per decade. Consequently, the ESO exhibits superior noise suppression capabilities when compared with the CF. In
Figure 4b, both the CF and ESO demonstrate high-pass characteristics for model information, but it is noteworthy that ESO passes more model information in the low-frequency range.
Furthermore, in order to compare the time–domain properties of the two angular acceleration estimators, simplified roll mode dynamics that do not consider the coupling effect are used. The roll mode dynamics can be expressed as
where
p is the roll angular rate;
is the aileron deflection angle; and
and
are the roll control parameter and damping parameter, respectively. The roll angular rate is measured by a gyroscope with its noise
and bias
[
32]. The aileron deflection angle is modeled as a sine function and can be expressed as
where
is the amplitude and
is the input frequency. The on-board model input
can be calculated as
where
is the gyroscope output with measurement noise.
The selected aileron input has an amplitude of 2 deg and a frequency of 2 Hz, and the estimation results are presented in
Figure 5. The estimated angular rates of ESO and CF are nearly identical, and both estimators exhibit excellent filtering performance, as shown in
Figure 5a. However, as depicted in
Figure 5b, the ESO provides a better estimate of the angular acceleration without introducing any angular rate measurement noise, which is consistent with the frequency domain analysis presented earlier. However, due to its positive gain in the high-frequency range, as depicted in
Figure 3, the CF amplifies the sensor noise in the estimation results of angular acceleration. Conversely, the ESO exhibits an attenuated gain for high-frequency noise, as shown in
Figure 4a. Therefore, it can be concluded that when accounting for measurement noise, ESO is capable of providing higher-quality angular acceleration estimation information in comparison with CF.
3.3. Reference Model with Pseudo-Control Hedging
The concept of pseudo-control hedging (PCH) was initially proposed as a means to enhance the performance of adaptive control [
33], specifically by modifying the reference model in situations where input position and rate saturation occur. In the presence of actuator saturation, it is possible for the control performance to degrade or even for the controller to become unstable [
34]. To address this issue, PCH is introduced. In comparison with the reference governor (RG) [
35], which is employed as a control reconfiguration strategy in fault-tolerant control, the PCH method offers the advantage of low computational cost, as it eliminates the need for computing the auxiliary reference (AR) sequence. In the ideal scenario, where the effect of actuator dynamics is not considered and the true values of angular acceleration and angular rate are used, the PCH signal
is obtained by subtracting the estimated saturated input
from the virtual command output
. The command signal
is utilized as a feedforward term to improve the tracking performance, as depicted in
Figure 6. In order to make the steady-state value of PCH equal to
, a control gain is introduced, which can be formulated as
where
is the PCH signal used to slow down the dynamics of the reference model when the saturation is triggered, and the gain
can be calculated as [
36]
where
and
are gain of the reference model and the error controller, respectively. The inequality
holds, because in general control practice, the reference model is designed to have slower dynamics compared with the error controller. This is implemented to prevent the error controller from being overloaded and to ensure a stable closed-loop system.
With regard to the roll mode dynamics described in Equation (25), the selected maximum absolute value for the aileron deflection angle is 5 degrees. The reference model and error controller gains are chosen as 8 and 10, respectively.
Figure 7 compares the response of the system states and the virtual control command in the hedged and unhedged simulations. During the first 3 s of the simulation, the results are nearly identical, indicating that the PCH is not activated when the control output is not saturated. However, at 3.5 s, the control effectiveness of the roll mode drops by 95%, resulting in the roll angular rate being unable to maintain the commanded value, as shown in
Figure 7a. At the activation of the PCH, depicted in
Figure 7b, the virtual control requirements are reduced by slowing down the reference signal. At the 4 s mark, when the command shifts to the opposite direction, the virtual command of the PCH is increased to minimize the tracking error, as illustrated in
Figure 7a. When the control output saturates, the PCH is capable of more rapid tracking of the command value.