Filter and sensor delay synchronization in incremental flight control laws

This paper introduces and investigates several methods of filtering and delay synchronization in incremental control law structures. It is shown that proper filter and delay synchronization is of great importance for both the performance and stability of the control laws. It is derived how to synthesize complementary filters in case of sensor dynamics, sensor delays and internal control law filtering such as roll-off filters and notch filters. The complementary filters are compared to common synchronization approaches previously applied in incremental control laws. It is shown that the proposed methods are beneficial for Multiple Input Multiple Output (MIMO) systems in case that different delays and filtering is present in the different measurement signals. In case of strongly coupled input effectiveness, the synchronization on the actuator feedback may become even unstable, where the complementary filter approach recovers the design closed-loop behavior. In addition, it is derived how to initialize the complementary filters for a transient-free engagement in non-steady-state conditions. The results are compared in simulation with a roll rate control law for a fixed wing aircraft.


Introduction
In the past decade, the control concept Incremental Nonlinear Dynamic Inversion (INDI) gained a lot of interest, especially in the domain of flight control. While control techniques such as Non-linear Dynamic Inversion (NDI) [1][2][3][4] and Backstepping (BS) [5] rely on accurate models of the 1. Novel aerial vehicle configurations might be complex to model or the models might be inaccurate or lacking the influence of dominating physical effects. 2. Aircraft models depend on aerodynamic parameters, which are expensive to identify, e.g., by performing extensive flight test campaigns; 3. Changes in the vehicle design require a repetition of the identification process, and therefore the flight test campaigns; 4. Robustness against a wide range of faults or structural damages; 5. Robustness against disturbances; 6. Suitability for automatic take-off and landing, especially for ultra-light aerial vehicles, where ground effects have a large impact and are difficult to model; 7. Unified control strategies without the necessity of switching or blending between different control laws for different flight phases.
Incremental non-linear flight control techniques, such as INDI, have been successfully demonstrated on several aerial platforms and for different applications: the Cessna Citation II [6,7], VTOL transition UAVs with movable tilts [8][9][10], quadrotor UAVs [10][11][12], hexacopter [14], a hybrid tail-sitter UAV [15,16], the automatic take-off and landing of a fixed wing tail-wheel aircraft [17], a helicopter [18], an airship [19], a piloted all attitude fixed wing turboprop demonstrator simulated aircraft [20], and a commercial civil aircraft simulation model [21,22]. It is commonly known that one major challenge of INDI is its sensitivity towards measurement delays in the output derivative feedback [23][24][25][26] and actuator command delay [EINDI]. In van't Veld et al. [27], the authors show that the stability of INDI controllers subject to measurement time delay is an issue, especially, when the transmission delay of the actuator measurements and that of the relevant output derivative measurement are not equal. Delays are usually encountered for several reasons: 1. Control signal transmission delay; 2. Sensor signal delays; 3. Processing and scheduling delays.
In van't Veld et al. [27], it is observed that INDI is more sensitive to output derivative measurement delay than actuator delay. Additionally, signal filtering can introduce lag to the system which might be an issue for the controller performance and stability as demonstrated in van't Veld et al. [27] as well. The use of signal filtering in control laws is of high importance, for performance, stability, safety and robustness. There are several reasons for introducing filtering of the used signals. The most common reasons are: 1. Anti aliasing filtering; 2. Roll-off filtering for noise attenuation; 3. Notch filtering for attenuation of structural mode excitation; 4. Filtering for estimation of certain variables, e.g., complementary filtering; 5. Filtering for bias removal.
Another common issue related to INDI for aircraft control applications is that measurements of the angular accelerations are often required. Although sensors for measuring the angular rate derivative exist, they are not part of the standard equipment of aerial vehicles. Hence, the rate derivative needs to be estimated in most applications. Obtaining the rate derivative by calculating the discrete derivative of the rate measurement using a differentiator, is not recom-mended because of its sensitivity to sensor noise leading to an amplification of noise on the control signal. In [24], the angular accelerations were estimated with this technique, using backward finite differences, and a first-order low-pass filter was introduced before the actuators to attenuate highfrequency oscillations caused by numerical noise. With the sensor dynamics and measurement delays present, a lowamplitude high-frequency oscillation still remained in the control commands. [28] proposed to filter the measurement, using a second-order low-pass filter, in combination with a discrete differentiation. This allows to estimate the derivative for low frequencies, while attenuating high-frequency sensor noise. Since this kind of second-order filter differentiator introduces lag and produces a delayed estimate of the relevant output derivative which degrades the controller performance and stability, several techniques were proposed to accommodate this issue. [29] proposes the use of an input scaling gain to increase the controller robustness to delays in the feedback loop, when using the second-order filter differentiator. Other approaches are based on the importance of delaying the actuator and state derivative measurements equally, which has been demonstrated in van't Veld et al. [27]. [6,11,27] synchronize the actuator measurement in time with respect to the angular acceleration by a synchronization filter. This synchronization corresponds to the second-order low-pass filter and a time delay, corresponding to the relative delay between the angular rates and actuator deflection measurements. Besides the second-order filter differentiator, complementary filters have been proposed for estimating the angular accelerations, where the measured rate is low-pass filtered and differentiated and the high-frequency part of the rate derivative is obtained using a high-pass filtered modelbased rate acceleration estimate [14,30]. The latter further proposed a synchronization scheme on the actuator feedback in combination with the complementary filter. In [23], a linear predictive filter is designed to predict the angular rate derivatives from angular rates and references thereof. The coefficients of the predictive filter are computed using least squares estimation [31] designs a nondelay differentiator based on a precise-delay differentiator, to acquire the output derivatives without delay. In this paper, it is investigated how to compensate for filter and sensor dynamics in the implementation of different types of incremental control laws. We start with the Non-Linear Dynamic Inversion (NDI) control law through the actuators introduced as Actuator-NDI (ANDI) in Steffensen et al. [32]. From this control law, several other derived control laws can be obtained by applying appropriate assumptions. The most basic form is Incremental Non-linear Dynamic Inversion (INDI) [28]. This control law does not explicitly take into account the actuator dynamics. The Extended INDI (E-INDI) [9] takes the actuators into account, but neglects the state dependent terms. The filter synchronization is derived ana-lytically for the ANDI control law, using a complementary filter exploiting relevant model information to compensate for the negative effects of the filters and sensors. Analysis is made for the initialization of the filter, to investigate methods of transient-free activation of the control law. An analytical comparison is made of synchronization techniques used in literature, and a re-formulation of these synchronization techniques is performed in terms of equivalent complementary filters with several benefits for MIMO systems when the filters applied to the measurements differ. The contribution of this paper consists of: Based on a general linear MIMO system, the closed-loop dynamics is analyzed analytically for (1) vanilla (perfect knowledge of all variables) INDI (2) INDI with complementary filter andẏ mdl (3) filter differentiator approach without synchronization (4) filter differentiator approach with synchronization (5) Hybrid INDI, i.e., complementary filter approach with synchronization. The analysis done in this paper further provides insights into the relations between these different approaches. Additionally, the cascaded complementary filter is demonstrated for an ANDI control law, introduced in Steffensen et al. [32], and compared to the E-INDI law [9].

Control laws
The following section provides a derivation of the ANDI (Actuators in Non-Linear Dynamics Inversion), E-INDI (Extended Incremental Non-linear Dynamic Inversion) and INDI (Incremental Non-linear Dynamic Inversion) control laws, as a basis for the investigation in Sects. 4 and 5.

Summary of ANDI control law
We consider the ANDI control law introduced in Steffensen et al. [32], as depicted in Fig. 1. The ANDI is based on NDI, hence includes the stability guarantees in terms of Lyapunov and exact inversion for tracking as well as specified error dynamics. With the assumption of first-order actuators from ANDI, the approximate concepts INDI and E-INDI can be deduced. Hence, the ANDI is taken as a starting point, and the concepts of synchronization will be extended directly to INDI and E-INDI. As was shown in Steffensen et al. [32], the ANDI control law corresponds to an INDI control law if the bandwidth of the actuators tends to infinity, and corresponds to the E-INDI [9] if the F xẋ term is neglected as explained later in this section. The ANDI control law allows to actively account for the actuator and state dependent dynamics and is summarized in the following. Consider the systeṁ with system state x ∈ R n , actuator state u ∈ R k , output y ∈ R m , f ∈ C (r +1) (R n × R k ; R n ) and h ∈ C (r +1) (R n ; R m ). Note, that the time dependence (t) was here and in the following omitted for better readability. The actuator dynamics are given by: with diagonal matrix ∈ R (k×k) with its entries representing the bandwidth of the different actuators. In Steffensen et al. [32] the ANDI control law is derived assuming the system to have a relative degree of r ∈ N, i.e., the r th derivative of y with respect to time is the first derivative of y which explicitly depends on u. For the sake of simplicity, we assume here r = 1: with F ∈ C 1 (R n × R k ; R m ). The second derivative of y is then where which is solved for u c , assuming F u to have full row rank and choosingÿ = ν as the virtual command: where (F u ) † denotes a right inverse matrix that solves the linear equation system given in Eq. (5). The virtual control input ν is designed using a linear error controller as By applying the control law to the system, the desired error dynamics are obtained as designed, because F u has full row rank such that (F u )(F u ) † = I m×m : The block diagram of the ANDI controller is depicted in Fig. 1.

ANDI control law with measured output
It is often not possible to directly measure the time derivativė y, which is needed to calculate the pseudo-control ν in Eq. (7), which is then used in the ANDI control law in Eq. (6). We propose to estimateẏ using the complementary filter which will be introduced in Sect. 3.3, that uses a model-based estimate ofẏ and the measured and filtered output signal y f as depicted in Fig. 2. In addition, due to the filtering of y, it is proposed to filter the reference signal y re f with the same filtering as the measurement to synchronize the error term. This filtering includes the filtering by F, which is any filtering applied to the signal, e.g., roll-off filters or notch filters, the measurement delay D y , as well as the sensor dynamics S. The control law is, hence, modified as such: In addition,ẋ needs to be estimated, which is covered in Sect. 4.6. Note, that in some casesx is the same asŷ, and the same complementary filtered signal can be used.

ANDI and INDI
To arrive at the INDI control law from the ANDI control law, a special choice of error dynamics have to be made as shown in Steffensen et al. [32]. The error dynamics shall be chosen as a product of actuator and system error dynamics as shown in Eq. (10). The actuator loop of the ANDI error dynamics, i.e., the second part of Eq. (10), shall be chosen with the same bandwidth as the actuators, ω. The assumption necessary on the system is that all actuators need to have the same bandwidth, that is = ωI k×k .
with the errors defined as: In the time domain, this can be expressed as: Then, the pseudo-control ν is chosen as: and by inserting ν into Eq. (6), the ANDI control law can be expressed as: which was first shown in Steffensen et al. [32]. If the actuators have a high bandwidth ω, the first term of Eq. (14) can be neglected, and the remaining part is equivalent to the INDI control law: Similar to Sect. 2.2, the INDI law can be implemented as depicted in Fig. 3 using a complementary filtered estimate oḟ y and synchronizing y re f and y f . It shall be noted that sometimes the first term in Eq. (14) is included using variables from a reference model [20,33], such that the command path exactly corresponds to the ANDI control law. If the system is (sufficiently) linear, this will only affect the feed-forward part of the system, and hence not change the closed-loop poles.

ANDI and E-INDI
In literature, several extensions to the vanilla INDI control law can be found. Two approaches that are taking explicitly the actuator dynamics into account are [9,11]. While [11] assumes that all actuators have the same bandwidth, i.e., = I k×k ω, [9] considers the more general case of actuators with different bandwidth. As was shown in Steffensen et al. [32], choosing the error dynamics given in Eq. (12), the respective control laws from [9,11], can be re-formulated as which in case of = I k×k ω, y = I m×m ω, can be expressed as This control law equals the ANDI control law in Equation (14), if F xẋ is neglected.

Complementary filters and initialization
In the following section, the complementary filter design and initialization is presented, and a formal initialization method is derived for transient-free engagement in non-steady-state conditions. As will be shown later, this design of the complementary filters will inherently contain the synchronization properties that have earlier been shown to be crucial for increasing performance and stability of INDI control laws. The use of this type of complementary filter design, allows for the use of different filters in the individual measurement channels, which is not directly achievable using common synchronization techniques on the actuator feedback. Figure 4 depicts a simple complementary filter, which will be used as an introductory example for the derivation of the initialization in non-steady-state conditions. First consider the zero-valued steady-state case. Then, the relation between the inputs and outputs of the filter can be

Basic complementary filter concept and initialization
where U 1 (s) = Y (s), U 2 (s) =Ẏ (s) and assuming that y(0) andẏ(0) are zero. In case that the modelẏ is accurate, the filter perfectly distributes the high-frequency part of theẏ estimate to the model-basedẏ and the low-frequency part to the differentiated y measurement.
In case of non-steady-state initial conditions, the filter needs to be carefully initialized. In the following, a method to obtain the correct initialization is derived. For now, consider H (s) as a first-order filter such that the differential equations describing the x 1 and x 2 dynamics, as depicted in Fig. 5, are given by: where u 1 = y and u 2 =ẏ. In the Laplace domain, accounting also for non-zero initial conditions, the following relation is obtained: The complementary filter calculatesẏ as shown in Fig. 5 bẏ In the Laplace domain, accounting for non-zero initial conditions, this results iṅ Substituting Eq. (20) into (22) results iṅ Substituting Initializing the integrators with x 1 (0) = y(0) = u 1 (0) and x 2 (0) = 0 as depicted in Fig. 5 by the red dotted lines, results iṅ Hence, in case of an accurate model ofẏ, the output of the filter will exactly match the trueẏ, also at the initial time instance when engaging the control law.

Complementary filter and sensor dynamics
Usually, the output y of the plant is measured by a sensor with corresponding sensor dynamics S(s). These dynamics may include an anti-aliasing filter because of the (down) sampling of the sensor signal for the flight control computer. For now consider these dynamics to be described by a first-order filter such that the measured plant output that is used in the complementary filter corresponds to The complementary filter in Fig. 6 is given bẏ To show the proper synchronization, now assume thatẏ mdl = y. The complementary filter for zero initial conditions results in: Hence, with an accurate model ofẏ, the filter again exactly distributes the high-frequency part of theẏ estimate to the model-basedẏ and the low-frequency part to the differentiated y in the presence of the sensor dynamics S(s).
To determine the initial values of the integrator state in the sensor filter model used in the complementary filter, we start with considering the differential equations of the sensor filter and the sensor filter model, as depicted in Fig. 7, given bẏ whereũ 1 = y andũ 2 =ẏ. The resulting relation in the Laplace domain is The estimate ofẏ is calculated by the complementary filter as depicted in Fig. 7 bẏ which results in the Laplace domain tȯ For X 1 (s) and X 2 (s), the relation in Eq. (20) is inserted. The input U 1 (s) and U 2 (s) are the states of the sensor filter and sensor filter model as depicted in Fig. 7, i.e., SubstitutingX 1 andX 2 in Eq. (33) by relation (30) results iṅ Initializing the integrators as discussed before with with x 1 (0) = u 1 (0) =x 1 (0) and x 2 (0) = 0 as depicted in Fig. 7 by the red dotted lines, results iṅ Hence, the sensor model state needs to be initialized tõ x 2 (0) =ω (y(0) −x 1 (0)) which corresponds to the time derivativeẋ 1 (0) of the sensor filter state as depicted in Fig.  7 by the red dotted lines. Note that in case the sensor statex 1 is initialized to y(0), x 1 (0) will be 0. This will be the case if the sensor is reset, and then the integrator statex 2 of the sensor model could be initialized with 0. Otherwise, alternatively as an approximation ofẋ 1 the model basedẏ could be used as initialization value forx 2 .

Complementary filter, sensor dynamics and additional filtering
Often the measured output y s of the plant is additionally filtered to attenuate noise, vibrations from engines, pickup of structural modes, etc. Commonly roll-off filters or notch filters are used for this purpose. Furthermore, transmission delays might be present on the measurements such that the where F(s) is the transfer function of the roll-off or notch filters, D(s) is the transmission delay of the measurement and S(s) is the sensor dynamics. The complementary filter needs then to be updated as shown in Fig. 8, such thaṫ The initialization of the filters can be derived in a similar manner as in the previous sections. In the following, we denote the series of the filters as The complementary filtered estimate of the derivative of the output signal y is, hence, given bŷ whereẎ mdl (s) is the model-based estimate ofẎ .

Synchronization techniques with related complementary filters
In the following, the proposed synchronization is compared to other synchronization approaches in literature. For the analysis, we consider a MIMO linear time invariant system: where r is the relative degree of the system, A is the system matrix, B is the input effectiveness, B d d(t) is the disturbance term, C the output matrix, and with the restriction that: where G A (s) are the actuator dynamics,Ḡ A (s) is a diagonal transfer function matrix, and the matrix C B has full row rank. Equation (42)  For the following, we consider without loss of generality a system with relative degree 1, which is common in baseline flight control laws. In addition it improves the readability. Substituting sY (s) with s r Y (s) for the following results recover the general case. Note that for better readability the arguments (s) and (t) are omitted. In the following, it is shown that the different synchronization techniques only vary in their influence on the state dependent term C AX of the closed-loop dynamics, summarized in Table 1. The influence on the direct disturbance term C B d D and pseudo-control term V is exactly the same for all synchronization techniques.
It is additionally shown that for methods 3 and 4 as proposed in [6,11,27] and [30], for MIMO systems, additional restrictive assumptions, e.g., same filtering of all measurements, are necessary to obtain the relations given in Table 1. The respective assumptions will be detailed in the related sections. A re-arrangement of the synchronization is proposed, such that the design closed-loop transfer function can be recovered for MIMO systems, relaxing the above-mentioned assumptions. This re-arrangement allows different filtering in the measurement feedback channels.

Vanilla INDI
The vanilla INDI with perfect knowledge of the state derivative is depicted in Fig. 9. The closed-loop transfer function is given by: as shown in Appendix A.1. It is seen thatẏ follows ν with the actuator dynamics, plus the state dynamics and disturbances filtered through a high-pass filter given by I −Ḡ A , i.e., if the actuator dynamics are much faster than the state dynamics and disturbances, the influence is filtered out.

INDI with complementary filter andẏ mdl
Ifẏ is estimated with a complementary filter as derived in Sect. 3.3, and depicted in Fig. 10, assuming that there are no model uncertainties or noise, thenẏ mdl =ẏ holds, and the closed-loop transfer function is given by: as shown in Appendix A.2, which shows that the complementary filter recovers the input-output dynamics of the vanilla INDI. It is seen that the disturbance is high-pass filtered by I −Ḡ A H F cy , such that disturbances below the cut-off of the combined actuator and filter dynamics are rejected. Note,ẏ mdl =ẏ might seem like a crude assumption but is made to highlight the synchronization effects i.e.,ẏ mdl gives an un-delayed estimate ofẏ. For exampleẏ mdl = C(Ax + Bu) would be such an estimate, ifx is obtained as an undelayed estimate, for example from a model or using again a complementary filter as depicted in Fig. 16.   [6,11,27] Note thatẏ mdl will not beẏ if there are model uncertainties (e.g., in A, B or C) or measurement noise. The cut-off frequency of the filter H in the complementary filter has to be chosen as a compromise, in general as high as possible but low enough such that the noise is not amplified by the derivative.

INDI with derivative filter and no synchronization
Ifẏ is estimated with a filter from a sensor measurement as depicted in Fig. 11, then the closed-loop transfer function is given by:

INDI with derivative filter and synchronization
Ifẏ is estimated, as proposed in [6,11,27], and depicted in Fig. 12, i.e., with a filter from a sensor measurement and synchronized with the same filtersH andF cy (with possibly different dimensions as indicated by the bar), as used for the derivative and sensor filtering in the control signal feedback path, then, as shown in Appendix A.4, the closed-loop transfer function is given by: Fig. 13 Re-formulation of INDI with derivative filter and synchronization assuming that: which hold if either of these additional assumptions are satisfied, respectively: Respective additions to the basic assumptions:  Fig. 13 is implemented instead, then the same closed-loop behavior as given in Eq. (46) is obtained, even without the additional assumptions given above.
It is additionally seen that the synchronization in Fig. 13 corresponds to the INDI with complementary filter depicted  [30] in Fig. 10 withẏ mdl = Bu, i.e., only the high-frequency part ofẏ is considered in the estimateẏ mdl . This re-formulation can be attractive if different filters in F cy are used in the separate control channels of y, e.g., that notch filters with different notch frequencies are used for the individual axes of the aircraft. Having the filter synchronization in the u feedback path does not allow for consistent separate filtering in the different axes.

Hybrid INDI
Ifẏ is estimated, as proposed in Kumtepe et al. [30], with a complementary filter and synchronized as depicted in Fig. 14 (with the On-board plant model (OBPM) beingẏ mdl = C Ax f +C Bu, then the closed-loop transfer function is given by: as shown in Appendix A.5, under the assumption that which holds if the basic assumptions and the additional assumptions mentioned in Sect. 4.4 are satisfied, respectively. If for example F cy and F cx are diagonal with the same elements on the diagonal, then: In this case, it can be seen that this filtering technique produces a high-pass filter on the states which is (I −Ḡ A F cy ) and hence with a cut-off frequency which is in between the cut-off frequencies of the high-pass filter in vanilla INDI (Sect. 4.1), The assumptions above can again be relaxed if the synchronization on the feedback is moved to the complementary filter and the filter chain F cy (s) is split into sensor filters S(s) and notch/roll-off filters F(s) as shown in Fig. 15. Different filtering F(s) in the control law can now be applied in the feedback paths. With this modification the only assumption is the sensor filters S x (s) and S y (s) are diagonal and identical in each channel in order to obtain the relation given by Eq. (50).

Un-delayed estimate of model-based output derivativeẏ mdl
In Section 4.2, the analysis was based on an un-delayed model estimate,ẏ mdl , ofẏ. This can for this example be obtained by a cascaded complementary filter forẋ and then x as depicted in Fig. 16. Sinceẏ = C(Ax + Bu), this provides the necessary estimate for the use in Section 4.2. Furthermore, this complementary filter can provide the estimatex for the ANDI control law.

Roll dynamics example
This subsection compares the different presented synchronization approaches for the simplified roll motion dynamics of an aircraft, given bẏ with roll damping L p = −2.7 1/s, effectiveness L ξ = −14 1/s 2 , control input ξ , which is the aileron deflection and measured output p, which is the roll rate. The actuator dynamics are given by with ω A = 50 rad/s. The sensor dynamics are given by with bandwidth ω S = 100 rad/s. The delay is given by with T = 0.03s. For this example, the filter F(s) = 1. F cy (s) is given by Eq. (39). The filter H is given by with bandwidth ω H = 30 rad/s. Figure 18 shows the desired roll acceleration signalṗ des , which shall be tracked, and the closed-loop roll acceleratioṅ p, for the vanilla INDI law depicted in Fig. 9, and the INDI with derivative filter and no synchronization depicted in Fig.  11. The pseudo-control is in both cases ν =ṗ des . Figure  18 shows that in case, the derivative of p is obtained by a differentiation filter without a synchronization technique, the resulting response might be oscillatory. With increasing phase loss of the filter F cy (s), the closed-loop system will become unstable as happened in this case. Figure 19 compares the responses of vanilla INDI (Fig. 9), INDI with derivative filter and synchronization (Fig. 12  However, if the sensor dynamics are fast and the measurement delay very low, the differences in the responses might be minor. In this case, the hybrid INDI or INDI with differentiation filter and synchronization might be the preferred solution because of its lower complexity. If the sensor dynamics are slow or the measurement delay non-negligible, the cascaded complementary filter might be preferred. Figure 20 compares the E-INDI control law given by Eq. (17), with the ANDI control law, given by Eq. (9), and depicted in Figs. 2 and 17. Both the E-INDI and ANDI are implemented with K 0 = k 0 = 0, i.e., without roll rate error feedback. Figure 20 reveals that E-INDI recovers the phase of the reference signal. This is because of the feed forward ofÿ re f in E-INDI. ANDI also uses this feed forward and additionally the model dependent term F xẋ , which corresponds to L pp in the considered example. The closed-loop response of ANDI In Fig. 21, the control input ξ are depicted for the different controller structures for the case that noise with a standard deviation of 0.1deg is added to the measured roll rate p. It can be seen that the resulting jitter on ξ is for all variants in a similar magnitude range, although it seems slightly less for ANDI.

Linearized lateral motion example
This subsection compares the different presented synchronization approaches for the linearized lateral dynamics of an aircraft given by  by a notch filter given by with parameters summarized in Table 2 and The actuator dynamics are given by  responses for the given command inṗ des . It can be seen that with the proposed cascaded complementary filter the response is equivalent to the vanilla INDI response. The other techniques slightly deteriorate the response, especially with regard to the decoupling of the axes. If a pure roll command is given the response in the r channel is desired to be as low as possible, but the excursion is much higher in the conventional synchronization schemes as shown in Fig. 23. The same holds for the responses to a yaw step command, which are depicted in Figs. 24 and 25.
The simulations done until now show that the cascaded filter has improvement over the synchronization on u, but the synchronization on u still works reasonably well. The synchronization approach does not require any additional model information; hence, it might be the preferred choice. The above simulations were made with a weakly coupled  47) and (49) almost hold. For conventional fixed wing aircraft, this is usually the case, but, for advanced configurations this is not necessarily the case. For example in a V-tail aircraft the control inputs are heavily coupled in the r and p channel. In this case, having the synchronization on u can have adverse effects as shown in the following. As derived in Sects. 4.4 and 4.5, re-formulations of the synchronization as complementary filters allows for separate filtering in the different channels. The following simulations are performed with the exact same lateral system only changing the C B matrix to be strongly coupled as given in Eq. (61). In Fig.  26 the synchronization from Fig. 12 is compared with the re-formulated filter from Fig. 13. It is seen that the synchronization technique on u is even unstable in this case, where the performance of the re-formulated complemen- tary filter approach is not affected by the couplings. Figure  26 also shows the same behavior for the Hybrid approach with synchronization in u as depicted in Fig. 14, and should be compared to the re-formulated Hybrid INDI depicted in Fig. 15.

Conclusion
In this paper, we investigated different ways to compensate and synchronize for filtering and delays of the measurements in several types of incremental control laws. It was shown how to synthesize complementary filters, taking into account the filtering and delays. It was shown that the use of complementary filters in contrast to synchronization on the feedback of u can be especially useful in MIMO systems, if separate filtering is necessary in the different measurements. It was shown that in case of strongly coupled input effectiveness, the synchronization on the actuator feedback may become even unstable, where the equivalent complementary filter formulation recovers the design closed-loop behavior. A method for deriving the initialization of the complementary filters was presented, which allows for transient-free engagement in non-steady state conditions. At last, it was shown that a cascaded complementary filter can be used to obtain an undelayed model-based derivative estimate. Future work will be focused on comparing the performance and robustness in practical applications. and replacing U by (A4) Using the assumption in Eq. (42) and rearranging the above results in Multiplying with I −Ḡ A from the left, results in

A.2 INDI with complementary filter from Section 3.3
Given the system (41) and the INDI control law with block diagram depicted in Fig. 10, then the closed-loop transfer function can be derived as follows, ifẏ mdl is calculated bẏ The transfer function from u to u in the Laplace domain can be deduced from the block diagram to be (A1). Substituting this relation into (A2) and replacing U by Adding and subtracting C B d D in the term fromẎ mdl and using the structure of sY from (A2) gives: Applying the assumption given in Eq. (42), arranging for sY and multiplying by (I −Ḡ A ) from the left gives: Finally, is obtained.

A.3 INDI with derivative filter and no synchronization
Given the system (41) and the INDI control law with block diagram depicted in Fig. 11, the closed-loop transfer function can be derived as follows. The transfer function from u to u in the Laplace domain can be deduced from the block diagram to be (A1). Substituting this relation into (A2) and replacing U by Using the assumption in Eq. (42) and rearranging the above results in A H F cy sY Finally, is obtained.

A.4 INDI with derivative filter and synchronization
Given the system (41) and the INDI control law with block diagram depicted in Fig. 12, the closed-loop transfer function can be derived as follows.
The transfer function from u to u in the Laplace domain can be deduced from the block diagram to be Substituting this relation into (A2) and replacing U by If the synchronization is implemented according to the block diagram in Fig. 13, then the closed-loop transfer function can be calculated as follows: Applying the assumption given in Eq.
Finally, the closed-loop relation can be obtained: Note that in contrast to the original implementation in Fig. 12, with the proposed modification in Fig. 13 it is not required to assume thatHF cy are square, only have elements on the diagonal and u and y have the same dimension, or that H F cy only has elements on the diagonal and all the elements are the same. This means that u and y can have different dimensions and different filtering can be applied to the different measurement signals.

A.5 Hybrid INDI with complementary filter using synchronization
Given the system (41) and the INDI control law with block diagram depicted in Fig. 14, the closed-loop transfer function can be derived as follows, ifẏ mdl is calculated bẏ where x f is x filtered by F cx (s). The transfer function from u to u in the Laplace domain can be deduced from the block diagram to be (A32) Assuming that F cy and F cx are diagonal with the same elements on the diagonal, then is obtained.