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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. Keywords Incremental control laws · Complementary ﬁlters · Sensor dynamics · Sensor delays 1 Introduction plant to be controlled, INDI simply uses measurements of derivatives of the output and the input effectiveness. This In the past decade, the control concept Incremental Non- is because the knowledge of the state dependent dynam- linear Dynamic Inversion (INDI) gained a lot of interest, ics can be approximated by an appropriate measurement. especially in the domain of ﬂight control. While control tech- INDI can, therefore, be interpreted as a sensor-based control niques such as Non-linear Dynamic Inversion (NDI) [1–4] method. This reduced model-dependency has been proven to and Backstepping (BS) [5] rely on accurate models of the be advantageous when controlling aerial vehicles because of multiple reasons: Rasmus Steffensen and Agnes Steinert have contributed equally to this work. 1. Novel aerial vehicle conﬁgurations might be complex to B Rasmus Steffensen model or the models might be inaccurate or lacking the rasmus.steffensen@tum.de inﬂuence of dominating physical effects. Agnes Steinert 2. Aircraft models depend on aerodynamic parameters, agnes.steinert@tum.de which are expensive to identify, e.g., by performing Zoe Mbikayi extensive ﬂight test campaigns; zoe.mbikayi@tum.de 3. Changes in the vehicle design require a repetition of the Stefan Raab identiﬁcation process, and therefore the ﬂight test cam- stefan.raab@tum.de paigns; Jorg Angelov 4. Robustness against a wide range of faults or structural jorg.angelov@tum.de damages; Florian Holzapfel 5. Robustness against disturbances; ﬂorian.holzapfel@tum.de 6. Suitability for automatic take-off and landing, especially for ultra-light aerial vehicles, where ground effects have Institute of Flight System Dynamics, Technical University of Munich, Boltzmannstrasse 15, Garching 85748, Germany a large impact and are difﬁcult to model; 123 Aerospace Systems 7. Uniﬁed control strategies without the necessity of switch- mended because of its sensitivity to sensor noise leading ing or blending between different control laws for differ- to an ampliﬁcation of noise on the control signal. In [24], ent ﬂight phases. the angular accelerations were estimated with this technique, using backward ﬁnite differences, and a ﬁrst-order low-pass Incremental non-linear ﬂight control techniques, such as ﬁlter was introduced before the actuators to attenuate high- INDI, have been successfully demonstrated on several aerial frequency oscillations caused by numerical noise. With the platforms and for different applications: the Cessna Citation sensor dynamics and measurement delays present, a low- II [6,7], VTOL transition UAVs with movable tilts [8–10], amplitude high-frequency oscillation still remained in the quadrotor UAVs [10–12], hexacopter [14], a hybrid tail-sitter control commands. [28] proposed to ﬁlter the measurement, UAV [15,16], the automatic take-off and landing of a ﬁxed using a second-order low-pass ﬁlter, in combination with a wing tail-wheel aircraft [17], a helicopter [18], an airship discrete differentiation. This allows to estimate the deriva- [19], a piloted all attitude ﬁxed wing turboprop demonstra- tive for low frequencies, while attenuating high-frequency tor simulated aircraft [20], and a commercial civil aircraft sensor noise. Since this kind of second-order ﬁlter differen- simulation model [21,22]. tiator introduces lag and produces a delayed estimate of the It is commonly known that one major challenge of INDI relevant output derivative which degrades the controller per- is its sensitivity towards measurement delays in the output formance and stability, several techniques were proposed to derivative feedback [23–26] and actuator command delay accommodate this issue. [29] proposes the use of an input [EINDI]. In van’t Veld et al. [27], the authors show that the scaling gain to increase the controller robustness to delays stability of INDI controllers subject to measurement time in the feedback loop, when using the second-order ﬁlter dif- delay is an issue, especially, when the transmission delay of ferentiator. Other approaches are based on the importance the actuator measurements and that of the relevant output of delaying the actuator and state derivative measurements derivative measurement are not equal. Delays are usually equally, which has been demonstrated in van’t Veld et al. encountered for several reasons: [27]. [6,11,27] synchronize the actuator measurement in time with respect to the angular acceleration by a synchronization 1. Control signal transmission delay; ﬁlter. This synchronization corresponds to the second-order 2. Sensor signal delays; low-pass ﬁlter and a time delay, corresponding to the rela- 3. Processing and scheduling delays. tive delay between the angular rates and actuator deﬂection measurements. Besides the second-order ﬁlter differentiator, In van’t Veld et al. [27], it is observed that INDI is more complementary ﬁlters have been proposed for estimating the sensitive to output derivative measurement delay than actu- angular accelerations, where the measured rate is low-pass ator delay. Additionally, signal ﬁltering can introduce lag to ﬁltered and differentiated and the high-frequency part of the the system which might be an issue for the controller perfor- rate derivative is obtained using a high-pass ﬁltered model- mance and stability as demonstrated in van’t Veld et al. [27] based rate acceleration estimate [14,30]. The latter further as well. The use of signal ﬁltering in control laws is of high proposed a synchronization scheme on the actuator feed- importance, for performance, stability, safety and robustness. back in combination with the complementary ﬁlter. In [23], There are several reasons for introducing ﬁltering of the used a linear predictive ﬁlter is designed to predict the angular signals. The most common reasons are: rate derivatives from angular rates and references thereof. The coefﬁcients of the predictive ﬁlter are computed using 1. Anti aliasing ﬁltering; least squares estimation [31] designs a nondelay differen- 2. Roll-off ﬁltering for noise attenuation; tiator based on a precise-delay differentiator, to acquire the 3. Notch ﬁltering for attenuation of structural mode excita- output derivatives without delay. tion; In this paper, it is investigated how to compensate for ﬁlter 4. Filtering for estimation of certain variables, e.g., comple- and sensor dynamics in the implementation of different types mentary ﬁltering; of incremental control laws. We start with the Non-Linear 5. Filtering for bias removal. Dynamic Inversion (NDI) control law through the actuators Another common issue related to INDI for aircraft control introduced as Actuator-NDI (ANDI) in Steffensen et al. [32]. applications is that measurements of the angular accelera- From this control law, several other derived control laws tions are often required. Although sensors for measuring can be obtained by applying appropriate assumptions. The most basic form is Incremental Non-linear Dynamic Inver- the angular rate derivative exist, they are not part of the standard equipment of aerial vehicles. Hence, the rate deriva- sion (INDI) [28]. This control law does not explicitly take into account the actuator dynamics. The Extended INDI (E-INDI) tive needs to be estimated in most applications. Obtaining the rate derivative by calculating the discrete derivative of [9] takes the actuators into account, but neglects the state the rate measurement using a differentiator, is not recom- dependent terms. The ﬁlter synchronization is derived ana- 123 Aerospace Systems lytically for the ANDI control law, using a complementary paper further provides insights into the relations between ﬁlter exploiting relevant model information to compensate these different approaches. Additionally, the cascaded com- for the negative effects of the ﬁlters and sensors. Analysis is plementary ﬁlter is demonstrated for an ANDI control law, made for the initialization of the ﬁlter, to investigate methods introduced in Steffensen et al. [32], and compared to the E- of transient-free activation of the control law. An analytical INDI law [9]. comparison is made of synchronization techniques used in literature, and a re-formulation of these synchronization tech- niques is performed in terms of equivalent complementary 2 Control laws ﬁlters with several beneﬁts for MIMO systems when the ﬁl- ters applied to the measurements differ. The following section provides a derivation of the ANDI The contribution of this paper consists of: (Actuators in Non-Linear Dynamics Inversion), E-INDI (Extended Incremental Non-linear Dynamic Inversion) and • Formulation of a complementary ﬁlter structure that INDI (Incremental Non-linear Dynamic Inversion) control can be used for ANDI, E-INDI and INDI control laws. laws, as a basis for the investigation in Sects. 4 and 5. The beneﬁts are 1) an inherent synchronization which 2.1 Summary of ANDI control law increases the robustness with regard to phase delays 2) It allows to apply different ﬁltering on the different mea- We consider the ANDI control law introduced in Steffensen surement signals. For example to attenuate structural modes in the various axes of the aircraft, different notch et al. [32], as depicted in Fig. 1. The ANDI is based on NDI, ﬁlters might be required on the different measurement hence includes the stability guarantees in terms of Lyapunov signals. and exact inversion for tracking as well as speciﬁed error • Formulation of a proper initialization method for these dynamics. With the assumption of ﬁrst-order actuators from complementary ﬁlters as used in incremental control ANDI, the approximate concepts INDI and E-INDI can be laws, for transient-free engagement in non-steady-state deduced. Hence, the ANDI is taken as a starting point, and conditions. the concepts of synchronization will be extended directly to • Derivation of alternative formulations of popular syn- INDI and E-INDI. As was shown in Steffensen et al. [32], the ANDI control law corresponds to an INDI control law if the chronization techniques in the actuator feedback path as modiﬁed complementary ﬁlters. These alternative formu- bandwidth of the actuators tends to inﬁnity, and corresponds lations have the beneﬁt that they can recover the design to the E-INDI [9]ifthe F x˙ term is neglected as explained closed-loop transfer function structure in MIMO systems later in this section. The ANDI control law allows to actively when different sensor ﬁltering in the various feedback account for the actuator and state dependent dynamics and is channels are used. It is shown that the synchronization in summarized in the following. Consider the system the feedback can become even unstable in case of strongly x˙ = f (x , u), coupled input effectiveness, where the equivalent com- plementary ﬁlter exactly recovers the design closed-loop y = h(x ), (1) behavior. n k • Application of appropriate ﬁlters on the output of the with system state x ∈ R , actuator state u ∈ R , output y ∈ m (r +1) n k n (r +1) n m reference model based on the ﬁltering on the measured R , f ∈ C (R × R ; R ) and h ∈ C (R ; R ). signals for use in the error controller feedback. Note, that the time dependence (t ) was here and in the fol- • Derivation of an un-delayed state estimate using a cas- lowing omitted for better readability. The actuator dynamics caded complementary ﬁlter taking into account signal are given by: ﬁltering and delays. This un-delayed state estimate can be used to obtain an un-delayed y ˙ . This cascaded ﬁlter mdl u ˙ = (u − u) , (2) additionally provides an x˙ estimate that can be used in (k×k) control laws based on ANDI. with diagonal matrix ∈ R with its entries representing the bandwidth of the different actuators. In Steffensen et al. Based on a general linear MIMO system, the closed-loop [32] the ANDI control law is derived assuming the system to dynamics is analyzed analytically for (1) vanilla (perfect have a relative degree of r ∈ N, i.e., the rth derivative of y knowledge of all variables) INDI (2) INDI with complemen- with respect to time is the ﬁrst derivative of y which explicitly tary ﬁlter and y ˙ (3) ﬁlter differentiator approach without depends on u. For the sake of simplicity, we assume here mdl synchronization (4) ﬁlter differentiator approach with syn- r = 1: chronization (5) Hybrid INDI, i.e., complementary ﬁlter approach with synchronization. The analysis done in this y ˙ = F (x , u), (3) 123 Aerospace Systems 1 n k m with F ∈ C (R × R ; R ). The second derivative of y is In addition, x˙ needs to be estimated, which is covered in ˆ ˆ then Sect. 4.6. Note, that in some cases x˙ is the same as y ˙, and the same complementary ﬁltered signal can be used. y ¨ = F x˙ + F u ˙, (4) x u 2.3 ANDI and INDI ∂ F (x ,u) ∂ F (x ,u) where F = and F = . Substituting Eq. (2) x u ∂ x ∂u To arrive at the INDI control law from the ANDI control into (4) results in law, a special choice of error dynamics have to be made as shown in Steffensen et al. [32]. The error dynamics shall be y ¨ = F x˙ + F (u − u) , (5) x u c chosen as a product of actuator and system error dynamics as shown in Eq. (10). The actuator loop of the ANDI error whichissolvedfor u , assuming F to have full row rank c u dynamics, i.e., the second part of Eq. (10), shall be chosen and choosing y ¨ = ν as the virtual command: with the same bandwidth as the actuators, ω. The assumption necessary on the system is that all actuators need to have the u = (F ) (ν − F x˙ ) + u, (6) c u x same bandwidth, that is = ω I . k×k where (F ) denotes a right inverse matrix that solves the E (s) (s + K ) (sI + ω I ) = 0, (10) y 0 linear equation system given in Eq. (5). The virtual control input ν is designed using a linear error controller as with the errors deﬁned as: e = y − y, y re f ν =¨ y + k (y ˙ −˙ y) + k (y − y). (7) ref 1 ref 0 ref (11) e ˙ =˙ y −˙ y. y re f By applying the control law to the system, the desired error In the time domain, this can be expressed as: dynamics are obtained as designed, because F has full row rank such that (F )(F ) = I : u u m×m e ¨ + k e ˙ + k e =¨ e + K e ˙ + ω(e ˙ + K e ) = 0. (12) y 1 y 0 y y 0 y y 0 y y ¨ = F x˙ + F (F ) (ν − F x˙ ) + u − u x u u x Then, the pseudo-control ν is chosen as: (8) =¨ y + k (y ˙ −˙ y) + k (y − y). ref 1 ref 0 ref ν =¨ y + K e ˙ + ω(e ˙ + K e ), (13) re f 0 y y 0 y The block diagram of the ANDI controller is depicted in Fig. and by inserting ν into Eq. (6), the ANDI control law can be expressed as: 2.2 ANDI control law with measured output u = (F ) y ¨ + K e ˙ − F x˙ c u re f 0 y x (14) It is often not possible to directly measure the time derivative † + (F ) e ˙ + K e + u, u y 0 y 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). which was ﬁrst shown in Steffensen et al. [32]. If the actuators We propose to estimate y ˙ using the complementary ﬁlter have a high bandwidth ω, the ﬁrst term of Eq. (14) can be which will be introduced in Sect. 3.3, that uses a model-based neglected, and the remaining part is equivalent to the INDI estimate of y ˙ and the measured and ﬁltered output signal y control law: as depicted in Fig. 2. In addition, due to the ﬁltering of y,it is proposed to ﬁlter the reference signal y with the same re f u = (F ) e ˙ + K e + u. (15) c u y 0 y ﬁltering as the measurement to synchronize the error term. This ﬁltering includes the ﬁltering by F, which is any ﬁltering Similar to Sect. 2.2, the INDI law can be implemented as applied to the signal, e.g., roll-off ﬁlters or notch ﬁlters, the depicted in Fig. 3 using a complementary ﬁltered estimate of measurement delay D , as well as the sensor dynamics S. y ˙ and synchronizing y and y . y re f f The control law is, hence, modiﬁed as such: It shall be noted that sometimes the ﬁrst term in Eq. (14) is included using variables from a reference model [20,33], u = (F ) (ν − F x˙ ) + u such that the command path exactly corresponds to the ANDI c u x control law. If the system is (sufﬁciently) linear, this will (9) = (F ) y ¨ + k (y ˙ − y ˙) u re f 1 re f only affect the feed-forward part of the system, and hence not change the closed-loop poles. + k (y − y ) − F x˙ + u. 0 re f , f f x 123 Aerospace Systems Fig. 1 ANDI with known outputs and state and output derivatives Fig. 2 Block diagram of ANDI with synchronization Fig. 3 Block diagram of INDI with synchronization 2.4 ANDI and E-INDI 3 Complementary ﬁlters and initialization In literature, several extensions to the vanilla INDI control In the following section, the complementary ﬁlter design and law can be found. Two approaches that are taking explicitly initialization is presented, and a formal initialization method the actuator dynamics into account are [9,11]. While [11] is derived for transient-free engagement in non-steady-state assumes that all actuators have the same bandwidth, i.e., = conditions. As will be shown later, this design of the comple- I ω,[9] considers the more general case of actuators with mentary ﬁlters will inherently contain the synchronization k×k different bandwidth. As was shown in Steffensen et al. [32], properties that have earlier been shown to be crucial for choosing the error dynamics given in Eq. (12), the respective increasing performance and stability of INDI control laws. control laws from [9,11], can be re-formulated as 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 u = (F ) y ¨ + K e ˙ + e ˙ + K e +u, (16) synchronization techniques on the actuator feedback. c u ref 0 y y y 0 y whichincaseof = I ω, = I ω, can be expressed k×k y m×m as 3.1 Basic complementary filter concept and initialization (17) u = (F ) y ¨ + K e ˙ + ω e ˙ + K e + u. c u re f 0 y y 0 y 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. This control law equals the ANDI control law in Equation First consider the zero-valued steady-state case. Then, the (14), if F x˙ is neglected. relation between the inputs and outputs of the ﬁlter can be 123 Aerospace Systems Fig. 4 Block diagram of complementary ﬁlter Fig. 6 Block diagram of complementary ﬁlter with sensor dynamics In the Laplace domain, accounting for non-zero initial con- ditions, this results in Y (s) = sX (s) − x (0) + U (s) − X (s). (22) 1 1 2 2 Substituting Eq. (20)into(22) results in ω 1 Y (s) = s U (s) + x (0) − x (0) + U (s) 1 1 1 2 s + ω s + ω ω 1 − U (s) + x (0) 2 2 s + ω s + ω sω ω = U (s) − x (0) Fig. 5 Block diagram of complementary ﬁlter with reset 1 1 s + ω s + ω ω 1 + U (s) − U (s) − x (0). 2 2 2 s + ω s + ω derived as follows: (23) ˙ ˙ Y (s) = (1 − H (s) + H (s))Y (s) (18) ˙ Substituting U (s) = Y (s) and U (s) = Y (s) = sY (s) − 1 2 = (1 − H (s)) Y (s) + H (s)sY (s), y(0), results in where U (s) = Y (s), U (s) = Y (s) and assuming that y(0) 1 2 sω ω Y (s) = Y (s) − x (0) and y ˙(0) are zero. In case that the model y ˙ is accurate, the s + ω s + ω ﬁlter perfectly distributes the high-frequency part of the y ˙ ω 1 + U (s) − (sY (s) − y(0)) − x (0) 2 2 estimate to the model-based y ˙ and the low-frequency part to s + ω s + ω the differentiated y measurement. ω ω 1 =− x (0) + U (s) + y(0) − x (0). 1 2 2 In case of non-steady-state initial conditions, the ﬁlter needs s + ω s + ω s + ω to be carefully initialized. In the following, a method to obtain (24) the correct initialization is derived. For now, consider H (s) as a ﬁrst-order ﬁlter such that the differential equations describ- Initializing the integrators with x (0) = y(0) = u (0) and 1 1 ing the x and x dynamics, as depicted in Fig. 5,are given 1 2 x (0) = 0 as depicted in Fig. 5 by the red dotted lines, results by: in ˙ ˙ x˙ = ω(u − x ), (19) Y (s) = U (s) = Y (s). (25) 1/2 1/2 1/2 Hence, in case of an accurate model of y ˙, the output of the where u = y and u =˙ y. In the Laplace domain, accounting 1 2 ﬁlter will exactly match the true y ˙, also at the initial time also for non-zero initial conditions, the following relation is instance when engaging the control law. obtained: ω 1 3.2 Complementary filter and sensor dynamics X (s) = U (s) + x (0). (20) 1/2 1/2 1/2 s + ω s + ω Usually, the output y of the plant is measured by a sensor The complementary ﬁlter calculates y ˙ as shown in Fig. 5 by with corresponding sensor dynamics S(s). These dynamics may include an anti-aliasing ﬁlter because of the (down) sam- y ˙ =˙ x + u − x . (21) pling of the sensor signal for the ﬂight control computer. For 1 2 2 123 Aerospace Systems now consider these dynamics to be described by a ﬁrst-order The estimate of y ˙ is calculated by the complementary ﬁlter ﬁlter such that the measured plant output that is used in the as depicted in Fig. 7 by complementary ﬁlter corresponds to y ˙ =˙ x +˜ u − x , (31) 1 2 2 Y (s) = S(s)Y (s). (26) The complementary ﬁlter in Fig. 6 is given by which results in the Laplace domain to ˙ ˙ Y (s) = (1 − H (s)S(s)) Y (s) + sH (s)Y (s). (27) mdl s ˙ ˜ Y (s) = sX (s) − x (0) + U (s) − X (s). (32) 1 1 2 2 To show the proper synchronization, now assume that y ˙ = mdl y ˙. The complementary ﬁlter for zero initial conditions results For X (s) and X (s), the relation in Eq. (20) is inserted. The 1 2 in: input U (s) and U (s) are the states of the sensor ﬁlter and 1 2 sensor ﬁlter model as depicted in Fig. 7, i.e., U (s) = X (s), 1 1 ˙ ˙ Y (s) = (1 − H (s)S(s)) Y (s) + sH (s)Y (s). (28) U (s) = X (s): 2 2 Hence, with an accurate model of y ˙, the ﬁlter again exactly ω 1 distributes the high-frequency part of the y ˙ estimate to the ˙ ˜ ˜ Y (s) = s X (s) + x (0) − x (0) + U (s) 1 1 1 2 s + ω s + ω model-based y ˙ and the low-frequency part to the differenti- ated y in the presence of the sensor dynamics S(s). ω 1 − X (s) + x (0) . (33) 2 2 To determine the initial values of the integrator state in the s + ω s + ω sensor ﬁlter model used in the complementary ﬁlter, we start with considering the differential equations of the sensor ﬁlter ˜ ˜ Substituting X and X in Eq. (33) by relation (30) results in and the sensor ﬁlter model, as depicted in Fig. 7, given by 1 2 x˜ =˜ ω(u ˜ −˜ x ), (29) 1/2 1/2 1/2 sω ω ˜ 1 s ˙ ˜ Y (s) = U (s) + x˜ (0) + x (0) 1 1 1 s + ω s +˜ ω s +˜ ω s + ω where u ˜ = y and u ˜ =˙ y. The resulting relation in the 1 2 − x (0) + U (s) 1 2 Laplace domain is ω ω ˜ 1 1 − U (s) + x˜ (0) − x (0). 2 2 2 s + ω s +˜ ω s +˜ ω s + ω ω ˜ 1 ˜ ˜ X (s) = U (s) + x˜ (0). (30) 1/2 1/2 1/2 (34) s +˜ ω s +˜ ω Fig. 7 Block diagram of complementary ﬁlter with reset 123 Aerospace Systems ˜ ˜ ˙ Using that U (s) = Y (s) and U (s) = Y (s) = sY (s) − y(0) 1 2 results in sω ω ˜ 1 s Y (s) = Y (s) + x˜ (0) + x (0) 1 1 s + ω s +˜ ω s +˜ ω s + ω − x (0) + Y (s) ω ω ˜ 1 − (sY (s) − y(0)) + x˜ (0) s + ω s +˜ ω s +˜ ω − x (0) s + ω Fig. 8 Block diagram of complementary ﬁlter with sensor dynamics sω s and ﬁltering = x˜ (0) + x (0) 1 1 (s + ω)(s +˜ ω) (s + ω) signal used by the complementary ﬁlter corresponds to s + ω − x (0) + Y (s) (s + ω) Y (s) = F (s)D (s)S(s)Y (s), (37) f y ω ω ˜ 1 1 − − y(0) + x˜ (0) − x (0) 2 2 s + ω s +˜ ω s +˜ ω s + ω where F (s) is the transfer function of the roll-off or notch sω = x˜ (0) 1 ﬁlters, D(s) is the transmission delay of the measurement (s + ω)(s +˜ ω) and S(s) is the sensor dynamics. The complementary ﬁlter ω(s +˜ ω) − x (0) + Y (s) 1 needs then to be updated as shown in Fig. 8, such that (s + ω)(s +˜ ω) ωω ˜ ω + y(0) − x˜ (0) 2 ˙ ˙ Y (s) = 1 − H (s)F (s)D (s)S(s) Y (s) (s + ω)(s +˜ ω) (s + ω)(s +˜ ω) + H (s)F (s)D (s)S(s)Y (s) − x (0). s + ω = 1 − H (s)F (s)D (s)S(s) Y (s) + sH (s)Y (s). y f (35) (38) Initializing the integrators as discussed before with with The initialization of the ﬁlters can be derived in a similar man- x (0) = u (0) =˜ x (0) and x (0) = 0 as depicted in Fig. 7 1 1 1 2 ner as in the previous sections. In the following, we denote by the red dotted lines, results in the series of the ﬁlters as ωω ˜ ˙ ˙ Y (s) = (y(0) −˜ x (0)) + Y (s) F (s) = F (s)D (s)S(s). (39) cy y (s + ω)(s +˜ ω) (36) − x˜ (0) The complementary ﬁltered estimate of the derivative of the (s + ω)(s +˜ ω) output signal y is, hence, given by Hence, the sensor model state needs to be initialized to ˙ ˙ x˜ (0) =˜ ω (y(0) −˜ x (0)) which corresponds to the time Y (s) = (1 − H (s)F (s))Y (s) + sH (s)Y (s), (40) 2 1 cy mdl f derivative x˜ (0) of the sensor ﬁlter state as depicted in Fig. ˙ ˙ where Y (s) is the model-based estimate of Y . 7 by the red dotted lines. mdl Note that in case the sensor state x˜ is initialized to y(0), x˜ (0) will be 0. This will be the case if the sensor is reset, 4 Synchronization techniques with related and then the integrator state x˜ of the sensor model could be initialized with 0. Otherwise, alternatively as an approxima- complementary ﬁlters tion of x˜ the model based y ˙ could be used as initialization In the following, the proposed synchronization is compared value for x˜ . to other synchronization approaches in literature. For the analysis, we consider a MIMO linear time invariant system: 3.3 Complementary filter, sensor dynamics and additional filtering x˙ (t ) = Ax (t ) + Bu(t ) + B d(t ), (41) (r −1) Often the measured output y of the plant is additionally y (t ) = Cx (t ), ﬁltered to attenuate noise, vibrations from engines, pickup of structural modes, etc. Commonly roll-off ﬁlters or notch where r is the relative degree of the system, A is the system ﬁlters are used for this purpose. Furthermore, transmission matrix, B is the input effectiveness, B d(t ) is the disturbance delays might be present on the measurements such that the term, C the output matrix, and with the restriction that: 123 Aerospace Systems Table 1 Effect of synchronization on CAX in closed-loop Synchronization CAX contribution Vanilla INDI I − G INDI with complementary ﬁlter and y ˙ I − G mdl A INDI with derivative ﬁlter and synchronization I − G HF A cy Fig. 9 Vanilla INDI control law Hybrid INDI (I − G F ) A cy −1 −1 † ¯ ¯ CB(I − G (s)) G (s)(CB) = I − G (s) G (s), A A A A (42) where G (s) are the actuator dynamics, G (s) is a diagonal A A transfer function matrix, and the matrix CB has full row rank. Fig. 10 INDI with complementary ﬁlter Equation (42)istrueifeither: Basic assumptions (either has to hold): 4.1 Vanilla INDI The vanilla INDI with perfect knowledge of the state deriva- 1. System is SISO, then G = G . A A tive is depicted in Fig. 9. The closed-loop transfer function 2. CB matrix and actuator transfer function matrix G are is given by: square, only have elements on the diagonal, and they have the same dimension, such that G = G . A A ¯ ¯ ¯ sY = I − G CAX + G V + I − G CB D, (43) 3. Actuator transfer function matrix G only has elements A A A d on the diagonal and all the elements are the same, i.e., as shown in Appendix A.1. It is seen that y ˙ follows ν with the the actuators operate independently and have the same actuator dynamics, plus the state dynamics and disturbances dynamics. Then, G is a diagonal transfer function ﬁltered through a high-pass ﬁlter given by I − G , i.e., if the matrix with elements equal to the elements in G but A actuator dynamics are much faster than the state dynamics the matrix have a different dimension. and disturbances, the inﬂuence is ﬁltered out. For the following, we consider without loss of generality a 4.2 INDI with complementary filter and y ˙ mdl system with relative degree 1, which is common in baseline ﬂight control laws. In addition it improves the readabil- If y ˙ is estimated with a complementary ﬁlter as derived in ity. Substituting sY (s) with s Y (s) for the following results Sect. 3.3, and depicted in Fig. 10, assuming that there are no recover the general case. Note that for better readability the model uncertainties or noise, then y ˙ =˙ y holds, and the mdl arguments (s) and (t ) are omitted. closed-loop transfer function is given by: In the following, it is shown that the different synchro- nization techniques only vary in their inﬂuence on the state ¯ ¯ ¯ sY = I − G CAX + G V + I − G HF CB D A A A cy d dependent term CAX of the closed-loop dynamics, summa- rized in Table 1. The inﬂuence on the direct disturbance term (44) CB D and pseudo-control term V is exactly the same for all synchronization techniques. as shown in Appendix A.2, which shows that the complemen- It is additionally shown that for methods 3 and 4 as pro- tary ﬁlter recovers the input–output dynamics of the vanilla posed in [6,11,27] and [30], for MIMO systems, additional INDI. It is seen that the disturbance is high-pass ﬁltered by restrictive assumptions, e.g., same ﬁltering of all measure- I − G HF , such that disturbances below the cut-off of A cy ments, are necessary to obtain the relations given in Table 1. the combined actuator and ﬁlter dynamics are rejected. The respective assumptions will be detailed in the related sec- Note, y ˙ =˙ y might seem like a crude assumption but is mdl tions. A re-arrangement of the synchronization is proposed, made to highlight the synchronization effects i.e., y ˙ gives mdl such that the design closed-loop transfer function can be an un-delayed estimate of y ˙. For example y ˙ = C ( Axˆ + mdl recovered for MIMO systems, relaxing the above-mentioned Bu) would be such an estimate, if xˆ is obtained as an un- assumptions. This re-arrangement allows different ﬁltering delayed estimate, for example from a model or using again in the measurement feedback channels. a complementary ﬁlter as depicted in Fig. 16. 123 Aerospace Systems Fig. 11 INDI with derivative ﬁlter Fig. 13 Re-formulation of INDI with derivative ﬁlter and synchroniza- tion ¯ ¯ ¯ sY = I − G HF CAX +G V+ I − G HF CB D, A cy A A cy d (46) Fig. 12 INDI with derivative ﬁlter and synchronization as used in [6, assuming that: 11,27] −1 −1 ¯ ¯ ¯ ¯ CB I − G H F G (CB) = I − G HF G , A cy A A cy A Note that y ˙ will not be y ˙ if there are model uncertain- mdl (47) ties (e.g., in A, B or C) or measurement noise. The cut-off frequency of the ﬁlter H in the complementary ﬁlter has to which hold if either of these additional assumptions are sat- be chosen as a compromise, in general as high as possible isﬁed, respectively: but low enough such that the noise is not ampliﬁed by the Respective additions to the basic assumptions: derivative. 1. No additional assumption if system is SISO. 4.3 INDI with derivative filter and no ¯ ¯ 2. H and F are square, only have elements on the diagonal cy synchronization and both have the same dimension as H and F , such cy ¯ ¯ that H F = HF . cy cy If y ˙ is estimated with a ﬁlter from a sensor measurement as 3. H and F only has elements on the diagonal and all the cy depicted in Fig. 11, then the closed-loop transfer function is elements of the respective matrix are the same, i.e., each given by: measurement channel is ﬁltered separately and equally. ¯ ¯ Then, H F is a diagonal transfer function matrix with cy −1 ¯ ¯ ¯ sY = I − G + G HF I − G CAX A A cy A elements equal to the elements in HF but the matrix cy −1 has a different dimension. ¯ ¯ ¯ (45) + I − G + G HF G V A A cy A −1 ¯ ¯ ¯ + I − G + G HF I − G CB D, A A cy A d It can be seen that this ﬁltering technique results in a closed- loop transfer behavior which is similar to the result of the as shown in Appendix A.3. It can be seen that this ﬁlter- vanilla INDI, i.e., y ˙ follows ν with the actuator dynam- ing technique does not correspond to the dynamics of the ics, plus the state dynamics ﬁltered through a high-pass vanilla INDI, i.e., INDI with perfect state derivative knowl- ﬁlter. While for vanilla INDI this high-pass ﬁlter is given edge. It is shown in Section 5 by simulation that having no ¯ ¯ by I − G , here the high pass results in (I − G HF ), A A cy synchronization can very easily lead to unstable closed-loop which allows signals with lower frequencies to pass through dynamics. compared to (I − G ), which means that more dynamics from the model dependent part CAX will pass through in 4.4 INDI with derivative filter and synchronization the closed-loop dynamics. The direct inﬂuence of the distur- bances are also high-pass ﬁltered by (I − G HF ), which A cy If y ˙ is estimated, as proposed in [6,11,27], and depicted in is similar to the complementary ﬁlter discussed in Sect. 4.2. Fig. 12, i.e., with a ﬁlter from a sensor measurement and In Appendix A.4, it is shown that if the block diagram in ¯ ¯ synchronized with the same ﬁlters H and F (with possi- Fig. 13 is implemented instead, then the same closed-loop cy bly different dimensions as indicated by the bar), as used for behavior as given in Eq. (46) is obtained, even without the the derivative and sensor ﬁltering in the control signal feed- additional assumptions given above. back path, then, as shown in Appendix A.4, the closed-loop It is additionally seen that the synchronization in Fig. 13 transfer function is given by: corresponds to the INDI with complementary ﬁlter depicted 123 Aerospace Systems Fig. 15 Re-formulation of hybrid INDI with complementary ﬁlter Fig. 14 Hybrid INDI with complementary ﬁlter and synchronization (I − G ), and the INDI with derivative ﬁlter and synchro- as proposed in Kumtepe et al. [30] nization (Sect. 4.4), (I − G HF ). If the state dynamics are A cy slow enough y ˙, hence follows ν with the actuator dynamics G . The direct inﬂuence of the disturbances are high-pass in Fig. 10 with y ˙ = Bu, i.e., only the high-frequency part A mdl ﬁltered by (I − G HF ), which is similar to the comple- A cy of y ˙ is considered in the estimate y ˙ . This re-formulation mdl mentary ﬁlter discussed in Sect. 4.2 and the derivative ﬁlter can be attractive if different ﬁlters in F areusedinthe cy with synchronization discussed in Sect. 4.4. separate control channels of y, e.g., that notch ﬁlters with The assumptions above can again be relaxed if the syn- different notch frequencies are used for the individual axes chronization on the feedback is moved to the complementary of the aircraft. Having the ﬁlter synchronization in the u feed- ﬁlter and the ﬁlter chain F (s) is split into sensor ﬁlters S(s) back path does not allow for consistent separate ﬁltering in cy and notch/roll-off ﬁlters F (s) as shown in Fig. 15. Different the different axes. ﬁltering F (s) in the control law can now be applied in the feedback paths. With this modiﬁcation the only assumption is 4.5 Hybrid INDI the sensor ﬁlters S (s) and S (s) are diagonal and identical in x y each channel in order to obtain the relation given by Eq. (50). If y ˙ is estimated, as proposed in Kumtepe et al. [30], with a complementary ﬁlter and synchronized as depicted in Fig. 4.6 Un-delayed estimate of model-based output 14 (with the On-board plant model (OBPM) being y ˙ = mdl derivative y ˙ mdl CAx +CBu, then the closed-loop transfer function is given by: In Section 4.2, the analysis was based on an un-delayed model estimate, y ˙ ,of y ˙. This can for this example be obtained by mdl ¯ ¯ sY = I − G HF CAX − G [I − H ] CAF X A cy A cx a cascaded complementary ﬁlter for x˙ and then x as depicted ¯ ¯ +G V + I − G HF CB D, (48) A A cy d in Fig. 16. Since y ˙ = C ( Ax + Bu), this provides the nec- essary estimate for the use in Section 4.2. Furthermore, this as shown in Appendix A.5, under the assumption that complementary ﬁlter can provide the estimate x˙ for the ANDI control law. −1 ¯ ¯ ¯ CB I − G H F − G (I − H ) G (CB) A cy A A −1 ¯ ¯ = I − G HF − G (I − H ) , (49) A cy A 5 Simulation which holds if the basic assumptions and the additional 5.1 Roll dynamics example assumptions mentioned in Sect. 4.4 are satisﬁed, respectively. If for example F and F are diagonal with the same ele- cy cx This subsection compares the different presented synchro- ments on the diagonal, then: nization approaches for the simpliﬁed roll motion dynamics of an aircraft, given by ¯ ¯ sY = (I − G F )CAX + G V A cy A + I − G HF CB D. (50) A cy d p ˙ = L p + L ξ, (51) p ξ In this case, it can be seen that this ﬁltering technique pro- with roll damping L =−2.71/s, effectiveness L =−14 p ξ duces a high-pass ﬁlter on the states which is (I −G F ) and 1/s , control input ξ , which is the aileron deﬂection and mea- A cy hence with a cut-off frequency which is in between the cut-off sured output p, which is the roll rate. The actuator dynamics frequencies of the high-pass ﬁlter in vanilla INDI (Sect. 4.1), are given by 123 Aerospace Systems Fig. 16 State complementary ﬁlter for un-delayed model estimate of x˙ and x Fig. 17 ANDI roll control G = , (52) (s + ω ) with ω = 50 rad/s. The sensor dynamics are given by S(s) = , (53) s + ω with bandwidth ω = 100 rad/s. The delay is given by −Ts D (s) = e , (54) with T = 0.03s. For this example, the ﬁlter F (s) = 1. F (s) cy is given by Eq. (39). The ﬁlter H is given by H (s) = , (55) s + ω H Fig. 18 Comparison of vanilla INDI and INDI with derivative ﬁlter without synchronization with bandwidth ω = 30 rad/s. Figure 18 shows the desired roll acceleration signal p ˙ , des which shall be tracked, and the closed-loop roll acceleration (Fig. 14), and INDI with the cascaded complementary ﬁl- p ˙, for the vanilla INDI law depicted in Fig. 9, and the INDI ter (Figs. 10 and 16). For all cases, the pseudo-control is with derivative ﬁlter and no synchronization depicted in Fig. again ν =˙ p . It is seen that INDI with derivative ﬁlter des 11. The pseudo-control is in both cases ν =˙ p . Figure and synchronization improves the response compared to the des 18 shows that in case, the derivative of p is obtained by INDI with derivative ﬁlter and without a synchronization, by a differentiation ﬁlter without a synchronization technique, removing the oscillations. The hybrid INDI with the com- the resulting response might be oscillatory. With increasing plementary ﬁlter and synchronization improves the response phase loss of the ﬁlter F (s), the closed-loop system will further (the resulting response is closer to the response cy become unstable as happened in this case. obtained by vanilla INDI) compared to the pure derivative Figure 19 compares the responses of vanilla INDI (Fig. 9), ﬁlter with synchronization. Compared to that, the INDI with INDI with derivative ﬁlter and synchronization (Fig. 12), the proposed cascaded complementary ﬁlter improves the hybrid INDI with complementary ﬁlter and synchronization response further by recovering the vanilla INDI response. 123 Aerospace Systems Fig. 21 Comparison of control signal with noise on the measurement Fig. 19 Comparison of vanilla INDI and INDI with different synchro- of y nization variants results in exact tracking of the reference signal p ˙ ,atthe re f cost of additional necessary model information. 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. 5.2 Linearized lateral motion example This subsection compares the different presented synchro- nization approaches for the linearized lateral dynamics of an aircraft given by Fig. 20 Veriﬁcation of the behavior of ANDI /E-INDI with cascaded ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ synchronization r˙ N N N 0 r N N r β p ξ ζ ⎢ ˙ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −1 + Y Y 0 β β Y Y ξ r β ξ ζ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ = + ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ p ˙ p L L ζ L L L 0 r β p ξ ζ However, if the sensor dynamics are fast and the measure- 00 1 0 00 ment delay very low, the differences in the responses might be ⎡ ⎤ ⎡ ⎤ minor. In this case, the hybrid INDI or INDI with differentia- −0.520 3.488 −0.628 0 r tion ﬁlter and synchronization might be the preferred solution ⎢ ⎥ ⎢ ⎥ −0.987 −0.199 0 0.130 β ⎢ ⎥ ⎢ ⎥ because of its lower complexity. If the sensor dynamics are ⎣ ⎦ ⎣ ⎦ 0.472 −14.408 −6.624 0 p slow or the measurement delay non-negligible, the cascaded 001 0 ⎡ ⎤ complementary ﬁlter might be preferred. 0.539 −2.005 Figure 20 compares the E-INDI control law given by Eq. (17), ⎢ ⎥ −0.012 0.040 ξ ⎢ ⎥ with the ANDI control law, given by Eq. (9), and depicted in + , (56) ⎣ ⎦ −10.700 2.899 ζ Figs. 2 and 17. Both the E-INDI and ANDI are implemented with K = k = 0, i.e., without roll rate error feedback. 0 0 Figure 20 reveals that E-INDI recovers the phase of the ref- T T erence signal. This is because of the feed forward of y ¨ in with y =[r , p] and x =[r,β, p,] . All states are re f E-INDI. ANDI also uses this feed forward and additionally ﬁltered by the sensor dynamics S(s) given by Eq. (53) the model dependent term F x˙, which corresponds to L p ˙ in with ω = 100 rad/s and are delayed with Eq. (54) with x p S the considered example. The closed-loop response of ANDI T = 0.03s. Only the rates p and r are additionally ﬁltered 123 Aerospace Systems Table 2 Values of notch ﬁlters ζ ω (rad/s) g N N min p 0.72 π ·20.1 r 0.72 π ·50.3 Fig. 23 Comparison of vanilla INDI and INDI with different synchro- nization variants ( p ˙ command to r˙ response) Fig. 22 Comparison of vanilla INDI and INDI with different synchro- nization variants ( p ˙ command to p ˙ response) by a notch ﬁlter given by 2 2 s + 2g ζ ω + ω min N N F (s) = , (57) s + 2ζ ω + ω N N with parameters summarized in Table 2 for p and r, respec- tively. The ﬁlter H is given by Eq. (55) with bandwidth ω = 30 rad/s. For the different INDI synchronization structures ν represents r˙ and p ˙. For the E-INDI and ANDI structures ν represents r¨ and p ¨. F = CB is given by Fig. 24 Comparison of vanilla INDI and INDI with different synchro- nization variants (r˙ command to p ˙ response) N N ξ ζ F = , (58) L L ξ ζ responses for the given command in p ˙ . It can be seen des that with the proposed cascaded complementary ﬁlter the and F = CA by response is equivalent to the vanilla INDI response. The other techniques slightly deteriorate the response, especially with N N N 0 r β p F = . (59) regard to the decoupling of the axes. If a pure roll command L L L 0 r β p is given the response in the r channel is desired to be as The actuator dynamics are given by low as possible, but the excursion is much higher in the con- ventional synchronization schemes as shown in Fig. 23.The −1 G = (sI + ) , (60) same holds for the responses to a yaw step command, which are depicted in Figs. 24 and 25. with = I ω and ω = 50rad/s. Figures 22 to 25 com- The simulations done until now show that the cascaded A A pare the response of ideal vanilla INDI (i.e., with perfect ﬁlter has improvement over the synchronization on u,but knowledge of y ˙ as depicted in Fig. 9) with the different syn- the synchronization on u still works reasonably well. The chronization techniques depicted in Figs. 12, 14 and 10, with synchronization approach does not require any additional the notch ﬁlters placed on the dominant feedback channel model information; hence, it might be the preferred choice. of the input. Figures 22 and 23 thereby show the p ˙ and r˙ The above simulations were made with a weakly coupled 123 Aerospace Systems Fig. 25 Comparison of vanilla INDI and INDI with different synchro- Fig. 27 Veriﬁcation of the behavior of ANDI /E-INDI with cascaded nization variants (r˙ command to r˙ response) synchronization ( p ˙ command to p ˙ response) Fig. 26 Comparison of INDI with synchronization on u with the re- Fig. 28 Veriﬁcation of the behavior of ANDI /E-INDI with cascaded formulated synchronization on the feedback ( p ˙ command to p ˙ response) synchronization ( p ˙ command to r˙ response) CB matrix; hence, the assumptions given in Eqs. (47) and tary ﬁlter approach is not affected by the couplings. Figure (49) almost hold. For conventional ﬁxed wing aircraft, this 26 also shows the same behavior for the Hybrid approach is usually the case, but, for advanced conﬁgurations this is with synchronization in u as depicted in Fig. 14, and should not necessarily the case. For example in a V-tail aircraft be compared to the re-formulated Hybrid INDI depicted in the control inputs are heavily coupled in the r and p chan- Fig. 15. nel. In this case, having the synchronization on u can have adverse effects as shown in the following. As derived in 1.8 −2 Sects. 4.4 and 4.5, re-formulations of the synchronization CB = . (61) coupled −10.78 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 CB At last, we compare the ANDI and E-INDI approaches used matrix to be strongly coupled as given in Eq. (61). In Fig. with the proposed cascaded complimentary ﬁlter in Figs. 26 the synchronization from Fig. 12 is compared with the 27, 28, 29 and 30. It can be seen that the response for the re-formulated ﬁlter from Fig. 13. It is seen that the syn- ANDI technique with cascaded complementary ﬁlter per- chronization technique on u is even unstable in this case, fectly tracks the reference dynamics and perfectly decouples where the performance of the re-formulated complemen- the roll and yaw channel. 123 Aerospace Systems in non-steady state conditions. At last, it was shown that a cascaded complementary ﬁlter can be used to obtain an un- delayed model-based derivative estimate. Future work will be focused on comparing the performance and robustness in practical applications. Acknowledgements The authors would like to thank Tijmen Pollack, Haichao Hong and Ewoud Smeur for valuable discussions and com- ments. The authors would also like to thank the reviewers for their very relevant comments. Funding This research is co-funded by the European Union in the scope of INCEPTION project, which has received funding from the EU Hori- zon 2020 Research and Innovation Programme under grant agreement No. 723515 Declarations Fig. 29 Veriﬁcation of the behavior of ANDI /E-INDI with cascaded synchronization (r˙ command to p ˙ response) Conﬂict of interest The authors declare that they have no conﬂict of interest. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adap- tation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indi- cate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copy- right holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. Appendix A Derivation of closed-loop trans- fer functions Fig. 30 Veriﬁcation of the behavior of ANDI /E-INDI with cascaded synchronization (r˙ command to r˙ response) Note that in the following the arguments (s) and (t ) are omit- ted for better readability. 6 Conclusion A.1 Vanilla INDI closed-loop transfer function In this paper, we investigated different ways to compensate and synchronize for ﬁltering and delays of the measurements Given the system in Eq. (41) and the vanilla INDI control law in several types of incremental control laws. It was shown with block diagram depicted in Fig. 9, then the closed-loop how to synthesize complementary ﬁlters, taking into account transfer function can be derived as follows. the ﬁltering and delays. It was shown that the use of comple- The transfer function from u to u in the Laplace domain mentary ﬁlters in contrast to synchronization on the feedback can be deduced from the block diagram to be of u can be especially useful in MIMO systems, if separate ﬁltering is necessary in the different measurements. It was −1 shown that in case of strongly coupled input effectiveness, the U = (I − G ) G U . (A1) A A synchronization on the actuator feedback may become even unstable, where the equivalent complementary ﬁlter formu- Substituting this relation into lation recovers the design closed-loop behavior. A method for deriving the initialization of the complementary ﬁlters was presented, which allows for transient-free engagement sY = C (AX + BU + B D), (A2) 123 Aerospace Systems ¯ ¯ and replacing U by sY = I − G CAX + I − G CB D A A d +G V + I − HF CB D . (A11) A cy d U = (CB) (V − sY ) (A3) Finally, results in ¯ ¯ ¯ sY = I − G CAX + G V + I − G HF CB D A A A cy d −1 † sY = CAX + CB(I − G ) G (CB) (V − sY ) A A (A12) +CB D. is obtained. (A4) A.3 INDI with derivative filter and no Using the assumption in Eq. (42) and rearranging the above synchronization results in −1 ¯ ¯ Given the system (41) and the INDI control law with block I + I − G G sY A A diagram depicted in Fig. 11, the closed-loop transfer function −1 ¯ ¯ = CAX + I − G G V + CB D. (A5) A A d can be derived as follows. The transfer function from u to u in the Laplace domain can be deduced from the block Multiplying with I − G from the left, results in diagram to be (A1). Substituting this relation into (A2) and replacing U by ¯ ¯ ¯ sY = I − G CAX + G V + I − G CB D. (A6) A A A d U = (CB) V − sH F Y (A13) cy A.2 INDI with complementary filter from Section 3.3 results in Given the system (41) and the INDI control law with block sY = CAX + CB D diagram depicted in Fig. 10, then the closed-loop transfer −1 † +CB(I − G ) G (CB) V − sH F Y . function can be derived as follows, if y ˙ is calculated by mdl A A cy (A14) y ˙ (t ) = C ( Ax (t ) + Bu(t )). (A7) mdl Usingthe assumptioninEq. (42) and rearranging the above The transfer function from u to u in the Laplace domain can results in be deduced from the block diagram to be (A1). Substituting −1 ¯ ¯ this relation into (A2) and replacing U by I + I − G G HF sY A A cy (A15) −1 ¯ ¯ = CAX + CB D + I − G G V. d A A U = (CB) V − I − HF (CAX + CBU ) cy −sH F Y (A8) cy Finally, −1 results in ¯ ¯ ¯ sY = I − G + G HF I − G CAX A A cy A −1 ¯ ¯ ¯ (A16) + I − G + G HF G V sY = CAX + CB D+ A A cy A −1 −1 † ¯ ¯ ¯ (A9) CB(I − G ) G (CB) + I − G + G HF I − G CB D A A A A cy A d (V − I − HF (CAX + CBU ) − sH F Y ). cy cy is obtained. Adding and subtracting CB D in the term from Y and d mdl A.4 INDI with derivative filter and synchronization using the structure of sY from (A2)gives: Given the system (41) and the INDI control law with block sY = CAX + CB D diagram depicted in Fig. 12, the closed-loop transfer function −1 † +CB(I − G ) G (CB) (V − sY A A can be derived as follows. + I − HF CB D). (A10) The transfer function from u to u in the Laplace domain cy d can be deduced from the block diagram to be Applying the assumption given in Eq. (42), arranging for sY −1 ¯ ¯ ¯ and multiplying by (I − G ) from the left gives: U = I − G H F G U . (A17) A A cy A 123 Aerospace Systems ¯ ¯ Substituting this relation into (A2) and replacing U by sY = I − G HF CAX + G V A cy A + I − G HF CB D. (A26) A cy d U = (CB) V − sH F Y (A18) cy Note that in contrast to the original implementation in results in Fig. 12, with the proposed modiﬁcation in Fig. 13 it is not ¯ ¯ required to assume that H F are square, only have elements cy −1 ¯ ¯ sY = CAX + CB I − G H F G (CB) A cy A on the diagonal and u and y have the same dimension, or (A19) V − sH F Y + CB D. that HF only has elements on the diagonal and all the cy d cy elements are the same. This means that u and y can have Under the assumption given by Eq. (47), we obtain different dimensions and different ﬁltering can be applied to the different measurement signals. −1 ¯ ¯ I + I − G HF G HF sY A cy A cy (A20) A.5 Hybrid INDI with complementary filter using −1 ¯ ¯ = CAX + I − G HF G V + CB D. A cy A d synchronization Finally, multiplying with I − G HF from the left results A cy Given the system (41) and the INDI control law with block in diagram depicted in Fig. 14, the closed-loop transfer function can be derived as follows, if y ˙ is calculated by mdl ¯ ¯ sY = I − G HF CAX + G V A cy A (A21) + I − G HF CB D. A cy d y ˙ (t ) = C ( Ax (t ) + Bu(t )), (A27) mdl f If the synchronization is implemented according to the where x is x ﬁltered by F (s). f cx block diagram in Fig. 13, then the closed-loop transfer func- The transfer function from u to u in the Laplace domain tion can be calculated as follows: can be deduced from the block diagram to be −1 † U = (I − G ) G (CB) A A −1 ¯ ¯ ¯ U = I − G H F − G (I − H ) G U . (A28) A cy A A V − sH F Y − I − HF CBU . (A22) cy cy Substituting the transfer function in Eq. (A28) into Eq. (A2) Applying the assumption given in Eq. (42), and adding/ and replacing U by subtracting the terms CAX and CB D results in U = (CB) V − [I − H ] CAF X cx −1 ¯ ¯ sY = CAX + CB D + I − G G d A A − [I − H ] CBU − sH F Y (A29) cy V − sH F Y − I − HF cy cy (A23) CBU + CAX + CB D ( ) results in + I − HF (CAX + CB D) . cy d sY = CAX + CB D Using the structure of sY from Eq. (A2)gives: −1 ¯ ¯ ¯ + CB I − G H F − G (I − H ) G (CB) A cy A A −1 ¯ ¯ V − [I − H ] CAF X cx sY = CAX + I − G G V − sY A A − [I − H ] CBU − sH F Y . cy + I − HF (CAX + CB D) + CB D. cy d d (A30) (A24) Using the assumption given in Eq. (49), adding and sub- Isolating for sY and multiplying by I − G from the left tracting [I − H ] CB and [I − H ] CAX and rearranging the results in above results in sY = I − G CAX −1 ¯ ¯ ¯ ¯ sY = CAX + CB D + I − G HF − G (I − H ) G +G V + I − HF CAX + CB D ( ) d A cy A A A cy d ¯ V − [I − H ] CAF X + [I − H ] CAX cx + I − G CB D. 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Aerospace Systems – Springer Journals
Published: Jun 1, 2023
Keywords: Incremental control laws; Complementary filters; Sensor dynamics; Sensor delays
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