Article

# Reconfigurable Flight Control Using Nonlinear Dynamic Inversion with a Special Accelerometer Implementation

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## Abstract

This paper presents an approach to on-line control design for aircraft that have suffered either actuator failure, missing effector surfaces, surface damage, or any combination. The approach is based on a modified version of nonlinear dynamic inversion. The approach does not require a model of the baseline vehicle (effectors at zero deflection), but does require feedback of accelerations and effector positions. Implementation issues are addressed and the method is demonstrated on an advanced tailless aircraft. An experimental simulation analysis tool is used to directly evaluate the nonlinear system's stability robustness.

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... By solving Eqs. (13), (16), (19), and (21) ...
... The WPID solution within the actuator constraints is given by [19,20] ...
... In previous works, the weights are unchanged and are determined either by maximum actuator position or by human experience [19][20][21][22][23][24][25] . In Eq. (26), the optimal object is an approximation of energy consumption, especially that of ballonets given in Eq. (21) of section B. The optimal solution is limited by constant weights. ...
Article
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A stratospheric airship flies at a working altitude of 20km when it takes off from the ground. During ascent and descent, the wind field and thermal environment are highly complex. The thermal environment affects altitude, whereas wind influences the horizontal position of the airship. At a low altitude, this horizontal position cannot be controlled by thrusts given the low thrust-to-weight ratio, especially under a large wind field. However, it may be controlled indirectly by the pitch angle during ascent and descent with a certain vertical velocity. This study therefore proposes ascending and descending schemes for a stratospheric airship based on the thermal model. In this model, altitude is determined by the net lift/weight, whereas the horizontal position is controlled by the thrust and pitch. The pitch angle is determined by ballonets and an elevator. To allocate pitch control between the ballonets and the elevator under different airspeeds, pseudo-inverse dynamics of varied weight are introduced. In horizontal position control, the method of chain allocation is then applied between a pitch angle and vectored thrust to control the position of a stratospheric airship during ascent/descent.
... In contrast to regular backstepping, this method is inherently implicit in the sense that desired closed-loop dynamics do not reside in some explicit model to be cancelled, but which results when the feedback loops are closed. Theoretical development of increments of nonlinear control action date back from the late nineties and started with activities concerning 'Implicit Dynamic Inversion' for DI-based flight control [24, 4], where the architectures considered in this paper were firstly described. Other designations for these developments found in the literature are 'Modified NDI' and 'Simplified NDI', but the designation 'Incremental NDI' is considered to describe the methodology and nature of these type of control laws better [9, 10, 11, 21] . ...
... Other designations for these developments found in the literature are 'Modified NDI' and 'Simplified NDI', but the designation 'Incremental NDI' is considered to describe the methodology and nature of these type of control laws better [9, 10, 11, 21] . INDI has been elaborated and applied theoretically in the past decade for flight control and space applications [21, 25, 4, 5, 6, 1]. The main motivation of this approach is to bring the implicitness of such sensorbased architectures with Lyapunov-based controller design such as backstepping for aerospace applications. ...
... Its design departure is from a stability and convergence viewpoint due to control Lyapunov function augmentations rather than forcing linear behaviour through conventional feedback linearization. Because of its advantage of stabilizing or tracking one or more loops within a single control command maintaining desired properties , the motivation for this approach also stems to the combined flexibility of this method over conventional approaches such as robust nonlinear dynamic inversion (NDI) [2, 3, 7, 12, 13, 15, 22, 23, 26, 27], and its adaptive [8, 19, 20] and incremental counterparts [1, 4, 5, 6, 9, 21, 24, 25]. For the discussion, we will consider physical systems or vehicle dynamics which are represented by the following strict-feedback second order cascaded form: ...
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... The present paper presents a method that is not conservative, but still incorporates all uncertainties, by feeding back angular accelerations. This concept is previously described by [5,6]. The control systems showed good performance when subjected to aerodynamic model mismatch. ...
... However, the robustness properties of angular acceleration feedback are not explicitly described by the authors. Also, [5,6] assume that the angular accelerations are readily available from measurements. ...
... In [6] the concept of feeding back angular accelerations is derived by first rewriting the rotational dynamic equations of motion into an incremental form and then applying regular NDI, resulting in incremental nonlinear dynamic inversion (INDI). It should be noted that INDI has been referred to by [5] as simplified dynamic inversion. ...
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This paper presents a flight control strategy based on nonlinear dynamic inversion. The approach presented, called incremental nonlinear dynamic inversion, uses properties of general mechanical systems and nonlinear dynamic inversion by feeding back angular accelerations. Theoretically, feedback of angular accelerations eliminates sensitivity to model mismatch, greatly increasing the robust performance of the system compared with conventional nonlinear dynamic inversion. However, angular accelerations are not readily available. Furthermore, it is shown that angular acceleration feedback is sensitive to sensor measurement time delays. Therefore, a linear predictive filter is proposed that predicts the angular accelerations, solving the time delay and angular acceleration availability problem. The predictive filter uses only references and measurements of angular rates. Hence, the proposed control method makes incremental nonlinear dynamic inversion practically available using conventional inertial measurement units.
... Incremental controllers arise from a simplified representation of the model dynamics: the Incremental Dynamics (ID) [5], [6]. From the ID formulation, control strategies have been proposed, such as the incremental nonlinear dynamics inversion (INDI) [7], the incremental backstepping (IBKS) [8], and the incremental sliding mode control (INDI-SMC) [9]. ...
... However, the state derivative includes quantities like the angular acceleration, which is not a standard measurement for aircraft. A second-order differentiator (SOD) may be included to mitigate the absence of state derivative measurements [6]. The SOD corresponds to a bandpass second-order filter providing state derivative estimation and filtering high-frequency noise [13], [14]. ...
Article
Incremental control strategies have been proposed recently to reduce the dependence on the aircraft model. However, the technique still requires modeling the actuator effectiveness, which depends on inertial and aerodynamic parameters. Therefore, it is of the utmost importance to analyze the robustness of the strategy regarding these parameters. This work focuses on evaluating the robustness of an incremental backstepping strategy applied for automatic control of a Boeing 747 (B747). A sensitivity analysis is carried out to quantify the impact of inertial parameters and flight condition in the input effectiveness, whose results are used to define scheduling strategies for the incremental controller. A robustness analysis is then performed using a B747 simulator applying different maneuvers in several flight conditions. The flight performance is evaluated according to the military standard MIL-DTL-9490E. Results show that the proposed controller is robust to mismatches in the input effectiveness model, evidencing that even without using a scheduling strategy the controller is able to perform within the requirements specified in the standard.
... Acceleration sensors can detect such disturbance forces and torques before they propagate to lower order states and therefore can act to reject those disturbances more rapidly when incorporated in feedback control. Incremental dynamic inversion has been shown in simulation in [19,20] to reject disturbance dynamics, given measurements of the acceleration states. Application of arrays of accelerometers in past works has focused on gyro-free angular velocity sensing and estimation, in so-called gyro-free inertial navigation systems. ...
... This control strategy is similar to the incremental dynamic inversion control suggested in [19,20]. As in these previous works, our proposed control needs only an estimate of the acceleration state of the vehiclex a , a sufficiently high-fidelity estimate of the control authority B, and the actuator command generated by the attitude controller u r . ...
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Rapid sensing of body motions is critical to stabilizing a flight vehicle in the presence of exogenous disturbances as well as providing high-performance tracking of desired control commands. This bandwidth requirement becomes more stringent as vehicle scale decreases. Many flying insects employ distributed networks of acceleration-sensitive sensors to provide information about body egomotion to rapidly detect forces and torques. In this work, a method for rapid sensing of force and torque using a distributed array of accelerometers, arbitrarily placed and rigidly affixed to a vehicle airframe, was developed. Simulations of the sensor array were performed to quantify the effects of sensor noise, sensor position error, and sensor number on acceleration state estimates. A hardware implementation of this distributed sensor array was designed and integrated into the avionics of a small quadrotor vehicle. The response of the array to induced acceleration stimuli was characterized. A linear state estimation matrix was derived from the calibration to directly estimate the total forces and torques exerted on the airframe. A force-Adaptive control law utilizing the force and torque estimates provided by the sensor network was implemented to improve tracking of reference states while rejecting exogenous force and torque disturbances. Successful rejection of disturbances in the form of internal actuator variation and external wind gusts was demonstrated on the quadrotor vehicle in flight. © Copyright 2015 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
... The incremental form of NDI, Incremental NDI or INDI, is less model dependent and more robust. It has been described in the literature since the late nineties [4] [5], sometimes referred to as simplified [6] or enhanced [7] NDI. Compared to NDI, instead of modeling the angular acceleration based on the state and inverting the actuator model to get the control input, the angular acceleration is measured and an increment of the control input is calculated based on a desired increment in angular acceleration. ...
... Moreover, disturbances cannot be predicted. Initially, a setup with multiple accelerometers was proposed by Ostroff and Bacon [5] to measure the angular acceleration. This setup has some drawbacks, because it is complex and the accelerometers are sensitive to structural vibrations. ...
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Incremental nonlinear dynamic inversion is a sensor-based control approach that promises to provide high-performance nonlinear control without requiring a detailed model of the controlled vehicle. In the context of attitude control of micro air vehicles, incremental nonlinear dynamic inversion only uses a control effectiveness model and uses estimates of the angular accelerations to replace the rest of the model. This paper provides solutions for two major challenges of incremental nonlinear dynamic inversion control: how to deal with measurement and actuator delays, and how to deal with a changing control effectiveness. The main contributions of this article are 1) a proposed method to correctly take into account the delays occurring when deriving angular accelerations from angular rate measurements; 2) the introduction of adaptive incremental nonlinear dynamic inversion, which can estimate the control effectiveness online, eliminating the need for manual parameter estimation or tuning; and 3) the incorporation of the momentum of the propellers in the controller. This controller is suitable for vehicles that experience a different control effectiveness across their flight envelope. Furthermore, this approach requires only very coarse knowledge of model parameters in advance. Real-world experiments show the high performance, disturbance rejection, and adaptiveness properties. Read More: http://arc.aiaa.org/doi/abs/10.2514/1.G001490
... The effect of choice of control variable weighting is demonstrated in these figures.Figure 6a shows responses for a control variable weighting of Q = diag([roll acceleration error weighting, pitch acceleration error weighting, yaw acceleration error weighting]) = diag([1,5,2]). As can be seen, the stability-axis roll rateFigure 6a – inner-loop p c , q c , and r c angular rate commands, Q = diag([1,5,2]).Figure 6b – inner-loop p c , q c , and r c angular rate commands, Q = diag([1,5,10]). ...
... V t c commands, Q = diag([1,5,2]). reference signal is more closely followed than the stability-axis yaw rate reference signal with low dutch roll damping as illustrated by the oscillatory sideslip response.Figure 6b shows responses for Q = diag([1,5,10]). This results in the stability-axis yaw rate reference signal being more closely followed than infigure 6a and a better damped sideslip response.Figure 7 shows time responses for outer-loop γ c ...
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Micro aerial vehicles have been the subject of continued interest and development over the last several years. The majority of current vehicle concepts rely on rigid fixed wings or rotors. An alternate design based on an aeroelastic membrane wing has also been developed that exhibits desired characteristics in flight test demonstrations, competition, and in prior aerodynamics studies. This paper presents a simulation model and an assessment of flight control characteristics of the vehicle. Linear state space models of the vehicle associated with typical trimmed level flight conditions and which are suitable for control system design are presented as well. The simulation is used as the basis for the design of a measurement based nonlinear dynamic inversion control system and outer loop guidance system. The vehicle/controller system is the subject of ongoing investigations of autonomous and collaborative control schemes. The results indicate that the design represents a good basis for further development of the micro aerial vehicle for autonomous and collaborative controls research.
... To improve the control performance of aircraft with nonlinear models, many nonlinear control methods have been studied; backstepping and feedback linearization are two common methods [10]. Liu and Sang proposed both backstepping and sliding mode backstepping control to realize trajectory doi: 10.1016/j.neucom.2021.08.069 3 tracking control of stratospheric airships [11][12][13][14][15]. Bacon and Ostroff expanded dynamic inversion into an incremental form by introducing an angular acceleration feedback [16]. Incremental nonlinear dynamic inversion (INDI), a method in which the dynamics are written in an incremental form, is a sensor-based control method that does not rely on accurate aircraft model, and it can realize aircraft flight under certain uncertainties and structural faults [17]. ...
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... is the available measurement (T s is the sampling time of the controller). Usually, angular acceleration measurements are used because of the model uncertainties in the rotational dynamics [22], but also other measurements can be used [12]. In order to reduce the necessary model knowledge, it is assumed that the term ?f ?x ...
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In this paper, the design of an Incremental Backstepping controller for fault tolerant control of a hexacopter system is presented. The proposed controller compensates for the modeling mismatches including faults by relying on sensor measurements. Outdoor flight test results show the better performance of the Incremental Backstepping controller compared to the Backstepping controller. Furthermore, robustness against faults is demonstrated for the case of unknown control degradation within the propulsion system.
... Their simulation results showed an absence of oscillations in the transient response of the roll command. Waszak et al. [9] used a modified nonlinear dynamic inversion technique [10] that does not require the plant model. They developed a controller for the MAV in which the control input is less than the control variables. ...
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This paper contains a study of two methods for use in a generic nonlinear simulation tool that could be used to determine achievable control dynamics and control power requirements while performing perfect tracking maneuvers over the entire flight envelope. The two methods are NDI (nonlinear dynamic inversion) and the SOFFT (Stochastic Optimal Feedforward and Feedback Technology) feedforward control structure. Equivalent discrete and continuous SOFFT feedforward controllers have been developed. These equivalent forms clearly show that the closed-loop plant model loop is a plant inversion and is the same as the NDI formulation. The main difference is that the NDI formulation has a closed-loop controller structure whereas SOFFT uses an open-loop command model. Continuous, discrete, and hybrid controller structures have been developed and integrated into the formulation. Linear simulation results show that seven different configurations all give essentially the same response, with the NDI...
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A method for real-time estimation of parameters in a linear dynamic state space model was developed and studied. The application is aircraft dynamic model parameter estimation from measured data in flight for indirect adaptive or reconfigurable control. Equation error in the frequency domain was used with a recursive Fourier transform for the real-time data analysis. Linear and nonlinear simulation examples and flight test data from the F-18 High Alpha Research Vehicle (HARV) were used to demonstrate that the technique produces accurate model parameter estimates with appropriate error bounds. Parameter estimates converged in less than 1 cycle of the dominant dynamic mode natural frequencies, using control surface inputs measured in flight during ordinary piloted maneuvers. The real-time parameter estimation method has low computational requirements, and could be implemented aboard an aircraft in real time. Nomenclature A,B,C,D system matrices EYd expectation operator g acceleration ...