Approximate Dynamic Inversion Research Articles

Inter-relationship between approximate dynamic inversion and MRAC augmented with proportional-integral controller

Approximate Dynamic Inversion (ADI) is basically an approximation of exact dynamic inversion or feedback linearisation, which converts a nonlinear system to an equivalent linear structure. This method can be widely applied for controlling minimum phase, nonaffine-in-control systems. For applying the ADI method, a fast dynamic subsystem for deriving explicit inversion of the nonaffine equation is required. With full state feedback, ADI may be expressed in the same way as a Proportional Integral (PI) controller with only knowledge of the sign of control effectiveness and also without any approximation. The Model Reference Adaptive Controller (MRAC) augmented with the PI method is an adaptive control technique where the PI parameters are updated/tuned as per the control methodology based on the MRAC-Massachusetts Institute of Technology (MIT) rule so that the plant is capable to follow the reference model. The main objective of this paper is to find the relationship between ADI and MRAC augmented with a PI controller.

Stability-Guaranteed Variable Impedance Control of Robots Based on Approximate Dynamic Inversion

Variable impedance control has been considered as one of the most important compliant control approaches for its abilities in improving compliance, safety, and efficiency in robot–environment interaction. However, existing variable impedance controllers have deficits in stability guarantee. This article proposes a stability-guaranteed variable impedance control approach for robots with modeling uncertainties based on approximate dynamic inversion (ADI). Novel constraints on variable impedance profiles are given to guarantee the exponential stability of the desired variable impedance dynamics. An ADI-based impedance control law is designed to achieve the desired variable impedance dynamics through the convergence of a variable impedance error. Based on the extended Tikhonovs theorem, it is proven that the closed-loop control system has semiglobal practical exponential stability. The proposed impedance controller can be implemented in a PID form and is appealing for its simple structure, easy implementation, and control stability guarantee. The effectiveness of the proposed variable impedance controller is illustrated by an illustrative example taken on a five-bar parallel robot.

Adaptive Learning-Based Observer With Dynamic Inversion for the Autonomous Flight of an Unmanned Helicopter

This article presents the development of an autonomous flight control system for a small-scale unmanned helicopter based on an online adaptive learning-based observer and model predictive control (MPC). The adaptive learning-based observer provides an approximate dynamic inversion for solving the coupled dynamic problem. The discrete-time MPC (DMPC) uses a prediction process to obtain a stable model. The Laguerre function, a state observer, and a recursive learning algorithm were integrated into DMPC, the core of the flight control system. Exponential data weighting was adopted to increase the stability of the designed flight control system. By integrating these approaches, an adaptive learning-based flight control system, which consists of the pitch, roll, yaw, and heave controllers, is presented to evaluate the stability and performance of the proposed autonomous flight control system of an unmanned helicopter. Moreover, an adaptive controller based on neural approximation for yaw motion is used to improve the performance of yawing control of the helicopter. In order to validate the feasibility of the proposed method, a hardware-in-the-loop simulation was performed in real time. The system included an embedded system, a self-developed users interface, and the X-plane flight simulator. The simulation results showed that the flight control system was able to maintain the position and the attitude of the helicopter in hover mode under the influence of wind gust and disturbance. The proposed method had better performance than a robust control system, in both ideal and disturbed environments.

Redesign of approximate dynamic inversion for pure-feedback nonaffine-in-control nonlinear systems via input saturation

In this paper, a redesigned approximate dynamic inversion (RADI) method is developed for a class of pure-feedback nonaffine-in-control nonlinear systems (PFNNSs). Recently, the approximate dynamic inversion (ADI)-based algorithms have been widely implemented for nonaffine-in-control nonlinear systems. To employ ADI-based methods, a fast dynamic subsystem has to be established to derive the explicit inversion of the nonaffine equation. However, the previous research on ADI rarely considers that the amplitude of actual input derived from the fast augmented subsystem can be extremely high, which maybe intolerable for practical systems. In this paper, we set up input saturation and view the difference between input with and without saturation as a perturbation. Then, an intermediate subsystem is constructed to compensate the influence of this perturbation by approximating its inversion. The exponential stability of the overall closed-loop system is ensured via singular perturbation theorem. Numerical examples are provided to demonstrate the effectiveness of the proposed method.

Approximate Dynamic Inversion for Nonaffine Nonlinear Systems With High-Order Mismatched Disturbances and Actuator Saturation

In this paper, a novel disturbance-observer-based approximate dynamic inversion (ADI) approach is developed for pure-feedback nonaffine-in-control nonlinear systems (PFNNSs) in the presence of both high-order mismatched disturbances and actuator saturation. Finite-time disturbance observers (FTDOs) are utilized to estimate the disturbances and their derivatives. Then, we rebuild the system with the outputs of FTDOs. Thereafter, ADI is employed to derive the desired virtual and actual control of the nominal system, where no disturbances and saturation are presented. Furthermore, an augmented intermediate subsystem is constructed for the reduced slow subsystem to compensate for the difference between inputs with and without saturation by approximating its inversion. The stability of the closed-loop system is studied using Tikhonov's theorem. The proposed method is applied to a numerical example and a one-link robotic system with a brush DC motor. The simulation results demonstrate the validity of the presented approach.

High Gain Observer-Based Saturated Controller for Feedback Linearizable System

The presence of pulse width modulation driven components, limiters, and electronic switches constrain the magnitude of control input in electrical and electro-mechanical systems. The presence of nonlinearities in these system models and unavailability of full state measurements, further complicate the controller design process. This brief introduces a novel saturated output feedback design to solve the tracking problem for a class of feedback linearizable circuits. The feedback scheme comprises of a modified approximate dynamic inversion controller in conjunction with a standard high gain observer. Unlike the Lyapunov-based approaches, the proposed contraction theory-based methodology details a quantitative framework for the convergence analysis of closed loop trajectories. This, in turn, serve the dual purpose of proving ultimate boundedness and quantifying the convergence bounds in terms of known parameters.

A singular perturbation approach to saturated controller design with application to bounded stabilization of wing rock phenomenon

This paper proposes a singular perturbation-based approach to design saturated controllers for a class of nonlinear systems and shows its effectiveness in suppressing wing rock phenomenon in aircraft. Approximate dynamic inversion technique is exploited to design a high-gain dynamic controller which enforces time-scale separation in the closed-loop system. The convergence analysis of closed-loop system is done using the framework of contraction theory. Explicit tracking error bounds are derived both in the presence and absence of disturbances in system dynamics. The derived bounds are valid beyond a restrictive range of perturbation parameter, unlike Lyapunov-based approaches.

Performance recovery of a class of uncertain non-affine systems with unmodelled dynamics: an indirect dynamic inversion method

ABSTRACTThis paper presents an output feedback indirect dynamic inversion (IDI) approach for a class of uncertain nonaffine systems with input unmodelled dynamics. Compared with previous approaches to achieve performance recovery, the proposed method aims at dealing with a broader class of nonaffine-in-control systems with triangular structure. An IDI state feedback law is designed first, in which less knowledge of the model plant is needed compared to earlier approximate dynamic inversion methods, thus yielding more robust performance. After that, an extended high-gain observer is designed to accomplish the task with output feedback. Finally, we prove that the designed IDI controller is equivalent to an adaptive proportional-integral (PI) controller, with respect to both time response equivalence and robustness equivalence. The conclusion implies that for the studied strict-feedback non-affine systems with unmodelled dynamics, there always exits a PI controller to stabilise the systems. The effectiveness and benefits of the designed approach are verified by three examples.

Acceleration measurement-based incremental nonlinear flight control for air-breathing hypersonic vehicles

In this paper, nonlinear flight control problem for Air-breathing Hypersonic Vehicles (AHVs) is addressed. Designing a high quality flight controller for AHVs is highly challenging due to model uncertainties caused by inherent structural/aerodynamic couplings, high nonlinearities within the large envelope. A nonlinear flight control approach named Sensor-Based Backstepping (SBB) is employed to design a highly robust controller for AHVs, and ensure that the controller does not suffer from large parameter variations and model uncertainties. The SBB approach is essentially an incremental type of approximate dynamic inversion (ADI) control scheme developed based on singular perturbation theory and multi-loop backstepping design methodology. The SBB uses angular accelerometers to capture dynamic changes in time derivatives of body angular rates, and requires nearly no accurate model information in its implementation. In this way, unlike model based adaptive nonlinear control laws such as adaptive nonlinear dynamic inversion (ANDI), dependency on accurate full model information and the associated identification workload is removed. Several key issues associated with inherent characteristics of AHVs before the SBB based AHV controller is developed. To validate the SBB flight controller, numerical simulations are performed on a nonlinear AHV model with sudden model parameter uncertainties introduced, and the controller is compared with the linear controller augmented Linear Quadratic Regulator (LQR) and the real-time model based ANDI controller. The results have demonstrated the superior tracking performance of the SBB approach, and highlighted its strong robustness to model uncertainties.

Adaptive neural network visual servo control for dynamic positioning of underwater vehicles

Dynamic positioning is an important technique to ensure the capability of a next generation underwater vehicle to fulfill complex intervention and observation tasks. This work proposes an adaptive neural network image-based visual servo (IBVS) controller for positioning an underwater vehicle with respect to a fixed target in camera-in-hand configuration. The visual servoing error is defined by projecting the image error to the vehicle space using a pseudo inverse of the estimated Jacobian matrix. In the proposed dual-loop system architecture, the velocity command is designed to stabilize the visual servoing error in the kinematic loop, and the desired acceleration is derived to track the velocity command in the dynamic loop. A single hidden layer (SHL) feedforward neural network, in conjunction with a conventional proportional-integral (PI) controller, is applied to the cascaded systems to compensate for the uncertainties in both the Jacobian matrix and the vehicle dynamics. The thruster control signals are obtained by using the approximate dynamic inversion (ADI) of the vehicle. The uniform ultimate boundedness of the visual servoing error and the neural network weight matrices are proved by Lyapunov theory. Simulation results are provided to verify the effectiveness of the controller with two candidate methods of estimating the image Jacobian matrix.

Proportional-Integral Controllers for Minimum-Phase Nonaffine-in-Control Systems

We show that stabilizing tracking proportional-integral (PI) controllers can be constructed for minimum-phase nonaffine-in-control systems. The constructed PI controller is an equivalent realization of an approximate dynamic inversion controller. This equivalence holds only for the time response when applied to the unperturbed system. Even when restricted to unperturbed minimum-phase linear time invariant systems, their closed loop robustness properties differ. This shows that in general, properties that do not define the equivalence relation for systems/controllers are not preserved under such equivalence transformations.

Discrete-Time Exact and Approximate Dynamic Inversion for Settle Performance

Single-track hard disk drive (HDD) seek performance is measured by settle time, ts, defined as the time from the arrival of a seek command until the measured position reaches and stays within an acceptable distance from the target track. In this paper, we show the effective use of feedforward dynamic inversion, coupled with an aggressive desired trajectory yd, to achieve high performance settle times. It is well known that the exact tracking solution for nonminimum phase (NMP) systems requires noncausal preactuation to maintain bounded internal signals. In the specific HDD operating modes of interest, anticipation of a seek command is unrealistic, and thus preactuation adds to the overall computation of settle time. Unlike many dynamic inversion tracking applications, this negative effect of preactuation leads to interesting trade-offs between preactuation delay, tracking accuracy, and achievable settle performance. We show that, surprisingly, very little preactuation is desirable when truncating the exact tracking solution and applying it to our NMP HDD model. For comparison, we also review the stable Taylor series approximate inverse, and show that a zero-order series’ settle performance is comparable to truncated exact inversion while being easier to compute and implement. We experimentally validate this conclusion on a Servo Track Writer (STW).

Dynamic Inversion for Nonaffine-in-Control Systems via Time-Scale Separation. Part I

This paper presents a new method for approximate dynamic inversion for nonaffine-in-control systems via time-scale separation. The control signal is sought as a solution of "fast" dynamics and is shown to asymptotically stabilize the original nonaffine system. Sufficient conditions are formulated, which are consistent with the assumptions of Tikhonov's theorem in singular perturbations theory. Several examples illustrate the theoretical results.

Adaptive Neural Control of Deep-Space Formation Flying

A novel nonlinear adaptive neural control methodology is presented for the challenging problem of deep-space spacecraft formation flying. When the framework of the circular restricted three-body problem with the sun and Earth as the primary gravitational bodies is utilized, a nonlinear model is developed that describes the relative formation dynamics. This model is not confined to the vicinity of the Lagrangian libration points but rather constitutes the most general nonlinear formulation. Then, a relative position controller is designed that consists of an approximate dynamic model inversion, linear compensation of the ideal feedback linearized model, and an adaptive neural-network-based element designed to compensate for the model inversion errors. The nominal dynamic inversion includes the gravitational forces, whereas the model inversion errors are assumed to stem from disturbances such as fourth-body gravitational effects and solar radiation pressure. The approach is illustrated by simulations, which confirm that the suggested methodology yields excellent tracking and disturbance rejection, thus, permitting submillimeter formation keeping precision.

Neural Networks and Adaptive Nonlinear Control of Agile Antiair Missiles

Research has shown that neural networks can be used to improve on approximate dynamic inversion for control ofuncertainnonlinearsystems.Inonearchitecture,theneuralnetworkadaptively cancelsinversionerrorsthrough on-linelearning. Suchlearning isaccomplishedby asimpleweightupdaterulederived from Lyapunov theory,thus assuring the stability of the closed-loop system. This methodology is reviewed and extended to incorporate an important classof neural networkswith onesigmoidal hidden layer. An agileantiair-missile autopilot is subsequently designed using this control scheme. A control law based on approximate inversion of the nonlinear dynamics is presented. This control system is augmented by the addition of a neural network with on-line learning. Numerical results from a nonlinear agile antiair-missile simulation demonstrate the effectiveness of the resulting autopilot.

Adaptive nonlinear control of agile antiair missiles using neural networks

Research has shown that neural networks can be used to improve upon approximate dynamic inversion controllers in the case of uncertain nonlinear systems. In one possible architecture, the neural network adaptively cancels linearization errors through online learning. Learning may be accomplished by a simple weight update rule derived from Lyapunov theory, thus assuring the stability of the closed-loop system. In the paper, the authors discuss the evolution of this methodology and its application in a bank-to-turn autopilot design for an agile antiair missile. First, a control scheme based on approximate inversion of the vehicle dynamics is presented. This nonlinear control system is then augmented by the addition of a feedforward neural network with online learning. Finally, the resulting control law is demonstrated in a nonlinear simulation and its performance is evaluated relative to a conventional gain-scheduled linear autopilot.