Linear Error Dynamics Research Articles

Game Theoretic Approach for Robust Trajectory Tracking by a Multirotor

In this article, the problem of precisely tracking a nominal trajectory by an under-actuated multirotor platform is formulated as a game against bounded external disturbances, measurement noise, and initial condition uncertainty. The nominal trajectory is composed of desired position and yaw angle, which is known to be a differentially flat output for the multirotor state dynamics. Using this property, a linearized error dynamics is obtained in the vicinity of the nominal trajectory. Using the linearized error dynamics, a linear-quadratic differential game approach is used to propose an integrated estimation and control method for trajectory tracking. A salient feature of this approach is that the drag coefficients are estimated online from the information of the tracking error. Moreover, there are only six parameters to be tuned for both estimation and control loops. The intuitive rationale behind the tuning of these parameters is also discussed. Simulations as well as experimental validations are presented to demonstrate the performance and applicability of the controllers.

Indirect Data-Driven Observer Design Using Neural Canonical Observer Structures

An indirect data-driven observer design approach for nonlinear discrete-time systems based on an input-output injection with neural canonical observer structures is proposed. An artificial neural network auto-encoder structure, trained with recorded state, input, and output data, is used for the identification of a system in a nonlinear Brunovsky observer form with output transformation. The neural approximations of the transformations and the input-output injection can be used to construct an observer with linear error dynamics using methods from linear control theory. The approach is demonstrated on two academic examples and on an industrially-motivated problem with a sampled continuous-time model.

Detecting pre-desynchronized states in oscillatory systems

This paper presents a strategy for detecting the onset of a critical transition from synchronization to uncorrelated behavior—called here pre-desynchronized state—occurring in networks composed of noisy oscillatory units. In the analysis, we exploit the fact that, at the critical point where the transition takes place, the covariance matrix of the linearized error dynamics becomes unbounded, i.e., at the pre-desynchronized state some of the synchronization errors will exhibit large and slow fluctuations. The critical point is characterized by the dominant eigenvalue of the synchronization error dynamics, which is assumed to be either real or a complex conjugated pair. In the analysis, we consider a generic family of linear non-autonomous oscillators and a network of FitzHug-Nagumo neurons as particular examples.

Functional observers with linear error dynamics for discrete-time nonlinear systems

This work deals with the problem of designing observers for the estimation of a single function of the states for discrete-time nonlinear systems. Necessary and sufficient conditions for the existence of lower order functional observers with linear dynamics and linear output map are derived. The results provide a direct generalization to Luenberger’s linear theory of functional observers.

Guidance for hypersonic re‐entry using receding horizon control with finite terminal weighting matrix

AbstractIn this article, nominal‐trajectory‐based guidance approach for hypersonic reentry vehicles is considered using the Vinh's equations. The nominal trajectory is generated via pseudo‐spectral method and then tracked by receding horizon control approach. Specially, several strategies are adopted to simplify the practical application. First, we formulate the receding optimal problem discretely using the approximated discrete linear state error dynamics. Thus, the differential matrix Riccati equation is substituted by a difference matrix Riccati equation. Second, we set the terminal state weighting matrix finite which brings flexibility of adjusting the sample period and prediction horizon. Third, the time‐varying state error dynamics is approximated by time invariant system. Lastly, instead of obtaining terminal weighting matrices which satisfy a linear matrix inequality using the LMI toolbox, we set them the same constant matrix. Naturally, numerical simulation is available to verify the validity and efficiency of the proposed approach.

Robust iterative learning control design based on iterative integral sliding mode for nonlinear systems with unrepeatable disturbance

For the repetitive tracking control tasks, the unrepeatable disturbance is an obstruction to learning control schemes. This paper discussed the robust iterative learning control (RILC) design based on the sliding mode control (SMC) technique for a class of nonlinear systems with unrepeatable disturbances. In order to make the tracking error converge to zero in the iteration domain under the sliding mode and eliminate the reaching phase of the SMC, an iterative integral sliding mode (IISM) surface design was proposed. The IISM leads to linear tracking error dynamics in the iteration domain, which guarantees the convergence if the sliding mode parameters meet the given condition. After that, the iterative learning control (ILC) based on the IISM was designed, which drove the sliding mode variable of the closed-loop system to reach the origin with iterations from the initial time instant, despite the effect of the unrepeatable disturbances. The robustness to the unrepeatable disturbance and the fast convergence were achieved. Finally, the proposed IISM method was applied to a one-link robotic manipulator, and the simulation results validated the performance of the proposed control design.

Synchronization in Dynamically Coupled Fractional-Order Chaotic Systems: Studying the Effects of Fractional Derivatives

This study presents the effectiveness of dynamic coupling as a synchronization strategy for fractional chaotic systems. Using an auxiliary system as a link between the oscillators, we investigate the onset of synchronization in the coupled systems and we analytically determine the regions where both systems achieve complete synchronization. In the analysis, the integration order is considered as a key parameter affecting the onset of full synchronization, considering the stability conditions for fractional systems. The local stability of the synchronous solution is studied using the linearized error dynamics. Moreover, some statistical metrics such as the average synchronization error and Pearson’s correlation are used to numerically identify the synchronous behavior. Two particular examples are considered, namely, the fractional-order Rössler and Chua systems. By using bifurcation diagrams, it is also shown that the integration order has a strong influence not only on the onset of full synchronization but also on the individual dynamic behavior of the uncoupled systems.

Boosting synchronization in chaotic systems: Combining past and present interactions

A classical interconnection for inducing synchronization in dynamical systems is static diffusive coupling, i.e., a coupling without dynamics that considers the output or state differences of neighboring systems. However, this type of coupling has some well-known limitations: 1) for some chaotic systems and certain input-output combinations, it is impossible to synchronize a pair or a network of identical systems, 2) the interval of coupling strength values for which synchronization is achieved is limited. Here, we show that these limitations may be removed, or at least considerably relaxed, by using a coupling with memory, i.e., a coupling composed by a standard diffusive-like term plus a delayed version of itself. As a particular example, we consider a pair of Rössler systems in a master-slave configuration, where we analyze the local stability of the synchronous solution using the linearized error dynamics in combination with the D-decomposition method. Also, analytic conditions for tuning the coupling strength values corresponding to the non-delayed and delayed terms of the overall coupling are provided. Furthermore, the applicability of the proposed coupling for boosting synchronization in other chaotic systems is also discussed. In particular, we show how the proposed coupling with memory is applicable for synchronizing a pair of uni-directionally coupled van der Pol oscillators with sinusoidal excitation and a pair of Hindmarsh-Rose neurons. Ultimately, the results presented in this paper suggest that the synchronizability of the coupled systems is enhanced when the present and past interactions are taken into account.

Optimal trajectory design accounting for the stabilization of linear time-varying error dynamics

This study is dedicated to the development of a direct optimal control-based algorithm for trajectory optimization problems that accounts for the closed-loop stability of the trajectory tracking error dynamics already during the optimization. Consequently, the trajectory is designed such that the Linear Time-Varying (LTV) dynamic system, describing the controller’s error dynamics, is stable, while additionally the desired optimality criterion is optimized and all enforced constraints on the trajectory are fulfilled. This is achieved by means of a Lyapunov stability analysis of the LTV dynamics within the optimization problem using a time-dependent, quadratic Lyapunov function candidate. Special care is taken with regard to ensuring the correct definiteness of the ensuing matrices within the Lyapunov stability analysis, specifically considering a numerically stable formulation of these in the numerical optimization. The developed algorithm is applied to a trajectory design problem for which the LTV system is part of the path-following error dynamics, which is required to be stable. The main benefit of the proposed scheme in this context is that the designed trajectory trades-off the required stability and robustness properties of the LTV dynamics with the optimality of the trajectory already at the design phase and thus, does not produce unstable optimal trajectories the system must follow in the real application.

Discrete-time flatness-based control of a gantry crane

This article addresses the design of a discrete-time flatness-based tracking control for a gantry crane and demonstrates the practical applicability of the approach by measurement results. The required sampled-data model is derived by an Euler-discretization with a prior state transformation in such a way that the flatness of the continuous-time system is preserved. Like in the continuous-time case, the flatness-based controller design is performed in two steps. First, the sampled-data system is exactly linearized by a discrete-time quasi-static state feedback. Subsequently, a further feedback enforces a stable linear tracking error dynamics. To underline its practical relevance, the performance of the novel discrete-time tracking control is compared to the classical continuous-time approach by measurement results from a laboratory setup. In particular, it turns out that the discrete-time controller is significantly more robust with respect to large sampling times. Moreover, it is shown how the discrete-time approach facilitates the design of optimal reference trajectories, and further measurement results are presented.

Functional observers with linear error dynamics for nonlinear systems

In this work, the problem of designing observers for estimating a single nonlinear functional of the state is formulated for general nonlinear systems. Notions of functional observer linearization are also formulated, in terms achieving exactly linear error dynamics in transformed coordinates and with prescribed rate of decay of the error. Necessary and sufficient conditions for the existence of a lower-order functional observer with linear dynamics are derived. The results provide a direct generalization of Luenberger’s linear theory of functional observers to nonlinear systems.

Feedback-linearization based robust relatively optimal trajectory tracking controller for 3-DOF helicopter

This paper proposes a feedback linearization-based robust trajectory tracking controller for the elevation and pitch axes of three degrees of freedom (3-DOF) helicopter. The 3-DOF helicopter is a benchmark system that operates on the same principle of operation as that of a tandem-rotor helicopter. Accurate control of this laboratory set up in the presence of external disturbances is a sophisticated problem as it is easily affected by external disturbances due to its low weight. Hence, to ensure efficient trajectory tracking performance in the presence of uncertainties and disturbances a nonlinear disturbance observer is integrated with a feedback linearization (FL) based relatively optimal trajectory tracking controller (ROTTC). Feedback linearization of the nonlinear model results in a decoupled linear error dynamics for each axis, which is used for the design of the ROTTC. The stability of the 3-DOF helicopter in the presence of estimation error and parameter uncertainties is analyzed using the Lyapunov method. A comparison study with an FL-based robust linear quadratic regulator (RLQR) method illustrates the superiority of the proposed method in terms of tracking error and disturbance estimation. The efficacy of the proposed controller to accommodate external disturbances is demonstrated with real-time implementation on the 3-DOF helicopter with a strong wind disturbance as well as an impulse disturbance.

State Observer for the Pacemaker Model Based on the Van der Pol Equation

A Dutch physiologist and a founder of electrocardiography V. Einthoven [10] proposed the first known model of the cardiac electrical activity. Later, van der Pol and van der Mark [11] developed a model of the heart, where the heartbeat is considered as a relaxation oscillation. From this point of view, to model the operation of pacemakers, the van der Pol equation [14,15,19] can be useful. The paper offers modeling of only one heart node that is the S-A (sinoatrial) node, which is the main heart pacemaker [20].Many control algorithms for dynamic systems are based on feedback, which involves the full state vector of a dynamic system. However, in practice, the full state vector is not always known. So, in the case of cardiac electrical activity, the potentials of the nodes rather than their changing rates are measured. To restore the full state vector from existing measurements, state observers are often used.In this paper, we solve the task of constructing an observer with linear error dynamics [22.25]. A necessary condition for the existence of such an observer is the system observability. The sufficient conditions can be formulated in the framework of the differential-geometric approach [25] using the ideas of duplicity [25,26]. Within this approach, an algorithm for observer construction can be developed. In the paper, a general problem to construct an observer for two-dimensional systems is solved and the results obtained are applied to the pacemaker model based on the Van der Pol oscillator. The numerical simulation enables us to illustrate operation of the observer developed.

Tube-Based Robust MPC Processor-in-the-Loop Validation for Fixed-Wing UAVs

Real systems, as Unmanned Aerial Vehicles (UAVs), are usually subject to environmental disturbances, which could compromise the mission accomplishment. For this reason, the main idea proposed in this research is the design of a robust controller, as autopilot control system candidate for a fixed-wing UAV. In detail, the inner loop of the autopilot system is designed with a tube-based robust model predictive control (TRMPC) scheme, able to handle additive noise. Moreover, the navigation outer loop is regulated by a proportional-integral-derivative controller. The proposed TRMPC is composed of two parts: (i) a linear nominal dynamics, evaluated online with an optimization problem, and (ii) a linear error dynamics,which includes a feedback gain matrix, evaluated offline. The key aspects of the proposed methodology are: (i) offline evaluation of the feedback gain matrix, and (ii) robustness to random, bounded disturbances. Moreover, a path-following algorithm is designated for the guidance task, which provides the reference heading angle as input to the control algorithm. Software-in-the-loop and processor-in-the-loop simulations have been performed to validate the proposed approach. The obtained performance have been evaluated in terms of tracking capabilities and computational load, assessing the real-time implementability compliance with the XMOS development board, selected as continuation of previous works.

Contact-aided invariant extended Kalman filtering for robot state estimation

Legged robots require knowledge of pose and velocity in order to maintain stability and execute walking paths. Current solutions either rely on vision data, which is susceptible to environmental and lighting conditions, or fusion of kinematic and contact data with measurements from an inertial measurement unit (IMU). In this work, we develop a contact-aided invariant extended Kalman filter (InEKF) using the theory of Lie groups and invariant observer design. This filter combines contact-inertial dynamics with forward kinematic corrections to estimate pose and velocity along with all current contact points. We show that the error dynamics follows a log-linear autonomous differential equation with several important consequences: (a) the observable state variables can be rendered convergent with a domain of attraction that is independent of the system’s trajectory; (b) unlike the standard EKF, neither the linearized error dynamics nor the linearized observation model depend on the current state estimate, which (c) leads to improved convergence properties and (d) a local observability matrix that is consistent with the underlying nonlinear system. Furthermore, we demonstrate how to include IMU biases, add/remove contacts, and formulate both world-centric and robo-centric versions. We compare the convergence of the proposed InEKF with the commonly used quaternion-based extended Kalman filter (EKF) through both simulations and experiments on a Cassie-series bipedal robot. Filter accuracy is analyzed using motion capture, while a LiDAR mapping experiment provides a practical use case. Overall, the developed contact-aided InEKF provides better performance in comparison with the quaternion-based EKF as a result of exploiting symmetries present in system.

On the stability of inverse dynamics control of flexible-joint parallel manipulators in the presence of modeling error and disturbances

Inverse dynamics control is considered for flexible-joint parallel manipulators in order to obtain a good trajectory tracking performance in the case of modeling error and disturbances. It is known that, in the absence of modeling error and disturbance, inverse dynamics control leads to linear fourth-order error dynamics, which is asymptotically stable if the feedback gains are chosen to make the real part of the eigenvalues of the system negative. However, when there are modeling errors and disturbances, a linear time-varying error dynamics is obtained whose stability is not assured only by keeping the real parts of the frozen-time eigenvalues of the system negative. In this paper, the stability of such systems is investigated and it is proved that the linear time-varying system can be rendered stable by selecting the feedback gains such that the variation of the system becomes sufficiently slow. To illustrate the performance of the control method, deployment motion of a 3-RPR planar parallel manipulator subject to impact is simulated. For the impact model, the impulse-momentum and the coefficient of restitution equations for the system are derived.

A Local Separation Principle via Dynamic Approximate Feedback and Observer Linearization for a Class of Nonlinear Systems

A separation principle for a class of nonlinear systems inspired by the techniques of feedback linearization and observer design with linear error dynamics is discussed. The output feedback construction combines strategies for approximate feedback linearization and observer design, which are of interest per se , yielding a dynamic control law that ensures a linear, spectrally assignable, behavior from the certainty equivalence input mismatch to the extended state of the system and the observer. The first ingredient, namely the approximate feedback linearization strategy, can be applied, under mild conditions, also to nonlinear systems that are linearly uncontrollable—or that do not possess a well-defined relative degree in the case of a given output function—yet providing a chain of integrators of length equal to the dimension of the state in the transformed coordinates. Interestingly, a systematically designed nonlinear inner loop enables the use of linear design techniques, $\text{e.g.}$ , pole placement. The observer design, on the other hand, employs an additional dynamic extension that allows us to assign the local dynamic behavior of the error dynamics independently from its zeros, differently from the classic high-gain observer design. The paper is concluded by presenting several numerical simulations, including an output tracking control problem for the Ball and Beam model that does not possess a well-defined relative degree.

Robust position control of hydraulic manipulators using time delay estimation and nonsingular fast terminal sliding mode

A simple and robust tracking controller based on time delay estimation and nonsingular fast terminal sliding mode is proposed for the position control of hydraulic manipulator. The proposed controller does not require the mathematical model of hydraulic manipulator dynamics, which ensures that the method is simple and model free. Two elements, time delay estimation and nonsingular fast terminal sliding mode, are implemented in the proposed controller. Time delay estimation estimates the complex dynamics of the hydraulic manipulator, and nonsingular fast terminal sliding mode is implemented as the nonlinear desired error dynamic to ensure fast convergence and high trajectory tracking accuracy. Experiments are performed to verify the tracking performance of the method on a hydraulic manipulator. The results demonstrate that this method achieves faster and higher precision position tracking performance than a conventional time delay control with linear error dynamics.

Linear Error Dynamics for Turbulent Flow in Urban Street Canyons

AbstractThe ability to make forecasts depends on atmospheric predictability and the growth of errors. It has recently been shown that the predictability of urban boundary layers differs in important respects from that of the free atmosphere on the mesoscale and larger; in particular, nonlinearity may play a less prominent role in the error evolution. This paper investigates the applicability of linear theory to the error evolution in turbulent street-canyon flow. Using large-eddy simulation, streamwise aspect ratios between 0.15 and 1.50, and identical-twin experiments, it is shown that the growth rate of the error kinetic energy can be estimated from Eulerian averages and that linear theory provides insight into the spatial structure of the error field after saturation. The results should be applicable to cities with deep and closely spaced canyons. Implications for data assimilation and modeling are discussed.

Functional Observers for Nonlinear Systems

Abstract: In this work, the problem of designing observers for estimating a single nonlinear functional of the state is formulated for general nonlinear systems. Notions of functional observer linearization are also formulated, in terms of achieving exactly linear error dynamics in transformed coordinates and with prescribed rate of decay of the error. Necessary and sufficient conditions for the existence of a lower-order functional observer with linear dynamics and linear output map are derived. The results provide a direct generalization of Luenberger’s linear theory of functional observers to nonlinear systems.