On algebraic time-derivative estimation and deadbeat state reconstruction

This note places into perspective the so-called algebraic time-derivative estimation method recently introduced by Fliess and co-authors with standard results from linear state-space theory for control systems. In particular, it is shown that the algebraic method can in a sense be seen as a special case of deadbeat state estimation based on the reconstructibility Gramian of the considered system.Comment: Maple-supplements available at https://www.tu-ilmenau.de/regelungstechnik/mitarbeiter/johann-reger

A novel attitude representation in view of spacecraft attitude reconstruction using temperature data

There are various different attitude representations that describe the orientation of a rigid body in space and allow the transformation between different coordinate systems. Among others, they differ in number of variables, uniqueness of the representation and their continuity. However, in spite of some of them being based on angles, none of them constitutes a simple representation of an angle between two arbitrary vectors. We tackle this issue by proposing a novel attitude representation that directly incorporates the desired angle. The usefulness of this representation is demonstrated in the attitude reconstruction from temperature data, which then leads to an order reduction of a non-linear system

Real‐Time Adaptive Optic System Using FPGAs

For “adaptive optics” (AO) that are used in a control loop, sensing of the wavefront is essential for achieving a good performance. One facet in this context is the delay introduced by the wavefront evaluation. This delay should be kept to a minimum. Since the problem can be split into multiple subproblems, field-programmable gate arrays (FPGAs) may beneficially be employed in view of the FPGAs’ power to compute many tasks in parallel. The evaluation of, e.g., a Shack-Hartmann wavefront sensor (SHWFS) may simply be seen as the evaluation of an image. Therefore, in general, image processing methods may be split into multiple assignments such as connected component labeling (CCL). In this chapter, a new method for real-time evaluation of an SHWFS is introduced. The method is presented in combination with a rapid-control prototyping (RCP) system that is based on real-time Linux operating system

Energy shaping and partial feedback linearization of mechanical systems with kinematic constraints

Traditionally, the energy shaping for mechanical systems requires the elimination of holonomic and nonholonomic constraints. In recent years, it was argued that such elimination might be unnecessary, leading to a possible simplification of the matching conditions in energy shaping. On the other hand, the partial feedback linearization (PFL) approach has been widely applied to unconstrained mechanical systems, but there is no general result for the constrained case. In this regard, this paper formalizes the PFL for mechanical systems with kinematic constraints and extends the energy shaping of such systems by including systems with singular inertia matrix and non-workless constraint forces, which can arise from the coordinate selection and PFL. We validated the proposed methodology on a 5-DoF portal crane via simulation

Robust adaptive tracking control for highly dynamic nanoprecision motion systems

This abstract focuses on the design and real-time implementation of advanced control strategies for motion systems with highly dynamic nanopositioning capabilities [1]. The key exemplar is the lifting and actuating unit (LAU), which integrates a pneumatic actuator for weight force compensation and a parallel electromagnetic drive to produce precision motion forces. Initial investigations cover the modeling and parametric identification of the overactuated nature of a single LAU [2]. This lifting module, integrated into a test bench, renders a 1D vertical motion system aimed to perform subnanometer positioning tasks while minimizing heat emission. To this end, we propose a control allocation strategy to assign (zero-mean) high-dynamic forces to the electromagnetic channel, producing a very low heat emission while the performance is fulfilled using an LQ-type controller plus an L1 adaptive augmentation [3]. This investigation closes with RMS positioning errors less than 0.25 nm and electrical currents less than 0.30 mA. Further investigations involve a 3D tilt-and-lift vertical motion system integrating three LAUs, each placed in each corner of a triangular payload. The key challenge of this configuration is to cope with the high cross-couplings between the degrees of freedom (DOF), i.e., vertical and rotational motion. The core of the decoupling task is the nominal LQG-type controller comprising disturbance-rejection-based observers aimed to fully compensate cross-couplings, while the L1 adaptive augmentation recovers the nominal performance in the presence of parametric uncertainties w.r.t. the input gain [4]. Given that the heat emission problem is fully solved for a single LAU (see [2] and [3]), we then focus on the performance and robustness of the 3D closed-loop system. Since full-state information of the cross-couplings is not simple to reconstruct, we adopt the output-feedback control architecture for the nominal controller and L1 adaptive augmentation [4]. The effectiveness of the proposed control strategy is verified via real-time experimentation rendering vertical RMS positioning errors of less than 0.25 nm and RMS rotational errors of less than 0.04 μrad while satisfying the heat emission constraint. The investigations conclude by exploring the outstanding performance/robustness trade-off of the L1 adaptive control theory for a nanometer planar positioning system with a travel range of ø200 mm (i.e., NPPS200) and the subsequent integration with the 3D vertical motion system, thereby transitioning to a full 6D system (i.e., NPPS200-6D) with 25 mmvertical stroke. Within this framework, the complexity of the controller design is higher because of the number of DOF, cross-couplings, external disturbances, and parametric perturbations. We completed our investigations through experimental validation with planar and vertical RMS positioning errors of less than 0.80 nm and RMS rotational errors of less than 0.05 μrad, as shown in Figure 1

Experimental Evaluation of Homogeneous Differentiators Applied to Hydraulic Stroke with Measurement Noise and Acceleration Disturbance

Author's accepted manuscript. © 2024 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.The design and experimental evaluation of continuous homogeneous differentiators are considered. The differentiator gains follow the high-gain setup with variable gain-scaling parameter L. This parameter results from a minimization of the effect of measurement noise and disturbance on the differentiation estimation error in terms of the homogeneous L2 -gain. We consider and compare various homogeneity degrees including the linear (high-gain) case. It turns out that an optimum exists not only w.r.t. L but also w.r.t. homogeneity degree d. For experimental evaluation, a hydraulic cylinder controlled by a servo-valve is excited over a frequency range and the noisy cylinder stroke measurements are differentiated. Via classical metrics (root-mean-square and maximum absolute error) and a precise full-order simulation model, we evaluate the velocity estimation accuracy. Applying a second differentiation to the stroke measurements, an acceleration disturbance can be detected from a comparison of its measurement and estimate.acceptedVersio

Dynamic extension for adaptive backstepping control of uncertain pure-feedback systems

An adaptive backstepping algorithm is developed for a class of uncertain systems in pure-feedback form. The control is based on a dynamic state feedback that allows to compensate for parametric uncertainties which enter linearly into the system. As possible in the nominal case, a dynamic extension of just order one is required, in addition to the dynamics of the identifiers for the adaptation. The regularity of the control law only requires standard assumptions

Quantitative robustness analysis of model following control for nonlinear systems subject to model uncertainties

We investigate a model following control (MFC) design for nonlinear minimumphase systems subject to model uncertainties. The model following control architecture is a two degrees-of-freedom structure consisting of two control loops. The model control loop (MCL) includes a nominal model of the process. The design of the process control loop (PCL) is based on the error system resulting from the nominal design and the actual process. Both control loops are designed using (partial) feedback linearisation. We analyse the robustness in view of the norm of the uncertainty and the region of attraction compared to a single-loop (partial) feedback linearisation control. It turns out that the proposed approach is able to stabilize significantly larger uncertainties, shows better tracking performance, and exhibits a larger region of attraction (based on a quadratic Lyapunov function)

Minimizing the Homogeneous L2\mathcal{L}_2-Gain of Homogeneous Differentiators

The differentiation of noisy signals using the family of homogeneous differentiators is considered. It includes the high-gain (linear) as well as robust exact (discontinuous) differentiator. To characterize the effect of noise and disturbance on the differentiation estimation error, the generalized, homogeneous $\mathcal{L}_2$-gain is utilized. Analog to the classical $\mathcal{L}_p$-gain, it is not defined for the discontinuous case w.r.t. disturbances acting on the last channel. Thus, only continuous differentiators are addressed. The gain is estimated using a differential dissipation inequality, where a scaled Lyapunov function acts as storage function for the homogeneous $\mathcal{L}_2$ supply rate. The fixed differentiator gains are scaled with a gain-scaling parameter similar to the high-gain differentiator. This paper shows the existence of an optimal scaling which (locally) minimizes the homogeneous $\mathcal{L}_2$-gain estimate and provides a procedure to obtain it. Differentiators of dimension two are considered and the results are illustrated via numerical evaluation and a simulation example