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

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)

Homogeneous Lp stability for homogeneous systems

The motivation of this paper comes from the fact that Lp−stability and Lp−gain, using the classical signal norms, is not well-defined for arbitrary continuous weighted homogeneous systems. However, using homogeneous signal norms it is possible to show that every internally stable homogeneous system has a globally defined finite homogeneous Lp−gain, for p sufficiently large. If the system has a homogeneous approximation, the homogeneous Lp−gain is inherited locally. Homogeneous Lp−stability can be characterized by a homogeneous dissipation inequality, which in the input affine case can be transformed to a homogeneous Hamilton-Jacobi inequality. An estimation of an upper bound for the homogeneous Lp−gain can be derived from these inequalities. Homogeneous L∞−stability is also considered and its strong relationship to Input-to-State stability is studied. These results are extensions to arbitrary homogeneous systems of the well-known situation for linear time-invariant systems, where the Hamilton-Jacobi inequality reduces to an algebraic Riccati inequality. A natural application of finite-gain homogeneous Lp−stability is in the study of stability for interconnected systems. An extension of the small-gain theorem for negative feedback systems and results for systems in cascade are derived for different homogeneous norms. Previous results in the literature use classical signal norms, hence, they can only be applied to a restricted class of homogeneous systems. The results are illustrated by several examples

Fractional-Order Partial Cancellation of Integer-Order Poles and Zeros

The key idea of this contribution is the partial compensation of non-minimum phase zeros or unstable poles. Therefore the integer-order zero/pole is split into a product of fractional-order pseudo zeros/poles. The amplitude and phase response of these fractional-order terms is derived to include these compensators into the loop-shaping design. Such compensators can be generalized to conjugate complex zeros/poles, and also implicit fractional-order terms can be applied. In the case of the non-minimum phase zero, its compensation leads to a higher phase margin and a steeper open-loop amplitude response around the crossover frequency resulting in a reduced undershooting in the step-response, as illustrated in the numerical example.publishedVersio