Nonlinear -controllers are designed for arbitrarily weighted, continuous homogeneous systems with a focus on systems affine in the control input. Based on the homogeneous -norm, the input-output behavior is quantified in terms of the homogeneous -gain as a generalization of the classical -gain (i.e., -norm in the linear case). The resulting memoryless state-feedback control law minimizes the upper estimate of the homogeneous -gain from disturbance to extended output. This estimate is obtained by constructing a solution, that is, storage function, to a Hamilton-Jacobi inequality (the homogeneous differential dissipation inequality) based on a control Lyapunov function. Both the continuous, homogeneously stabilizing control law and closed-loop homogeneous -gain estimate depend on the choice of storage function. The design procedure and resulting performance are illustrated with examples and simulations.

Whenever time-derivatives of measurements are necessary for observation, fault detection or control, these signals need to be estimated, e.g. using differentiators. In general, large differentiator gains increase the effect of measurement noise on the differentiation estimation error whereas small gains yield insufficient attenuation of higher-order derivatives (of the base signal to be differentiated). This work deals with the optimal gain design for the family of continuous weighted homogeneous differentiators. That is, we design, apply and evaluate homogeneous differentiators (with negative homogeneity degree) based on a theoretical framework developed recently. The design follows a high-gain setup, meaning that stabilizing differentiator parameters are fixed and the only variable tuning parameter is a gain-scaling. This is chosen to minimize the estimated effect of noise and higher-order derivative on the differentiation estimation error in terms of the homogeneous L-p-gain for p = 3 and p = infinity. Due to their practical relevance, hydromechanic drives are considered where the differentiator is applied to the noisy position measurements in order to estimate the cylinder piston's acceleration. The experimental setup includes an accelerometer rendering the differentiation estimation error available for evaluation. Several harmonic excitation signals are considered (sinusoidal, chirp, multi-sine) and the performance is evaluated via standard metrics (root-mean-squared error), the transient response and frequency-domain characteristics. The differentiators are compared to a high-gain differentiator with optimal gains and a straight-forward approach. The evaluation underlines the promising theoretical results that it is based on.

We consider the set-point control problem for nonlinear systems with flat output that are subject to perturbations. The nonlinear dynamics as well as the perturbations are locally Lipschitz. We apply the model-following control (MFC) approach which consists of a model control loop (MCL) for a feedforward generation and a process control loop (PCL) that compensates the perturbations using high-gain feedback. We analyse the resulting closed-loop system and discuss its relation to a standard flatness-based high-gain approach. In particular we analyse the estimated region of attraction provided by a quadratic Lyapunov function. A case study illustrates the approach and quantifies the region of attraction obtained for each control approach. Using the initial condition of the model control loop as tuning parameter for the MFC design, provides that a significantly larger set of initial process states can be stabilised compared to a conventional single-loop high-gain design.

Continuous homogeneous approximations of the robust exact filtering differentiators are considered. These constitute homogeneous input-output mappings, and their differentiation estimation error dynamics are investigated by means of the homogeneous ℒp-gain for p ≥ 1, and for any filtering and differentiation orders. The effect of noise and a higher-order derivative on the differentiation estimation error of interest is estimated by constructing a solution, i.e. storage function, to the (homogeneous) differential dissipation inequality (a Hamilton-Jacobi inequality). Thanks to homogeneity, any strict homogeneous Lyapunov function for the error dynamics qualifies as a storage function if properly scaled. Within a high-gain setup, an optimal gain-scaling is proved to exist that globally minimizes the homogeneous ℒp-gain estimate, thereby minimizing the effect of noise and higher-order derivative on the differentiation estimation error of interest. These theoretical results are evaluated numerically and compared to the standard homogeneous differentiator.

We extend the direct adaptive control approach by adding feedthrough of an error term to its update law. This reduces undesired oscillations of the calculated weights which can arise due to high adaptation rates. We show robustness (here: boundedness of all signals) of the closed loop against sufficiently fast, unmodeled dynamics and identify a class of harmless unmodeled dynamics for which adding feedthrough to the update law does not compromise the robustness margin if the damping injected into the adaptation process by the σ-modification is reduced as the gain of the feedthrough is increased. For harmless unmodeled dynamics, frequency-domain analysis of the purely linear case shows that such margin-preserving parameter adjustment simultaneously improves bandwidth, low-frequency disturbance rejection and attenuation of undesired oscillations.

We analyse the stability of sliding-mode control designs with a nonlinear dynamic sliding variable, i.e. the sliding variable is given by the output of a nonlinear dynamical system. In particular, we consider a nonlinear system in Byrnes-Isidori form with matched unbounded perturbations satisfying local bounds on a subset of the state space, and apply continuous first-order sliding-mode control with a dynamic sliding variable for trajectory tracking. We propose a Lyapunov-based stability analysis, which geometrically characterises the invariance properties of the closed loop with positively invariant sets given by a Lyapunov function. In doing so, we show that the solution of the closed loop remains in a tube along the reference trajectory. We illustrate our results with an example.

We consider the analytical control design for a pair of switched linear multiple-input multiple-output (MIMO) systems that are subject to arbitrary switching signals. A state feedback controller design method is proposed to obtain an eigenstructure assignment that ensures that the closed-loop switched system is globally asymptotically stable, and the outputs achieve the non-overshooting tracking of a step reference. Our analysis indicates whether non-overshooting or even monotonic tracking is achievable for the given system and considered outputs and provides a choice of possible eigenstructures to be assigned to the constituent subsystems. We derive a structural condition that verifies the feasibility of the chosen assignment. A constructive algorithm to obtain suitable feedback matrices is provided, and the method is illustrated with numerical examples.

We analyze actuator chattering in a scalar integrator system subject to second-order actuator dynamics with an unknown time constant and first-order sliding-mode control, using both a conventional static sliding manifold and a dynamic sliding manifold. Using the harmonic balance method we proof that it is possible to adjust the parameters of the dynamic sliding manifold so as to reduce the amplitude of the chattering in comparison to the static manifold. The proof of concept is illustrated with an example.

This paper proposes an algebraic observer-based modulating function approach for linear time-variant systems and a class of nonlinear systems with discrete measurements. The underlying idea lies in constructing an observability transformation that infers some properties of the modulating function approach for designing such algebraic observers. First, we investigate the algebraic observer design for linear time-variant systems under an observable canonical form for continuous-time measurements. Then, we provide the convergence of the observation error in an L2-gain stability sense. Next, we develop an exponentially stable sampled-data observer which relies on the design of the algebraic observer and an output predictor to achieve state estimation from available measurements and under small inter-sampling periods. Using a trajectory-based approach, we prove the convergence of the observation error within a convergence rate that can be adjusted through the fixed time-horizon length of the modulating function and the upper bound of the sampling period. Furthermore, robustness of the sampled-data algebraic observer, which yields input-to-state stability, is inherited by the modulating kernel and the closed-loop output predictor design. Finally, we discuss the implementation procedure of the MF-based observer realization, demonstrate the applicability of the algebraic observer, and illustrate its performance through two examples given by linear time-invariant and linear time-variant systems with nonlinear input-output injection terms.

Continuous higher order sliding mode (CHOSM) controllers represent an efficient tool for disturbance rejection. For the systems with relative degree , CHOSM approaches provide theoretically exact compensation of the matched Lipschitz perturbation, ensuring the finite-time convergence to the -th sliding-mode set, by using only information on the sliding output and its derivatives up to the order . In this paper, we investigate the disturbance rejection properties of a PID-like CHOSM controller, as the simplest and most intuitively clear example which incorporates nonlinear actions on the output error, its derivative, and integration of its sign. We use the harmonic balance approach and develop an analysis of the propagation of the matched Lipschitz perturbation through the control loop in frequency domain. The resulting solution appears in the form of the Bode-like loci, which also depend on the amplitude of the harmonic disturbances. Such amplitude-frequency characteristics allow certain comparability with standard disturbance sensitivity functions of a linear PID-controlled system in the frequency domain. Also, a simple design procedure for the robust linear PID controller targeting the second-order system plants under investigation is provided for benchmarking. Additional (parasitic) actuator dynamics, which can lead to self-induced steady oscillations, that is, chattering, is also respected. A detailed experimental case study, accomplished on an electro-mechanical actuator in a laboratory setting, highlights and makes the pros and cons of both PID and CHOSM controllers comparable for broadband disturbance rejection.

This paper addresses the tracking control problem for a dual-steering omnidirectional robot subject to rolling constraints, where the control action is applied along the admissible direction of motion. To this end, a cascade controller is developed from the reduced-order model in virtual-velocity coordinates. The outer loop regulates the admissible component of the tracking error, while the inner loop regulates the forward velocities of the drive modules. The resulting formulation is consistent with the constrained model and is defined on a singularity-free domain. Based on the reduced-order dynamics, the closed-loop design guarantees exponential convergence of the inner velocity error and local convergence of the admissible tracking error.

This paper presents a novel approach to slip ratio control on roads with differing friction coefficients on either side. The proposed methodology uses super-twisting control with an extended sliding variable that considers lateral acceleration in order to manage lateral vehicle movement effectively. Through simulation studies we demonstrate the importance of this extended sliding variable in scenarios without alternative lateral control mechanisms. Our results show that including lateral acceleration in the control strategy improves vehicle performance.

This paper addresses the modeling of an omnidirectional mobile robot equipped with dual steerable drive modules that are subject to nonholonomic rolling constraints. This class of mobile robots with steerable driving units is nowadays attractive for tasks that demand locomotion versatility and steering reorientation capability. However, their motion is constrained by rolling conditions that explicitly depend on the steering configuration, which increases the complexity of the modeling problem compared to conventional fixed-wheel platforms. The proposed formulation includes planar kinematics, rigid-body dynamics under rolling constraints, and a reduced virtual-velocity parameterization that characterizes admissible motions in the null space of the constraints. The resulting model offers a compact and physically consistent basis for simulation and subsequent control design.

The rolling resistance coefficient is a crucial parameter in vehicle dynamics. It directly affects the resistance to motion and, consequently, energy consumption and emissions in any vehicle, i.e. for combustion engine vehicles, hybrid or electric vehicles. It varies with factors such as vehicle speed, temperature, road surface type, tire pressure, and weather conditions. This variability significantly influences the vehicle performance, particularly in terms of fuel consumption and pollutant emissions, for engine vehicles. Besides energy consumption, in case of electric vehicles it also affects the recovery through regenerative braking. Therefore its estimation is an important area of research.In this work, based on a typical vehicle motion model, we compare three algorithms for estimating this coefficient: Recursive Least Squares (RLS), Super Twisting Algorithm (STA), and Extended Kalman Filter (EKF). To assess their effectiveness in a realistic scenario, we use experimental data from three vehicle tests under different maneuvers.

Considering a nonlinear system in Byrnes-Isidori form that is subject to unbounded perturbations, we apply Lyapunov redesign via feedback linearisation for trajectory tracking. Leveraging the ideas of tube-based geometric characterisation of the invariance properties of the closed loop, we generalise the classical stability criterion from the literature from constant to nonconstant reference trajectories. The proposed analysis is tailored to the Lyapunov redesign and the tracking problem insofar as we incorporate the reference trajectory and the transient decrease of the tracking error enforced by the controller. In particular, we exploit that the Lyapunov function of the tracking error satisfies a differential inequality, thereby guaranteeing that the solution of the closed loop remains in a contracting tube along the reference trajectory.

Differentiation of noisy signals is a relevant and challenging task. Widespread approaches are the linear high-gain observer acting as a differentiator and Levant's robust exact differentiator with a discontinuous right-hand side. We consider the family of arbitrary-order homogeneous differentiators, which includes these special cases. The effect of noise and disturbance on the differentiation estimation error of interest is estimated in terms of the homogeneous -gain. For this purpose, we construct a solution to the homogeneous differential dissipation inequality based on a Lyapunov function. Analogous to the classical -gain, the homogeneous -gain is not defined for the discontinuous case when a disturbance acts on the last channel. Hence, only continuous differentiators are considered. In a high-gain setup, we show not only the existence of an optimal gain-scaling that globally minimizes the homogeneous -gain estimate for any admissible , including the worst-case considerations , but also how to obtain it. The theoretical results are underlined with extensive numerical evaluation and simulation examples.

This paper presents an output feedback discrete-time sliding mode control for an uncertain sampled-data system using a novel multi-rate estimator. In the existing works, the exact state estimation of an uncertain system is not possible due to the presence of disturbance in the multi-rate estimator. Hence, the desired performance is not achieved under multi-rate output feedback sliding mode control. In this paper, we propose a novel multi-rate estimation algorithm based on the decoupling approach that estimates the unknown plant states exactly despite the disturbance. Here, we transform the system into cascaded subsystems under the assumption that each of these subsystems is observable. This assumption on (stronger) observability is well known in the continuous-time framework, and it guarantees the estimation of the unknown (partial) state vector in one time step. Using these estimated values, an output feedback discrete-time sliding mode control is designed using Gao's reaching law to guarantee the same desired performance as that of the state-feedback controller. The simulation results are presented to demonstrate the system performance under the proposed algorithm.

The update law in adaptive control schemes can be extended to include feedthrough of an error term. This reduces undesired oscillations of the calculated weights. When the sigma-modification is used for achieving robustness against unstructured uncertainty, the gain of the feedthrough in the update law cannot be chosen arbitrarily without losing the guarantee for closed-loop stability provided by a Lyapunov-based analysis. Compared to our previous result, we show stability of the closed loop for a larger parameter-range for the gain of the feedthrough in the update law. This parameter-range includes a configuration for which the influence of the integration in the update law diminishes over time, i.e. for which the adaptation for large times is governed solely by the feedthrough in the update law. By initializing at zero, this allows for removing the integration from the update law, resulting in a static update law. For the purely linear case, the adaptation acts like a disturbance observer. Frequency-domain analysis of the closed loop with a second order plant shows that removing the integration from the update law with sigma-modification and feedthrough affects how precisely disturbances in the low-frequency band are observed. If the damping injected into the adaptation process by the sigma-modification exceeds certain bounds, then the precision is increased by using the static update law.

This paper presents the practical implementation of an adaptive control strategy for a motion system subject to disturbances primarily caused by the fundamental frequency of the periodic reference signal. To this end, we devise a model for the motion platform which adequately describes the rigid-body dynamics. This physical model has been validated using measured data in the frequency domain to illustrate that it adequately captures the real dynamics of the motion platform. Using this model, we derive a standard motion control architecture with an adaptive augmentation. The control strategy consists of a feedforward controller based on model inversion and a baseline feedback controller for regulation plus an adaptive component resorting to L-1 control theory. The adaptive term is responsible for the rapid rejection of harmonic disturbances while recovering the nominal performance even in the presence of parametric perturbations. The effectiveness of the proposed control strategy is verified through real-time experiments featuring outstanding fast adaptation and rapid disturbance rejection.