Funnel control for systems with relative degree two

PublishedJournal ArticleTracking of reference signals yref (·) by the output y(·) of linear (as well as a considerably large class of nonlinear) single-input, single-output systems is considered. The system is assumed to have strict relative degree two with (weakly) stable zero dynamics. The control objective is tracking of the error e = y - yref and its derivative e within two prespecified performance funnels, respectively. This is achieved by the so-called funnel controller u(t) = -k0(t)2e(t)-k 1(t)e(t), where the simple proportional error feedback has gain functions k0 and k1 designed in such a way to preclude contact of e and e with the funnel boundaries, respectively. The funnel controller also ensures boundedness of all signals. We also show that the same funnel controller (i) is applicable to relative degree one systems, (ii) allows for input constraints provided a feasibility condition (formulated in terms of the system data, the saturation bounds, the funnel data, bounds on the reference signal, and the initial state) holds, (iii) is robust in terms of the gap metric: if a system is sufficiently close to a system with relative degree two, stable zero dynamics, and positive high-frequency gain, but does not necessarily have these properties, then for small initial values the funnel controller also achieves the control objective. Finally, we illustrate the theoretical results by experimental results: the funnel controller is applied to a rotatory mechanical system for position control. © 2013 Society for Industrial and Applied Mathematics

Indirect adaptive higher-order sliding-mode control using the certainty-equivalence principle

Seit den 50er Jahren werden große Anstrengungen unternommen, Algorithmen zu entwickeln, welche in der Lage sind Unsicherheiten und Störungen in Regelkreisen zu kompensieren. Früh wurden hierzu adaptive Verfahren, die eine kontinuierliche Anpassung der Reglerparameter vornehmen, genutzt, um die Stabilisierung zu ermöglichen. Die fortlaufende Modifikation der Parameter sorgt dabei dafür, dass strukturelle Änderungen im Systemmodell sich nicht auf die Regelgüte auswirken. Eine deutlich andere Herangehensweise wird durch strukturvariable Systeme, insbesondere die sogenannte Sliding-Mode Regelung, verfolgt. Hierbei wird ein sehr schnell schaltendes Stellsignal für die Kompensation auftretender Störungen und Modellunsicherheiten so genutzt, dass bereits ohne besonderes Vorwissen über die Störeinflüsse eine beachtliche Regelgüte erreicht werden kann. Die vorliegende Arbeit befasst sich mit dem Thema, diese beiden sehr unterschiedlichen Strategien miteinander zu verbinden und dabei die Vorteile der ursprünglichen Umsetzung zu erhalten. So benötigen Sliding-Mode Verfahren generell nur wenige Informationen über die Störung, zeigen jedoch Defizite bei Unsicherheiten, die vom Systemzustand abhängen. Auf der anderen Seite können adaptive Regelungen sehr gut parametrische Unsicherheiten kompensieren, wohingegen unmodellierte Störungen zu einer verschlechterten Regelgüte führen. Ziel dieser Arbeit ist es daher, eine kombinierte Entwurfsmethodik zu entwickeln, welche die verfügbaren Informationen über die Störeinflüsse bestmöglich ausnutzt. Hierbei wird insbesondere Wert auf einen theoretisch fundierten Stabilitätsnachweis gelegt, welcher erst durch Erkenntnisse der letzten Jahre im Bereich der Lyapunov-Theorie im Zusammenhang mit Sliding-Mode ermöglicht wurde. Anhand der gestellten Anforderungen werden Regelalgorithmen entworfen, die eine Kombination von Sliding-Mode Reglern höherer Ordnung und adaptiven Verfahren darstellen. Neben den theoretischen Betrachtungen werden die Vorteile des Verfahrens auch anhand von Simulationsbeispielen und eines Laborversuchs nachgewiesen. Es zeigt sich hierbei, dass die vorgeschlagenen Algorithmen eine Verbesserung hinsichtlich der Regelgüte als auch der Robustheit gegenüber den konventionellen Verfahren erzielen.Since the late 50s, huge efforts have been made to improve the control algorithms that are capable of compensating for uncertainties and disturbances. Adaptive controllers that adjust their parameters continuously have been used from the beginning to solve this task. This adaptation of the controller allows to maintain a constant performance even under changing conditions. A different idea is proposed by variable structure systems, in particular by the so-called sliding-mode control. The idea is to employ a very fast switching signal to compensate for disturbances or uncertainties. This thesis deals with a combination of these two rather different approaches while preserving the advantages of each method. The design of a sliding-mode controller normally does not demand sophisticated knowledge about the disturbance, while the controller's robustness against state-dependent uncertainties might be poor. On the other hand, adaptive controllers are well suited to compensate for parametric uncertainties while unstructured influence may result in a degraded performance. Hence, the objective of this work is to design sliding-mode controllers that use as much information about the uncertainty as possible and exploit this knowledge in the design. An important point is that the design procedure is based on a rigorous proof of the stability of the combined approach. Only recent results on Lyapunov theory in the field of sliding-mode made this analysis possible. It is shown that the Lyapunov function of the nominal sliding-mode controller has a direct impact on the adaptation law. Therefore, this Lyapunov function has to meet certain conditions in order to allow a proper implementation of the proposed algorithms. The main contributions of this thesis are sliding-mode controllers, extended by an adaptive part using the certainty-equivalence principle. It is shown that the combination of both approaches results in a novel controller design that is able to solve a control task even in the presence of different classes of uncertainties. In addition to the theoretical analysis, the advantages of the proposed method are demonstrated in a selection of simulation examples and on a laboratory test-bench. The experiments show that the proposed control algorithm delivers better performance in regard to chattering and robustness compared to classical sliding-mode controllers

Adaptive vehicle control by combined DYC and FWS

textVehicle stability is an important consideration in vehicle design. When driver intervention is insufficient, safety can be improved by the addition of vehicle stability control (VSC). Typical vehicle stability controllers are designed using a linearized vehicle model and an assumed set of parameters. However, some parameters like mass and inertial properties may not be constant between operations. To recover controller performance in the presence of unknown parameters, adaptive estimates can be developed. This thesis seeks to implement a model reference adaptive controller for yaw rate and side slip control and to evaluate any implementation issues that may arise. A linearized vehicle model is used for controller design via a Lyapunov approach and a combined front wheel steering (FWS) and direct yaw control (DYC) controller is developed. The combined FWS+DYC controller is tested in a low friction double lane change with initial parameter estimation error. The FWS+DYC controller was found to be robust to parameter changes, and the adaptive parameter estimates did not provide any noticeable improvement over the non-adaptive case. A four wheel steering (4WS) controller is developed by a similar approach and tested under the same conditions. Both controllers were found to be effective at stabilizing the vehicle. An unexpected finding was that though the combined FWS+DYC controller was effective even in low friction conditions with parameter errors, the required motor torque was very large and oscillated rapidly. This was diminished through the addition of a low pass filter on the controller yaw moment output, but could not be removed entirely.Mechanical Engineerin

Adaptive and Neural Network-Based Aircraft Tracking Control with Synthetic Jet Actuators

Wing-embedded synthetic jet actuators (SJA) can be used to achieve maneuvering control in aircraft by delivering controllable airflow perturbations near the wing surface. Trajectory tracking control design for aircraft equipped with SJA is particularly challenging, since the controlling actuator itself has an uncertain dynamic model. These challenges necessitate advanced nonlinear control design methods to achieve desirable performance for SJA-based aircraft (e.g., micro air vehicles (MAVs)). In this research, adaptive and neural-network based control methods are investigated, which are specifically designed to compensate for the SJA dynamic model uncertainty and unpredictable operating conditions characters tic of real-world MAV applications. The control design methods discussed in this thesis are rigorously developed to achieve a prescribed level of trajectory tracking control performance, and numerical simulation results are presented to demonstrate the performance of the controllers in the presence of adversarial operating conditions

A Control Lyapunov Perspective on Episodic Learning via Projection to State Stability

The goal of this paper is to understand the impact of learning on control synthesis from a Lyapunov function perspective. In particular, rather than consider uncertainties in the full system dynamics, we employ Control Lyapunov Functions (CLFs) as low-dimensional projections. To understand and characterize the uncertainty that these projected dynamics introduce in the system, we introduce a new notion: Projection to State Stability (PSS). PSS can be viewed as a variant of Input to State Stability defined on projected dynamics, and enables characterizing robustness of a CLF with respect to the data used to learn system uncertainties. We use PSS to bound uncertainty in affine control, and demonstrate that a practical episodic learning approach can use PSS to characterize uncertainty in the CLF for robust control synthesis

Funnel control for systems with relative degree two

Tracking of reference signals yref(·) by the output y(·) of linear (as well as a considerably large class of nonlinear) single-input, single-output system is considered. The system is assumed to have strict relative degree two with ("weak") stable zero dynamics. The control objective is tracking of the error e = y − yref and its derivative e˙ within two prespecified performance funnels, resp. This is achieved by the so called 'funnel controller': u(t) = −k0(t)2e(t) − k1(t)e˙(t), where the simple proportional error feedback has gain functions k0 and k1 designed in such a way to preclude contactof e and e˙ with the funnel boundaries, resp. The funnel controller also ensures boundedness of all signals. We also show that the same funnel controller is (i) applicable to relative degree one systems, (ii) allows for input constraints provided a feasibility condition (formulated in terms of the system data, the saturation bounds, the funnel data, bounds on the reference signal and the initial state) holds, (iii) is robust in terms of the gap metric: if a system is sufficiently close to a system with relative degree two, stable zero dynamics and positive high-frequency gain, but does not necessarily have these properties, then for small initial values the funnel controller also achieves the control objective. Finally, we illustrate the theoretical results by experimental results: the funnel controller is applied to a rotatory mechanical system for position control

A Robust Nonlinear Model Reference Adaptive Control for Disturbed Linear Systems: An LMI Approach

International audienceIn this paper a robust nonlinear Model Reference Adaptive Control (MRAC) is proposed for disturbed linear systems, i.e., linear systems with parameter uncertainties, and external time-dependent perturbations or nonlinear unmodeled dynamics matched with the control input. The proposed nonlinear control law is composed of two nonlinear adaptive gains. Such adaptive gains allow the control to counteract the effects of some perturbations and nonlinear unmodeled dynamics ensuring asymptotic convergence of the tracking error to zero, and the boundedness of the adaptive gains. The nonlinear controller synthesis is given by a constructive method based on the solution of Linear Matrix Inequalities (LMIs). Besides, the simulation results show that, due to the nonlinearities, the rate of convergence of the proposed algorithm is faster than the provided by a classic MRAC