Backstepping and Sequential Predictors for Control Systems

We provide new methods in mathematical control theory for two significant classes of control systems with time delays, based on backstepping and sequential prediction. Our bounded backstepping results ensure global asymptotic stability for partially linear systems with an arbitrarily large number of integrators. We also build sequential predictors for time-varying linear systems with time-varying delays in the control, sampling in the control, and time-varying measurement delays. Our bounded backstepping results are novel because of their use of converging-input-converging-state conditions, which make it possible to solve feedback stabilization problems under input delays and under boundedness conditions on the feedback control. Our sequential predictors work is novel in its ability to cover time-varying measurement delays and sampling which were beyond the scope of existing sequential predictor methods for time-varying linear systems, and in the fact that the feedback controls that we obtain from our sequential predictors do not contain any distributed terms

Backstepping for Uncertain Nonlinear Systems with a Delay in the Control

International audienceThe recent new backstepping control design strategy based on the introduction of artificial delays and/or dynamic extensions is adapted to a family of systems. That way, globally asymptotically stabilizing control laws for fundamental systems which cannot be handled by other techniques are determined

On Observer-Based Control of Nonlinear Systems

Filtering and reconstruction of signals play a fundamental role in modern signal processing, telecommunications, and control theory and are used in numerous applications. The feedback principle is an important concept in control theory. Many different control strategies are based on the assumption that all internal states of the control object are available for feedback. In most cases, however, only a few of the states or some functions of the states can be measured. This circumstance raises the need for techniques, which makes it possible not only to estimate states, but also to derive control laws that guarantee stability when using the estimated states instead of the true ones. For linear systems, the separation principle assures stability for the use of converging state estimates in a stabilizing state feedback control law. In general, however, the combination of separately designed state observers and state feedback controllers does not preserve performance, robustness, or even stability of each of the separate designs. In this thesis, the problems of observer design and observer-based control for nonlinear systems are addressed. The deterministic continuous-time systems have been in focus. Stability analysis related to the Positive Real Lemma with relevance for output feedback control is presented. Separation results for a class of nonholonomic nonlinear systems, where the combination of independently designed observers and state-feedback controllers assures stability in the output tracking problem are shown. In addition, a generalization to the observer-backstepping method where the controller is designed with respect to estimated states, taking into account the effects of the estimation errors, is presented. Velocity observers with application to ship dynamics and mechanical manipulators are also presented

Backstepping controller synthesis and characterizations of incremental stability

Incremental stability is a property of dynamical and control systems, requiring the uniform asymptotic stability of every trajectory, rather than that of an equilibrium point or a particular time-varying trajectory. Similarly to stability, Lyapunov functions and contraction metrics play important roles in the study of incremental stability. In this paper, we provide characterizations and descriptions of incremental stability in terms of existence of coordinate-invariant notions of incremental Lyapunov functions and contraction metrics, respectively. Most design techniques providing controllers rendering control systems incrementally stable have two main drawbacks: they can only be applied to control systems in either parametric-strict-feedback or strict-feedback form, and they require these control systems to be smooth. In this paper, we propose a design technique that is applicable to larger classes of (not necessarily smooth) control systems. Moreover, we propose a recursive way of constructing contraction metrics (for smooth control systems) and incremental Lyapunov functions which have been identified as a key tool enabling the construction of finite abstractions of nonlinear control systems, the approximation of stochastic hybrid systems, source-code model checking for nonlinear dynamical systems and so on. The effectiveness of the proposed results in this paper is illustrated by synthesizing a controller rendering a non-smooth control system incrementally stable as well as constructing its finite abstraction, using the computed incremental Lyapunov function.Comment: 23 pages, 2 figure

Adaptive control for time-varying systems: congelation and interconnection

This thesis investigates the adaptive control problem for systems with time-varying parameters. Two concepts are developed and exploited throughout the thesis: the congelation of variables, and the active nodes. The thesis first revisits the classical adaptive schemes and explains the challenges brought by the presence of time-varying parameters. Then, the concept of congelation of variables is introduced and its use in combinations with passivity-based, immersion-and-invariant, and identification-based adaptive schemes are discussed. As the congelation of variables method introduces additional interconnection in the closed-loop system, a framework for small-gain-like control synthesis for interconnected systems is needed.\vspace{2ex} To this end, the thesis proceeds by introducing the notion of active nodes. This is instrumental to show that as long as a class of node systems that possess adjustable damping parameters, that is the active nodes, satisfy certain graph-theoretic conditions, the desired small-gain-like property for the overall system can be enforced via tuning these adjustable parameters. Such conditions for interconnected systems with quadratic, nonlinear, and linearly parametrized supply rates, respectively, are elaborated from the analysis and control synthesis perspectives. The placement and the computation/adaptation of the damping parameters are also discussed. Following the introduction of these two fundamental tools, the thesis proceeds by discussing state-feedback designs for a class of lower-triangular nonlinear systems. The backstepping technique and the congelation of variables method are combined for passivity-based, immersion-and-invariance, and identification-based schemes. The notion of active nodes is exploited to yield simple and systematic proofs. Based on the results established for lower-triangular systems, the thesis continues to investigate output-feedback adaptive control problems. An immersion-and-invariance scheme for single-input single-output linear systems and a passivity-based scheme for nonlinear systems in observer form are proposed. The proof and interpretation of these results are also based on the notion of active nodes. The simulation results show that the adaptive control schemes proposed in the thesis have superior performance when compared with the classical schemes in the presence of time-varying parameters. Finally, the thesis studies two applications of the theoretical results proposed. The servo control problem for serial elastic actuators, and the disease control problem for interconnected settlements. The discussions show that these problems can be solved efficiently using the framework provided by the thesis.Open Acces

Delay-Adaptive Control of First-order Hyperbolic PIDEs

We develop a delay-adaptive controller for a class of first-order hyperbolic partial integro-differential equations (PIDEs) with an unknown input delay. By employing a transport PDE to represent delayed actuator states, the system is transformed into a transport partial differential equation (PDE) with unknown propagation speed cascaded with a PIDE. A parameter update law is designed using a Lyapunov argument and the infinite-dimensional backstepping technique to establish global stability results. Furthermore, the well-posedness of the closed-loop system is analyzed. Finally, the effectiveness of the proposed method was validated through numerical simulation

Safety-Aware Longitudinal and Lateral Control of Autonomous Vehicles

Safety is undoubtedly the most critical design requirement regarding autonomous vehicle controllers. This research considers an autonomous vehicle to keep a desired distance from the leader vehicle, as well as stay centered within the lane. To achieve this, the lateral control problem and the combined longitudinal and lateral control problem were studied. Adaptive control laws were proposed with the aid of the backstepping technique and the barrier function technique. Simulation was done to verify the effectiveness of the proposed control laws