The problem of partial stabilization for nonlinear control systems described by the Ito stochastic differential equations is considered. For these systems, we propose a constructive control design method which leads to establishing the asymptotic stability in probability of the trivial solution of the closed-loop system with respect to a part of state variables. Mechanical examples are presented to illustrate the efficiency of the obtained controllers.
In this paper we consider the problem of stabilization and tracking of desired state trajectory for a wide range of nonlinear control problems with disturbances. We present the sufficient conditions for the existence of $C^k$ state feedback controllers and the process of their mathematical designing is described.
We study the output feedback exponential stabilization of a one-dimensional unstable wave equation, where the boundary input, given by the Neumann trace at one end of the domain, is the sum of the control input and the total disturbance. The latter is composed of a nonlinear uncertain feedback term and an external bounded disturbance. Using the two boundary displacements as output signals, we design a disturbance estimator that does not use high gain. It is shown that the disturbance estimator can estimate the total disturbance in the sense that the estimation error signal is in $L^2[0,\infty)…
This paper is concerned with the output feedback exponential stabilization for a flexible beam with tip mass. When there is no disturbance, it is shown that only one non-collocated measurement is enough to exponentially stabilize the original system by constructing an infinite-dimensional Luenberger state observer to track the state and designing an estimated state based output feedback control law. This essentially improves the existence result in [F. Conrad and O. Mörgül, SIAM J. Control Optim., 36 (1998), 1962-1986] where two collocated measurements including high order feedback were adopte…
This paper develops systematically the output feedback exponential stabilization for a one-dimensional unstable/anti-stable wave equation where the control boundary suffers from both internal nonlinear uncertainty and external disturbance. Using only two displacement signals, we propose a disturbance estimator that not only can estimate successfully the disturbance in the sense that the error is in $L^2(0,\infty)$ but also is free high-gain. With the estimated disturbance, we design a state observer that is exponentially convergent to the state of original system. An observer-based output feed…
In this expository paper, which covers material presented at the NATO Advanced Study Institute "Nonlinear Analysis, Differential Equations, and Control" (Montreal, Jul/Aug 1998), we deal with several questions related to stability and stabilization of nonlinear finite-dimensional continuous-time systems. We review the basic problem of feedback stabilization, placing an emphasis upon relatively new areas of research which concern stability with respect to "noise" (such as errors introduced by actuators or sensors). The table of contents is as follows: Review of Stability and Asymptotic Controll…
In solving the problem of asymptotic tracking and disturbance rejection, it has been long always assumed that the reference to be tracked and the disturbance to be rejected must be generated by an exosystem such as a finite dimensional exosystem with pure imaginary eigenvalues. The objective of this paper is to solve such a tracking problem for the unstable Burgers' equation without this assumption. Our treatment of this problem is straightforward. Using the method of variable transform, the tracking problem is split into two separate problems: a simple Neumann boundary stabilization problem a…
In this paper, we are concerned with the stabilization of linear port-Hamiltonian systems of arbitrary order $N \in \mathbb{N}$ on a bounded $1$-dimensional spatial domain $(a,b)$. In order to achieve stabilization, we couple the system to a dynamic boundary controller, that is, a controller that acts on the system only via the boundary points $a,b$ of the spatial domain. We use a nonlinear controller in order to capture the nonlinear behavior that realistic actuators often exhibit and, moreover, we allow the output of the controller to be corrupted by actuator disturbances before it is fed ba…
Adaptive stabilization of a leaderless bearing-constrained formation with disturbances
East Asia University of Technology（东亚理工大学）；University of Southern California（南加州大学）；Ghost Robotics（幽灵机器人公司）；Hanoi University of Science and Technology（河内科技大学）
Yet Another Tutorial of Disturbance Observer: Robust Stabilization and Recovery of Nominal Performance
ASRI, Department of Electrical and Computer Engineering, Seoul National University（首尔国立大学电气与计算机工程系自动系统研究室）；Department of Electrical Engineering and Computer Science, University of Central Florida（中佛罗里达大学电气与计算机科学系）；School of Robotics, Kwangwoon University（光云大学机器人学院）；Department of Electrical Engineering, Soongsil University（崇实大学电气工程系）
We consider nonlinear control systems of the so-called generalized triangular form (GTF) with time-varying and periodic dynamics which linearly depends on some external disturbances. Our purpose is to construct a feedback controller which provides the global input-to-state stability of the corresponding closed-loop w.r.t. the disturbances. To do this, we combine the method proposed in the earlier work \cite{pavlichkov_ge_2009} devoted the the global asymptotic stabilization of the GTF systems without disturbances with the ISS theory for time-varying systems proposed in \cite{wang}. Following t…
The method of controlled Lagrangians for discrete mechanical systems is extended to include potential shaping in order to achieve complete state-space asymptotic stabilization. New terms in the controlled shape equation that are necessary for matching in the discrete context are introduced. The theory is illustrated with the problem of stabilization of the cart-pendulum system on an incline. We also discuss digital and model predictive control.
We study the output feedback exponential stabilization for a 1-d wave PDE with dynamic boundary. With only one measurement, we construct an infinite-dimensional state observer to trace the state and design an estimated state based controller to exponentially stabilize the original system. This is an essentially important improvement for the existence literature [Ö. Morgül, B.P. Rao and F. Conrad, IEEE Transactions on Automatic Control, 39(10) (1994), 2140-2145] where two measurements including the high order angular velocity feedback were adopted. When a control matched nonlinear internal unce…
This work addresses the problem of transient stabilization of a power grid, following a destabilizing disturbance. The model considered is the cascade interconnection of seven New England test models with the disturbance (e.g., a powerline failure) occurring in the first grid and propagating forward, emulating a wide-area blackout. We consider a data-driven control framework based on the Koopman operator theory, where a linear predictor, evolving on a higher dimensional (embedded) state-space, is built from observed data and subsequently used within a model predictive control (MPC) framework, …
In this paper, the control problem for underactuated systems in the presence of external disturbances and model uncertainties is considered. An adaptive fuzzy sliding mode controller (AFSMC) is proposed to solve the problem, satisfying the robustness properties against uncertainties and disturbances. The designed controller can be applied to a wide class of underactuated systems and have fewer restrictions, compared with many pervious works. To evaluate the performance of proposed algorithm, it is applied to an underactuated inverted pendulum and a Translational Oscillator/Rotational Actuator …
Optimal Universal Controllers for Roll Stabilization
Oceaneering AGV Systems B.V.（Oceaneering AGV系统有限公司）；University of Groningen（格罗宁根大学）；Politecnico di Torino（都灵理工大学）；Institute for Problems of Mechanical Engineering（机械工程学院问题研究所）
Controlled Lagrangian and matching techniques are developed for the stabilization of relative equilibria and equilibria of discrete mechanical systems with symmetry as well as broken symmetry. Interesting new phenomena arise in the controlled Lagrangian approach in the discrete context that are not present in the continuous theory. In particular, to make the discrete theory effective, one can make an appropriate selection of momentum levels or, alternatively, introduce a new parameter into the controlled Lagrangian to complete the kinetic matching procedure. Specifically, new terms in the cont…
In this paper input-to-state practically stabilizing control laws for retarded, control-affine, nonlinear systems with actuator disturbance are investigated. The developed methodology is based on the Arstein's theory of control Liapunov functions and related Sontag's formula, extended to retarded systems. If the actuator disturbance is bounded, then the controller yields the solution of the closed-loop system to achieve an arbitrarily fixed neighborhood of the origin, by increasing a control tuning parameter. The considered systems can present an arbitrary number of discrete as well as distrib…
This paper deals with the exponential input-to-state stabilization with respect to boundary disturbances of a class of diagonal infinite-dimensional systems via delay boundary control. The considered input delays are uncertain and time-varying. The proposed control strategy consists of a constant-delay predictor feedback controller designed on a truncated finite-dimensional model capturing the unstable modes of the original infinite-dimensional system. We show that the resulting closed-loop system is exponentially input-to-state stable with fading memory of both additive boundary input perturb…
A mechanical system consisting of a rigid body and attached Kirchhoff plates under the action of three independent controls torques is considered. The equations of motion of such model are derived in the form of a system of coupled nonlinear ordinary and partial differential equations. The operator form of this system is represented as an abstract differential equation in a Hilbert space. A feedback control law is constructed such that the corresponding infinitesimal generator is dissipative.
In this paper, we investigate the problem of semi-global minimal time robust stabilization of analytic control systems with controls entering linearly, by means of a hybrid state feedback law. It is shown that, in the absence of minimal time singular trajectories, the solutions of the closed-loop system converge to the origin in quasi minimal time (for a given bound on the controller) with a robustness property with respect to small measurement noise, external disturbances and actuator noise.
In this note we study the generation of attractive oscillations of a class of mechanical systems with underactuation one. The proposed design consists of two terms, i.e., a partial linearizing state feedback, and an immersion and invariance orbital stabilization controller. The first step is adopted to simplify analysis and design, however, bringing an additional difficulty that the model loses its Euler-Lagrange structure after the collocated pre-feedback. To address this, we propose a constructive solution to the orbital stabilization problem via a smooth controller in an analytic form, and …
In this paper, a novel control strategy namely disturbance observer-based control is first applied to stabilization and disturbance rejection for an anti-stable stochastic heat equation with Neumann boundary actuation and unknown boundary external disturbance generated by an exogenous system. A disturbance observer-based boundary control is designed based on the backstepping approach and estimation/cancellation strategy, where the unknown disturbance is estimated in real time by a disturbance observer and rejected in the closed-loop, while the in-domain multiplicative noise whose intensity is …
One approach to robust control for linear plants with structured uncertainty as well as for linear parameter-varying (LPV) plants (where the controller has on-line access to the varying plant parameters) is through linear-fractional-transformation (LFT) models. Control issues to be addressed by controller design in this formalism include robust stability and robust performance. Here robust performance is defined as the achievement of a uniform specified $L^{2}$-gain tolerance for a disturbance-to-error map combined with robust stability. By setting the disturbance and error channels equal to z…
We extend the method of controlled Lagrangians with kinetic shaping to those mechanical systems on semidirect product Lie groups with broken symmetry, more specifically to the Euler--Poincaré equations with advected parameters. We find a matching condition for the controlled Lagrangian for such systems whose configuration manifold is a general semidirect product Lie group $\mathsf{G} \ltimes V$. Our motivating examples are a bottom-heavy underwater vehicle and a top spinning on a movable base. Their configuration space is the special Euclidean group $\mathsf{SE}(3) = \mathsf{SO}(3) \ltimes \ma…
This study focuses on the problem of optimal mismatched disturbance rejection control for uncontrollable linear discrete-time systems. In contrast to previous studies, by introducing a quadratic performance index such that the regulated state can track a reference trajectory and minimize the effects of disturbances, mismatched disturbance rejection control is transformed into a linear quadratic tracking problem. The necessary and sufficient conditions for the solvability of this problem over a finite horizon and a disturbance rejection controller are derived by solving a forward-backward diffe…
We construct a patchy feedback for a general control system on $\R^n$ which realizes practical stabilization to a target set $Σ$, when the dynamics is constrained to a given set of states $S$. The main result is that $S$--constrained asymptotically controllability to $Σ$ implies the existence of a discontinuous practically stabilizing feedback. Such a feedback can be constructed in "patchy" form, a particular class of piecewise smooth controls which ensure the existence of local Carathéodory solutions to any Cauchy problem of the control system and which enjoy good robustness properties with r…
We apply the method of controlled Lagrangians by potential shaping to Euler--Poincaré mechanical systems with broken symmetry. We assume that the configuration space is a general semidirect product Lie group $\mathsf{G} \ltimes V$ with a particular interest in those systems whose configuration space is the special Euclidean group $\mathsf{SE}(3) = \mathsf{SO}(3) \ltimes \mathbb{R}^{3}$. The key idea behind the work is the use of representations of $\mathsf{G} \ltimes V$ and their associated advected parameters. Specifically, we derive matching conditions for the modified potential exploiting t…
Interconnection and damping assignment passivity-based control (IDA-PBC) is an excellent method to stabilize mechanical systems in the Hamiltonian formalism. In this paper, several improvements are made on the IDA-PBC method. The skew-symmetric interconnection submatrix in the conventional form of IDA-PBC is shown to have some redundancy for systems with the number of degrees of freedom greater than two, containing unnecessary components that do not contribute to the dynamics. To completely remove this redundancy, the use of quadratic gyroscopic forces is proposed in place of the skew-symmetri…
In this work, we propose a methodology for the expression of necessary and sufficient Lyapunov-like conditions for the existence of stabilizing feedback laws. The methodology is an extension of the well-known Control Lyapunov Function (CLF) method and can be applied to very general nonlinear time-varying systems with disturbance and control inputs, including both finite- and infinite-dimensional systems. The generality of the proposed methodology is also reflected upon by the fact that partial stability with respect to output variables is addressed. In addition, it is shown that the generalize…
This paper discusses the boundary feedback stabilization of a reaction-diffusion equation with Robin boundary conditions and in the presence of a time-varying state-delay. The proposed control design strategy is based on a finite-dimensional truncated model obtained via a spectral decomposition. By an adequate selection of the number of modes of the original infinite-dimensional system, we show that the design performed on the finite-dimensional truncated model achieves the exponential stabilization of the original infinite-dimensional system. In the presence of distributed disturbances, we sh…
Transverse linearization-based approaches have become among the most prominent methods for orbitally stabilizing feedback design in regards to (periodic) motions of underactuated mechanical systems. Yet, in an $n$-dimensional state-space, this requires knowledge of a set of $(n-1)$ independent transverse coordinates, which can be nontrivial to find and whose definitions might vary for different motions (trajectories). In this paper, we consider instead a generic set of $excessive$ transverse coordinates which are defined in terms of a particular parameterization of the motion and a projection …
Robust global stabilization of nonlinear systems by observer-based feedback controllers is a challenging task. This article investigates the problem of designing observer-based stabilizing controllers for incrementally quadratic nonlinear systems with external disturbances. The nonlinearities considered in the system model satisfy the incremental quadratic constraints, which are characterized by incremental multiplier matrices and encompass many common nonlinearities. The simultaneous search for the observer and the controller gain matrices is formulated as a feasibility problem of linear matr…
We give a method for designing a mechanical impedance to suppress the propagation of disturbances along a chain of masses. The key feature of our method is that it is scale free. This means that it can be used to give a single, fixed, design, with provable performance guarantees in mass chains of any length. We illustrate the approach by designing a bidirectional control law in a vehicle platoon in a manner that is independent of the number of vehicles in the platoon.
We investigate the problem of stabilizing an unknown networked linear system under communication constraints and adversarial disturbances. We propose the first provably stabilizing algorithm for the problem. The algorithm uses a distributed version of nested convex body chasing to maintain a consistent estimate of the network dynamics and applies system level synthesis to determine a distributed controller based on this estimated model. Our approach avoids the need for system identification and accommodates a broad class of communication delay while being fully distributed and scaling favorabl…
In this note, it is shown by counterexamples, theoretical analysis and simulation tests that the two types of sliding-mode controllers presented in the paper [1] fail to solve the stabilization problem of a class of under-actuated mechanical systems.
The task of inducing, via continuous static state-feedback control, an asymptotically stable heteroclinic orbit in a nonlinear control system is considered in this paper. The main motivation comes from the problem of ensuring convergence to a so-called point-to-point maneuver in an underactuated mechanical system. Namely, to a smooth curve in its state--control space, which is consistent with the system dynamics and connects two (linearly) stabilizable equilibrium points. The proposed method uses a particular parameterization, together with a state projection onto the maneuver as to combine tw…
The prescribed-time stabilization problem for a general class of nonlinear systems with unknown input gain and appended dynamics (with unmeasured state) is addressed. Unlike the asymptotic stabilization problem, the prescribed-time stabilization objective requires convergence of the state to the origin in a finite time that can be arbitrarily picked (i.e., prescribed) by the control system designer irrespective of the initial condition of the system. The class of systems considered is allowed to have general nonlinear uncertain terms throughout the system dynamics as well as uncertain appended…
