Stabilizing Feedback Laws Research Articles

Finite- and Fixed-Time Nonovershooting Stabilizers and Safety Filters by Homogeneous Feedback

Non-overshooting stabilization is a form of safe control where the setpoint chosen by the user is at the boundary of the safe set. In this paper we develop homogeneous feedback laws for fixed-time nonovershooting stabilization for nonlinear systems that are input-output linearizable with a full relative degree, i.e., for systems that are diffeomorphically equivalent to the chain of integrators. These homogeneous feedback laws can also assume the secondary role of ‘fixed-time safety filters’ (FxTSf filters) which keep the system within the closed safe set for all time but, in the case where the user's nominal control commands approach to the unsafe set, allow the system to reach the boundary of the safe set no later than a desired time that is independent of nominal control and independent of the value of the state at the time the nominal control begins to be overridden.

On the fixed-time extinction based nonlinear control and systems decomposition: applications to bilinear systems

This paper deals with new criteria of the fixed-time extinction of nonlinear dynamical systems. We show, by comparison principal, that Gronwall-like integral inequalities leading to the extinction of the energy system in fixed-time; this helps us to build fixed-time stabilizing feedback laws for bilinear control systems in Hilbert space. In addition, in order to relax some existing results on the fixed-time stability of systems by Lyapunov function, we prove that the decomposition of the system into two subsystems so that one is finite-time stable and the other is polynomially stable, leads to fixed-time extinction of the whole system. Our results are applied to the double integrator where new fixed-time stabilizing feedbacks, with various aspects, are constructed.

Prescribed-Time Mean-Nonovershooting Control under Finite-Time Vanishing Noise

We develop a prescribed-time mean-nonovershooting stabilizing feedback law for stochastic nonlinear systems with noise that vanishes in finite time. The prescribed time of stabilization must be strictly after the noise vanishes, but it may occur as early as the user desires after the noise vanishes. In fact, more generally, the feedback is stabilizing and mean-overshoot free for any noise that, given the prescribed time of stabilization, satisfies certain decay rate properties which require, in particular, that no noise component vanish slower than linearly in the “time to go” until the prescribed time. In contrast to the existing stochastic prescribed-time designs where only multiplicative noise is allowed, our design can deal with multiplicative and additive noise simultaneously. A new controller is designed to guarantee that the mean of the system output prescribed time tracks a given trajectory without overshooting, that the fourth moment of the tracking error between states and derivatives of the reference trajectory converges to zero in prescribed time, and that the controller and all of the states are mean-square bounded. Finally, a simulation example is given to illustrate the prescribed-time mean-nonovershooting design.

Spill-Free Transfer and Stabilization of Viscous Liquid

This article studies the feedback stabilization problem of the motion of a tank that contains an incompressible, Newtonian, viscous liquid. The control input is the force applied on the tank and the overall system consists of two nonlinear partial differential equations and two ordinary differential equations. Moreover, a spill-free condition is required to hold. By applying the control Lyapunov functional methodology, a set of initial conditions (state space) is determined for which spill-free motion of the liquid is possible by applying an appropriate control input. Semi-global stabilization of the liquid and the tank by means of a simple feedback law is achieved, in the sense that for every closed subset of the state space, it is possible to find appropriate controller gains, so that every solution of the closed-loop system initiated from the given closed subset satisfies specific stability estimates. The closed-loop system exhibits an exponential convergence rate to the desired equilibrium point. The proposed stabilizing feedback law does not require measurement of the liquid level and velocity profiles inside the tank and simply requires measurements of 1) the tank position error and tank velocity, 2) the total momentum of the liquid, and 3) the liquid levels at the tank walls. The obtained results allow an algorithmic solution of the problem of the spill-free movement and slosh-free settlement of a liquid in a vessel of limited height (such as water in a glass) by a robot to a prespecified position, no matter how full the vessel is.

Integration of inventory control into the port‐Hamiltonian framework for dissipative stabilization of chemical reactors

AbstractThis work integrates passivity‐based inventory control with the port‐Hamiltonian framework to design stabilizing feedback laws for lumped parameter chemically reacting systems. A static state feedback law with admissible controls is developed using an inventory‐related storage function as a closed loop Hamiltonian function. The control gain matrix is derived from the interconnection and damping matrices of the resulting Hamiltonian representation. Consequently, the proposed controller globally and exponentially stabilizes the system at a desired set‐point (including the open loop unstable equilibrium point) without restrictions on the reaction kinetics. Numerical simulations illustrate the application of the theory to a nonisothermal, continuous stirred tank reactor with the steady‐state multiplicity. The proposed approach is compared with the thermodynamic availability‐based control approach in terms of amplitude and variation rate to show its performance and effectiveness.

Gradient-augmented Supervised Learning of Optimal Feedback Laws Using State-Dependent Riccati Equations

A supervised learning approach for the solution of large-scale nonlinear stabilization problems is presented. A stabilizing feedback law is trained from a dataset generated from State-dependent Riccati Equation solvers. The training phase is enriched by the use of gradient information in the loss function, which is weighted through the use of hyperparameters. High-dimensional nonlinear stabilization tests demonstrate that real-time sequential large-scale Algebraic Riccati Equation solvers can be substituted by a suitably trained feedforward neural network.

New Features of P3δ Software. Insights and Demos

This paper presents the software entitled “Partial Pole Placement via Delay Action,” or “P3δ” for short. P3δ is a Python software with a friendly user interface for the design of parametric stabilizing feedback laws with time-delays for dynamical systems. After recalling the theoretical foundation of the so-called “Partial Pole Placement” methodology we propose as well the main features of the current version of P3δ. We illustrate its use in feedback stabilization of several control systems operating under time delays.

On the geometric construction of a stabilizing time-invariant state feedback controller for the nonholonomic integrator

The paper presents a rather natural and elementary geometric construction of a stabilizing time-invariant state feedback law for the nonholonomic integrator. The key features of the particular construction are the direct inclusion of certain classical optimality considerations pertaining to the geodesics of the nonholonomic integrator, the confinement of all discontinuities in the feedback law to the z-axis, as well as a uniform exponential convergence result for the closed-loop system. The results of this treatment also have interesting implications for the control of nonholonomic systems in general, e.g., they highlight that for nonholonomic systems even the most natural seeming stabilizing feedback laws may not be amenable to a closed-form expression and may need to be formulated in more elaborate and implicit terms.

New Features of P3δ software: Partial Pole Placement via Delay Action

This paper presents the software Partial Pole Placement via Delay Action, or P3$\delta$ for short. P3$\delta$ is a Python software with a friendly user interface for the design of parametric stabilizing feedback laws with time-delays, thanks to two properties of the distribution of quasipolynomials' zeros, called multiplicity-induced-dominancy and coexisting real roots-induced-dominancy. After recalling recent theoretical results on these properties and their use for the feedback stabilization of control systems operating under time delays, the paper presents the main features of the current version of P3$\delta$. We detail, in particular, the assignable admissible region (the set of allowable dominant roots and the corresponding delay), which helps the user in the choice of input information, allowing a reliable stabilizing delayed feedback. We also present the newly set online version of P3$\delta$.

Some results on the internal finite‐time stabilization of vibrating strings with interior mass

In this paper, the problem of internal finite‐time stabilization for 1‐D coupled wave equations with interior point mass is handled. The nonlinear stabilizing feedback law leads, in closed‐loop, to nonlinear evolution equations where Kato theory is used to prove the well‐posedness. In addition, it is showed that in some cases, the solution of this hybrid system is constant in finite‐time if we use Neumann boundary conditions. This result can be improved (in complete finite‐time stability sense) if we change the above feedback.

A Variational Approach to Local Asymptotic and Exponential Stabilization of Nonlinear Systems

Local asymptotic stabilizability is a topic of great theoretical interest and practical importance. Broadly, if a system \(\dot {x} = f(x,u)\) is locally asymptotically stabilizable, we are guaranteed a feedback controller u(x) that forces convergence to an equilibrium for trajectories initialized sufficiently close to it. A necessary condition was given by Brockett: such controllers exist only when f is open at the equilibrium. Recently, Gupta, Jafari, Kipka, and Mordukhovich considered a modification to this condition, replacing Brockett’s topological openness by the linear openness property of modern variational analysis. In this paper, we show that, under the linear openness assumption, there exists a local diffeomorphism rendering the system exponentially stabilizable by means of continuous stationary feedback laws. Introducing a transversality property and relating it to the above diffeomorphism, we prove that linear openness and transversality on a punctured neighborhood of an equilibrium is sufficient for local exponential stabilizability of systems with a rank-deficient linearization. The main result goes beyond the usual Kalman and Hautus criteria for the existence of exponential stabilizing feedback laws, since it allows us to handle systems for which exponential stabilization is achieved through higher order terms. However, it is implemented so far under a rather restrictive row-rank conditions. This suggests a twofold approach to the use of these properties: a pointwise version is enough to ensure stability via the linearization, while a local version is enough to overcome deficiencies in the linearization.

Nonlinear Observer-Based Control of an Under-Actuated Hovercraft Vehicle

This paper presents a model-based approach to the nonlinear tracking control for the body-fixed velocities of an under-actuated hovercraft vehicle. To enable a corresponding state feedback with accurate velocity signals, an observer-based sensor fusion is envisaged using acceleration measurements as well as data from an optical flux sensor. The horizontal and the vertical motion of the vehicle are modeled accordingly, and decentralized state-space representations are used for a subsequent nonlinear control design, where flatness-based techniques are employed for simplicity. To ensure steady-state accuracy, integral parts are introduced in the stabilizing feedback laws. The performance of the proposed control structure is investigated by simulation using an identified model of a corresponding experimental vehicle. In addition, also first experimental results are provided.

Stabilization of Nonlinear Control-Affine Systems With Multiple State Constraints

This paper considers the synthesis of stabilizing controllers for nonlinear control-affine systems under multiple state constraints. A new control Lyapunov-barrier function approach is introduced for solving the considered problem. Assuming a classical control Lyapunov function, two possible methods for constructing new control Lyapunov-barrier functions are discussed. Sufficient conditions for the existence of new control Lyapunov-barrier functions are derived. With modifying the Sontag’s formula, an explicit state-constrained stabilizing feedback law is presented. Finally, two numerical examples are provided to illustrate the obtained theoretical results.

Fundamental Limitations of the Decay of Generalized Energy in Controlled (Discrete–Time) Nonlinear Systems Subject to State and Input Constraints

Abstract This paper is devoted to the analysis of fundamental limitations regarding closed-loop control performance of discrete-time nonlinear systems subject to hard constraints (which are nonlinear in state and manipulated input variables). The control performance for the problem of interest is quantified by the decline (decay) of the generalized energy of the controlled system. The paper develops (upper and lower) barriers bounding the decay of the system’s generalized energy, which can be achieved over a set of asymptotically stabilizing feedback laws. The corresponding problem is treated without the loss of generality, resulting in a theoretical framework that provides a solid basis for practical implementations. To enhance understanding, the main results are illustrated in a simple example.

Feedback Control of Nonlinear Hyperbolic PDE Systems Inspired by Traffic Flow Models

This paper investigates and provides results, including feedback control, for a nonlinear, hyperbolic, one-dimensional partial differential equation (PDE) system on a bounded domain. The considered model consists of two first-order PDEs with a dynamic boundary condition on the one end and actuation on the other. It is shown that, for all positive initial conditions, the system admits a globally defined, unique, classical solution that remains positive and bounded for all times; these properties are important, for example for traffic flow models. Moreover, it is shown that global stabilization can be achieved for arbitrary equilibria by means of an explicit boundary feedback law. The stabilizing feedback law depends only on collocated boundary measurements. The efficiency of the proposed boundary feedback law is demonstrated by means of a numerical example of traffic density regulation.

Smooth output feedback stabilization for a class of high-order switched nonlinear systems

In this paper, the problem of global stabilization via smooth output feedback, for a class of high-order switched nonlinear systems with subsystems whose Jacobian linearization neither controllable nor observable, is addressed. By virtue of adding a power integrator approach, a common Lyapunov function is found and state stabilizing feedback laws are designed simultaneously. Reduced-order observers for the switched nonlinear systems are constructed to estimate the unmeasurable states. A machinery is developed to make the observer gains assignment in an iterative way possible. Coupling the state feedback controllers with the developing nonlinear observers, global asymptotical stabilization of the switched nonlinear system under arbitrary switchings is achieved by the output feedback control. An example is employed to verify the efficiency of the proposed method.

Cone-bounded feedback laws for m-dissipative operators on Hilbert spaces

This work studies the influence of some constraints on a stabilizing feedback law. It is considered an abstract nonlinear control system for which we assume that there exists a linear feedback law that makes the origin of the closed-loop system globally asymptotically stable. This controller is then modified via a cone-bounded nonlinearity. A well-posedness and a stability theorems are stated. The first theorem is proved thanks to the Schauder fixed-point theorem, the second one with an infinite-dimensional version of LaSalle's Invariance Principle. These results are illustrated on a linear Korteweg-de Vries equation by some simulations and on a nonlinear heat equation.

Parameter Tuning for Prediction-based Quadcopter Trajectory Planning using Learning Automata

This paper presents a target tracking technique for a quadcopter based on Model Predictive Control (MPC) tuned using machine learning. Specifically, it uses learning automata to select the weighting parameters of the objective function such that they minimize tracking error. It develops an approximate linear state-space model for the quadcopter dynamics by linearizing around a hover condition. The optimum sequence of control actions is expressed as perturbations on a stabilizing feedback law expanded over a finite prediction horizon. Simulation results demonstrate the learned weighting parameters can be used to provide optimized trajectories when implemented as receding horizon MPC. Furthermore, a comparison with previous work demonstrates improved tracking performance.

Planar Inequality Constraints for Stable, Collision-free Model Predictive Control of a Quadcopter

This paper presents a stable, suboptimal, collision-free target tracking technique for a quadcopter based on convex Model Predictive Control. It develops an approximate linear state-space model for the quadcopter dynamics by linearizing around a hover condition. The quadcopter’s path is constrained by a sequence of planes tangent to the surface of obstacles. When implemented in a receding horizon, the orientation of these planes adapt to changes in the environment. A softened terminal constraint is used to improve stability characteristics while avoiding feasibility errors. The sequence of control actions are expressed as perturbations on a stabilizing feedback law expanded over a finite prediction horizon. Simulations demonstrate the technique can be used to avoid spherical obstacles in a target tracking scenario.

Stabilization of nonlinear stochastic systems without unforced dynamics via time-varying feedback

In this paper we give sufficient conditions under which a nonlinear stochastic differential system without unforced dynamics is globally asymptotically stabilizable in probability via time-varying smooth feedback laws. The technique developed to design explicitly the time-varying stabilizers is based on the stochastic Lyapunov technique combined with the strategy used to construct bounded smooth stabilizing feedback laws for passive nonlinear stochastic differential systems. The interest of this work is that the class of stochastic systems considered in this paper contains a lot of systems which cannot be stabilized via time-invariant feedback laws.