Ellipsoidal Invariant Sets Research Articles

Capability Exploration of Extended-State Observer-Based Control Under the Uncertain Case of Disturbance and Actuator Saturation

Considering a class of nonlinear systems subject to actuator saturation and uncertain control input, this article explored the control capability of an extended-state observer-based active disturbance rejection controller in the presence of disturbances and uncertainties. First, the actuator saturation is handled with the convex hull in the extended-state observer-based control scheme, and an ellipsoid invariant set inside the domain of attraction for control systems is obtained using the Lyapunov method. Second, the enlargement problem of the invariant set is also studied, which illustrates that the convex hull of the ellipsoid invariant sets is also an enlargement invariant set. Finally, comprehensive comparisons with existing methods on a brushless dc motor demonstrate the stronger capability of the designed controller on disturbance rejection and actuator saturation. The results show that the disturbance offset and performance index (integrated time and absolute error and integral absolute error) are about half of the classical proportional–integral–derivative(PID) and double-integral sliding-mode control, or the disturbance rejection capability is about twice in this case.

Discrete gain scheduling control approach to elliptical orbit rendezvous system with actuator saturation

AbstractThis paper studies the discrete gain scheduling control design problem of elliptical orbit spacecraft rendezvous system with actuator saturation. Due to the presence of actuator saturation, the dynamic performance of the spacecraft rendezvous system degrades significantly. In order to improve the dynamic performance of the system, a discrete gain scheduling control approach is adopted to construct a group of time‐invariant ellipsoidal invariant sets, which can be used to determine the switching points of the discrete gain scheduling control. By choosing some discrete parameter values, the discrete gain scheduling control is obtained from a solution of a periodic Riccati matrix differential equation. Under the control obtained, the dynamic performance of the system is much improved while accomplishing successfully the rendezvous mission of the spacecraft. Finally, a practical example is provided to show the effectiveness of the proposed control design approach.

An offline predictive control with ellipsoid invariant set for time-variant system

An offline predictive control with ellipsoid invariant set for time-variant system

An offline predictive control with ellipsoid invariant set for time-variant system

An offline predictive control with ellipsoid invariant set for time-variant system

Distributed Safe Learning using an Invariance-based Safety Framework

In large-scale networks of uncertain dynamical systems, where communication is limited and there is a strong interaction among subsystems, learning local models and control policies offers great potential for designing high-performance controllers. At the same time, the lack of safety guarantees, here considered in the form of constraint satisfaction, prevents the use of data-driven techniques to safety-critical distributed systems. This paper presents a safety framework that guarantees constraint satisfaction for uncertain distributed systems while learning. The framework considers linear systems with coupling in the dynamics and subject to bounded parametric uncertainty, and makes use of robust invariance to guarantee safety. In particular, a robust non-convex invariant set, given by the union of multiple ellipsoidal invariant sets, and a nonlinear backup control law, given by the combination of multiple stabilizing linear feedbacks, are computed offline. In presence of unsafe inputs, the safety framework applies the backup control law, preventing the system to violate the constraints. As the robust invariant set and the backup stabilizing controller are computed offline, the online operations reduce to simple function evaluations, which enables the use of the proposed framework on systems with limited computational resources. The capabilities of the safety framework are illustrated by three numerical examples.

Constrained Uncertain System Stabilization with Enlargement of Invariant Sets

An enhanced method able to perform accurate stability of constrained uncertain systems is presented. The main objective of this method is to compute a sequence of feedback control laws which stabilizes the closed-loop system. The proposed approach is based on robust model predictive control (RMPC) and enhanced maximized sets algorithm (EMSA), which are applied to improve the performance of the closed-loop system and achieve less conservative results. In fact, the proposed approach is split into two parts. The first is a method of enhanced maximized ellipsoidal invariant sets (EMES) based on a semidefinite programming problem. The second is an enhanced maximized polyhedral set (EMPS) which consists of appending new vertices to their convex hull to minimize the distance between each new vertex and the polyhedral set vertices to ensure state constraints. Simulation results on two examples, an uncertain nonisothermal CSTR and an angular positioning system, demonstrate the effectiveness of the proposed methodology when compared to other works related to a similar subject. According to the performance evaluation, we recorded higher feedback gain provided by smallest maximized invariant sets compared to recently studied methods, which shows the best region of stability. Therefore, the proposed algorithm can achieve less conservative results.

Interpolation based predictive control by ellipsoidal invariant sets

A new interpolating control scheme for uncertain linear systems is developed. The interpolation is done between two saturated control laws, each given with its associated invariant set. A linear search is performed to compute a sub-optimal interpolation coefficient. Two algorithms are suggested — each incorporating a different optimization objective. The method guarantees robust stability and recursive feasibility, also in presence of persistent disturbances.

Stability Guarantees for Translational Thrust-Propelled Vehicles Dynamics Through NMPC Designs

We propose two different nonlinear model predictive control (NMPC) schemes (with and without terminal cost and constraints) for stabilizing the translational dynamics of thrust-propelled vehicles. Both approaches make use of an elaborated nonlinear feedback linearization controller and its associated ellipsoidal invariant set under restrictive input constraints, hence guaranteeing the closed-loop asymptotic stability. The terminal constraint set of the corresponding NMPC design is easy to tune due to its clear formulation expressed directly in terms of the tuning variables, while for the NMPC scheme without terminal constraint, the design allows to stabilize the system with a significantly shorter prediction horizon in comparison with the existing method in the literature. Simulation and experimental tests over a nano-drone platform validate the proposed approaches.

Shifting H∞ Linear Parameter Varying State-Feedback Controllers Subject to Time-Varying Input Saturations

This paper establishes a methodology based on linear matrix inequalities (LMIs) to design a shifting H∞ linear parameter varying (LPV) state-feedback controller for systems affected by time-varying input saturations. By means of the shifting paradigm, the instantaneous saturation values are linked to a scheduling parameter vector. Then, the disturbance rejection is dealt with the quadratic boundedness concept and the shifting H∞ methodology. The design conditions are obtained within the LPV framework using ellipsoidal invariant sets, thus obtaining an LMI-based feasibility problem that can be solved via available solvers. Finally, the main characteristics of the proposed approach are validated by means of an illustrative example.

Stabilization of Switched Linear Systems Subject to Actuator Saturation via Invariant Semiellipsoids

This article presents new stabilization conditions for the state feedback control of switched linear systems subject to actuator saturation. Based on the semiellipsoidal approach and the multiple Lyapunov functions method, state feedback controllers and a state-dependent switching law are designed. The presented stabilization conditions are less conservative than those obtained by the ellipsoidal method and simpler than those established by the polyhedral method. Moreover, the union of all positively invariant semiellipsoids is constructed to approximate the region of attraction. Each semiellipsoidal set is the intersection of a positively invariant ellipsoidal set and a constraint set. Constructive solutions of the presented conditions are obtained based on two different techniques: the sum-of-squares technique and the nonlinear matrix inequality technique. Two illustrative examples are given to demonstrate the effectiveness of the main results.

Estimation of the Attraction Domain for an Affine System with Constrained Vector Control Closed by the Linearizing Feedback

Nonlinear affine systems with constrained vector control that are represented in a canonical (normal) form and are closed by feedbacks linearizing the system in a neighborhood of the origin, are considered. For the nonlinear closed-loop system, the problem is set to construct an estimate of the attraction domain of an equilibrium position. A method for constructing an estimate of the attraction domain, which is based on results of absolute stability theory, is suggested. The estimate is sought as a Cartesian product of positive invariant sets of the subsystems composing the system. In the case of ellipsoidal invariant sets, construction of the estimate reduces to solving a system of linear matrix inequalities. The discussion is illustrated by numerical examples.

A Polytopic Invariant Set Based Iterative Learning Model Predictive Control

Model predictive control (MPC) is often combined with iterative learning control (ILC), which results in the so-called iterative learning model predictive control (ILMPC), to control batch processes with constraints. It is a long standing and challenging problem that how to simultaneously guarantee system stability and constraint satisfaction in ILMPC design. Several invariant set-based methods, such as the zero-terminal state and the ellipsoidal invariant set, have been proposed to solve this problem. However, these methods are often restrictive with conservative control performance and limited applicability. In this paper, we propose a polytopic invariant set based ILMPC method to reduce conservativeness. Specifically, a polytopic invariant set is designed based on geometric computation and proved to be convex and compact. An iterative algorithm is proposed to compute the maximal one. Numerical simulations are provided to demonstrate its effectiveness.

State space realizations robust to overloading for discrete-time LTI systems

Abstract In digital signal processing, overloading of a value occurs if the amplitude of a signal exceeds a predetermined limit. The overloading should be avoided, since it may cause a catastrophic effect on the system. To mitigate the overloading, the l2 norm scaling has been utilized for many years. However, the l2 norm scaling does not always work well, since it is not directly related to the amplitude of the signal. This paper presents a design method of state space realizations for digital discrete-time LTI systems that are robust to overloading. Upper bounds of the amplitudes of state variables are evaluated by using the invariant set theory. When different numbers of bits are allocated to represent the values of state variables, our realization is given by minimizing the volume of an ellipsoidal invariant set. On the other hand, when an equal number of bits is assigned to each state variable, an upper bound on the magnitude of the state variables is minimized and the optimal realization is obtained by the spherical invariant set. Numerical examples are provided to show that our realizations are robust to overloading.

Interpolation Model Predictive Control of Nonlinear Systems Described by Quasi-LPV Model

This paper investigates the interpolation model predictive control (MPC) algorithm for nonlinear discrete-time systems, which can be represented by affine linear parameter varying (LPV) model. The general nonlinear model is transformed into the quasi-LPV model, then the equivalent polytopic LPV model and disturbed Linear time-invariant (LTI) model are obtained. Therefore, a finite-horizon interpolation MPC algorithm based ellipsoidal invariant set (EIS) is proposed. For comparison, the existing zero-horizon interpolation MPC algorithm, based on EIS, is also described to display the advantages of proposed algorithm. By virtue of the finite-horizon technique, the feasible region of proposed algorithm is much larger than zero-horizon interpolation MPC algorithm. An illustrative example is given to verify the effectiveness of proposed algorithms.

Offline Robust Model Predictive Control for Lipschitz Non‐Linear Systems Using Polyhedral Invariant Sets

AbstractIn this paper the concept of maximal admissible set (MAS) for linear systems with polytopic uncertainty is extended to non‐linear systems composed of a linear constant part followed by a non‐linear term. We characterize the maximal admissible set for the non‐linear system with unstructured uncertainty in the form of polyhedral invariant sets. A computationally efficient state‐feedback RMPC law is derived off‐line for Lipschitz non‐linear systems. The state‐feedback control law is calculated by solving a convex optimization problem within the framework of linear matrix inequalities (LMIs), which leads to guaranteeing closed‐loop robust stability. Most of the computational burdens are moved off‐line. A linear optimization problem is performed to characterize the maximal admissible set, and it is shown that an ellipsoidal invariant set is only an approximation of the true stabilizable region. This method not only remarkably extends the size of the admissible set of initial conditions but also greatly reduces the on‐line computational time. The usefulness and effectiveness of the method proposed here is verified via two simulation examples.

Gain scheduling consensus of multi-agent systems subject to actuator saturation

ABSTRACT This paper presents a gain scheduling approach for achieving the consensus tracking of multi-agent systems with actuator saturation. We first construct a series of nesting ellipsoid invariant sets associated with consensus errors. When the consensus errors stay between the two ellipsoid invariant sets, the feedback gains keep constant, but when the consensus errors enter into the smaller ellipsoid invariant set, the feedback gains abruptly become larger. By combining this gain scheduling technique and the parametric Lyapunov equations, we, respectively, design state and output feedback gain scheduling protocols. Their main advantage, in comparison with the fixed case, is that the convergence rate of consensus tracking can be enhanced by scheduling the gain parameters. Numerical simulations verify the effectiveness of theoretical analysis.

An invariant-set approach for constraining gas exchange dynamics during cycling

This paper proposes a method for finding the maximal mechanical power, that can be developed for a cyclist without excess of carbon dioxide production. The method is based on a recently proposed discrete-time linear parameter-varying model of gas exchange dynamics. In addition, the problem is solved by using an ellipsoidal invariant set which characterizes all possible trajectories of the system state in presence of a bounded pedal power. The latter allows us to tackle the problem as a geometrical optimization problem involving an inequality constraint. The numerical example concerns two scenarios, which cover moderate and high intensity exercises.

Reference tracking controller design for anesthesia

Abstract This paper addresses the problem of tracking a constant BIS reference during anesthesia taking into account control input saturation, multiple time scale dynamics and inter-patient variability. LMIs conditions are proposed to compute a state feedback involving an integral action to ensure a perfect output tracking. These conditions guarantee that the trajectories of the closed-loop system remain in an invariant ellipsoidal set. The theoretical conditions are numerically illustrated on a set of adult patients as a proof of principle.

Model predictive control synthesis algorithm based on polytopic terminal region for Hammerstein-Wiener nonlinear systems

An improved model predictive control algorithm is proposed for Hammerstein-Wiener nonlinear systems. The proposed synthesis algorithm contains two parts: offline design the polytopic invariant sets, and online solve the min-max optimization problem. The polytopic invariant set is adopted to replace the traditional ellipsoid invariant set. And the parameter-correlation nonlinear control law is designed to replace the traditional linear control law. Consequently, the terminal region is enlarged and the control effect is improved. Simulation and experiment are used to verify the validity of the wind tunnel flow field control algorithm.

Set-membership observer design based on ellipsoidal invariant sets

This paper presents a new set-membership observer design method for linear discrete-time systems. The real process is assumed to be perturbed by unknown but bounded disturbances. The set-membership observer provides a deterministic state interval that is build as the sum of the estimated system states and its corresponding estimation errors bounds. The proposed approach is based on the solutions of a few number of Linear Matrix Inequalities that are suitable modified to provide both the observer parameters and ellipsoidal Robustly Positive Invariant sets. The latter are used to frame the estimation error in a very simple and accurate way. The enhanced precision on the computation of the estimation error bounds has been possible thanks to the use of the a posteriori calculated covariance matrix that allows, in a second time, a more suitable description of the dissipation equation used in the Bounded-real lemma formulation. A numerical example illustrates the behavior of such observer and discuss its easy implementation.