Large Domain Of Attraction Research Articles

A competition system with nonlinear cross-diffusion: exact periodic patterns

Our concern in this paper is to shed some additional light on the mechanism and the effect caused by the so called cross-diffusion. We consider a two-species reaction–diffusion (RD) system. Both “fluxes” contain the gradients of both unknown solutions. We show that–for some parameter range– there exist two different type of periodic stationary solutions. Using them, we are able to divide into parts the (eight-dimensional) parameter space and indicate the so called Turing domains where our solutions exist. The boundaries of these domains, in analogy with “bifurcation point”, called “bifurcation surfaces”. As it is commonly believed, these solutions are limits as t goes to infinity of the solutions of corresponding evolution system. In a forthcoming paper we shall give a detailed account about our numerical results concerning different kind of stability. Here we also show some numerical calculations making plausible that our solutions are in fact attractors with a large domain of attraction in the space of initial functions.

PID Control of Planar Nonlinear Uncertain Systems in the Presence of Actuator Saturation

This paper investigates PID control design for a class of planar nonlinear uncertain systems in the presence of actuator saturation. Based on the bounds on the growth rates of the nonlinear uncertain function in the system model, the system is placed in a linear differential inclusion. Each vertex system of the linear differential inclusion is a linear system subject to actuator saturation. By placing the saturated PID control into a convex hull formed by the PID controller and an auxiliary linear feedback law, we establish conditions under which an ellipsoid is contractively invariant and hence is an estimate of the domain of attraction of the equilibrium point of the closed-loop system. The equilibrium point corresponds to the desired set point for the system output. Thus, the location of the equilibrium point and the size of the domain of attraction determine, respectively, the set point that the output can achieve and the range of initial conditions from which this set point can be reached. Based on these conditions, the feasible set points can be determined and the design of the PID control law that stabilizes the nonlinear uncertain system at a feasible set point with a large domain of attraction can then be formulated and solved as a constrained optimization problem with constraints in the form of linear matrix inequalities (LMIs). Application of the proposed design to a magnetic suspension system illustrates the design process and the performance of the resulting PID control law.

Componentwise Hölder Inference for Robust Learning-Based MPC

This article presents a novel learning method based on componentwise Hölder continuity, which allows one to consider independently the contribution of each input to each output of the function to be learned. The method provides a bounded prediction error, and its learning property is proven. It can be used to obtain a predictor for a nonlinear robust learning-based predictive controller for constrained systems. The resulting controller achieves better closed loop performance and larger domains of attraction than learning methods that only consider nonlinear set membership, as illustrated by a case study.

Nonlinear state estimation for inertial navigation systems with intermittent measurements

This paper considers the problem of simultaneous estimation of the attitude, position and linear velocity for vehicles navigating in a three-dimensional space. We propose two types of hybrid nonlinear observers using continuous angular velocity and linear acceleration measurements as well as intermittent landmark position measurements. The first type relies on a fixed-gain design approach based on an infinite-dimensional optimization, while the second one relies on a variable-gain design approach based on a continuous–discrete Riccati equation. For each case, we provide two different observers with and without the estimation of the gravity vector. The proposed observers are shown to be exponentially stable with a large domain of attraction. Simulation and experimental results are presented to illustrate the performance of the proposed observers.

Robust observer-based control for discrete-time semi-Markov jump systems with actuator saturation

<p style='text-indent:20px;'>This paper investigates the control synthesis for discrete-time semi-Markov jump systems with nonlinear input. Observer-based controllers are designed in this paper to achieve a better performance and robustness. The nonlinear input caused by actuator saturation is considered as a group of linear controllers in the convex hull. Moreover, the elapse time and mode dependent Lyapunov functions are investigated and sufficient conditions are derived to guarantee the <inline-formula><tex-math id="M1">\begin{document}$ H_\infty $\end{document}</tex-math></inline-formula> performance index. The largest domain of attraction is estimated as the existing saturation in the system. Finally, a numerical example is utilized to verify the effectiveness and feasibility of the developed strategy.</p>

Second-Order Sliding Mode Controller Design with a Larger Domain of Attraction

In this paper, by combining a second-order sliding mode (SOSM) algorithm with the saturation technique, a novel SOSM control scheme has been presented. The feature of the proposed SOSM controller lies that there is a saturation function imposed on the sliding variable, which could significantly enlarge the domain of attraction for the closed-loop system. The geometric method has been utilized to prove that all the sliding variables will be steered to the origin in a finite time. Meanwhile, the relation between the control parameters and the shape of the phase trajectory has also been discussed. Finally, the proposed method has been applied to the tracking control problem for a robotic manipulator.

Heterogeneously parameterized tube model predictive control for LPV systems

This paper presents a heterogeneously parameterized tube-based model predictive control (MPC) design applicable to linear parameter-varying (LPV) systems. In a heterogeneous tube, the parameterizations of the tube cross sections and the associated control laws are allowed to vary along the prediction horizon. Two extreme cases that can be described in this framework are scenario MPC (high complexity, larger domain of attraction) and homothetic tube MPC with a simple time-invariant control parameterization (low complexity, smaller domain of attraction). In the proposed framework, these extreme parameterizations, as well as other parameterizations of intermediate complexity, can be combined within a single tube. By allowing for more flexibility in the parameterization design, one can influence the trade-off between computational cost and the size of the domain of attraction. Sufficient conditions on the parameterization structure are developed under which recursive feasibility and closed-loop stability are guaranteed. A specific parameterization that combines the principles of scenario and homothetic tube MPC is proposed and it is shown to satisfy the required conditions. The properties of the approach, including its capability of achieving improved complexity/performance trade-offs, are demonstrated using two numerical examples.

Ellipsoidal Lyapunov-based hybrid model predictive control for mixed logical dynamical systems with a recursive feasibility guarantee

This paper proposes an ellipsoidal hybrid model predictive control approach to solve the robust stability problem of uncertain hybrid dynamical systems modelled by the mixed logical dynamical framework. In this approach, the traditional terminal equality constraint is replaced by an ellipsoid that results in a maximal positive invariant set for the closed-loop system. Then, a Lyapunov decreasing condition along with the robustness criterion is introduced to the optimization problem to achieve the robust stability of the closed-loop system. As the main advantages, the ellipsoidal terminal set proposed in this paper attains a larger domain of attraction along with the recursive feasibility guarantee. Moreover, the stability and robustness constraints are achieved by a lower prediction horizon, which leads to a smaller dimension optimization problem. In addition, to reduce the computational complexity of the corresponding optimization problem, a suboptimal version of the proposed algorithm is introduced. Finally, numerical and car suspension system examples show the capabilities of the proposed method.

Improving the Region of Attraction of a Non-Hyperbolic Point in Slow-Fast Systems With One Fast Direction

Through recent research combining the geometric desingularization or blow-up method and classical control tools, it has been possible to locally stabilize non-hyperbolic points of singularly perturbed control systems. In this letter we propose a simple method to enlarge the region of attraction of a non-hyperbolic point in the aforementioned setting by expanding the geometric analysis around the singularity. In this way, we can synthesize improved controllers that stabilize non-hyperbolic points within a large domain of attraction. Our theoretical results are showcased in a couple of numerical examples.

A linear programming approach for stabilization of positive Markovian jump systems with a saturated single input

This paper proposes a linear programming approach for stabilization of positive Markovian jump systems (PMJSs) with a saturated single input. The proposed approach first derives a sufficient condition for stabilization of PMJSs with input saturation based on the linear co-positive Lyapunov function. By introducing an intermediate scalar whose absolute value is less than the absolute value of product of nonnegative vector of the linear co-positive Lyapunov function and input matrix and constructing a special form of the controller gains, this approach obtains a modified condition applicable for the linear programming. Finally, four numerical examples show that the proposed approach gives the larger domain of attraction than the existing approach based on the quadratic Lyapunov function.

Enlarging the basin of attraction by a uniting output feedback controller

We consider a system for which two predesigned stabilizing output feedback controllers with bounded domains of attraction are known. One renders the system asymptotically stable with some desired performance, and the other provides ultimate boundedness with larger domain of attraction. Assuming that two subsets of the domains of attraction are known, one larger than the other, this work states the problem of combining both controllers with the goal of guaranteeing asymptotic stability properties in the largest subset while the desired performance is locally achieved. We design a switching logic between the controllers that solves the problem, based on the existence of a local tunable observer. The resulting control law is defined by a hybrid output feedback controller. The effectiveness of the proposed solution is illustrated by a numerical example.

Output feedback control design to enlarge the domain of attraction of a supercavitating vehicle subject to actuator saturation

To enlarge the domain of attraction of a supercavitating vehicle subject to actuator saturation, this paper presents a new output feedback control design in consideration of the immeasurable vertical speed. The dive-plane dynamics of a supercavitating vehicle are considered. By introducing the sector condition of the planing force, a new output feedback control law that locally stabilizes the closed-loop system is proposed. The design of the controller that maximizes the vehicle’s domain of attraction is then formulated and solved as an optimization problem with linear matrix inequality (LMI) constraints. Simulations are conducted for systems under saturation-oriented and non-saturation-oriented controllers. The results show that the proposed design can achieve a much larger domain of attraction than do conventional, non-saturation-oriented approaches.

An Inertial-Aided Homography-Based Visual Servo Control Approach for (Almost) Fully Actuated Autonomous Underwater Vehicles

A nonlinear inertial-aided image-based visual servo control approach for the stabilization of (almost) fully actuated autonomous underwater vehicles (AUVs) is proposed. It makes use of the homography matrix between two images of a planar scene as feedback information while the system dynamics are exploited in a cascade manner in a control design: An outer-loop control defines a reference setpoint based on the homography matrix and an inner-loop control ensures the stabilization of the setpoint by assigning the thrust and torque controls. Unlike conventional solutions that only consider the system kinematics, the proposed control scheme is novel in considering the full system dynamics (incorporating all degrees of freedom, nonlinearities, and couplings, as well as interactions with the surrounding fluid) and in not requiring information of the relative depth and normal vector of the observed scene. Augmented with integral corrections, the proposed controller is robust with respect to model uncertainties and disturbances. The almost global asymptotic stability of the closed-loop system is demonstrated, which is the largest domain of attraction one can achieve by means of continuous feedback control. Simulation results illustrating these properties on a realistic AUV model subjected to a sea current are presented and finally experimental results on a real AUV are reported.

Regional robust stabilisation and domain-of-attraction estimation for MIMO uncertain nonlinear systems with input saturation

ABSTRACTThe regional robust stabilisation by means of linear time-invariant state feedback control for a class of uncertain MIMO nonlinear systems with parametric uncertainties and control input saturation is investigated. The nonlinear systems are described in a differential algebraic representation and the regional stability is handled considering the largest ellipsoidal domain-of-attraction (DOA) inside a given polytopic region in the state space. A novel set of sufficient Linear Matrix Inequality (LMI) conditions with new auxiliary decision variables are developed aiming to design less conservative linear state feedback controllers with corresponding larger DOAs, by considering the polytopic description of the saturated inputs. A few examples are presented showing favourable comparisons with recently published similar control design methodologies.

Stabilizing model predictive control using parameter-dependent dynamic policy for nonlinear systems modeled with neural networks

A class of parameter-dependent dynamic control policies is explored for its use in a model predictive control (MPC) algorithm for a nonlinear system modeled with a feedforward neural network (NN). The NN-modeled system is expressed as a polytopic quasi-linear-parameter-varying (quasi-LPV) system over a region of the state-input space for a range of operating points, and the dynamics of the proposed policy, which are optimized off-line to enlarge the region of attraction, are allowed to depend on a time-varying parameter of the polytopic quasi-LPV system model such that the resulting control involves a continuous gain-scheduling that leads to reduced conservativeness. A complete MPC algorithm using the dynamic policy as the terminal policy ensures stabilization and improved control performance over a larger domain of attraction without a larger horizon length. Simulation examples with tank and tubular reactor systems illustrate the effective performance of the proposed approach in practical applications.

Robust State Feedback Stabilization of Uncertain Discrete-Time Switched Linear Systems Subject to Actuator Saturation

The robust stabilization problem is investigated for a class of discrete-time switched linear systems with time-varying norm-bounded uncertainties and saturating actuators by using the multiple Lyapunov functions method. A switching law and a state feedback law are designed to asymptotically stabilize the system with a large domain of attraction. Based on the multiple Lyapunov functions method, sufficient conditions are obtained for robust stabilization. Furthermore, when some parameters are given in advance, the state feedback controllers and the estimation of domain of attraction are presented by solving a convex optimization problem subject to a set of linear matrix inequalities (LMI) constraints. A numerical example is given to show the effectiveness of the proposed technique.

Robust Stability and Stabilization of Positive Interval Systems Subject to Actuator Saturation

AbstractThis paper addresses the stabilization problem for a class of uncertain positive linear systems (PLSs) in the presence of saturating actuators. The objective is to obtain sufficient conditions for the robust stability of PLSs and to design robust state feedback control laws such that the closed‐loop uncertain system is asymptotically stable and positive at the origin with a large domain of attraction. Several sufficient conditions for robust stabilization and positivity are derived via the Lyapunov function approach and convex analysis method for both the discrete‐time and the continuous‐time cases, respectively. The state feedback controller design and the estimation of the domain of attraction are presented by solving a convex optimization problem with linear matrix inequalities (LMIs) constraints. A numerical example is given to show the effectiveness of the proposed methods.

Dynamic anti-windup design for anticipatory activation: enlargement of the domain of attraction

Traditional anti-windup compensators are designed for activation immediately at the occurrence of actuator saturation. Recently, anti-windup compensators designed for actuation either after the saturation has reached a certain level or in anticipation of its occurrence. In the case of static anti-windup compensators, it has been shown that an anti-windup compensator designed for activation in anticipation of actuator saturation would lead to better performance than those designed for immediate or delayed activation could, both in terms of transient performance and the size of the domain of attraction. More recently, it has been shown that a dynamic anti-windup compensator designed for anticipatory activation would also result in better transient performance than those designed for immediate or delayed activation could. In this paper, we design dynamic anti-windup compensators for the enlargement of the domain of attraction. These compensators are designed respectively for immediate, delayed and anticipatory activation. We will show by simulation that a dynamic anti-windup compensator designed for anticipatory activation would result in a larger domain of attraction than a dynamic anti-windup compensator designed for immediate or delayed activation could.

Balancing a Legged Robot Using State-Dependent Riccati Equation Control

In this paper, we propose a nonlinear control approach for balancing underactuated legged robots. For the balancing task, the robot is modeled as a generalized version of a Segway. The control design is based on the State-Dependent Riccati Equation (SDRE) approach. The domain of attraction of the SDRE controller is compared to the domain of attraction of a linear quadratic controller. Using a simulation example of a four-legged robot balancing on its hind legs, we show that the SDRE controller gives a reasonably large domain of attraction, even with realistic level constraints on the control input, while the linear quadratic controller is unable to stabilize the system.

On improving the convergence speed of linear continuous-time systems under constrained control: Sylvester equation approach

A controller design methodology is proposed for linear continuous-time systems with asymmetrical constrained control. Because of the trade-off between the size of the domain of attraction and convergence speed, one can improve the convergence speed during the system evolution. The closed-loop system starts with bad performances and a large domain of attraction and ends with good performances near the origin. The objective of this work is to overcome some discontinuities resulting from a piecewise linear control law previously proposed to improve the convergence speed of a linear system during its evolution. The proposed method is based on the use of the Sylvester equation together with a quadratic Lyapunov function. It guaranties the improvement of the speed of convergence without any discontinuities on the control vector components and respecting always the imposed asymmetric constraints without saturations.