Set Invariance Research Articles

Control of discrete‐time periodic linear systems with input saturation via multi‐step periodic invariant sets

SUMMARYThis paper studies local control of discrete‐time periodic linear systems subject to input saturation by using the multi‐step periodic invariant set approach. A multi‐step periodic invariant set refers to a set from which all trajectories will enter a periodic invariant set after finite steps, remain there forever, and eventually converge to the origin as time approaches infinity. The problems of (robust) estimation of the domain of attraction, (robust) local stabilization (with bounded uncertainties), and disturbance rejection are considered. Compared with the conventional periodic invariant set approach, which has been used in the literature for local stability analysis and stabilization of discrete‐time periodic linear systems subject to input saturation, this new invariant set approach is capable of significantly reducing the conservatism by introducing additional auxiliary variables in the set invariance conditions. Moreover, the new approach allows to design (robust) stabilizing periodic controller, in the presence of norm bounded uncertainties, whose period is the same as the open‐loop system and is different from the existing periodic enhancement approach by which the period of the controller is multiple times of the period of the open‐loop system. Several numerical examples are worked out to show the effectiveness of the proposed approach. Copyright © 2013 John Wiley & Sons, Ltd.

On the positive invariance of polyhedral sets in fractional-order linear systems

In this paper, we derive conditions for a given polyhedral set to be a positively invariant set with respect to fractional-order linear time-invariant (FO-LTI) systems with fractional order of 0<α<1. FO-LTI systems are described using Riemann–Liouville operator with initialization response. Then, the conditions are obtained from Farkas’ lemma and the definition of the Mittag-Leffler function. Furthermore, we apply these conditions to the constrained stabilization.

Distributed model predictive control for polytopic uncertain systems subject to actuator saturation

In this paper, we present a distributed model predictive control (MPC) algorithm for polytopic uncertain systems subject to actuator saturation. The global system is decomposed into several subsystems. A set invariance condition for polytopic uncertain system with input saturation is identified and a min–max distributed MPC strategy is proposed. The distributed MPC controller is designed by solving a linear matrix inequalities (LMIs) optimization problem. An iterative algorithm is developed for making coordination among subsystems. Case studies are carried out to illustrate the effectiveness of the proposed algorithm.

Interpretation of Different Approaches to Sensitivity Analysis in Cash Flow Problem

Cash is the driving power of all business and cash-flow statement is one of the major issues of institutions, especially in crisis. Optimal cash-flow plan of a company could be one of the most important indicators of that business's financial health and can be considered as its financial analysts' ability and skill. Linear optimization (LO) is one of the mathematical tools in modeling the cash-flow problem and its rich literature helps analysts to device the optimal one when the situation satisfies the requirements of the LO model. However, the situation is always due to variation and the optimal solution arisen from the LO model have to be analyzed according to measurable variation of input data. Sensitivity analysis and parametric programming is the tool to this analysis. Using the Simplex method to find a basic optimal solution and having multiple optimal solutions is one of the reasons that different solvers lead to different optimal solutions. In these situations, sensitivity analysis may produce confusing results. Moreover, there are different points of views to sensitivity analysis such as optimal basis invariancy, optimal partition invariancy, support set invariancy to name some examples. Here, we briefly review different approaches to sensitivity analysis in LO and a short term cash-flow problem of a dummy institution is modeled as an LO problem. It is shown that the problem has multiple optimal solutions which are degenerate, the situation that usually occurs in practice and causes of ambiguous and unclear results. The confusing results in analyzing the sensitivity of these solutions are highlighted in this example. Then, a strictly complementary optimal solution is provided and its useful interpretation in sensitivity analysis is mentioned in a nutshell. In the sequel, the concept of the results arising in different points of views to sensitivity analysis is analyzed. Keywords: Cashflow Problem; Linear Optimization; Sensitivity Analysis. 2010 Mathematics Subject Classification: 62P05

Reliable piecewise control design for systems with actuator saturation and fault

This paper proposes a reliable piecewise control (RPC) law for input-saturated systems with norm-bounded external disturbances and faults, whose feature is that the following three goals can be simultaneously achieved: improving the disturbance attenuation performance, maximising the region of attraction, and totally removing the influence of the actuator fault. Here, the RPC law is developed to be dependent on several switching sets with the set invariance property, where the size of the most inner switching set is minimised and the size of the outer switching sets is maximised to fulfil the first two goals, respectively. Moreover, for the third goal, this paper introduces a norm-bounded representation method for the actuator fault and incorporates a variable structure control scheme into the RPC law.

Constrained control for linear time-varying systems in the presence of disturbances

The aim of this paper is twofold. In the first part, we revisit the robust invariance for ellipsoidal sets with respect to discrete-time linear time-varying and uncertain systems with bounded additive disturbances. In this context we provide an extension of the invariance condition of the previously published results in the literature. In the second part a novel robust interpolation based control method involving several local unconstrained robust optimal controls is proposed. Among them, one controller plays the role of a performance candidate, while the remaining controllers are used in the interpolation scheme for enlarging the domain of attraction. By using an appropriate objective function, recursive feasibility and a robustly asymptotically stable closed loop behavior are guaranteed.

Sliding Mode Reference Coordination of Constrained Feedback Systems

This paper addresses the problem of coordinating dynamical systems with possibly different dynamics (e.g., linear and nonlinear, different orders, constraints, etc.) to achieve some desired collective behavior under the constraints and capabilities of each system. To this end, we develop a new methodology based on reference conditioning techniques using geometric set invariance and sliding mode control: the sliding mode reference coordination (SMRCoord). The main idea is to coordinate the systems references. Starting from a general framework, we propose two approaches: a local one through direct interactions between the different systems by sharing and conditioning their own references and a global centralized one, where a central node makes decisions using information coming from the systems references. In particular, in this work we focus in implementation on multivariable systems like unmanned aerial vehicles (UAVs) and robustness to external perturbations. To show the applicability of the approach, the problem of coordinating UAVs with input constraints is addressed as a particular case of multivariable reference coordination with both global and local configuration.

H ∞ output-feedback LPV control for systems with input saturation

This paper investigates the problem of designing a dynamic H∞ output-feedback controller under an L∞ performance representing componentwise input constraints. In order to derive a less conservative stabilization condition therein, this paper introduces a Lyapunov function-based polytopic control law and proposes a method capable of applying the information on interpolation parameters appearing in the procedure of representing saturation nonlinearity as convex polytope. Through this paper, the resultant H∞/L∞ problem is efficiently solved based on the set invariance condition formulated in terms of linear matrix inequalities (LMIs).

Stability, positive invariance and design of constrained regulators for networked control systems

In this article, the stability analysis, the positive invariance of polyhedral sets and the design of state-feedback regulators for networked control systems (NCS) with bounded transmission delays, constant and unknown or time-varying, are investigated. The dynamics of the NCS is described by autoregressive-moving-average (ARMA) models. Contrary to former approaches based on quadratic Lyapunov functions, in this article polyhedral Lyapunov functions are used for both stability and positive invariance analysis and state-feedback synthesis. Then, based on the property that the exponential of a matrix can be expressed as a weighted sum of its constituent matrices, it is proven that the problems of determination of stability margins or the design of stabilising controllers can be reduced to linear programming optimisation problems. The use of ARMA models allows the development of methods for the design of state-feedback controllers satisfying state constraints or convergence rate specifications defined on the NCS state space and not on the state of an augmented state space representation.

Robust fault-tolerant control against time-varying actuator faults and saturation

A set invariance condition for a piecewise continuous time-varying system is developed and applied to the fault-tolerant control problem against the actuator effectiveness loss for linear systems subject to system uncertainties and actuator saturation in this study. Linear matrix inequality conditions are proposed to compute invariant ellipsoidal sets. Discussions on system performance optimisation are given including reduction on computational complexity, maximisation of the domain of attraction and disturbance rejection. The proposed design techniques are applied to an unstable multi-input and multi-output system and to the flight control of a non-linear unmanned aerial vehicle model subject to the control effectiveness loss fault in the elevator.

UAV reference conditioning for formation control via set invariance and sliding modes

A novel methodology is proposed for formation control of UAVs. The scheme is based on the sliding mode reference conditioning technique in a sort of supervisory level. The main idea is to shape the UAVs references in order to keep them coordinated. This implies a virtual leader which has the information of the formation structure, position of the agents in the formation and bounds on the distance between its reference and the ones of the agents. Geometric set invariance techniques together with sliding mode control is used for this purpose in order to make some set, defined from the reference constraints, invariant. To show the applicability of the approach, the problem of coordination and formation a number UAVs in cartesian motion is illustrated through simulation results using non-linear identified models of the UAVs.

Cyclic invariance for discrete time-delay systems

This technical communique introduces a new concept of set invariance with respect to linear discrete time dynamics affected by delay. We are interested in the definition and characterization of sequences of cyclically invariant subsets in the state space. The algebraic conditions established in the late ’80s for linear dynamics are generalized to invariance analysis in the presence of delays for given sequences of polyhedral sets.

Robust H ∞ static output feedback stabilization of T-S fuzzy systems subject to actuator saturation

This paper proposes a method for designing robust H∞ static output feedback stabilization of Takagi-Sugeno (T-S) fuzzy systems under actuator saturation. In this paper, the input saturation is represented by a polytopic model and the modeling error is assumed a norm-bounded uncertainty. A set invariance condition for robust H∞ static output feedback system under actuator saturation is first established. Then, the estimation of the largest domain of attraction for the system is formulated and solved as a Linear Matrix Inequality (LMI) optimization problem. Two examples are used to demonstrate the effectiveness of the proposed design method.

Further results on the construction of invariant families of sets for linear systems with delay

Almost all synthesis methods for the construction of invariant sets for delay difference equations (DDEs) suffer either from computational complexity, i.e., those based on the Krasovskii approach, or come with considerable conservatism, i.e., those based on the Razumikhin approach. Only recently, the notion of an invariant family of sets was introduced in order to allow for a suitable tradeoff between conceptual generality and computational simplicity. In this paper, the construction of such families is further elaborated. In particular, a parametrization of families of sets is proposed that leads to tractable synthesis algorithms while preserving several advantages over the Razumikhin approach. For polyhedral sets, the corresponding synthesis algorithms can be implemented via an algebraic recursion and parametric linear programming. Thus, the invariant family of sets becomes a computationally tractable set invariance notion for DDEs, the advantages of which are illustrated via several examples.

Model Predictive Control of Impulsive Dynamical Systems

We propose an MPC framework to address impulsive dynamical systems, i.e. systems in which the state trajectories are permitted to have discontinuities (jumps) in response to impulsive controls. We describe how previous results on sampled-data trajectories, invariance of sets, optimal control, and stability in the context of impulsive systems can be used and adapted to a sampled-data MPC framework. We provide sufficient conditions on the design parameters of the optimal control problems that guarantee: (i) feasibility of the sequence of optimal control problems, and (ii) stability of the controlled system.

Convexity Invariance of Fuzzy Sets under the Extension Principles

We discuss the convexity invariance of fuzzy sets under the extension principles. Particularly, we give a necessary and sufficient condition for a mapping to be an inverse *-convex transformation, and also obtain some sufficient conditions for a mapping to be an *-convex transformation. Two applications are given to illustrate the obtained results. Finally, we give some applications of the main results to the hyperstructure convexity invariance of type 2 fuzzy sets under hyperalgebra operations, and to the convexity invariance of fuzzy numbers under basic arithmetic operations.

A study on the state set invariance for a class of discrete-time linear control systems

This paper considers the state set invariance for a class of linear control systems over the discrete time horizon. Given a state set and a control set, the author proposes the methodology to design a state feedback controller and its corresponding maximum invariant subset of the state set. The corresponding invariant subset means that with the controller any state starting from the subset remains in the subset at all times. Fixing the controller, the necessary and sufficient condition for the maximum invariant subset to be the state set itself is first derived. For the systems not satisfying the condition, the author characterizes the corresponding maximum invariant subset based on the spectral analysis of matrices. Numerical examples for different cases are given to illustrate the proposed computation techniques.

Invariance properties of interval dynamical systems

This article analyses the flow (positive) invariance of different types of time-dependent sets with respect to the dynamics of interval systems (state-space models described by interval matrices) in both discrete and continuous time. The investigated sets have general shapes, defined in terms of Hőlder vector p-norms, . The time dependence of the sets is separately considered arbitrary and exponentially decreasing. We develop procedures that explore set invariance by using (i) a single test matrix, introduced as a majorant of the interval matrix and (ii) all the distinct vertices of the matrix polytope corresponding to the interval matrix. In case (i) we provide sufficient conditions for all p, , whereas the necessity is guaranteed only for ; in case (ii) we formulate necessary and sufficient conditions valid for all p, . Our analysis shows that the existence of invariant sets (with arbitrary or exponential time-dependence, defined by a p-norm) approaching the state-space origin for infinite time horizon implies the asymptotic stability. Some examples are presented to provide intuitive support for the theoretical concepts and to illustrate the applicability of the main results.

An [formula omitted]th-step set invariance approach to the analysis and design of discrete-time linear systems subject to actuator saturation

For a discrete-time linear system subject to actuator saturation, an analysis and design approach based on N th-step set invariance is proposed. An N th-step invariant set is a set from which all trajectories will enter a contractively invariant set in N steps and thus can be used as an estimate of the domain of attraction. Such an estimate is less conservative than an estimate obtained on the basis of contractive invariance of ellipsoids, which excludes initial conditions from which the trajectories “overshoot” outside the ellipsoid before converging to the origin. In comparison with the existing approach based on conventional set invariance, the proposed approach results in a larger domain of attraction and better performance, both in the analysis of the closed-loop system under a given feedback gain and in the design of feedback gain. In comparison with the saturated and nonsaturated (SNS) invariant set approach, which is an analysis approach, the proposed approach can be used for control design even in the presence of plant uncertainties.

Positive invariance, monotonicity and comparison of nonlinear systems

In this paper, the interrelation between positive invariance, monotonicity and comparison of iterated nonlinear systems defined in partially ordered sets is studied. First, necessary and sufficient conditions guaranteeing the positive invariance of sets defined by relations of the form v ( x ) ≤ w with respect to nonlinear systems are established. Then, various characterizations of monotone nonlinear systems are developed and a connection between positive invariance and monotonicity is established. Finally, necessary and sufficient conditions for a, not necessarily monotone, nonlinear system to be a comparison system are established. No specific algebraic structure is necessary for developing the main results of this work. Numerical examples illustrating the applicability of all these results to the case of discrete-time dynamical systems are also given.