Positive Continuous-time Linear Systems Research Articles

On controllability of fractional positive continuous-time linear systems with delay

On controllability of fractional positive continuous-time linear systems with delay

Non-overshoot tracking control for positive continuous-time linear systems

This article focuses on state and output tracking control for positive continuous-time linear systems without overshoot. Linear state feedback controllers are designed respectively to ensure both the system state, and output can unilateral track the reference state and output. In addition, using Lyapunov direct method, the exponential convergence rate of tracking system is calculated. A numerical example is given to demonstrate the effectiveness of the proposed method.

Robust [formula omitted] and [formula omitted] control for positive continuous-time uncertain linear systems

This paper addresses the problems of H2 and H∞ control design for positive continuous-time linear systems with uncertain parameters belonging to a polytope. Although it is well known that the optimal H∞ state-feedback gain can be obtained from the bounded real lemma condition with a diagonal Lyapunov matrix without any conservatism for precisely known positive linear systems, the same is not true for the H2 norm, neither for output-feedback control. When uncertainties affect the plant, a fixed diagonal structure for the Lyapunov or any other matrix variable involved in the conditions can produce poor results in terms of H2 and H∞ guaranteed costs, specially if the matrix constrained to be diagonal is involved in the computation of the control gain. In this sense, sufficient linear matrix inequalities (LMIs) for the existence of a stabilizing gain and an H2 (or H∞) guaranteed cost are proposed. As main characteristic, the gain and the Lyapunov matrix can be treated as independent optimization variables in the conditions. Using relaxations in the stability of the system, an algorithm is constructed to iteratively solve the LMIs. Therefore, no change of variable is necessary and, differently from other LMI-based methods for positive systems from the literature, there is no need to impose a diagonal structure on the matrix used to recover the gain to enforce closed-loop positivity. Moreover, static output- or state-feedback controllers, including bounded entries or decentralized structures, can be computed in a straightforward way. With an appropriate initialization, the iterative procedure, implemented in two phases, is assured to end up, at the first step, with a finite bound for the real part of the eigenvalues of the closed-loop positive system. Whenever a stabilizing gain is obtained, the second step provides non-increasing H2 (or H∞) guaranteed costs along the iterations. Numerical examples show that the proposed technique provides good estimates for the H2 and H∞ norms when compared with other methods available in the literature.

Stability of interval positive continuous-time linear systems

Stability of interval positive continuous-time linear systems

Comparison of Different Models of Positive Fractional Continuous-Time Linear Systems and Electrical Circuits Based on the Step Responses

The step responses of positive fractional continuous-time linear systems and electrical circuits described by different models will be investigated and comparison of the models will be performed. Solutions to the state equations of continuous-time linear systems described by Caputo and Caputo-Fabrizio derivatives and necessary and sufficient conditions of the positivity of fractional systems will be recalled. Considerations will be illustrated by numerical examples.

Stability of fractional positive continuous-time linear systems with state matrices in integer and rational powers

AbstractThe stability of fractional standard and positive continuous-time linear systems with state matrices in integer and rational powers is addressed. It is shown that the fractional systems are asymptotically stable if and only if the eigenvalues of the state matrices satisfy some conditions imposed on the phases of the eigenvalues. The fractional standard systems are unstable if the state matrices have at least one positive eigenvalue.

Minimum energy control of fractional positive continuous-time linear systems using Caputo-Fabrizio definition

Abstract The Caputo-Fabrizio definition of the fractional derivative is applied to minimum energy control of fractional positive continuous- time linear systems with bounded inputs. Conditions for the reachability of standard and positive fractional linear continuous-time systems are established. The minimum energy control problem for the fractional positive linear systems with bounded inputs is formulated and solved.

Stochastic stability analysis and [formula omitted]-gain controller design for positive Markov jump systems with time-varying delays

This paper is concerned with stochastic stability analysis and L∞-gain (also called L∞-induced norm) controller design for positive Markov jump systems (MJSs) in continuous-time context by means of a linear programming (LP) technique. Firstly, by introducing an equivalent deterministic positive continuous-time linear system, we propose necessary and sufficient conditions of stochastic stability with a prescribed L∞-gain performance in LP form for the underlying system. Then based on the obtained results, an effective method for designing a desired L∞-gain controller is established. All the proposed conditions are solvable in terms of LP with additional parameters. Finally, a numerical example is given to illustrate the effectiveness of the present method.

Positive time-varying continuous-time linear systems and electrical circuits

AbstractThe positivity of time-varying continuous-time linear systems and electrical circuits are addressed. Necessary and sufficient conditions for the positivity of the systems and electrical circuits are established. It is shown that there exists a large class of positive electrical circuits with time-varying parameters. Examples of positive electrical circuits are presented.

Minimum energy control of fractional positive continuous-time linear systems with bounded inputs

Abstract A minimum energy control problem for fractional positive continuous-time linear systems with bounded inputs is formulated and solved. Sufficient conditions for the existence of a solution to the problem are established. A procedure for solving the problem is proposed and illustrated with a numerical example.

Reachability, observability and minimum energy control of fractional positive continuous-time linear systems with two different fractional orders

The reachability, observability and minimum energy control problems for the fractional positive continuous-time linear systems with two different fractional orders are formulated. Necessary and sufficient conditions for the reachability and observability are established. Solution to the minimum energy control problem is derived and demonstrated on example of electrical circuit.

Minimum energy control of positive continuous-time linear systems with bounded inputs

Abstract The minimum energy control problem for the positive continuous-time linear systems with bounded inputs is formulated and solved. Sufficient conditions for the existence of solution to the problem are established. A procedure for solving of the problem is proposed and illustrated by a numerical example

Minimum energy control of positive continuous-time linear systems with bounded inputs

Abstract The minimum energy control problem for positive continuous-time linear systems with bounded inputs is formulated and solved. Sufficient conditions for the existence of a solution to the problem are established. A procedure for solving the problem is proposed and illustrated with a numerical example.

Minimum energy control of fractional positive continuous-time linear systems

Abstract The minimum energy control problem for the fractional positive continuous-time linear systems is formulated and solved. Sufficient conditions for the existence of solution to the problem are established. A procedure for solving of the problem is proposed and illustrated by a numerical example.

Decoupling zeros of positive continuous-time linear systems

Abstract Necessary and sufficient conditions for the reachability and observability of the positive continuous-time linear systems are established. Definitions of the input-decoupling zeros, output-decoupling zeros and input-output decoupling zeros are proposed. Some properties of the decoupling zeros are discussed.

Robust stability of positive continuous-time linear systems with delays

The paper is devoted to the problem of robust stability of positive continuous-time linear systems with delays with structured perturbations of state matrices. Simple necessary and sufficient condi...

Robust stability of positive continuous-time linear systems with delays

Robust stability of positive continuous-time linear systems with delaysThe paper is devoted to the problem of robust stability of positive continuous-time linear systems with delays with structured perturbations of state matrices. Simple necessary and sufficient conditions for robust stability in the general case and in the case of systems with a linear uncertainty structure in two sub-cases: (i) a unity rank uncertainty structure and (ii) nonnegative perturbation matrices are established. The problems are illustrated with numerical examples.

Stability of positive continuous-time linear systems with delays

The asymptotic stability of positive continuous-time linear systems with delays is addressed. It is shown that: 1) the asymptotic stability of the positive systems with delays is independent of their delays, 2) the checking of the asymptotic stability of the positive systems with delays can be reduced to checking of the asymptotic stability of positive systems without delays. Simple stability conditions for the positive systems with delays are give and illustrated by numerical examples.

Fractional Positive Continuous-Time Linear Systems and Their Reachability

Fractional Positive Continuous-Time Linear Systems and Their ReachabilityA new class of fractional linear continuous-time linear systems described by state equations is introduced. The solution to the state equations is derived using the Laplace transform. Necessary and sufficient conditions are established for the internal and external positivity of fractional systems. Sufficient conditions are given for the reachability of fractional positive systems.

On the reachability in any fixed time for positive continuous-time linear systems

This paper deals with the reachability of continuous-time linear positive systems. The reachability of such systems, which we will call here the strong reachability, amounts to the possibility of steering the state in any fixed time to any point of the positive orthant by using nonnegative control functions. The main result of this paper essentially says that the only strongly reachable positive systems are those made of decoupled scalar subsystems. Moreover, the strongly reachable set is also characterized.