Positive Linear Time-invariant Systems Research Articles

Robust state feedback stabilisation of positive LTI systems with polytopic uncertainty

ABSTRACT This paper deals with the robust state feedback stabilizability problem of linear time-invariant (LTI) systems in the presenceof polytopic uncertainty. The paper contributes two major results involving (i) necessary and sufficient conditions to simultaneouslycheck positivity and robust stability of an uncertain LTI system (ii) the development of a design algorithm to solve the mainstabilizability problem. This problem is not easy to deal with because it is not a convex problem. To cope with this issue, a designalgorithm is proposed that checks the design space of the stabilizer parameters, divides it into smaller subspaces, checks thefeasibility of the corner points of the design subspaces, removes the detected total infeasible design subspaces, and shrinks theremaining parts to solve the design problem, iteratively. The proposed algorithm is applied to the cancer chemotherapy applicationand the results demonstrate its efficacy to control the model considering the positivity condition.

Stability and [formula omitted]-gain characterization of positive linear systems on time scales

The stability and Lp-gain (p∈{1,∞}) analysis problems are investigated for positive linear systems on arbitrary time scales in this paper. Firstly, by virtue of the linear copositive Lyapunov function, the uniform stability and uniformly exponential stability conditions are presented for positive linear time-varying systems on time scales. Then, we conclude that the uniformly exponential stability condition of positive linear time-invariant systems is independent of time scales. Meanwhile, it is further proved that the Lp-gain (p∈{1,∞}) conditions of these systems are also independent of time scales and can be fully characterized by system matrices. Finally, a numerical example is given to demonstrate the validity of our conclusions.

The ${L}_{{q}}/{L}_{{p}}$ Hankel Norms of Positive Systems

This letter is concerned with the $L_{q}/L_{p}$ Hankel norms of linear time-invariant positive systems, where the Hankel norms are defined as the induced norms from vector-valued $L_{p}$ -past inputs to vector valued $L_{q}$ -future outputs. The $L_{q}/L_{p}$ Hankel norms are studied in detail for general (nonpositive) LTI systems, and closed-form formulas have been derived for their characterization. However, some of them unavoidably include the absolute integral of impulse responses and/or implicit functions in their characterization and hence there remain difficulties in practical computation. In this letter we show that such difficulties are circumvented for positive systems, where positivity drastically facilitates the characterization of the past input that attains the Hankel norm. We also provide linear-programming- and semidefinite-programming-based characterizations of the $L_{q}/L_{p}$ Hankel norms of positive systems, which are useful in the case where we analyze the $L_{q}/L_{p}$ Hankel norms of positive systems affected by parametric uncertainties.

Analysis of Positive Systems Using Copositive Programming

In the field of control, a wide range of analysis and synthesis problems of linear time-invariant (LTI) systems are reduced to semidefinite programming problems (SDPs). On the other hand, in the field of mathematical programming, a class of conic programming problems, so called the copositive programming problem (COP), is actively studied. COP is a convex optimization problem on the copositive cone, and the completely positive cone, the doubly nonnegative cone, and the Minkowski sum of the positive semidefinite cone and the nonnegative cone are also closely related to COP. These four cones naturally appear when we deal with optimization problems described by nonnegative vectors. In this letter, we show that the stability, the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> and the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performances of LTI positive systems are basically characterized by the feasibility/optimization problems over these four cones. These results can be regarded as the generalization of well-known LMI/SDP-based results on the positive semidefinite cone. We also clarify that in some performances such direct generalization is not possible due to inherent properties of the copositive or the completely positive cone. We thus capture almost entire picture about how far we can generalize the SDP-based results for positive systems to those on the four cones related to COP.

Stability radius formulation of L σ ‐gain in positive stabilisation of regular and time‐delay systems

This study initially considers the relationship between stability radius and -gain of linear time-invariant positive systems. The -, -, and -gains of an asymptotically stable positive system are characterised in terms of stability radii and useful bounds are derived. The authors show that the structured perturbation of a stable matrix can be regarded as a closed-loop system with uncertainty structure represented by the unknown static output feedback. This makes it possible to relate the -gains in terms of closed-form expression available for stability radii of Metzler matrices. The authors generalise the above connection for positive-delay systems as well. Performance characterisation and computation of -gains are also given based on linear programming for and linear matrix inequality (LMI) for . The importance of this characterisation becomes evident when state feedback controllers are designed for regular and time-delay systems with positivity constraints. In particular, they show that positive stabilisation with maximum stability radius for the case of can be considered as an -gain minimisation, which can be solved by LMI. This inherently achieves the performance criterion and establishes a link to the reported iterative convex optimisation approaches that have been developed for the cases of and . A significant result of this study is the derivation of bounds for -gains and the unique commonality among the optimal state feedback gain matrices in obtaining -gains of the stabilised system.

Eigenvalue assignment for positive observers and its feasibility

In this paper, we discuss a new problem of establishing feasibility conditions for regional eigenvalue assignment of positive observers. Previous results on regional eigenvalue assignment for observers of linear time-invariant positive systems are also improved. Unlike observable dynamical systems whose closed-loop eigenvalues can be assigned arbitrarily, eigenvalues of positive observers are indicated unable to be assigned into any arbitrary region. We derive feasibility conditions in the form of constrained convex programming under which the regional eigenvalue assignment is possible. Moreover, we propose a new method for solving the regional eigenvalue problem of positive observers once the feasibility conditions are satisfied. Numerical examples are given to show the efficacy of the proposed method.

On reachable sets for positive linear systems under constrained exogenous inputs

This paper focuses on positive linear time-invariant systems with constant coefficients and specific exogenous disturbance. The problem of finding a hyper-pyramid to bound the set of the states that are reachable from the origin in the Euclidean space is addressed, subject to inputs whose (1,1)-norm or (∞,1)-norm is bounded by a prescribed constant. The Lyapunov approach is applied and a bounding hyper-pyramid is obtained by solving a set of inequalities. Iterative procedures (with an adjustable parameter) for reducing the hyper-volume of the bounding hyper-pyramid for the reachable set are proposed.

Dominant pole and eigenstructure assignment for positive systems with state feedback

In this paper, the dominant pole assignment problem, the dominant eigenstructure assignment problem and the robust dominant pole assignment problem for linear time-invariant positive systems with state feedback are considered. The dominant pole assignment problem is formulated as a linear programming problem, and the dominant eigenstructure problem is formulated as a quasiconvex optimisation problem with linear constraints. The robust dominant pole assignment problem is formulated as a non-convex optimisation problem with non-linear constraints which is solved using particle swarm optimisation (PSO) with an efficient scheme which employs the dominant eigenstructure assignment technique to accelerate the convergence of the PSO procedure. Each of the three problems can be further constrained by requiring that the controller has a pre-specified structure, or the gain matrix have both elementwise upper and lower bounds. These constraints can be incorporated into the proposed scheme without increasing the complexity of the algorithms. Both the continuous-time case and the discrete-time case are treated in the paper.

Frequency-Limited &lt;inline-formula&gt; &lt;tex-math notation="TeX"&gt;${\mmb H}_{\infty}$&lt;/tex-math&gt;&lt;/inline-formula&gt; Model Reduction for Positive Systems

In this paper, the problem of frequency-limited H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> model reduction for positive linear time-invariant systems is investigated. Specifically, our goal is to find a stable positive reduced-order model for a given positive system such that the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> norm of the error system is bounded over a frequency interval of interest. A new condition in terms of matrix inequality is developed for characterizing the frequency-limited H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance. Then an equivalent parametrization of a positive reduced-order model is derived, based on which, an iterative algorithm is constructed for optimizing the reduced-order model. The algorithm utilizes coarse reduced-order models resulting from (generalized) balanced truncation as the initial value. Both continuous- and discrete-time systems are considered in the same framework. Numerical examples clearly show the effectiveness and advantages of the proposed model reduction method.

On the Distinguishability of Positive Linear Time-Invariant Systems with Affine Parametric Uncertainties K. Motchon

Studies of distinguishability have focused on the case of Linear Time-Invariant systems without uncertainties. In this work, distinguishability is studied for Positive Linear Time-Invariant systems with affine parametric uncertainties in the state space model. We propose a definition of distinguishability adapted to this new context and give a characterization of this notion. The approach used is based on the estimate of the reachable output space of the systems. Under suitable assumptions, a sufficient condition for distinguishability is established.

A New Approach to H∞ Model Reduction for Positive Systems

In this paper, we investigate model reduction for positive linear time-invariant systems in the H∞ sense. First, the bounded real lemma is further developed to obtain a new characterization of the H∞ performance, which does not include any product term between the Lyapunov matrix and the parameter of the reduced-order model. Then a new parameterization of a positive reduced-order model is proposed, and accordingly, an iterative algorithm is constructed, which makes use of the coarse reduced-order models resulting from the (generalized) balanced truncation as the initial value to search for a desired positive one. Both continuous- and discrete-time systems are considered in the same framework. Numerical examples clearly show the effectiveness and advantages of the proposed method.

The Bounded Real Lemma for Internally Positive Systems and H-Infinity Structured Static State Feedback

We consider the bounded real lemma for internally positive linear time-invariant systems. We show that the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$H_{\infty}$</tex> </formula> norm of such systems can be evaluated by checking the existence of a certain diagonal quadratic storage function. Taking advantage of this fact, the problem of designing a structured static state feedback controller achieving internal stability, contractiveness, and internal positivity in closed loop becomes convex and tractable.

D-Stability and Delay-Independent Stability of Homogeneous Cooperative Systems

We introduce a nonlinear definition of D-stability, extending the usual concept for positive linear time-invariant systems. We show that globally asymptotically stable, cooperative systems, homogeneous of any order with respect to arbitrary dilation maps are D-stable. We also prove a strong stability result for delayed cooperative homogeneous systems. Finally, we show that both of these results also hold for planar cooperative systems without the restriction of homogeneity.

Stability radii for positive linear time-invariant systems on time scales

We deal with dynamic equations on time scales, where we characterize the positivity of a system. Uniform exponential stability of a system is determined by the spectrum of its matrix. We investigate the corresponding stability radii with respect to structured perturbations and show that, for positive systems, the complex and the real stability radius coincide.

On the excitability of a class of positive continuous time-delay systems

This article is on the excitability of positive linear time-invariant systems subject to internal point delays. It is proved that excitability independent of delay is guaranteed if an auxiliary delay-free system is excitable. Necessary and sufficient conditions for excitability and transparency independent of the delay size are formulated in terms of the parameterization of the dynamics and control matrices . Some particular results are also given for the properties being dependent on the size of the point delay and for any possible finite values of the delay. The same formulation is given in parallel in terms of strict positivity of a matrix of an associate system obtained from the influence graph of the original system. The excitability and transparency properties are both testable through simple algebraic tests involving a moderate computational effort that is directly related to the system's order.

On some testable conditions for excitability and transparency of a class of time-delay systems with point delays

This article is concerned with the excitability of positive linear time-invariant systems subject to internal point delays. It is proved that the excitability independent of delay is guaranteed if an auxiliary delay-free system is excitable. Necessary and sufficient conditions for excitability and transparency are formulated in terms of the parameterization of the dynamics and control matrices and, equivalently, in terms of strict positivity of a matrix of an associate system obtained from the influence graph of the original system. Such conditions are testable through simple algebraic tests involving moderate computational effort.

Identification of positive linear systems with Poisson output transformation

We study the identification problem for third-order linear time invariant positive systems in experiments where the output is a Poisson process. The problem well-posedness is investigated when the input–output model is described by a sum of real exponentials. A maximum likelihood procedure is then proposed and the admissible set for the unknown parameters is characterized. The novelty of the approach consists in solving a constrained maximum likelihood problem to estimate residues and eigenvalues based on theoretical results on minimality of positive realizations recently obtained in the literature. Numerical results are also provided.