Invariant properties of positive linear systems with integer and fractional orders

Some specific properties of positive standard and fractional linear systems with interval state matrices are analyzed. The stability, positivity and transfer matrices of positive different orders fractional continuous-time linear systems are considered. New necessary and sufficient conditions for the asymptotic stability of positive different orders fractional linear systems are established. It is shown that the transfer matrices of positive asymptotically stable different orders fractional linear systems have only nonnegative coefficients. New conditions for the interval stability of positive standard and fractional linear systems are established. It is shown that the adjoint matrix of a singular Metzler matrix with zero sum of entries of each row (column) has all equal entries.

The minimum energy control problem of positive fractional-discrete time linear systems is addressed. Necessary and sufficient conditions for the reachability of the system are established. Sufficient conditions for the solvability of the minimum energy control of the positive fractional discrete-time systems are given. A procedure for computation of the optimal sequence of inputs minimizing the quadratic performance index is proposed. INTRODUCTION In positive systems inputs, state variables and outputs take only non-negative values. Examples of positive systems are industrial processes involving chemical reactors, heat exchangers and distillation columns, storage systems, compartmental systems, water and atmospheric pollution models. A variety of models having positive linear systems behaviour can be found in engineering, management science, economics, social sciences, biology and medicine, etc. Positive linear systems are defined on cones and not on linear spaces. Therefore, the theory of positive systems is more complicated and less advanced. An overview of state of the art in standard positive systems is given in the monographs (Farina and Rinaldi 2000; Kaczorek 2002). The realization problem for positive standard and singular continuous-time systems with delays was formulated and solved in (Kaczorek 2007c, 2007d). The reachability, controllability and minimum energy control of positive linear discrete-time systems with time-delays have been considered in (Buslowicz and Kaczorek 2004; Xie and Wang 2003). The realization problem for cone systems has been addressed in (Kaczorek 2006). The reachability and controllability to zero of positive fractional linear systems have been investigated in (Kaczorek 2008a, 2007a, 2007b, 2008b; Klamka 2002; Klamka 2005). Mathematical fundamentals of fractional calculus are given in the monographs (Miller and Ross 1993; Nishimoto 1984; Oldham and Spanier 1974; Oustalup 1993; Podlubny 1999). The fractional order controllers have been developed in (Oustalup 1993). A generalization of the Kalman filter for fractional order systems has been proposed in (Sierociuk and Dzielinski 2006). Some other applications of fractional order systems can be found in (Ferreira and Machado 2003; Moshrefi-Torbati and Hammond 1998; Ortigueira 1997; Ostalczyk 2000, 2004a, 2004b; Podlubny 2002; Samko et al. 1993; Vinagre et al. 2002; Vinagre and Feliu 2002; Galkowski and Kummert 2005). The minimum energy control problem has been solved for different classes of linear systems in (Klamka 1991, 1976, 1983; Kaczorek and Klamka 1986). In this paper the minimum energy control problem will be addressed for positive fractional discrete-time linear systems. The paper is organized as follows. In section 2 the solution of the state equation and the necessary and sufficient conditions for the positivity of the fractional systems are recalled. Necessary and sufficient conditions for the reachability of the positive fractional systems are established in section 3. The main result of the paper is presented in section 4 in which the minimum energy control problem is formulated and solved. Concluding remarks are given in section 5. To the best knowledge of the author the minimum energy control problem for the positive fractional discrete-time linear systems have not been considered yet. POSITIVE FRACTIONAL SYSTEMS Let be the set of real matrices and The set of matrices with nonnegative entries will be denoted by and The set of nonnegative integers will be denoted by n m × R n m × 1 : n n× R = R . . m n × m n × + R 1 : n n× + + R = R Z+ and the identity matrix by n n × . n I In this paper definition of the fractional difference of the form (Kaczorek 2007a) 0 ( 1) , k j k j x j α α − = ⎛ ⎞ Δ = − ⎜ ⎟ ⎝ ⎠ ∑ k j x

The asymptotic stability of interval positive continuous-time linear systems of integer and fractional orders is investigated. The classical Kharitonov theorem is extended to the interval positive continuous-time linear systems of integer and fractional orders. It is shown that: (1) The interval positive linear system is asymptotically stable if and only if the matrices bounding the state matrix are Hurwitz Metzler. (2) The interval positive fractional system is asymptotically stable if and only if bounding the state matrix are Hurwitz Metzler. (3) The interval positive of integer and fractional orders continuous-time linear systems with interval characteristic polynomials are asymptotically stable if and only if their lower bounds of the coefficients are positive.

The stability and the angles between state matrices of positive continuous-time and discrete-time linear systems are addressed. It is shown that: 1) The angles between matrices can be useful tool for analysis of the stability of positive continuous-time and discrete-time linear systems; 2) The positive linear system is asymptotically stable if and only if the symmetrical part of the state matrix is Hurwitz for continuous-time systems and Schur for discrete-time systems; 3) Using the angles between matrices necessary and sufficient conditions are established for the asymptotic stability of the positive linear systemst.

The positivity and asymptotic stability of descriptor linear continuous-time and discrete-time systems with interval state matrices and interval polynomials are investigated. Necessary and sufficient conditions for the positivity of descriptor continuous-time and discrete-time linear systems are established. It is shown that the convex linear combination of polynomials of positive linear systems is also the Hurwitz polynomial. The Kharitonov theorem is extended to the positive descriptor linear systems with interval state matrices. Necessary and sufficient conditions for the asy mptotic stability of descriptor positive linear systems have been also established. The considerations have been illustrated by numerical examples.

Notations of the practical stability and of the asymptotic stability of positive and cone fractional 1D and 2D linear systems are introduced. Necessary and sufficient conditions for the practical stability and the asymptotic stability of positive and cone fractional 1D and 2D linear systems are established. It is shown that the checking of the practical stability and asymptotic stability of positive 2D linear systems can be reduced to testing the stability of corresponding 1D positive linear systems. Three LMI approaches are proposed for checking the stability of positive fractional linear systems. LMI approach is applied to compute gain matrices of state-feedbacks such that closed-loop systems are positive and asymptotically stable. The proposed methods are illustrated on numerical examples.

The problem of stability of non-degenerate linear systems admitting a first integral in the form of a non-degenerate quadratic form is considered. New algebraic criteria for stability, as well as complete instability of such systems, have been established in the form of equality to zero of traces of products of matrices, which include an additional symmetric matrix. These conditions are closely related to the symplectic geometry of the phase space, which is determined by the matrix of the original linear system and the symmetric matrix defining the first integral. General results are applied to finding conditions for complete instability of linear gyroscopic systems.

This paper presents new results on the analysis of positive linear time-invariant systems with non-negative state variables and output data for non-negative initial conditions and inputs. The positive linear time-invariant systems are characterized by the state-space equations which offer a formal mathematical structure of form $$\frac{dx(t)}{dt} = Ax(t)$$ where the system matrix $A\in \mathbb{R}^{n,n}$ is Metzler, with non-negative off-diagonal components. These systems exhibit essential characteristics such as monotonicity, stability, and non-negativity, making them fundamental in applications such as biological systems, mathematical economics, chemical reaction networks, and transportation models. We present the theoretical foundations utilizing a mathematical framework from algebraic systems, matrix theory, and stability analysis to investigate stability, $\mathfrak{D}$-stability, and strong $\mathfrak{D}$-stability of positive linear time-invariant systems in the presence of Metzler and Hurwitz matrices. The numerical testing supports the spectrum analysis and $\epsilon$-pseudospectrum of Metzler matrices.

New tests for checking asymptotic stability of positive 1D continuous-time and discrete-time linear systems without and with delays and of positive 2D linear systems described by the general and the Roesser models are proposed. Checking of the asymptotic stability of positive 2D linear systems is reduced to checking of suitable corresponding 1D positive linear systems. It is shown that the stability tests can be also applied to checking the asymptotic stability of fractional discrete-time linear systems with delays. Effectiveness of the tests is shown on numerical examples.

Asymptotic stability of positive 2D linear systems with delays New necessary and sufficient conditions for the asymptotic stability of positive 2D linear systems with delays described by the general model, Fornasini-Marchesini models and Roesser model are established. It is shown that checking of the asymptotic stability of positive 2D linear systems with delays can be reduced to the checking of the asymptotic stability of corresponding positive 1D linear systems without delays. The efficiency of the new criterions is demonstrated on numerical examples.

Robust stability of positive discrete-time linear systems of fractional orderThe paper is devoted to the problem of robust stability of linear positive discrete-time systems of fractional order with structured perturbations of state matrices. Simple necessary and sufficient conditions for robust stability in the general case and in the case of linear uncertainty structure with unity rank uncertainty structure and with non-negative perturbation matrices, are established. It is shown that robust stability of the positive discrete-time fractional system is equivalent to: 1) robust stability of the corresponding positive discrete-time system of natural order - in the general case, 2) robust stability of the corresponding finite family of positive discrete-time systems of natural order - in the case of linear unity rank uncertainty structure, 3) asymptotic stability of only one corresponding positive discrete-time system of natural order - in the case of linear uncertainty structure with non-negative perturbation matrices. Moreover, simple necessary and sufficient condition for robust stability of the positive interval discrete-time linear systems of fractional order is given. The considerations are illustrated by numerical examples.

This paper considers the problem of the semi-global asymptotic stabilisation of fractional-order (FO) linear systems subject to actuator saturation by output feedback. To solve the problem, a family of observer-based linear output feedback laws, parameterised in a positive scalar, is proposed by means of the low gain feedback design technique. The design applies to FO linear systems that are stabilisable and detectable, but not exponentially unstable. For such an FO system under the proposed observer-based linear low gain output feedback, the peak value of the control input for a given initial condition can be made arbitrarily small to avoid actuator saturation by decreasing the value of the parameter towards zero and thus semi-global asymptotic stabilisation is achieved. To obtain these results, we establish the properties of low gain feedback, derive asymptotic expansions and the bounds of high-order derivatives of the Mittag-Leffler (ML) functions to estimate the state responses of FO linear systems, and explicitly construct a Hermitian matrix to satisfy a linear matrix inequality (LMI) stability condition for FO linear systems. The results in this paper extend the corresponding results for integer-order (IO) linear systems.

A new concept of the practical stability of the positive fractional 2D linear systems is proposed. Necessary and sufficient conditions for the practical stability and the asymptotic stability of the positive fractional 2D systems are established. It is shown that the checking of the practical stability and the asymptotic stability of positive fractional 2D linear systems can be reduced to testing the stability of corresponding 1D positive linear systems.

A new concept of the practical stability of the positive fractional 2D linear systems is proposed. Necessary and sufficient conditions for the practical stability and the asymptotic stability of the positive fractional 2D systems are established. It is shown that the checking of the practical stability and the asymptotic stability of positive fractional 2D linear systems can be reduced to testing the stability of corresponding 1D positive linear systems. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Positive descriptor fractional discrete-time linear systems with fractional different orders are addressed in the paper. The decomposition of the regular pencil is used to extend necessary and sufficient conditions for positivity of the descriptor fractional discrete-time linear system with different fractional orders. A method for finding the decentralized controller for the class of positive systems is proposed and its effectiveness is demonstrated on a numerical example.