New Stability Tests Of Positive 1D And 2D Linear Systems

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. Effectiveness of the tests is shown on numerical examples. INTRODUCTION A dynamical system is called positive if its trajectory starting from any nonnegative initial state remains forever in the positive orthant for all nonnegative inputs. An overview of state of the art in positive theory is given in the monographs (Farina and Rinaldi 2000; Kaczorek 2002). Variety of models having positive behavior can be found in engineering, economics, social sciences, biology and medicine, etc.. New stability conditions for discrete-time linear systems have been proposed in (Busłowicz 2008) and next have been extended to robust stability of fractional discretetime linear systems in (Busłowicz 2010). The stability of positive continuous-time linear systems with delays has been addressed in (Kaczorek 2009c) The independence of the asymptotic stability of positive 2D linear systems with delays of the number and values of the delays has been shown in (Kaczorek 2009d). The asymptotic stability of positive 2D linear systems without and with delays has been considered in (Kaczorek 2009a and 2009b). The stability and stabilization of positive fractional linear systems by state-feedbacks have been analyzed in (Kaczorek 2010). In this paper 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 will be proposed. It will be shown that the checking of the asymptotic stability of positive 2D linear systems can be reduced to checking of stability of suitable corresponding 1D positive linear systems. The paper is organized as follows. In section 2 new stability tests for positive continuous-time linear systems are proposed. An extension of these tests for positive discrete-time linear systems is given in section 3. Application of the tests to checking the asymptotic stability of positive 1D linear systems with delays is given in section 4. In section 5 the tests are applied to positive 2D linear systems described by the general and Roesser models. Concluding remarks are given in section 6. The following notation will be used: R the set of real numbers, m n× R the set of m n× real matrices, m n× + R the set of m n× matrices with nonnegative entries and 1 × + + R = R n n , n M the set of n n× Metzler matrices (real matrices with nonnegative off-diagonal entries), n I the n n× identity matrix. CONTINUOUS-TIME LINEAR SYSTEMS Consider the continuous-time linear system ) ( ) ( t Ax t x = & (2.1) where n t x R ∈ ) ( is the state vector and n n A × R ∈ . The system (2.1) is called (internally) positive if n t x + R ∈ ) ( , 0 ≥ t for any initial conditions n x x + R ∈ = 0 ) 0 ( (Farina and Rinaldi 2000; Kaczorek 2002). Theorem 2.1. (Farina and Rinaldi 2000; Kaczorek 2002) The system (2.1) is positive if and only if A is a Metzler matrix. The positive system is called asymptotically stable if 0 lim ) ( lim 0 = = ∞ → ∞ → x e t x At t t for all n x + R ∈ 0 Theorem 2.2. (Farina and Rinaldi 2000; Kaczorek 2002) The positive system (2.1) is asymptotically stable if and only if all principal minors n i M i ,..., 1 , = of the matrix –A are positive, i.e. 0 ] det[ ,..., 0 , 0 22 21 12 11 2 11 1 > − = > − − − − = > − =