Realizations of Nonlinear Control Systems on Time Scales

Bartosiewicz, Zbigniew; Pawłuszewicz, Ewa

doi:10.1109/tac.2007.914243

Cited by 65 publications

(54 citation statements)

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“…An equivalent reformulation is that solutions of ∆x(t n ) = Ax(t n ), A :=Ã − I, (I. 3) are exponentially stable if and only if spec(A) ⊂ {z ∈ C : |1+ z| < 1}. Here, ∆ denotes the forward difference operator.…”

Section: Exponential Stability On R and Zmentioning

confidence: 99%

“…For dynamical systems on these general time domains T, understanding the structure of S plays a key role in various aspects of both the control theory and control applications [2], [3], [9], [21]. However, since S can be difficult to compute on general time scales, other more tractable sufficient conditions for the exponential stability of (II.1) have been explored [25], [14], [10].…”

Section: Exponential Stability On R and Zmentioning

confidence: 99%

See 1 more Smart Citation

Regions of exponential stability for LTI systems on nonuniform discrete domains

Davis

Gravagne

Marks

et al. 2011

2011 IEEE 43rd Southeastern Symposium on System Theory

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Abstract-For LTI systems on a class of nonuniform discrete domains, we establish a region in the complex plane for which pole placement is a necessary and sufficient condition for exponential stability of solutions of the system. We study the interesting geometry of this region, comparing and contrasting it with the standard geometry of the regions of exponential stability for ODE systems on R and finite difference/recursive equations on Z. This work connects other results in the literature on the topic and explains the connection geometrically using time scales theory.Index Terms-exponential stability, pole placement, time scales I. EXPONENTIAL STABILITY ON R AND ZLet A ∈ R n×n . A basic result concerning the continuous, linear time invariant (LTI) systeṁis that solutions are exponentially stable if and only if spec(A) ⊂ C − . A similar basic result for discrete LTI systemsis that solutions are exponentially stable if and only if spec(A) ⊂ {z ∈ C : |z| < 1}. An equivalent reformulation is that solutions ofare exponentially stable if and only if spec(A) ⊂ {z ∈ C : |1+ z| < 1}. Here, ∆ denotes the forward difference operator. For reasons that will soon become apparent, we will use (I.3) rather than (I.2) as the canonical discrete LTI system throughout this paper. Thus, the regions of exponential stability for (I.1) and (I.3) are quite straightforward. This simple geometry is exploited frequently in pole placement arguments for exponential stability . In this paper, we explore the following question: What is the geometry of the region of exponential stability for an LTI system defined on a nonuniform discrete domain?II. EXPONENTIAL STABILITY OF NONUNIFORM DISCRETE SYSTEMS This question can be efficiently handled using time scales theory [5]; see the Appendix for a brief overview. Let T be a nonuniform, discrete time scale that is unbounded above, and consider the LTI systemor its equivalent recursive formDefinition II.1.[25] For t, t 0 ∈ T and x 0 ∈ R n , the systemis exponentially stable if there exists a constant α > 0 such that for every t 0 ∈ T there exists a K ≥ 1 withwith K being chosen independently of t 0 . Here, Φ A (t, t 0 ) denotes the unique solution to (II.3), also called the transition matrix for (II.3); see the Appendix.The following theorem due to Pötzsche, Siegmund, and Wirth [25] provides a spectral characterization of the region of exponential stability of (II.2) for scalar problems.Theorem II.1.[25] Let T be a time scale which is unbounded above. Fix t 0 ∈ T and let λ ∈ C. Then the scalar equationis exponentially stable if and only if either of the following holds:(C2) For every T ∈ T, there exists a t ∈ T with t > T such that 1 + µ(t)λ = 0, where we use the convention log 0 = −∞ in (C1).Definition II.2.[25] Given a time scale T which is unbounded above, for arbitrary t 0 ∈ T, define the sets S C (T) := {λ ∈ C : (C1) holds}, S R (T) := {λ ∈ R : (C2) holds}.

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Section: Exponential Stability On R and Zmentioning

confidence: 99%

Section: Exponential Stability On R and Zmentioning

confidence: 99%

Regions of exponential stability for LTI systems on nonuniform discrete domains

Davis

Gravagne

Marks

et al. 2011

2011 IEEE 43rd Southeastern Symposium on System Theory

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“…Similar characterization of positive observability has been established in [16]. Standard controllability of linear systems on time scales has been studied in [17,18], where standard Gram matrices on time scales have been used. Realizations of linear positive systems on time scales have been considered in [19].…”

Section: Introductionmentioning

confidence: 92%

On positive reachability of time-variant linear systems on time scales

Bartosiewicz¹

2013

Bulletin of the Polish Academy of Sciences: Technical Sciences

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Abstract. Positive reachability of time-variant linear positive systems on arbitrary time scales is studied. It is shown that the system is positively reachable if and only if a modified Gram matrix corresponding to the system is monomial. The general criterion is then specified for particular cases of continuous-time systems and various classes of discrete-time systems. It is shown that in the case of continuous-time systems with analytic coefficients the conditions for positive reachability are very restrictive, similarly as for time-invariant systems.

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“…[3,4]. As the unification of both cases with classical tools one can consider systems on time scales [5,6]. The theory of q-difference linear control systems is developed separately as a special kind of systems on time scales, see e.g.…”

Section: Introductionmentioning

confidence: 99%