Regions of Exponential Stability for LTI Systems on Nonuniform Discrete Domains John M. Davis Department of Mathematics Baylor University Waco, TX 76798 Email: John M Davis@baylor.edu Ian A. Gravagne and Robert J. Marks II Department of Electrical and Computer Engineering Baylor University Waco, TX 76798 Email: Ian Gravagne@baylor.edu, Robert Marks@baylor.edu Billy J. Jackson Department of Mathematics Saint Xavier University Chicago, IL 60655 Email: bjackson@sxu.edu 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 ex- ponential 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 Z Let A ∈ R n×n . A basic result concerning the continuous, linear time invariant (LTI) system ˙ x(t)= Ax(t), (I.1) is that solutions are exponentially stable if and only if spec(A) ⊂ C − . A similar basic result for discrete LTI systems x n+1 = ˜ Ax n , (I.2) is that solutions are exponentially stable if and only if spec(A) ⊂{z ∈ C : |z| < 1}. An equivalent reformulation is that solutions of Δx(t n )= Ax(t n ), A := ˜ A − I, (I.3) are exponentially stable if and only if spec(A) ⊂{z ∈ C : |1+ z| < 1}. Here, Δ 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 sta- bility . 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 system x Δ (t)= Ax(t), (II.1) or its equivalent recursive form x n+1 =(I + Aµ n )x n . (II.2) Definition II.1. [25] For t, t 0 ∈ T and x 0 ∈ R n , the system x Δ (t)= Ax(t), x(t 0 )= x 0 , (II.3) is exponentially stable if there exists a constant α> 0 such that for every t 0 ∈ T there exists a K ≥ 1 with ‖Φ A (t, t 0 )‖≤ Ke −α(t−t0) for t ≥ t 0 , with 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 P¨ otzsche, 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 equation x Δ (t)= λx(t), x(t 0 )= x 0 , is exponentially stable if and only if either of the following holds: (C1) lim sup T →∞ 1 T − t 0  T t0 lim sցμ(t) log |1+ sλ| s Δt< 0, (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}. 978-1-4244-9593-1/11/$26.00 ｩ2011 IEEE 37 IEEE 43rd Southeastern Symposium on System Theory Auburn University Auburn, AL, USA, March 14-16, 2011