The generalised singular perturbation approximation for bounded real and positive real control systems

The generalised singular perturbation approximation (GSPA) is considered as a model reduction scheme for bounded real and positive real linear control systems. The GSPA is a state-space approach to truncation with the defining property that the transfer function of the approximation interpolates the original transfer function at a prescribed point in the closed right half complex plane. Both familiar balanced truncation and singular perturbation approximation are known to be special cases of the GSPA, interpolating at infinity and at zero, respectively. Suitably modified, we show that the GSPA preserves classical dissipativity properties of the truncations, and existing a priori error bounds for these balanced truncation schemes are satisfied as well

A circle criterion for strong integral input-to-state stability

We present sufficient conditions for integral input-to-state stability (iISS) and strong iISS of the zero equilibrium pair of continuous-time forced Lur'e systems, where by strong iISS we mean the conjunction of iISS and small-signal ISS. Our main results are reminiscent of the complex Aizerman conjecture and the well-known circle criterion. We derive a number of corollaries, including a result on stabilisation by static feedback in the presence of input saturation. In particular, we identify classes of forced Lur'e systems with saturating nonlinearities which are strongly iISS, but not ISS

Non-dissipative boundary feedback for Rayleigh and Timoshenko beams

We show that a non-dissipative feedback that has been shown in the literature to exponentially stabilize an EulerBernoulli beam makes a Rayleigh beam and a Timoshenko beam unstable

A counter-example to “Positive realness preserving model reduction with H-\infty norm error bounds”

We provide a counter example to the H∞ error bound for the difference of a positive real transfer function and its positive real balanced truncation stated in “Positive realness preserving model reduction with H-\infty norm error bounds” IEEE Trans. Circuits Systems I Fund. Theory Appl. 42 (1995), no. 1, 23–29. The proof of the error bound is based on a lemma from an earlier paper “A tighter relative-error bound for balanced stochastic truncation.” Systems Control Lett. 14 (1990), no. 4, 307–317, which we also demonstrate is false by our counter example. The main result of this paper was already known in the literature to be false. We state a correct H-\infty error bound for the difference of a proper positive real transfer function and its positive real balanced truncation and also an error bound in the gap metric

Bounded real and positive real balanced truncation for infinite-dimensional systems

Bounded real balanced truncation for infinite-dimensional systems is considered. This provides reduced order finite-dimensional systems that retain bounded realness. We obtain an error bound analogous to the finite-dimensional case in terms of the bounded real singular values. By using the Cayley transform a gap metric error bound for positive real balanced truncation is subsequently obtained. For a class of systems with an analytic semigroup, we show rapid decay of the bounded real and positive real singular values. Together with the established error bounds, this proves rapid convergence of the bounded real and positive real balanced truncations

Semi-global incremental input-to-state stability of discrete-time Lur'e systems

We present sufficient conditions for semi-global incremental input-to-state stability of a class of forced discrete-time Lur'e systems. The results derived are reminiscent of well-known absolute stability criteria such as the small gain theorem and the circle criterion. We derive a natural sufficient condition which guarantees that asymptotically (almost) periodic inputs generate asymptotically (almost) periodic state trajectories. As a corollary, we obtain sufficient conditions for the converging-input converging-state property to hold

Low-gain Integral Control for Multi-Input, Multi-Output Linear Systems with Input Nonlinearities

We consider the inclusion of a static anti-windup component in a continuous-time low-gain integral controller in feedback with a multi-input multi-output stable linear system subject to an input nonlinearity (from a class of functions that includes componentwise diagonal saturation). We demonstrate that the output of the closed- loop system asymptotically tracks every constant reference vector which is “feasible” in a natural sense, provided that the integrator gain is sufficiently small. Robustness properties of the proposed control scheme are investigated and three examples are discussed in detail

A necessary condition for dispersal driven growth of populations with discrete patch dynamics

We revisit the question of when can dispersal-induced coupling between discrete sink populations cause overall population growth? Such a phenomenon is called dispersal driven growth and provides a simple explanation of how dispersal can allow populations to persist across discrete, spatially heterogeneous, environments even when individual patches are adverse or unfavourable. For two classes of mathematical models, one linear and one non-linear, we provide necessary conditions for dispersal driven growth in terms of the non-existence of a common linear Lyapunov function, which we describe. Our approach draws heavily upon the underlying positive dynamical systems structure. Our results apply to both discrete- and continuous-time models. The theory is illustrated with examples and both biological and mathematical conclusions are drawn