Partial-realization theory for linear switched systems —A formal power series approach

The paper presents partial-realization theory and a realization algorithm for linear switched systems. The results are similar to partial-realization theory of linear and bilinear systems. Our main tool is the theory of rational formal power series.     

Feedback control of nonlinear stochastic systems for targeting a specified stationary probability density

In the present paper, an innovative procedure for designing the feedback control of multi-degree-of-freedom (MDOF) nonlinear stochastic systems to target a specified stationary probability density function (SPDF) is proposed based on the technique for obtaining the exact stationary solutions of the dissipated Hamiltonian systems. First, the control problem is formulated as a controlled, dissipated Hamiltonian system together with a target SPDF. Then the controlled forces are split into a conservative part and a dissipative part. The conservative control forces are designed to make the controlled system and the target SPDF have the same Hamiltonian structure (mainly the integrability and resonance). The dissipative control forces are determined so that the target SPDF is the exact stationary solution of the controlled system. Five cases, i.e., non-integrable Hamiltonian systems, integrable and non-resonant Hamiltonian systems, integrable and resonant Hamiltonian systems, partially integrable and non-resonant Hamiltonian systems, and partially integrable and resonant Hamiltonian systems, are treated respectively. A method for proving that the transient solution of the controlled system approaches the target SPDF as t→∞ is introduced. Finally, an example is given to illustrate the efficacy of the proposed design procedure.     

Stochastic system transformation using generalized orthonormal basis functions with applications to continuous-time system identification

This paper studies the system transformation using generalized orthonormal basis functions that include the Laguerre basis as a special case. The transformation of the deterministic systems is studied in the literature, which is called the Hambo transform. The aim of the paper is to develop a transformation theory for stochastic systems. The paper establishes the equivalence of continuous and transformed-discrete-time stochastic systems in terms of solutions. The method is applied to the continuous-time system identification problem. It is shown that using the transformed signals the PO-MOESP subspace identification algorithm yields consistent estimates for system matrices. An example is included to illustrate the efficacy of the proposed identification method, and to make a comparison with the method using the Laguerre filter.     

Set-valued Lyapunov functions for difference inclusions

The paper relates set-valued Lyapunov functions to pointwise asymptotic stability in systems described by a difference inclusion. Pointwise asymptotic stability of a set is a property which requires that each point of the set be Lyapunov stable and that every solution to the inclusion, from a neighborhood of the set, be convergent and have the limit in the set. Weak set-valued Lyapunov functions are shown, via an argument resembling an invariance principle, to imply this property. Strict set-valued Lyapunov functions are shown, in the spirit of converse Lyapunov results, to always exist for closed sets that are pointwise asymptotically stable.     

Equilibrium-independent passivity: A new definition and numerical certification

We extend the traditional notion of passivity to a forced system whose equilibrium is dependent on the control input by defining equilibrium-independent passivity, a system property characterized by a dissipation inequality centered at an arbitrary equilibrium point. We provide a necessary input/output condition which can be tested for systems of arbitrary dimension and sufficient conditions to certify this property for scalar systems. An example from network stability analysis is presented which demonstrates the utility of this new definition. We then proceed to numerical certification of equilibrium-independent passivity using sum-of-squares programming. Finally, through numerical examples we show that equilibrium-independent passivity is less restrictive than incremental passivity.     

The spring paradigm in tracking control of simple mechanical systems

An intrinsic formulation of geometric proportional–derivative tracking control for fully actuated mechanical systems is developed. The region of stability is determined directly from the size of the system’s injectivity radius and, for a restricted set of control problems, the system’s locus of cut points about a desired reference point. Exponential stability is obtained under certain boundedness conditions. For controlled motion along a geodesic, the proffered scheme yields a particularly simple and elegant manifestation of the underlying use of the mass–spring–damper paradigm in the control design methodology.     

Nonlinear minimum-time control with pre- and post-actuation

This article studies the time-optimal output transition problem to change the system output, from an initial value (for all time ) to a final value (for all time ), for invertible nonlinear systems. The main contribution of the article is to show that the use of pre- and post-actuation input outside the transition interval I_T=[0,T] can reduce the transition time T beyond the standard bang–bang-type inputs for optimal state transition. The advantage of using pre- and post-actuation is demonstrated with an illustrative nonlinear example.     

A second-order maximum principle for discrete-time bilinear control systems with applications to discrete-time linear switched systems

A powerful approach for analyzing the stability of continuous-time switched systems is based on using optimal control theory to characterize the “most unstable” switching law. This reduces the problem of determining stability under arbitrary switching to analyzing stability for the specific “most unstable” switching law. For discrete-time switched systems, the variational approach received considerably less attention. This approach is based on using a first-order necessary optimality condition in the form of a maximum principle (MP), and typically this is not enough to completely characterize the “most unstable” switching law. In this paper, we provide a simple and self-contained derivation of a second-order necessary optimality condition for discrete-time bilinear control systems. This provides new information that cannot be derived using the first-order MP. We demonstrate several applications of this second-order MP to the stability analysis of discrete-time linear switched systems.     

Spacecraft formation reconfiguration with collision avoidance

In this paper we present a behavioral control solution for reconfiguration of a spacecraft formation using the Null-Space Based (NSB) concept. The solution is task based, and aims to reconfigure and maintain a rigid formation while avoiding collisions between spacecraft. A model of relative translation is derived, together with a passivity-based sliding surface controller which globally stabilizes the equilibrium point of the closed-loop system. The NSB control method is implemented by giving each task different priorities and then calculating desired velocity and a Jacobian matrix for each spacecraft and each task. The velocity vector for each task is then projected into the null-space for higher prioritized tasks to remove conflicting velocity components. Simulation results are presented, showing that each spacecraft moves into the predefined formation without breaking any rules for the higher priority tasks, and all collisions are avoided.     

An internal model principle is necessary and sufficient for linear output synchronization

Output synchronization of a network of heterogeneous linear state–space models under time-varying and directed interconnection structures is investigated. It is shown that, assuming stabilizability and detectability of the individual systems and imposing very mild connectedness assumptions on the interconnection structure, an internal model requirement is necessary and sufficient for synchronizability of the network to polynomially bounded trajectories. The resulting dynamic feedback couplings can be interpreted as a generalization of existing methods for identical linear systems.