For periodic linear control systems with bounded control range, an autonomized system is introduc... more For periodic linear control systems with bounded control range, an autonomized system is introduced by adding the phase to the state of the system. Here, a unique control set (i.e., a maximal set of approximate controllability) with nonvoid interior exists. It is determined by the spectral subspaces of the homogeneous part which is a periodic linear differential equation. Using the Poincaré sphere, one obtains a compactification of the state space allowing us to describe the behavior “near infinity” of the original control system. Furthermore, an application to quasi-affine systems yields a unique control set with nonvoid interior.
Realistic models of mechanical systems often depend on various parameters, such as controlled inp... more Realistic models of mechanical systems often depend on various parameters, such as controlled inputs, material constants, tunable parameters, and uncertainties. Uncertain parameters can be (time varying) deterministic perturbations or stochastic excitations whose influence on the system depends on the perturbation dynamics (multiplicative or additive), the perturbation range, and its statistics in the stochastic case. For a given operating region of the system, i.e. for a set of state space constraints, the behavior of the system within this region depends strongly on the type of perturbation dynamics and on its range. We present some basic theory for additively and multiplicatively perturbed systems, where the uncertainty can be a family of time varying functions, or a Markov diffusion process. The uncertainty range plays the role of a bifurcation parameter and determines concepts like discontinuities of control sets and supports of invariant measures, stability radii, and invariance radii with respect to the constraint set. It turns out that in many instances the stochastic and the deterministic bifurcation scenarios agree, and the cases in which they differ are related to a nonuniform behavior of the stochastically perturbed system. The example of a model of ship roll motion is treated in detail, revealing some of the fundamental agreements and disagreements of the two bifurcation scenarios.
The region of asymptotic null controllability of bilinear systems with control constraints is cha... more The region of asymptotic null controllability of bilinear systems with control constraints is characterized using Lyapunov exponents. It is given by the cone over the region of attraction of the maximal control set in projective space containing zero in its spectral interval.
Methods from learning theory are used in the state space of linear dynamical and control systems ... more Methods from learning theory are used in the state space of linear dynamical and control systems in order to estimate the system matrices. An application to stabilization via algebraic Riccati equations is included. The approach is illustrated via a series of numerical examples.
For a class of piecewise deterministic Markov processes, the supports of the invariant measures a... more For a class of piecewise deterministic Markov processes, the supports of the invariant measures are characterized. This is based on the analysis of controllability properties of an associated deterministic control system. Its invariant control sets determine the supports.
IMA Journal of Mathematical Control and Information, 2015
For nonlinear systems described by ordinary di¤erential equations, the paper studies W-control se... more For nonlinear systems described by ordinary di¤erential equations, the paper studies W-control sets which are de…ned as maximal subsets of complete approximate controllability within a safe region or world W in the state space. In particular, their relative invariance properties and their behavior under parameter variations are characterized. An application to invariance entropy shows that the information needed to keep a system in a subset of the state space is determined by the relatively invariant W-control sets.
Volume 3A: 15th Biennial Conference on Mechanical Vibration and Noise — Vibration of Nonlinear, Random, and Time-Varying Systems
The stability behavior of time varying systems can be studied using the concept of Lyapunov expon... more The stability behavior of time varying systems can be studied using the concept of Lyapunov exponents and their corresponding Lyapunov subspaces. For linear time varying systems the entire Lyapunov spectrum can be approximated by the Floquet exponents of periodic systems. This leads to a variety of stability results, including the characterization of stability radii. Furthermore, a structural stability type theorem shows that stability features of time varying hyperbolic systems persist under small perturbations. For nonlinear time varying systems a stable manifold theorem allows us to interpret the linear results for the nonlinear system locally around an equilibrium point.
This paper classi…es continuous linear ‡ows using concepts and techniques from topological dynami... more This paper classi…es continuous linear ‡ows using concepts and techniques from topological dynamics. Speci…cally, the concepts of equivalence and conjugacy are adapted to ‡ows in vector bundles and the Lyapunov decomposition is characterized using the induced ‡ows on the Grassmann and the ‡ag bundles. These results are then applied to bilinear control systems for which their behavior in R d , on the projective space P d 1 and on the Grassmannians is characterized.
For linear flows on vector bundles, it is analyzed when subbundles in the Selgrade decomposition ... more For linear flows on vector bundles, it is analyzed when subbundles in the Selgrade decomposition yield chain transitive subsets for the induced flow on the associated Poincaré sphere bundle.
Controllability properties are studied for control-affine systems depending on a parameter α and ... more Controllability properties are studied for control-affine systems depending on a parameter α and with constrained control values. The uncontrolled systems in dimension two and three are subject to a homoclinic bifurcation. This generates two families of control sets depending on a parameter in the involved vector fields and the size of the control range. A new parameter β given by a split function for the homoclinic bifurcation determines the behavior of these control sets. It is also shown that there are parameter regions where the uncontrolled equation has no periodic orbits, while the controlled systems have periodic solutions arbitrarily close to the homoclinic orbit.
ASME 1995 Design Engineering Technical Conferences collocated with the ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium, Sep 17, 1995
This paper discusses topological conjugacy for skew product ows on topological group bundles and... more This paper discusses topological conjugacy for skew product ows on topological group bundles and semidirect products. One one hand, our results are generalizations of contributions to topological conjugacy of linear and a¢ ne skew product ows, which were mainly motivated by bilinear control systems; see Ayala, Colonius and Kliemann [2] and Colonius and Santana [5], as well as Kawan, Rocio and Santana [7] for ows on Lie groups. On the other hand, there are close analogies to classical (and much more general) results by Lind [9] in the framework of ergodic skew product systems and splitting theorems. Our results may be viewed as topological versions of some of Linds results. The contents of this paper is as follows: In Section 2 some basic properties of topological groups and semidirect products are collected. Section 3 discusses ows on semidirect products, in particular, of Lie groups and topological conjugacy on theses groups. Section 4 generalizes the discussion of topological...
For a class of linear switched systems in continuous time a controllability condition implies tha... more For a class of linear switched systems in continuous time a controllability condition implies that state feedbacks allow to achieve almost sure stabilization with arbitrary exponential decay rates. This is based on the Multiplicative Ergodic Theorem applied to an associated system in discrete time. This result is related to the stabilizability problem for linear persistently excited systems.
Recently, considerable progress has been made in the numerical computation of reachable sets and ... more Recently, considerable progress has been made in the numerical computation of reachable sets and control sets. It is the purpose of this paper to survey some of these developments. In general, computation of reachable sets and control sets is a very difficult problem, since the objects which are to be computed have full dimension in the state space. In turns out that certain reformulated problems which in many cases give the desired objects are numerically easier to handle.
Systems & Control: Foundations & Applications, 2000
The structure of one-dimensional control systems is particularly easy to analyze. One obtains a c... more The structure of one-dimensional control systems is particularly easy to analyze. One obtains a complete picture of the system behavior by studying the dynamics for constant control values, using some calculus. But the class of one-dimensional systems is sufficiently rich to furnish interesting examples for much of the theory developed in the previous chapters, including some not so obvious counterexamples. The concepts of global theory and linearization theory can be characterized explicitly for one-dimensional systems, which allows us to compute control sets, chain control sets, domains of attraction, spectral intervals, and invariant manifolds directly in terms of the system dynamics. With these characterizations we can go one step further and analyze the (local and global) bifurcation structure completely-something that is only partially possible in higher dimensions (see Sections 9.2 and 9.4).
Systems & Control: Foundations & Applications, 2000
In this chapter, the exponential growth behavior of bilinear control systems on vector bundles is... more In this chapter, the exponential growth behavior of bilinear control systems on vector bundles is analyzed. Roughly, these are linear differential equations where the coefficients are determined by a nonlinear control system. Section 6.1 describes the main examples of these systems, in particular, the linearized control systems; introduces the relevant notions; and presents the main results. Section 6.2 describes the Morse spectrum of bilinear control flows on vector bundles and discusses the Floquet spectrum. Section 6.3 characterizes the Lya-punov spectrum under an inner pair condition, and Section 6.4 presents results on local stable manifolds.
Control systems Call be viewed as dynamical systems over (infmite)• dimensional state spaces. Fro... more Control systems Call be viewed as dynamical systems over (infmite)• dimensional state spaces. From this point of view the long tenn behavior of control systems, such as limit sets, Mone sets. approximations on the entire time axis, ergodicity, Lyapunov exponents, stable and unstable manifolds etc. becomes accessible. This paper presents some of the underlying theory, as well as applications 1.0 the global characterization of control systems with bounded control range that are not completely controllable, to control of chaotic systems, and to exponential stability of uncertain systems.
For periodic linear control systems with bounded control range, an autonomized system is introduc... more For periodic linear control systems with bounded control range, an autonomized system is introduced by adding the phase to the state of the system. Here, a unique control set (i.e., a maximal set of approximate controllability) with nonvoid interior exists. It is determined by the spectral subspaces of the homogeneous part which is a periodic linear differential equation. Using the Poincaré sphere, one obtains a compactification of the state space allowing us to describe the behavior “near infinity” of the original control system. Furthermore, an application to quasi-affine systems yields a unique control set with nonvoid interior.
Realistic models of mechanical systems often depend on various parameters, such as controlled inp... more Realistic models of mechanical systems often depend on various parameters, such as controlled inputs, material constants, tunable parameters, and uncertainties. Uncertain parameters can be (time varying) deterministic perturbations or stochastic excitations whose influence on the system depends on the perturbation dynamics (multiplicative or additive), the perturbation range, and its statistics in the stochastic case. For a given operating region of the system, i.e. for a set of state space constraints, the behavior of the system within this region depends strongly on the type of perturbation dynamics and on its range. We present some basic theory for additively and multiplicatively perturbed systems, where the uncertainty can be a family of time varying functions, or a Markov diffusion process. The uncertainty range plays the role of a bifurcation parameter and determines concepts like discontinuities of control sets and supports of invariant measures, stability radii, and invariance radii with respect to the constraint set. It turns out that in many instances the stochastic and the deterministic bifurcation scenarios agree, and the cases in which they differ are related to a nonuniform behavior of the stochastically perturbed system. The example of a model of ship roll motion is treated in detail, revealing some of the fundamental agreements and disagreements of the two bifurcation scenarios.
The region of asymptotic null controllability of bilinear systems with control constraints is cha... more The region of asymptotic null controllability of bilinear systems with control constraints is characterized using Lyapunov exponents. It is given by the cone over the region of attraction of the maximal control set in projective space containing zero in its spectral interval.
Methods from learning theory are used in the state space of linear dynamical and control systems ... more Methods from learning theory are used in the state space of linear dynamical and control systems in order to estimate the system matrices. An application to stabilization via algebraic Riccati equations is included. The approach is illustrated via a series of numerical examples.
For a class of piecewise deterministic Markov processes, the supports of the invariant measures a... more For a class of piecewise deterministic Markov processes, the supports of the invariant measures are characterized. This is based on the analysis of controllability properties of an associated deterministic control system. Its invariant control sets determine the supports.
IMA Journal of Mathematical Control and Information, 2015
For nonlinear systems described by ordinary di¤erential equations, the paper studies W-control se... more For nonlinear systems described by ordinary di¤erential equations, the paper studies W-control sets which are de…ned as maximal subsets of complete approximate controllability within a safe region or world W in the state space. In particular, their relative invariance properties and their behavior under parameter variations are characterized. An application to invariance entropy shows that the information needed to keep a system in a subset of the state space is determined by the relatively invariant W-control sets.
Volume 3A: 15th Biennial Conference on Mechanical Vibration and Noise — Vibration of Nonlinear, Random, and Time-Varying Systems
The stability behavior of time varying systems can be studied using the concept of Lyapunov expon... more The stability behavior of time varying systems can be studied using the concept of Lyapunov exponents and their corresponding Lyapunov subspaces. For linear time varying systems the entire Lyapunov spectrum can be approximated by the Floquet exponents of periodic systems. This leads to a variety of stability results, including the characterization of stability radii. Furthermore, a structural stability type theorem shows that stability features of time varying hyperbolic systems persist under small perturbations. For nonlinear time varying systems a stable manifold theorem allows us to interpret the linear results for the nonlinear system locally around an equilibrium point.
This paper classi…es continuous linear ‡ows using concepts and techniques from topological dynami... more This paper classi…es continuous linear ‡ows using concepts and techniques from topological dynamics. Speci…cally, the concepts of equivalence and conjugacy are adapted to ‡ows in vector bundles and the Lyapunov decomposition is characterized using the induced ‡ows on the Grassmann and the ‡ag bundles. These results are then applied to bilinear control systems for which their behavior in R d , on the projective space P d 1 and on the Grassmannians is characterized.
For linear flows on vector bundles, it is analyzed when subbundles in the Selgrade decomposition ... more For linear flows on vector bundles, it is analyzed when subbundles in the Selgrade decomposition yield chain transitive subsets for the induced flow on the associated Poincaré sphere bundle.
Controllability properties are studied for control-affine systems depending on a parameter α and ... more Controllability properties are studied for control-affine systems depending on a parameter α and with constrained control values. The uncontrolled systems in dimension two and three are subject to a homoclinic bifurcation. This generates two families of control sets depending on a parameter in the involved vector fields and the size of the control range. A new parameter β given by a split function for the homoclinic bifurcation determines the behavior of these control sets. It is also shown that there are parameter regions where the uncontrolled equation has no periodic orbits, while the controlled systems have periodic solutions arbitrarily close to the homoclinic orbit.
ASME 1995 Design Engineering Technical Conferences collocated with the ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium, Sep 17, 1995
This paper discusses topological conjugacy for skew product ows on topological group bundles and... more This paper discusses topological conjugacy for skew product ows on topological group bundles and semidirect products. One one hand, our results are generalizations of contributions to topological conjugacy of linear and a¢ ne skew product ows, which were mainly motivated by bilinear control systems; see Ayala, Colonius and Kliemann [2] and Colonius and Santana [5], as well as Kawan, Rocio and Santana [7] for ows on Lie groups. On the other hand, there are close analogies to classical (and much more general) results by Lind [9] in the framework of ergodic skew product systems and splitting theorems. Our results may be viewed as topological versions of some of Linds results. The contents of this paper is as follows: In Section 2 some basic properties of topological groups and semidirect products are collected. Section 3 discusses ows on semidirect products, in particular, of Lie groups and topological conjugacy on theses groups. Section 4 generalizes the discussion of topological...
For a class of linear switched systems in continuous time a controllability condition implies tha... more For a class of linear switched systems in continuous time a controllability condition implies that state feedbacks allow to achieve almost sure stabilization with arbitrary exponential decay rates. This is based on the Multiplicative Ergodic Theorem applied to an associated system in discrete time. This result is related to the stabilizability problem for linear persistently excited systems.
Recently, considerable progress has been made in the numerical computation of reachable sets and ... more Recently, considerable progress has been made in the numerical computation of reachable sets and control sets. It is the purpose of this paper to survey some of these developments. In general, computation of reachable sets and control sets is a very difficult problem, since the objects which are to be computed have full dimension in the state space. In turns out that certain reformulated problems which in many cases give the desired objects are numerically easier to handle.
Systems & Control: Foundations & Applications, 2000
The structure of one-dimensional control systems is particularly easy to analyze. One obtains a c... more The structure of one-dimensional control systems is particularly easy to analyze. One obtains a complete picture of the system behavior by studying the dynamics for constant control values, using some calculus. But the class of one-dimensional systems is sufficiently rich to furnish interesting examples for much of the theory developed in the previous chapters, including some not so obvious counterexamples. The concepts of global theory and linearization theory can be characterized explicitly for one-dimensional systems, which allows us to compute control sets, chain control sets, domains of attraction, spectral intervals, and invariant manifolds directly in terms of the system dynamics. With these characterizations we can go one step further and analyze the (local and global) bifurcation structure completely-something that is only partially possible in higher dimensions (see Sections 9.2 and 9.4).
Systems & Control: Foundations & Applications, 2000
In this chapter, the exponential growth behavior of bilinear control systems on vector bundles is... more In this chapter, the exponential growth behavior of bilinear control systems on vector bundles is analyzed. Roughly, these are linear differential equations where the coefficients are determined by a nonlinear control system. Section 6.1 describes the main examples of these systems, in particular, the linearized control systems; introduces the relevant notions; and presents the main results. Section 6.2 describes the Morse spectrum of bilinear control flows on vector bundles and discusses the Floquet spectrum. Section 6.3 characterizes the Lya-punov spectrum under an inner pair condition, and Section 6.4 presents results on local stable manifolds.
Control systems Call be viewed as dynamical systems over (infmite)• dimensional state spaces. Fro... more Control systems Call be viewed as dynamical systems over (infmite)• dimensional state spaces. From this point of view the long tenn behavior of control systems, such as limit sets, Mone sets. approximations on the entire time axis, ergodicity, Lyapunov exponents, stable and unstable manifolds etc. becomes accessible. This paper presents some of the underlying theory, as well as applications 1.0 the global characterization of control systems with bounded control range that are not completely controllable, to control of chaotic systems, and to exponential stability of uncertain systems.
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Papers by Fritz Colonius