Differential periodic Riccati equations: Existence and uniqueness of nonnegative definite solutions

In this paper we consider the differential periodic Riccati equation. All the periodic nonnegative definite solutions are characterized in the more general case, providing a method for constructing them. The method is obtained from the study of the invariant subspaces of the monodromy matrix of the associated Hamiltonian system, and from the relations between these invariant subspaces and the controllability and unobservability subspaces. Finally, the method is applied to obtain necessary and sufficient conditions for the existence of any periodic nonnegative definite solution and to study the existence and uniqueness of minimal, maximal, stabilizing, and strong solutions.This work has been partially supported by Spanish DGICYT Grant No. PB91-O535.     

Solving the infinite-dimensional discrete-time algebraic riccati equation using the extended symplectic pencil

In this paper we present results about the algebraic Riccati equation (ARE) and a weaker version of the ARE, the algebraic Riccati system (ARS), for infinite-dimensional, discrete-time systems. We introduce an operator pencil, associated with these equations, the so-called extended symplectic pencil (ESP). We present a general form for all linear bounded solutions of the ARS in terms of the deflating subspaces of the ESP. This relation is analogous to the results of the Hamiltonian approach for the continuous-time ARE and to the symplectic pencil approach for the finite-dimensional discrete-time ARE. In particular, we show that there is a one-to-one relation between deflating subspaces with a special structure and the solutions of the ARS. Using the relation between the solutions of the ARS and the deflating subspaces of the ESP, we give characterizations of self-adjoint, nonnegative, and stabilizing solutions. In addition we give criteria for the discrete-time, infinite-dimensional ARE to have a maximal self-adjoint solution. Furthermore, we consider under which conditions a solution of the ARS satisfies the ARE as well.     

Almost invariant subspaces: An approach to high gain feedback design--Part I: Almost controlled invariant subspaces

In a previous paper [1] we have introduced the notion of "almost controlled invariant subspaces" which are subspaces to which one can steer the state of a linear system arbitrarily close. In the present paper we will show how these subspaces my be viewed as ordinary controlled invariant subspaces when one allows distributional inputs, or as those subspaces which can be approximated by controlled invariant subspaces. The results are applied to a number of control synthesis problems, i.e., disturbance decoupling, robustness, noisy gain stabilization, and cheap control. Part II of the paper will treat the dual theory of almost conditionally invariant subspaces.     

Measures of controllability and observability and residues

The relationship between two measures of controllability and observability is derived, given a linear time-invariant system. Both of the measures are based on angles between various subspaces of the state space. It is shown that these measures can be used to bound the norm of the residues of the transfer function     

Stabilization of exponentially unstable linear systems withsaturating actuators

We study the problem of stabilizing exponentially unstable linear systems with saturating actuators. The study begins with planar systems with both poles exponentially unstable. For such a system, we show that the boundary of the domain of attraction under a saturated stabilizing linear state feedback is the unique stable limit cycle of its time-reversed system. A saturated linear state feedback is designed that results in a closed-loop system having a domain of attraction that is arbitrarily close to the null controllable region. This design is then utilized to construct state feedback laws for higher order systems with two exponentially unstable poles     

Structural Invariant Subspaces of Singular Hamiltonian Systems and Nonrecursive Solutions of Finite-Horizon Optimal Control Problems

This note introduces an analytic, nonrecursive approach to the solution of finite-horizon optimal control problems formulated for discrete-time stabilizable systems. The procedure, which adapts to handle both the case where the final state is weighted by a generic quadratic function and the case where the final state is an admissible, sharply assigned one, provides the optimal control sequences, as well as the corresponding optimal state trajectories, in closed form, as functions of time, by exploiting an original characterization of a pair of structural invariant subspaces associated to the singular Hamiltonian system. The results hold on the fairly general assumptions which guarantee the existence and uniqueness of the stabilizing solution of the corresponding discrete algebraic Riccati equation and, as a consequence, solvability of an appropriately defined symmetric Stein equation. Some issues to be considered in the numerical implementation of the proposed approach are mentioned. The application of the suggested methodology to $mmb H_2$ optimal rejection with preview is also discussed.      

Dynamic Output Feedback Consensualization of Uncertain Swarm Systems with Time Delays

Consensus analysis and design problems of high‐dimensional discrete‐time swarm systems in directed networks with time delays and uncertainties are dealt with by using output information. Two subspaces are introduced, namely a consensus subspace and a complement consensus subspace. By projecting the state of a swarm system onto the two subspaces, a necessary and sufficient condition for consensus is presented, and based on different influences of time delays and uncertainties, an explicit expression of the consensus function is given which is very important in applications of swarm systems. A method to determine gain matrices of consensus protocols is proposed. Numerical simulations are presented to demonstrate theoretical results.     

Sufficient lyapunov-like conditions for stabilization

In this paper we study the stabilizability problem for nonlinear control systems. We provide sufficient Lyapunov-like conditions for the possibility of stabilizing a control system at an equilibrium point of its state space. The stabilizing feedback laws are assumed to be smooth except possibly at the equilibrium point of the system.     

Hybrid Linear Modeling via Local Best-Fit Flats

We present a simple and fast geometric method for modeling data by a union of affine subspaces. The method begins by forming a collection of local best-fit affine subspaces, i.e., subspaces approximating the data in local neighborhoods. The correct sizes of the local neighborhoods are determined automatically by the Jones?? ?? ₂ numbers (we prove under certain geometric conditions that our method finds the optimal local neighborhoods). The collection of subspaces is further processed by a greedy selection procedure or a spectral method to generate the final model. We discuss applications to tracking-based motion segmentation and clustering of faces under different illuminating conditions. We give extensive experimental evidence demonstrating the state of the art accuracy and speed of the suggested algorithms on these problems and also on synthetic hybrid linear data as well as the MNIST handwritten digits data; and we demonstrate how to use our algorithms for fast determination of the number of affine subspaces.     

Overapproximating Reachable Sets by Hamilton-Jacobi Projections

In earlier work, we showed that the set of states which can reach a target set of a continuous dynamic game is the zero sublevel set of the viscosity solution of a time dependent Hamilton-Jacobi-Isaacs (HJI) partial differential equation (PDE). We have developed a numerical tool—based on the level set methods of Osher and Sethian—for computing these sets, and we can accurately calculate them for a range of continuous and hybrid systems in which control inputs are pitted against disturbance inputs. The cost of our algorithm, like that of all convergent numerical schemes, increases exponentially with the dimension of the state space. In this paper, we devise and implement a method that projects the true reachable set of a high dimensional system into a collection of lower dimensional subspaces where computation is less expensive. We formulate a method to evolve the lower dimensional reachable sets such that they are each an overapproximation of the full reachable set, and thus their intersection will also be an overapproximation of the reachable set. The method uses a lower dimensional HJI PDE for each projection with a set of disturbance inputs augmented with the unmodeled dimensions of that projection's subspace. We illustrate our method on two examples in three dimensions using two dimensional projections, and we discuss issues related to the selection of appropriate projection subspaces.     

Ultimate bound and optimal measurement for estimation of coupling constant in Tavis–Cummings model

We seek to study the problem of estimating the atoms-field coupling constant in Tavis–Cummings model for interaction between two atoms and an electromagnetic field by means of local estimation theory. We calculate the quantum Fisher information (QFI) for the most general pure probe state that undergoes evolution generated by the Hamiltonian of the Tavis–Cummings model; then, proper probe states which maximize the QFI are determined. Furthermore, we consider subspaces separately and show that QFI for atomic subspace (contains both qubits) and cavity field subspace can reach the maximum value of QFI in the whole space by choosing proper initial state. Finally, the optimal measurement that saturates the Cramer–Rao bound, i.e., the measurement with Fisher information equal to QFI, for considered states are determined in the whole space and the subspaces, separately.     

Algebraic decompositions of DP problems with linear dynamics

We consider Dynamic Programming (DP) problems in which the dynamics are linear, the cost is a function of the state, and the state-space is finite dimensional but defined over an arbitrary field. Starting from the natural decomposition of the state-space into a direct sum of invariant subspaces consistent with the rational canonical form, and assuming the cost functions exhibit an additive structure compatible with this decomposition, we extract from the original problem two distinct families of smaller DP problems associated with the invariant subspaces. Each family constitutes a decomposition of the original problem when the optimal policy and value function can be reconstructed from the optimal policies and value functions of the smaller problems. We derive necessary and sufficient conditions for these decompositions to exist, propose a readily verifiable sufficient condition for the first decomposition, and establish a hierarchy relating the two notions of decomposition.     

Nash equilibria in stabilizing systems

The objective of this paper is three-fold. First, we specify what it means for a fixed point of a stabilizing distributed system to be a Nash equilibrium. Second, we present methods that can be used to verify whether or not a given fixed point of a given stabilizing distributed system is a Nash equilibrium. Third, we argue that in a stabilizing distributed system, whose fixed points are all Nash equilibria, no process has an incentive to perturb its local state, after the system reaches one fixed point, in order to force the system to reach another fixed point where the perturbing process achieves a better gain. If the fixed points of a stabilizing distributed system are all Nash equilibria, then we refer to the system as perturbation-proof. Otherwise, we refer to the system as perturbation-prone. We identify four natural classes of perturbation-(proof/prone) systems. We present system examples for three of these classes of systems, and show that the fourth class is empty.     

Robust Detectability From the Measurements Plus State Feedback Stabilization Imply Semiglobal Stabilization From the Measurements

We study the problem of stabilizing with large regions of attraction a general class of nonlinear system consisting of a linear nominal system plus uncertainties. A similar result was given by the same author in previous works; in this note, we prove that what was referred in these works to as “nonlinear coupling condition” can be reformulated in the control design as a “nonlinear rescaling” of the Lyapunov functions of the closed-loop system plus the requirement for a suitably faster convergence of the state estimation error. We obtain a paradigm very similar to the linear case, for which if a couple of Riccati-like inequalities (state feedback and observer design) are satisfied then a measurement feedback stabilizing controller can be readily found. Examples are given for showing improvements over the existing literature.     

Stabilization of Markovian jump linear system over networks with random communication delay

This paper is concerned with the stabilization problem for a networked control system with Markovian characterization. We consider the case that the random communication delays exist both in the system state and in the mode signal which are modeled as a Markov chain. The resulting closed-loop system is modeled as a Markovian jump linear system with two jumping parameters, and a necessary and sufficient condition on the existence of stabilizing controllers is established. An iterative linear matrix inequality (LMI) approach is employed to calculate a mode-dependent solution. Finally, a numerical example is given to illustrate the effectiveness of the proposed design method.     

Coprime factorization and robust stabilization for discrete-time infinite-dimensional systems

We solve the problem of robust stabilization with respect to right-coprime factor perturbations for irrational discrete-time transfer functions. The key condition is that the associated dynamical system and its dual should satisfy a finite-cost condition so that two optimal cost operators exist. We obtain explicit state space formulas for a robustly stabilizing controller in terms of these optimal cost operators and the generating operators of the realization. Along the way we also obtain state space formulas for Bezout factors.     

A New Approach to Two-View Motion Segmentation Using Global Dimension Minimization

We present a new approach to rigid-body motion segmentation from two views. We use a previously developed nonlinear embedding of two-view point correspondences into a 9-dimensional space and identify the different motions by segmenting lower-dimensional subspaces. In order to overcome nonuniform distributions along the subspaces, whose dimensions are unknown, we suggest the novel concept of global dimension and its minimization for clustering subspaces with some theoretical motivation. We propose a fast projected gradient algorithm for minimizing global dimension and thus segmenting motions from 2-views. We develop an outlier detection framework around the proposed method, and we present state-of-the-art results on outlier-free and outlier-corrupted two-view data for segmenting motion.