Kenji Kashima

Model Reduction and Clusterization of Large-Scale Bidirectional Networks

This paper proposes two model reduction methods for large-scale bidirectional networks that fully utilize a network structure transformation implemented as positive tridiagonalization. First, we present a Krylov-based model reduction method that guarantees a specified error precision in terms of the H∞-norm. Positive tridiagonalization allows us to derive an approximation error bound for the input-to-state model reduction without computationally expensive operations such as matrix factorization. Second, we propose a novel model reduction method that preserves network topology among clusters, i.e., node sets. In this approach, we introduce the notion of cluster uncontrollability based on positive tridiagonalization, and then derive its theoretical relation to the approximation error. This error analysis enables us to construct clusters that can be aggregated with a small approximation error. The efficiency of both methods is verified through numerical examples, including a large-scale complex network.

Brief paper: Controlled invariant feasibility - A general approach to enforcing strong feasibility in MPC applied to move-blocking

Strong feasibility of MPC problems is usually enforced by constraining the state at the final prediction step to a controlled invariant set. However, such terminal constraints fail to enforce strong feasibility in a rich class of MPC problems, for example when employing move-blocking. In this paper a generalized, least restrictive approach for enforcing strong feasibility of MPC problems is proposed and applied to move-blocking MPC. The approach hinges on the novel concept of controlled invariant feasibility. Instead of a terminal constraint, the state of an earlier prediction step is constrained to a controlled invariant feasible set. Controlled invariant feasibility is a generalization of controlled invariance. The convergence of well-known approaches for determining maximum controlled invariant sets, and j-step admissible sets, is formally proved. Thus an algorithm for rigorously approximating maximum controlled invariant feasible sets is developed for situations where the exact maximum cannot be determined.

System theory for numerical analysis

Many numerical schemes can be suitably studied from a system theoretic point of view. This paper studies the relationship between the two disciplines, that is, numerical analysis and system theory. We first see that various iterative solution schemes for linear and nonlinear equations can be suitably transformed into the form of a closed-loop feedback system, and show the crucial role of the internal model principle in such a context. This leads to new stability criteria for Newtons method. We then study Runge-Kutta type methods for solving differential equations, and also derive new stability criteria based on recent results on LMI. A numerical example is given to illustrate the advantage of the present theory.

A new characterization of invariant subspaces of H 2 and applications to the optimal sensitivity problem

This paper gives a new equivalent characterization for invariant subspaces of H 2 , when the underlying structure is specified by the so-called pseudorational transfer functions. This plays a fundamental role in computing the optimal sensitivity for a certain important class of infinite-dimensional systems, including delay systems. A closed formula, easier to compute than the well-known Zhou–Khargonekar formula, is given for optimal sensitivity for such systems. An example is given to illustrate the result.

Clustered model reduction of positive directed networks

This paper proposes a clustered model reduction method for semistable positive linear systems evolving over directed networks. In this method, we construct a set of clusters, i.e., disjoint sets of state variables, based on a notion of cluster reducibility, defined as the uncontrollability of local states. By aggregating the reducible clusters with aggregation coefficients associated with the Frobenius eigenvector, we obtain an approximate model that preserves not only a network structure among clusters, but also several fundamental properties, such as semistability, positivity, and steady state characteristics. Furthermore, it is found that the cluster reducibility can be characterized for semistable systems based on a projected controllability Gramian that leads to an a priori H 2 -error bound of the state discrepancy caused by aggregation. The efficiency of the proposed method is demonstrated through an illustrative example of enzyme-catalyzed reaction systems described by a chemical master equation. This captures the time evolution of chemical reaction systems in terms of a set of ordinary differential equations.

Clustering-based ℌ 2 -state aggregation of positive networks and its application to reduction of chemical master equations

In this paper, based on a notion of network clustering, we propose a state aggregation method for positive systems evolving over directed networks, which we call positive networks. In the proposed method, we construct a set of clusters (i.e., disjoint sets of state variables) according to a kind of local uncontrollability of systems. This method preserves interconnection topology among clusters as well as stability and some particular properties, such as system positivity and steady-state characteristic (steady-state distribution). In addition, we derive an ℌ2-error bound of the state discrepancy caused by the aggregation. The efficiency of the proposed method is shown through the reduction of a chemical master equation representing the time evolution of the Michaelis-Menten chemical reaction system.

Control of Quantum Systems Despite Feedback Delay

Feedback control (based on the quantum continuous measurement) of quantum systems inevitably suffers from estimation delays. In this technical note we give a delay-dependent stability criterion for a wide class of nonlinear stochastic systems including quantum spin systems. We utilize a semi-algebraic problem approach to incorporate the structure of density matrices. To show the effectiveness of the result, we derive a globally stabilizing control law for a quantum spin-1/2 systems in the face of feedback delays.

On standard H∞ control problems for systems with infinitely many unstable poles

Abstract In this paper, H ∞ control for a class of linear time invariant systems with infinitely many unstable poles is studied. An example of such a plant is a high gain system with delayed feedback. We formulate the problem via a generalized plant which consists of a rational transfer matrix and the inverse of a scalar (possibly irrational) inner function. It is shown that the problem can be decomposed into a finite-dimensional H ∞ control problem and an additional rank condition.

A Hamiltonian-based solution to the mixed sensitivity optimization problem for stable pseudorational plants

This paper considers the mixed sensitivity optimization problem for a class of infinite-dimensional stable plants. This problem is reducible to a two- or one-block H ∞ control problem with structured weighting functions. We first show that these weighting functions violate the genericity assumptions of existing Hamiltonian-based solutions such as the well-known Zhou–Khargonekar formula. Then, we derive a new closed form formula for the computation of the optimal performance level, when the underlying plant structure is specified by a pseudorational transfer function.

Model reduction of multi-input dynamical networks based on clusterwise controllability

This paper proposes a model reduction method for a multi-input linear system evolving on large-scale complex networks, called dynamical networks. In this method, we construct a set of clusters (i.e., disjoint subsets of state variables) based on a notion of clusterwise controllability that characterizes a kind of local controllability of the state-space. The clusterwise controllability is determined through a basis transformation with respect to each input. Aggregating the constructed clusters, we obtain a reduced model that preserves interconnection topology of the clusters as well as some particular properties, such as stability, steady-state characteristic and system positivity. In addition, we derive an H∞-error bound of the state discrepancy caused by the aggregation. The efficiency of the proposed method is shown by a numerical example including a large-scale complex network.