Takayuki Ishizaki

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.

Hierarchical distributed control for networked linear systems

In this paper, we propose a design method of hierarchical distributed controllers for networked linear systems. The hierarchical distributed controller has an advantage that an L2-performance of the closed-loop system improves as improving a performance of local controllers that stabilize disjoint subsystems individually. Towards systematic design, we utilize state-space expansion that enables us to deal with the state variables associated with disjoint subsystems and those associated with interference among hierarchically clustered subsystems in a tractable manner. Moreover, by the integration of a hierarchical distributed observer having good compatibility with the structured controller, we build a framework to implement an observer-based hierarchical distributed control. The efficiency of the proposed control system is shown through an example of power networks.

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.

Glocal (global/local) control synthesis for hierarchical networked systems

This paper is concerned with how to develop a new system control theory for providing systematic design methods of hierarchical networked systems composed of various kinds of subsystems from the glocal (global/local) control view point. Through examinations of energy network control we first explain the idea and concept of glocal control and propose a structure of glocal control system consisting multi-resolved sensor, glocal predictor, and glocal controller. We then provide methods of designing glocal predictor and glocal controller which fit the proposed structure with numerical examples.

Spatiotemporally Multiresolutional Optimization Toward Supply-Demand-Storage Balancing Under PV Prediction Uncertainty

Large-scale penetration of photovoltaic (PV) power generators and storage batteries is expected in recently constructed power systems. For the realization of smart energy management, we need to make an appropriate day-ahead schedule of power generation and battery charge cycles based on the prediction of demand and PV power generation, which inevitably involves nontrivial prediction errors. With this background, a novel framework is proposed to maintain the balance among the total amounts of power generation, demand, and battery charging power with explicit consideration of the prediction uncertainty, assuming that consumer storage batteries are not directly controllable by a supplier. The proposed framework consists of the following three steps: 1) the day-ahead scheduling of the total amount of generation power and battery charging power; 2) the day-ahead scheduling of utility energy consumption requests to individual consumers, which aim to regulate battery charging cycles on the consumer side; and 3) the incentive-based management of the entire power system on the day of interest. In this paper, we especially focus on the day-ahead scheduling problems in steps 1) and 2), and show that they can be analyzed in a manner originating from spatiotemporal aggregation. Finally, we demonstrate the validity of the proposed framework through numerical verification of the power system management.

Planning of Optimal Daily Power Generation Tolerating Prediction Uncertainty of Demand and Photovoltaics.

Abstract The concern with renewable energy has been growing. Large-scale installation of photovoltaic (PV) generation and electricity storage is expected to be installed into the power system in Japan. In this situation, we need to keep supply-demand balance by systematically using not only traditional power generation systems but also the PV generation and storage equipment. Towards this balancing, a number of prediction methods for PV generation and demand have been developed in literature. However, prediction-based balancing is not necessarily easy. This is because the prediction of PV generation and the demand forecasting inevitably includes some uncertainty. Against this background, we formulate a problem to plan battery charge pattern while minimizing the fuel cost of generators with explicit consideration of prediction uncertainty. In this problem, given as interval quadratic programming, the prediction uncertainty is described as a parameter in constraint condition. Furthermore, we propose a method to find a solution to this problem from the viewpoint of monotonicity analysis. Finally, by numerical analysis based on this problem and its solution method, we discuss the relation between the minimal regulating capacity and the required battery charge/discharge pattern to tolerate a given amount of prediction uncertainty.

Structured model reduction of interconnected linear systems based on singular perturbation

This paper proposes a singular perturbation approximation that preserves system passivity and an interconnection topology among subsystems. In the first half of this paper, we develop a singular perturbation approximation valid for stable linear systems. Using the relation between the singular perturbation and the reciprocal transformation, we derive a tractable expression of the error system in the Laplace domain, which provides a novel insight to regulate the approximating quality of reduced models. Then in the second half, we develop a structured singular perturbation approximation that focuses on a class of interconnected systems. This structured approximation provides a reduced model that not only possesses fine approximating quality, but also preserves the original interconnection topology and system passivity.

Reaction-diffusion clustering of single-input dynamical networks

A novel clustering method for single-input dynamical networks is proposed, where we aggregate state variables that behave similarly for any input signals. This clustering method is based on the Reaction-Diffusion transformation, which can be applied to large-scale networks, and preserves the stability as well as a kind of network structure of the original system. In addition, the upper bound of the state discrepancy caused by the clustering is evaluated in terms of H∞-norm.

Dissipativity-Preserving Model Reduction for Large-Scale Distributed Control Systems

We propose a dissipativity-preserving structured model reduction method for distributed control systems. As a fundamental tool to develop structured model reduction, we first establish dissipativity-preserving model reduction for general linear systems on the basis of a singular perturbation approximation. To this end, by deriving a tractable expression of singular perturbation models, we characterize dissipativity preservation in terms of a projection-like transformation of storage functions, and we show that the resultant approximation error is relevant to the sum of neglected eigenvalues of an index matrix. Next, utilizing this dissipativity-preserving model reduction, we develop a structured controller reduction method for distributed control systems. The major significance of this method is to preserve the spatial distribution of dissipative controllers and to provide an a priori bound for the performance degradation of closed-loop systems in terms of the H2-norm. The efficiency of the proposed method is verified through a numerical example of vibration suppression control for interconnected second-order systems.