Second-order Linear Time-invariant Systems Research Articles

Model reduction for second-order systems with inhomogeneous initial conditions

In this paper, we consider the problem of finding surrogate models for large-scale second-order linear time-invariant systems with inhomogeneous initial conditions. For this class of systems, the superposition principle allows us to decompose the system behavior into three independent components. The first behavior corresponds to the transfer between the input and output having zero initial conditions. In contrast, the other two correspond to the transfer between the initial position or the initial velocity and the output when no input is applied. Based on this superposition of systems, our goal is to propose model reduction schemes that allow to preserve the second-order structure in the surrogate models. To this aim, we introduce tailored second-order Gramians for each system component and compute them numerically, solving Lyapunov equations. As a consequence, two methodologies are proposed. The first one consists in reducing each of the components independently using a suitable balanced truncation procedure. The sum of these reduced systems provides an approximation of the original system. This methodology allows flexibility on the order of the reduced-order model. The second proposed methodology consists in extracting the dominant subspaces from the sum of Gramians to build the projection matrices leading to a surrogate model. Additionally, we discuss error bounds for the overall output approximation. Finally, the proposed methods are illustrated by means of benchmark problems.

Improved DC-Link Voltage Regulation Strategy for Grid-Connected Converters

In this article, an improved dc-link voltage regulation strategy is proposed for grid-connected converters applied in dc microgrids. For the inner loop of the grid-connected converter, a voltage modulated direct power control is employed to obtain two second-order linear time-invariant systems, which guarantees that the closed-loop system is globally exponentially stable. For the outer loop, a sliding mode control strategy with a load current sensor is employed to maintain a constant dc-link voltage even in the presence of constant power loads at the dc-side, which adversely affect the system stability. Furthermore, an observer for the dc-link current is designed to remove the dc current sensor at the same time improving the reliability and decreasing the cost. From both simulation and experimental results obtained from a 15-kVA prototype setup, the proposed method is demonstrated to improve the transient performance of the system and has robustness properties to handle parameter mismatches compared with the input-output linearization method.

Tuning of a time‐delayed controller for a general class of second‐order linear time invariant systems with dead‐time

In this study, the problem of σ -stabilising a general class of second-order linear time invariant systems with dead-time using a proportional–integral–retarded (PIR) controller is considered. The σ -stability of a system determines the exponential decay in its response. Here the σ -stability of the closed-loop system is ensured by assigning up to four dominant real roots at − σ . For the tuning of the controller gains an extensive analysis of all the possible allocations of the gains according to the response of the closed-loop system is presented. The D-partition method is used to provide important insight into the problem. As a consequence of this analysis, to achieve the desired decay rate, exact analytic expressions for tuning the PIR controller parameters are given. To illustrate the theoretical results obtained, the under-actuated mechanical system called inverted pendulum is used.

Distributed fault detection for interconnected second-order systems

In this paper, the existence of unknown input observers for networks of interconnected second-order linear time invariant systems is studied. Two classes of distributed control systems of large practical relevance are considered. It is proved that for these systems, one can construct a bank of unknown input observers, and use them to detect and isolate faults in the network. The result presents a distributed implementation. In particular, by exploiting the system structure, this work provides further insight into the design of UIO for networked systems. Moreover, the importance of certain network measurements is shown. Infeasibility results with respect to available measurements and faults are also provided, as well as methods to remove faulty agents from the network. Applications to power networks and robotic formations are presented. It is shown how the developed methodology apply to a power network described by the swing equation with a faulty bus. For a multi-robot system, it is illustrated how a faulty robot can be detected and removed.

1423 線形時不変システムのハイブリッド安定化 : 円錐切替え法を用いたアプローチ(GS-10 制御手法,研究発表講演)

For second-order linear time-invariant systems which are not stabilizable via a single static output feedback, we consider a hybrid stabilization strategy. More precisely, we design two controllers and a conic switching law between the controllers so that the entire closed-loop system is asymptotically stable. We demonstrate the design method with various examples, and show that in some cases the stabilizability depends on the region of the initial state.

Second-order balanced truncation

We consider second-order linear time-invariant systems. The objective of this paper is to present a new method for constructing a reduced system by preserving the second-order structure of the original system. This new model reduction method uses a variant of the well-known balanced truncation technique applied to second-order gramians. We also compare it with another existing technique.

Hybrid static output feedback stabilization of second-order linear time-invariant systems

For second-order linear time-invariant (LTI) systems which are not stabilizable via a single static output feedback, we study the open problem whether there exists a finite-state hybrid static output feedback to asymptotically stabilize the system. We show that the answer to this question is affirmative for a class of second-order LTI systems, by constructing a 2-state static output feedback incorporated with a conic switching law.

Common quadratic Lyapunov-like functions with associated switching regions for two unstable second-order LTI systems

In this paper we establish quadratic Lyapunov-like functions for qualitative analysis of switched systems. Specifically, for a class of switched systems consisting of two unstable second-order LTI (linear time-invariant) subsystems, we explore in detail some necessary and sufficient conditions for the existence of common (weakly) quadratic Lyapunov-like functions with associated switching regions in R 2 plane. The existence conditions and the construction of such quadratic Lyapunov-like functions are established by using the conic switching laws.