A graph-theoretic method to find decentralized fixed modes of LTI systems

This paper deals with the decentralized pole assignability of interconnected systems by means of linear time-invariant (LTI) controllers. A simple graph-theoretic approach is proposed to identify the distinct decentralized fixed modes (DFMs) of the system, i.e., the unrepeated modes which cannot be moved by means of a LTI decentralized controller. The state-space representation of the system is transformed to the decoupled form using a proper change of coordinates. For any unrepeated mode, a matrix is then computed which resembles the transfer function matrix of the system at some point in the complex plane. A bipartite graph is constructed accordingly in terms of the computed matrix. Now, the problem of verifying if this mode is a DFM of the system reduces to checking if the constructed graph has a complete bipartite subgraph with a certain property. The sole restriction of this work is that it is only capable of identifying the distinct DFMs of a system. However, it is axiomatic that most of the modes of the real-world systems are normally distinct. The primary advantage of the present paper is its simplicity, compared to the existing ones which often require evaluating the rank of several matrices.     

Algebraic characterization of decentralized fixed modes

An input-output, frequency-domain characterization of decentralized fixed modes is given in this paper, using only standard block-diagram algebra, well-known determinantal expansions and the Binet-Cauchy formula.     

Characterizations of decentralized fixed modes for interconnected systems

A study of the characterization of decentralized fixed modes for large-scale interconnected systems is made. This is done by obtaining a recursive characterization of decentralized fixed modes, in which the existence of fixed modes of a ν-control agent system is expressed in terms of the existence of fixed modes of ν ? 1 control agent systems. An interpretation of these conditions is then made in the frequency domain. Simple conditions, in terms of the controllability of the system's subsystems, are then found for a composite system consisting of ν interconnected subsystems to have no decentralized fixed modes. This result clarifies some recent discussion on the fixed modes of interconnected systems; in particular, it proves that Fessas's conjecture is true for systems with two control agents, but false for systems with more than two control agents.     

Discrete-time control of continuous systems with approximate decentralized fixed modes

In this paper, the discrete-time control of decentralized continuous-time systems, which have approximate decentralized fixed modes, is studied. It is shown that under certain conditions, discrete-time controllers can improve the overall performance of the decentralized control system, when a linear time-invariant continuous-time controller is ineffective. In order to obtain these conditions, a quantitative measure for different types of approximate fixed modes in a decentralized system is given. In this case, it is shown that discrete-time zero-order hold (ZOH) controllers, and in particular, that generalized sampled-data hold functions (GSHF), can significantly improve the overall performance of the resultant closed-loop system. The proposed sampled-data controller is, in fact, a linear time-varying controller for the continuous-time system.     

Decomposition and decentralized control of systems with multi-overlapping structure

This paper considers decomposition and decentralized control of systems with multi-overlapping structure. It is demonstrated, using the inclusion principle, how the systems with longitudinal, loop and radial topologies can be expanded, and how the results can be used for designing controllers under information structure constraints. The proposed methodology is applied to automatic generation control (AGC) of an electric power system.     

Decentralised fixed modes of networked MIMO systems

In this paper, decentralised fixed modes (DFMs) of a networked system are studied. The network topology is directed and weighted and the nodes are higher-dimensional linear time-invariant (LTI) dynamical systems. The effects of the network topology, the node-system dynamics, the external control inputs, and the inner interactions on the existence of DFMs for the whole networked system are investigated. A necessary and sufficient condition for networked multi-input/multi-output (MIMO) systems in a general topology to possess no DFMs is derived. For networked single-input/single-output (SISO) LTI systems in general as well as some typical topologies, some specific conditions for having no DFMs are established. It is shown that the existence of DFMs is an integrated result of the aforementioned relevant factors which cannot be decoupled into individual DFMs of the node-systems and the properties solely determined by the network topology.     

Control of continuous-time LTI systems by means of structurally constrained controllers

This paper deals with the characterization of the fixed modes of multi-channel systems with respect to linear time-invariant (LTI) structurally constrained controllers. Fixed modes can be found numerically for any LTI system with respect to any given control structure, using a random number generator. The existing analytical methods, however, are not capable of characterizing the fixed modes in the most general case of non-strictly proper systems with non-block diagonal (i.e., overlapping) control structure, efficiently. The notion of a decentralized overlapping fixed mode (DOFM) is introduced in this paper to address the above problem in the most general case. To this end, the knowledge of the overlapping control structure is translated into a bipartite graph, whose vertices correspond to the input and output vectors of various control channels. An efficient technique is applied to the obtained graph to identify the DOFMs of the system. It is to be noted that a system is stabilizable via an appropriate LTI decentralized overlapping controller if and only if it does not have any unstable DOFM. Moreover, it is shown how those modes which are not DOFMs can be placed freely in the complex plane using a proper LTI decentralized overlapping controller. The efficacy of this work is demonstrated through an example.     

Periodic compensation of a class of decentralized systems with fixed modes

This paper first finds that the high frequency (ω) periodic controller suggested in Lee, Meerkov, and Runolfsson (1987) for fixed mode removal, and pole and zero placement, of a class of decentralized systems (A,B_i,C_i) all the strongly interconnected channels of which satisfy C_iB_i=0, causes large, O(ω), oscillations in the outputs. Next it proposes an alternative controller, a dual of the above, that achieves the compensation with oscillations of O(1) only.     

Stabilization of distributed control systems with delay

In this paper the asymptotic stabilization of linear distributed parameter control systems with delay is considered. Specifically, we are concerned with the class of control systems described by the equation x^′(t)=Ax(t)+L(x_t)+Bu(t), where A is the infinitesimal generator of a strongly continuous semigroup on a Banach space X. Assuming appropriate conditions, we will show that the usual spectral controllability assumption implies the feedback stabilization of the system. Applications to systems described by partial differential equations with delay are given.     

Stabilization of decentralized control systems by means of periodic feedback

This paper deals with structurally constrained periodic control design for interconnected systems. It is assumed that the system is linear time-invariant (LTI), observable and controllable, and that its modes are distinct and nonzero. It is shown that the notions of a quotient fixed mode (QFM) and a structured decentralized fixed mode (SDFM) are equivalent for this class of systems. Then, it is proved that if the system is decentrally stabilizable, then one candidate for the decentralized stabilizing controller is a time-varying one consisting of a decentralized LTI discrete-time compensator and a zero-order hold. More specifically, the non-quotient fixed modes of the system will be eliminated via sampling for almost all sampling periods, while any QFM will still remain a fixed mode. The results obtained are ultimately extended to the case when the system has some repeated modes, none of which is a DFM.     

Stabilizing decentralized model predictive control of nonlinear systems

This note presents a stabilizing decentralized model predictive control (MPC) algorithm for nonlinear discrete time systems. No information is assumed to be exchanged between local control laws. The stability proof relies on the inclusion of a contractive constraint in the formulation of the MPC problem.     

Stabilization of positive linear systems

We consider stabilization of equilibrium points of positive linear systems which are in the interior of the first orthant. The existence of an interior equilibrium point implies that the system matrix does not possess eigenvalues in the open right half plane. This allows to transform the problem to the stabilization problem of compartmental systems, which is known and for which a solution has been proposed already. We provide necessary and sufficient conditions to solve the stabilization problem by means of affine state feedback. A class of stabilizing feedbacks is given explicitly.     

Stabilization of continuous-time switched nonlinear systems

The paper considers three problems for continuous-time nonlinear switched systems. The first result of this paper is a open-loop stabilization strategy based on dwell time computation. The second considers a state switching strategy for global stabilization. The strategy is of closed loop nature (trajectory dependent) and is designed from the solution of what we call nonlinear Lyapunov–Metzler inequalities from which the stability condition is expressed. Finally, results on the stabilization of nonlinear time varying polytopic systems are provided.     

Stabilization of nonholonomic chained systems via nonregular feedback linearization

This paper addresses the problem of feedback stabilization of nonholonomic chained systems within the framework of nonregular feedback linearization. Firstly, the nonsmooth version of nonregular feedback linearization is formulated, and a criterion for nonregular feedback linearization is provided. Then, it is proved that the chained form is linearizable via nonregular feedback control, thus enable us to design feedback control laws using standard techniques for linear systems. The obtained discontinuous control laws guarantee convergence of the closed-loop system with exponential rates. Finally, simulation results are presented to show the effectiveness of the approach.     

Stabilization of linear discrete-time systems with actuator saturation

This paper investigates the stabilization problem of linear discrete-time systems with actuator saturation. A simple method recently developed for analyzing the elemental perturbation bound of robust pole-assignment is extended to obtain the stabilization condition. The state feedback controller is designed to satisfy the requirement of stability under the nonlinear saturation. An example is given to demonstrate the use of the proposed approach.     

Stabilization of linear systems over networks with bounded packet loss

This paper is concerned with the stabilization problem of networked control systems where the main focus is the packet-loss issue. Two types of packet-loss processes are considered. One is the arbitrary packet-loss process, the other is the Markovian packet-loss process. The stability conditions of networked control systems with both arbitrary and Markovian packet losses are established via a packet-loss dependent Lyapunov approach. The corresponding stabilizing controller design techniques are also given based upon the stability conditions. These results are also extended to the unit time delay case. Finally, the numerical example and simulations have demonstrated the usefulness of the developed theory.     

On robust stability of LTI fractional-order delay systems of retarded and neutral type

This paper deals with the analysis of robust BIBO-stability of LTI fractional order delay systems in the presence of real parametric uncertainties. Two large classes of these systems, namely retarded and neutral types, are considered. Two theorems are given to check the robust BIBO-stability of these two families of fractional order systems. One of these theorems provides necessary and sufficient conditions for the case of retarded type and another one presents only sufficient conditions for the case of neutral type. Furthermore, upper and lower bounds (cutoff frequencies) are provided for drawing the value sets. To illustrate the results, two numerical examples are presented.     

Stabilization of nonlinear systems: A bilinear approach

This paper gives a necessary and sufficient condition for the stabilization of planar bilinear systems. Local stabilization results are obtained for other systems.