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.     

An algebraic characterization of quotient decentralized fixed modes

In the control of linear time-invariant (LTI) decentralized systems, a decentralized fixed mode (DFM) is one which is immovable using an LTI decentralized controller. However, some DFMs can be moved using more complicated decentralized controllers; the ones which cannot are labelled quotient DFMs (QDFMs), since they arise from a related quotient system. The classical algorithm used to compute the QDFMs requires two steps: a partitioning of the sub-systems, using graph theory, followed by the application of standard tools from decentralized control. Here the goal is to provide a more direct approach to computing them.     

On the quantitative characterization of approximate decentralized fixed modes using transmission zeros

The spectrum of a linear time-invariant multivariable system, using decentralized linear time-invariant controllers, can only be assigned to a symmetric set of complex numbers that include the decentralized fixed modes (DFM). Hence only systems with stable DFM can be stabilized. Although the concept of DFM characterizes when a decentralized controller can stabilize a system, it gives no indication of howhard it is to effect such a stabilization. A system is considered hard to stabilize if large controller gains are required. Modes that are hard to shift are termedapproximate decentralized fixed modes. In this paper two new assignability measures which quantify the difficulty of shifting a mode are derived. The first is coordinate invariant and is based on the distance between a mode and a set of transmission zeros. The second is coordinate dependent and is based on the minimum singular value of a set of transmission zero matrices. This work has been supported by the Natural Sciences are Engineering Research Council of Canada under Grant No. A4396.     

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.     

Overlapping control systems with optimal information exchange

This paper deals with the stabilizability of interconnected systems via linear time-invariant (LTI) decentralized controllers. Given a controllable and observable system with some distinct decentralized fixed modes (DFMs), it is desired to find a suitable control structure (in terms of information flow) for it. Since a decentralized controller consists of a number of non-interacting local controllers, the objective here is to establish certain interactions between the local controllers in order to eliminate the undesirable DFMs. This objective is achieved by translating the knowledge of the system into some bipartite graphs. Then, the notions of minimal sets and maximal subgraphs are introduced, which lead to a simple combinatorial algorithm for solving the underlying problem. The efficacy of the results obtained is demonstrated by an illustrative example.     

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.     

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.     

Overlapping control design for multi-channel systems

This paper deals with the decentralized overlapping control of interconnected systems. The notion of a quotient overlapping fixed mode (QOFM) is first introduced and it is shown that a mode of an interconnected linear time-invariant system can be shifted by means of a general decentralized overlapping controller if and only if it is not a QOFM. It is then asserted that any interconnected system with no unstable QOFM can be stabilized by using an appropriate finite-dimensional linear time-varying controller. It is also shown how the existing results aiming at designing a decentralized controller of a certain type such as generalized sampled-data hold function, finite-dimensional linear time-varying, and sampled-data can be utilized to design a decentralized overlapping controller of a desired form, in order to achieve any design specification. The efficacy of the results is elucidated through two numerical examples.     

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.     

Decentralized switching control for hierarchical systems

In this paper, a decentralized switching scheme is introduced for uncertain interconnected systems with a structure which is “approximately hierarchical”, and where the switching controller acts on the control agents independently of each other. In this problem, it is assumed that the plant at any point of time can be described by a finite set of linear time-invariant (LTI) finite-dimensional models, but that the switching controller does not require any knowledge of these plant models; the only requirement made is that there exists a known finite set of decentralized controllers, containing at least one controller which can stabilize and regulate the actual physical plant at any time. Simulation results obtained for the proposed decentralized switching controller are given.     

Decentralized control of symmetric systems

This paper is concerned with the stabilizability problem of symmetric systems by means of decentralized controllers. It is shown that the set of decentralized fixed modes of a symmetric system is equal to the set of uncontrollable and unobservable modes of the system and implies that the stabilizability of a state-space symmetric system by means of a decentralized controller is equivalent to the stabilizability for the system by means of a centralized controller.     

Special decentralized control problems in discrete-time interconnected systems composed of two subsystems

In this paper, some special decentralized control problems are addressed for discrete-time interconnected systems. First it is pointed out that some subsystems must be unstable to ensure stability of the overall system in some special cases. Then a special kind of decentralized control problem is studied. This kind of problem can be viewed as harmonic control among independent subsystems. Research results show that two unstable systems can generate a stable system through some effective cooperations. In addition, a decentralized controller design method based on linear matrix inequality is also given by using parameter-dependent Lyapunov function method developed for the study of robust stability. A special linear star coupled dynamical network is also considered. The central subsystem must be unstable to stabilize the whole network under a special coupling. Several examples are given to illustrate the results.     

Fundamental limits in robustness and performance for unstable,underactuated systems

We derive bounds on the ℋ_∞ norm of a weighted output sensitivity function and a weighted output complementary sensitivity function for an underactuated, multiple-input-multiple-output (MIMO), linear, time-invariant (LTI), unstable plant in feedback with an internally stabilizing, LTI controller. These bounds indicate a limit to the achievable robustness and performance for such systems and thus, are valuable tools for performing system design tradeoffs     

Low-Order Stabilization of LTI Systems With Time Delay

This paper considers the problem of stabilizing a single-input-single-output (SISO) linear time-invariant (LTI) plant with known time delay using a low-order controller, such as a Proportional (P), a Proportional-Integral (PI), or a proportional-integral-derivative (PID) controller. For the SISO LTI system with time delay, the closed-loop characteristic function is a quasipolynomial that possesses the following features: all its infinite roots are located on the left of certain vertical line of the complex plane, and the number of its unstable roots is finite. Necessary and sufficient conditions for the stability of LTI systems with time delay are first presented by employing an extended Hermite-Biehler Theorem applicable to quasi-polynomials. Based on the conditions, analytical algorithms are then proposed to compute the stabilizing sets of P, PI and PID controllers. The resulting characterizations of the stabilizing sets for P, PI and PID controllers are analogous to the Youla parameterization of all stabilizing controllers for plants without time delay. Numerical examples are provided to illustrate the proposed algorithm.     

On the destabilizing effect of cross-coupling in decentralized adaptive control

The stability of square multivariable linear time-invariant (LTI) systems coupled with an adaptive controller for which the process cross-coupling terms have been neglected is studied in this paper. Our aim is to find conditions over the reference signals richness and the process transfer matrix frequency response such that, for sufficiently small adaptation rates, the stability of the adaptive system is insured. It is shown that if the adaptive system without interconnections is stable then stability of the overall system is preserved if either: (1) The frequency bands of the reference signals are non-overlapping. (2) When there is a spectrum overlap each output of the LTI system to input signals in this spectrum has an average energy where the contribution from its corresponding input is larger than the one due to the other inputs. Global results are obtained for a simple single-parameter adaptation scheme. The extension to general adaptive controllers is straightforward in a local context [5].     

Simultaneous LQ control of a set of LTI systems using constrained generalized sampled-data hold functions

In this paper, sampled-data control of a set of continuous-time LTI systems is considered. It is assumed that a predefined guaranteed continuous-time quadratic cost function, which is, in fact, the sum of the performance indices for all systems, is given. The main objective here is to design a decentralized periodic output feedback controller with a prespecified form, e.g., polynomial, piecewise constant, exponential, etc., which minimizes the above mentioned guaranteed cost function. This problem is first formulated as a set of matrix inequalities, and then by using a well-known technique, it is reformulated as a LMI problem. The set of linear matrix inequalities obtained provides necessary and sufficient conditions for the existence of a decentralized optimal simultaneous stabilizing controller with the prespecified form (rather than a general form). Moreover, an algorithm is presented to solve the resultant LMI problem. Finally, the efficiency of the proposed method is demonstrated in two numerical examples.     

A measure for the decentralized assignability of eigenvalues

The results of [7] are generalized to produce a continuous measure of how ‘close’ a mode is to being a decentralized fixed mode [13]. Implications of this measure on controller design are also explored.     

Decentralized H-infinity state feedback control for discrete-time singular large-scale systems

The decentralized H-infinity control problem for discrete-time singular large-scale systems is considered. Based on the bounded real lemma of discrete-time singular systems, a sufficient condition for the existence of decentralized H-infinity controller for discrete-time singular large-scale systems is presented in terms of the solvability to a certain system of linear matrix inequalities by linear matrix inequality (LMI) approach, and the feasible solutions to the system of LMIs provide a parameterized representation of a set of decentralized H-infinity controller. The given example shows the application of the method.     

连续互联系统的模糊分散控制及新的稳定性条件

针对一类连续模糊互联系统,提出一种模糊分散控制器的设计方法,并给出了保证控制系统稳定的更为宽松的充分条件.应用Lyapunov函数法和线性矩阵不等式,证明了模糊分散控制系统的稳定性.仿真结果进一步验证了所提出的模糊分散控制方法的有效性.