Model-Matching type-methods and Stability of Networks consisting of non-Identical Dynamic Agents

Many recent approaches of distributed control over networks of dynamical agents rely on the assumption of identical agent dynamics. In this paper we propose a systematic method for removing this assumption, leading to a general approach for distributed-control stabilization of networks of non-identical dynamics. Local agents are assumed to share a minimal set of structural properties, such as input dimension, state dimension and controllability indices, which are generically satisfied for parametric families of systems. Our approach relies on the solution of certain model-matching type problems using local state-feedback and input matrix transformations which map the agent dynamics to a target system, selected to minimize the joint control effort of the local feedback-control schemes. By adapting a well-established distributed LQR control design methodology to our framework, the stabilization problem for a network of non-identical dynamical agents is solved. The applicability of our approach is illustrated via a simple UAV formation control problem

Distributed LQR Methods for Networks of Non-Identical Plants

Two well-established complementary distributed linear quadratic regulator (LQR) methods applied to networks of identical agents are extended to the non-identical dynamics case. The first uses a top-down approach where the centralized optimal LQR controller is approximated by a distributed control scheme whose stability is guaranteed by the stability margins of LQR control. The second consists of a bottom-up approach in which optimal interactions between self-stabilizing agents are defined so as to minimize an upper bound of the global LQR criterion. In this paper, local state-feedback controllers are designed by solving model-matching type problems and mapping all the agents in the network to a target system specified a priori. Existence conditions for such schemes are established for various families of systems. The single-input and then the multi-input case relying on the controllability indices of the plants are first considered followed by an LMI approach combined with LMI regions for pole clustering. Then, the two original top-down and bottom-up methods are adapted to our framework and the stability problem for networks of non-identical dynamical agents is solved. The applicability of our approach for distributed network control is illustrated via a simple example

Distributed LQR design for identical dynamically coupled systems: Application to load frequency control of multi-area power grid

The paper proposes a distributed LQR method for the solution to regulator problems of networks composed of dynamically dependent agents. It is assumed that these dynamical couplings among agents can be expressed in a state-space form of a certain structure. Following a top-down approach we approximate a centralized LQR optimal controller by a distributed scheme the stability of which is guaranteed via a stability test applied to convex combination of Hurwitz matrices. The method is applied to N-identical-area power grid where a distributed state-feedback Load Frequency Controller (LFC) is proposed to achieve frequency regulation under power demand variations. An illustrative numerical example demonstrates the applicability of the method

Distributed Model Predictive Load Frequency Control of multi-area Power Grid: A Decoupling Approach

A model-predictive scheme for load frequency control of a multi-area power system is proposed. The method depends on a decoupling technique which allows for a control design with a distributed architecture. Treating the total power inflows of each area as input variables, a decoupled linearized model for each area is derived. This allows for the formulation and solution of a model predictive control problem with a quadratic performance index and input saturating constraints on the individual tie-line power flows, along with an overall equality constraint to address the energy balance of the network. It is assumed that the interconnection topology (tie-lines) coincides with the communication topology of the network. The only information which needs to be shared between interconnected areas is the local frequency variables. The effectiveness of the method is illustrated via a simulation study of a three-area network. Future work will attempt to establish formally the stability of the control scheme and to enhance the versatility of the method by including constraints on the state variables

Cooperative distributed LQR control for longitudinal flight of a formation of non-identical low-speed experimental UAV's

In this paper, an established distributed LQR control methodology applied to identical linear systems is extended to control arbitrary formations of non-identical UAV's. The nonlinear model of a low-speed experimental UAV known as X-RAE1 is utilized for simulation purposes. The formation is composed of four dynamically decoupled X-RAE1 which differ in their masses and their products of inertia about the xz plane. In order to design linear controllers the nonlinear models are linearized for horizontal flight conditions at constant velocity. State-feedback, input and similarity transformations are applied to solve model-matching type problems and compensate for the mismatch in the linearized models due to mass and symmetry discrepancies among the X-RAE1 models. It is shown that the method is based on the controllability indices of the linearized models. Distributed LQR control employed in networks of identical linear systems is appropriately adjusted and applied to the formation of the nonidentical UAV's. The applicability of the approach is illustrated via numerous simulation results

The use of management science techniques by business organisations with special emphasis on the use in different functional areas : survey in Greek companies

This paper presents an ongoing survey project researching into the extent and nature of use of management science techniques by business organisations in Greece. The survey shows that the usage of MS-techniques in Greek companies is relatively high compared to other European countries. However, the usage of more advanced techniques is very low in Greek firms and the application of MStechniques is concentrated on a few traditional techniques. The reason for the low degree of applicability of more advanced techniques seems to be the general lack of understanding for the importance of these techniques.peer-reviewe

Approximate zero polynomials of polynomial matrices and linear systems

This paper introduces the notions of approximate and optimal approximate zero polynomial of a polynomial matrix by deploying recent results on the approximate GCD of a set of polynomials [1] and the exterior algebra [4] representation of polynomial matrices. The results provide a new definition for the "approximate", or "almost" zeros of polynomial matrices and provide the means for computing the distance from non-coprimeness of a polynomial matrix. The computational framework is expressed as a distance problem in a projective space. The general framework defined for polynomial matrices provides a new characterization of approximate zeros and decoupling zeros [2], [4] of linear systems and a process leading to computation of their optimal versions. The use of restriction pencils provides the means for defining the distance of state feedback (output injection) orbits from uncontrollable (unobservable) families of systems, as well as the invariant versions of the "approximate decoupling polynomials"

Effects of dynamic and non-dynamic element changes in RC and RL networks

The paper deals with the redesign of passive electric networks by changes of single dynamic and non-dynamic elements which may retain or affect the natural topology of the network. It also deals with the effect of such changes on the natural dynamics of the network, the natural frequencies. The impedance and admittance modeling for passive electrical networks is used which provides a structured, symmetric, integral-differential description, which in the special cases of RC and RL networks is reduced to matrix pencil descriptions. The transformations on the network are expressed as those preserving, or modifying the two natural topologies of the network, the impedance graph and the admittance graph topologies. For the special cases of RC and RL networks, we consider the problem of the effect of changes of a single dynamic or non-dynamic element on the natural frequencies. Using the Determinantal Assignment Framework, it is shown that the family of single parameter variation problems is reduced to equivalent Root Locus problems with the possibility of fixed modes. An explicit characterization of the fixed modes is given, and a number of interesting properties of the spectrum are derived such as the interlacing property of poles and zeros for the entire family of Root Locus problems

Approximating distributional behaviour of LTI differential systems using Gaussian function and its derivatives

The paper is concerned with defining families of smooth functions that can be used for the approximation of impulsive types of solutions for linear systems. We review the different types of approximations of distributions in terms of smooth functions and explains their significance in the characterization of system properties where impulses were used for their characterisation. For controllable systems, we establish an interesting relation between the time t and sigma (volatility) in the approximation of distributional solutions. An algorithm is then proposed for the calculation of the coefficients of the input required to minimize the distance of our desired target state before and after approximation is proposed. The optimal choice of sigma is derived for a pre-determined time t for the state transition

Strong stability of internal system descriptions

The paper introduces a new notion of stability for internal autonomous system descriptions that is referred to as “strong stability” and characterizes the case of avoiding overshoots for all initial conditions within a given sphere. This is a stronger notion of stability compared to alternative definitions (asymptotic, Lyapunov), which allows the analysis and design of control systems described by natural coordinates to have no overshooting response for arbitrary initial conditions. For the case of LTI systems necessary and sufficient conditions for strong stability are established in terms of the negative definiteness of the symmetric part of the state matrix. The invariance of strong stability under orthogonal transformations is established and this enables the characterisation of the property in terms of the invariants of the Schur form of the state matrix. Some interesting relations between strong stability and special forms of coordinate frames are examined and links between the skewness of the eigenframe and the violation of strong stability are derived. The characterisation of properties of strong stability given here provides the basis for the development of feedback theory for strong stabilisation