Controllability Subspaces Research Articles

Decentralized control of interconnected electric power systems in competitive markets

This paper investigates the prospects of decentralized control in the emerging competitive electricity markets. The weaknesses of current control performance criteria for dealing with such markets are discussed and it is hypothesized that decentralized control, providing full control over economically independent electrically interconnected areas, is necessary in such markets. To study the feasibility of such a decentralization rigorously, a state space decomposition method is introduced and applied to the model of an interconnected system. As a result it is shown that, in electric power systems, regardless of the number and location of the tie lines and implemented control schemes, the frequency and phase of each area are in the controllable subspaces of all areas in the interconnection. This shows that, if decentralization is to be achieved, flexible links which decouple the phases of areas must be utilized. The side effects of such implementations are also discussed.

Controllability Indices for Bond Graph Models

A new methodology is proposed for the characterization of the controllability indices of linear multivariate systems. Related to the state space representation, a new symbolism dealing only with numbers associated with the position of non null terms of matrices is proposed. This symbolism associated with the graphical representation which is either a digraph or a bond graph model allows to highlight, from a structural point of view, a list of dimensions of controllable subspaces corresponding one to one with the list of controllability indices. The purpose of this paper is two-fold. The first part is devoted to the characterization of the controllability indices of bond graph models from a structural point of view. In the second part the authors discuss possibilities to associate an input variable to a controllabilty indice.

Cyclic dimensions, controllability subspaces and Gohberg-Kaashoek numbers

A theorem of Macaev and Olshevsky on cyclic dimensions of a matrix is placed in a system theoretic context. Two alternative proofs of the theorem, emerging from ideas from system theory, are presented.

Oblique Production Methods for Large Scale Model Reduction

The aim of this paper is to consider approximating a linear transfer function $F( s )$ of McMillan degree N, by one of McMillan degree m in which $N \gg m$ and where N is large. Krylov subspace methods are employed to construct bases to parts of the controllability and observability subspaces associated with the state space realisation of $F( s )$. Low rank approximate grammians are computed via the solutions to low dimensional Lyapunov equations and computable expressions for the approximation errors incurred are derived. We show that the low rank approximate grammians are the exact grammians to a perturbed linear system in which the perturbation is restricted to the transition matrix, and furthermore, this perturbation has at most rank = 2. This paper demonstrates that this perturbed linear system is equivalent to a low dimensional linear system with state dimension no greater than m. Finally, exact low dimensional expressions for the $\mathcal{L}^\infty $ norm of the errors are derived. The model reduction of discrete time linear systems is considered via the use of the same Krylov schemes. Finally, the behaviour of these algorithms is illustrated on two large scale examples.

Analysis of camera movement errors in vision-based vehicle tracking

When a camera is used to provide the navigational parameters in autonomous vehicle operations, it is subjected to unexpected movements or vibrations of the mounting platform. This paper presents a framework for analyzing the effect of uncontrollable camera movements on the navigational parameters, in particular on the range and-heading angle in vision-based vehicle tracking. The noise introduced by the platform movements is modeled in two ways: camera noise approach and image noise approach. The parameter space of the camera is divided into a controllable subspace consisting of its height and depression angle, and an uncontrollable subspace consisting of the tracked object coordinates and rotation angle errors. A consistent detectable region is then obtained such that the tracked object is always seen by the camera. Based on this region, a reliable region consisting of no singularity points is established so that the range error does not become infinity. The optimum parameters of the controllable subspace with respect to the uncontrollable subspace are found by employing two estimation schemes: (a) the mini-max estimator to provide the worst case effect, and (b) the minimum-mean-square estimator to provide the average or overall effect. From the results obtained, it is shown how an optimum imaging geometry of a monocular vision-based tracking system can be designed in order to satisfy prescribed levels of range and heading angle errors.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Model reduction and process control of thermomechanical behaviour of non‐linear material deformation

AbstractThis paper describes a design procedure for metal forming processes by using the controllable subspace of the full system. The velocity profile of the moving die is designed using the reduced order system. The metal forming processes are simulated using non‐linear finite element methods based on the rigid viscoplastic flow formulation. The balanced model reduction technique is applied to reduce the full state space model to a reduced order model that retains the controllable subspace of the thermomechanical system. The linear quadratic regulator theory with output tracking is used as an off‐line design tool to design the die velocity schedule. The process design is carried out to maintain the strain rate of the critical portion of the billet at a desired value. The procedure for designing the process parameters is demonstrated using two case studies.

Differential periodic Riccati equations: Existence and uniqueness of nonnegative definite solutions

In this paper we consider the differential periodic Riccati equation. All the periodic nonnegative definite solutions are characterized in the more general case, providing a method for constructing them. The method is obtained from the study of the invariant subspaces of the monodromy matrix of the associated Hamiltonian system, and from the relations between these invariant subspaces and the controllability and unobservability subspaces. Finally, the method is applied to obtain necessary and sufficient conditions for the existence of any periodic nonnegative definite solution and to study the existence and uniqueness of minimal, maximal, stabilizing, and strong solutions.

Method for optimal actuator and sensor placement for large flexible structures

A method of finding the optimal sensor and actuator locations for the control of flexible structures is presented. The method is based on the orthogonal projection of structural modes into the intersection subspace of the controllable and observable subspaces corresponding to an actuator/sensor pair. The controllability and observability grammians are then used to weight the projections to reflect the degrees of controllabili ty and observability. This method produces a three-dimensio nal design space wherein sets of optimal actuators and sensors may be selected. A novel parameter is introduced that is potentially useful for studying the problem of the number of actuators and sensors, in addition to their optimal locations. UPPOSE a specific number of actuators and sensors is given and they are placed at specific locations on a flexible structure such that the effectiveness of the chosen actuator and sensor locations could be analyzed. If it turns out that the a priori chosen number and locations for the actuators and sensors are not sufficiently effective, the question naturally arises as to how the locations could be changed to improve the system. Furthermore, it is possible that the a priori number of actuators and/or sensors used is insufficient or redundant. Thus, there is clearly a need for a computationall y feasible technique that is capable of determining an optimal set of locations and the minimal number of actuators and sensors. In general, there will be many more candidate locations (perhaps an order of magnitude more) than the number of actuators and sensors actually available. If the number of actuators and sensors is known a priori, all possible combina- tions could be evaluated, and in principle, the global optimum could then be found. Unfortunately, the number of possible combinations increases factorially, and therefore an exhaus- tive search for a global optimal is usually computationally infeasible, while nonlinear programming based techniques typically produce local minimums. In the past, various definitions of the degree of controllabil- ity and observability have been used in guiding the search for optimal actuator and sensor locations. Among these, the de- gree of controllabili ty defined by scalar measures of recovery regions appears useful for the purpose of actuator and sensor placement.1'3 A second approach4 uses the projection magni- tudes of eigenvectors into the input and output matrices to define gross measures of modal controllability and observabil- ity. However, only little attention is given to the development of a systematic search strategy for actually solving for an optimal set of actuators and sensors, and most attention has been directed toward defining what constitutes most suitable actuator and sensor locations. In this paper, the problem of defining and obtaining the optimal actuator and sensor locations is addressed. A method that is based on the controllability and observability of an actuator/sensor pair is introduced. An outline of the present paper follows. First, the model of actuator and sensor loca- tions for a linear, second-order dynamical system is presented. The basic assumption is that we are given a set of significant modes whose control is desired via feedback. In the next section, controllable, observable, and their intersection sub- spaces are presented, which forms the basis of the method presented in the sequel. The following section presents the main results of this paper. A cost function that is based on the weighted projection of structural modes into the intersection subspace of the controllable and observable subspaces is intro- duced, and a simple interpretation in terms of balanced coor- dinates is given. A novel method for selecting optimal sensor and actuator locations based on the preceding cost function is outlined. The weighted projections of the structural modes can be viewed as a scalar field in three-dimensi onal design space wherein a designer can easily select a set of actuators and sensors based on his or her own criteria without resorting to elaborate nonlinear programming strategies. The method also allows for the comparison of many actuator and sensor candi- date locations since the computational effort depends only on the product of the number of actuator and sensor location candidates rather than combinatorially based search strategies whpse computational effort is in the order of factorials. In the next section, the method of finding optimal locations is ap- plied to an existing laboratory structure to demonstrate the algorithm. Finally, a few concluding remarks are given.

The geometric approach for linear periodic discrete-time systems

Recently, the disturbance localization problem and the problems of designing disturbance decoupled observers and of giving a geometric characterization of invariant zeros were solved for linear periodic discrete-time systems through an extension of the geometric approach, based on the notions of controlled invariant and conditionally invariant subspaces, to periodic ones. Here the existing periodic geometric theory is supplemented with the notions of outer reachable subspace and controllability subspace, and with some further results on the previously introduced notions of inner reachable (controllable) subspace and outer controllable subspace. In addition, it is shown that any such periodic geometric theory can be restated in an equivalent time-invariant form, which is just the same theory written for a suitable time-invariant system having an extended state space.

On the generic dimension of controllable subspaces

It is proved that the generic dimension of the controllable subspace remains unchanged even if the free parameters in a linear system are allowed to vary with time. More precisely, it is proved that the generic rank of the controllability matrix equals the generic rank of a controllability matrix for a discrete, time-varying system. As a by-product, a new proof of S. Hosoe's (1980) characterization of the generic dimension of the controllable subspace is obtained. >

Block noninteracting control with stability via static state feedback

Necessary and sufficient conditions for the existence of a static state feedback that achieves noninteraction and internal stability are obtained. This is accomplished by first characterizing the set of all controllability subspaces (distributions) that can arise as solutions to the noninteracting control problem. This characterization is then used to identify a fixed internal dynamics that is common to every noninteractive closed loop. The stability properties of this dynamics is shown to be the key factor in the problem of achieving noninteracting control with internal stability.

On the zeros and poles of a transfer function

The poles and zeros of a linear transfer function can be studied by means of the pole module and the transmission zero module. These algebraic constructions yield finite dimensional vector spaces whose dimensions are the number of poles and the number of multivariable zeros of the transfer function. In addition, these spaces carry the structure of a module over a ring of polynomials, which gives them a dynamical or state space structure. The analogous theory at infinity gives finite dimensional spaces which are modules over the valuation ring of proper rational functions. Following ideas of Wedderburn and Forney, we introduce new finite dimensional vector spaces which measure generic zeros which arise when a transfer function fails to be injective or surjective. A new exact sequence relates the global spaces of zeros, the global spaces of poles, and the new generic zero spaces. This sequence gives a structural result which can be summarized as follows: “The number of zeros of any transfer function is equal to the number of poles (when everything is counted appropriately).” The same result unifies and extends a number of results of geometric control theory by relating global poles and zeros of general (possibly improper) transfer functions to controlled invariant and controllability subspaces (including such spaces at infinity).

Algebraic characterization of controllability subspaces in the current base and its practical computation

From the concepts of input chain and controllability chain introduced by Heymann and in view of the generalized eigenvector assignment, a new algebraic characterization of controllability subspaces (CSs) is developed. Compared with the previous ones based on matrix pencil theory, it presents both the advantages of being stated in the current base and of being derived without resorting to use of canonical forms. A practical algebraic algorithm is provided in order to compute the controllable structure of a given pair (A, B). This approach is then applied to characterize CSs contained in a given subspace and span CSs via eigenvector assignment.

Generalized Reachability Subspaces for Singular Systems

One of the most important concepts of the geometric theory for proper linear systems is that of the controllability subspaces of $(A,B)$. In an attempt to extend this concept to singular systems, it became clear that the nonequivalence of reachability and controllability for singular systems makes it necessary to define both controllability and reachability subspaces of $(E,A,B)$ as different, though naturally related, entities. A subspace ${\bf R}$ (respectively, ${\bf C}$) is defined to be a generalized reachability subspace (respectively, generalized controllability subspace) if there exist linear maps F and G, such that ${\bf R}$ (respectively, ${\bf C}$) is the reachable (respectively, controllable) subspace of a regular singular system $E\dot x = (A + BF)x + BGv$. It is proved that (1) ${\bf R}$ is a generalized reachability subspace if and only if it is a (generalized) $(A,E,{\bf B})$-invariant almost reachability subspace; (2) every subspace ${\bf K}$ contains a unique supremal generalized reachability subspace ${\bf R}^ * ({\bf K})$; and (3) a subspace ${\bf C} \subset {\operatorname{Im}}{\bf E}$ is a generalized controllability subspace if and only if ${\bf C} = E{\bf R}^ * (E^{ - 1} {\bf K})$. In the case where ${\bf R}$ is a generalized reachability subspace, the F and G matrices that make ${\bf R}$ the reachable subspace of the closed-loop system are constructed explicitly. Spectral assignability properties of generalized reachability and controllability subspaces are also treated. For completeness, all results are finally extended to the nonregular case (i.e., $\det (sE - A) = 0$).

Disturbance Decoupling, $(f,g)$-Invariant and Controllability Subspaces of a Class of Homogeneous Polynomial Systems

This paper discusses control systems of the type \[\dot x = f(x) + \sum_{j = 1}^m {b_i u_i + \sum_{j = 1}^k {d_j \omega _j ,\quad x \in \mathbb{R}^n } } \] where $f(x)$ is a homogeneous polynomial vector field, $b_i $ and $d_i $ are constant vectors, and the output function is linear. It is shown that the well-known theory of disturbance decoupling of linear systems extends to this class in a very natural way. The resulting theory is far simpler than the general nonlinear theory. More important, all computations needed can be done using very simple algorithms, which require only a finite number of computations and use only methods from linear algebra.

Asymptotic orders of reachability in perturbed linear systems

A framework for studying asymptotic orders of reachability in perturbed linear, time-invariant systems is developed. The systems of interest are defined by matrices that have asymptotic expansions in powers of a perturbation parameter epsilon about the point 0. The reachability structure is exposed by means of the Smith form of the reachability matrix. This approach is used to provide insight into the kinds of inputs needed to reach weakly reachable target states, into the structure of high-gain feedback for pole placement, and into the types of inputs that steer trajectories arbitrarily close to almost (A,B)-invariant subspaces and almost (A,B)-controllability subspaces. >

On almost invariant subspaces of structural systems and decentralized control

Graph-theoretic conditions are obtained for a structured system to have the property that the supremal L/sub /p-almost invariant (controllability) subspace is generically the entire state space and that the infimal /sup L//sub /p-almost conditional invariant (complementary observability) subspace is generically the zero subspace. The conditions are used to determine stabilizability of structured interconnected systems by means of decentralized feedback control. Although the obtained graph-theoretic conditions are conservative, they are considered satisfactory to the extent that the benefits of easy testability involving only binary calculations outweigh the conservativeness in the results. >

Zeros and poles of linear periodic multivariable discrete-time systems

In this paper the notions of transmission zero, invariant zero, and structural zero (that is the geometric notion of zero) are extended to linear periodic discrete-time systems, as well as the notion of pole. Their meaning is clarified and their relationships stressed, thus extending the time-invariant theory. In particular, the invariant zeros and the structural zeros are shown to coincide, together with their multiplicities, and the former are shown to be independent of a linear state feedback. Moreover, the nonzero transmission zeros, the nonzero invariant zeros, and the nonzero poles are shown to be independent of time, together with their multiplicities. Some of these results are obtained with the help of the notions of reachability subspace and inner controllable subspace, which constitute a further development of the geometric theory for this class of systems.

Remarks on the periodic orbits of the matrix Riccati equation

We investigate certain questions concerning the periodic structure of the matrix Riccati differential equation with constant coefficients. A closed-form expression for the periodic solutions is obtained for both the cases involving distinct or repeated eigenvalues in the associated linear hamiltonian system. Previous results are extended by establishing that periodic solutions are bounded if and only if the span of their range does not intersect the orthogonal complement of the controllable subspace of the associated linear system.

Disturbance localization by measurement feedback for linear periodic discrete-time systems

The recent notions of inner controllable, outer controllable and outer reconstructible subspaces, together with the extension of the notion of ( C, A, B)-pair of subspaces to the class of linear periodic discrete-time systems, are the main tools used in this paper in order to study the disturbance localization problem by measurement feedback for the same class of systems. By means of these concepts the necessary and sufficient solvability conditions of such a problem and synthesis procedures of the solutions are found for three different cases, i.e. with and without an additional requirement of output or state dead-beat control.