A complement about almost controllability subspaces

We show that, for any almost controllability subspace, there exist a maximal input chain and a family of state feedbacks which lead to the well known ‘feedback-chain’ expression for this subspace.     

Design of a minimal precompensator for the decoupling problem

For right invertible linear time-invariant systems, which cannot be decoupled row by row using regular static state feedback laws, we propose a new algorithmic procedure for designing a minimal precompensator leading to a compensated system which can be decoupled by regular static state feedback.     

Some results on the genericity of the pole placement problem

In this note, the pole placement problem for a linear MIMO systems with p outputs and m inputs is studied from the algebraic point of view. A formulation is proposed, that allows to analyze both theoretical and numerical aspects of the case min(m,p)=2 with more sharpness. Moreover, here it is shown that arbitrary pole placement by static output feedback of unitary rank is generically not possible even if m+p>n holds true.     

Unknown-state, unknown-input reconstruction in discrete-time nonminimum-phase systems: Geometric methods

The complete solution to the unknown-state, unknown-input reconstruction problem in systems with invariant zeros is inherently conditioned by the fact that, for any invariant zero, at least one initial state exists, such that the output is not affected when the mode of the invariant zero is properly injected into the system. Despite this intrinsic limitation, the problem of reconstructing the initial state and the inaccessible inputs from the available measurements has recently attracted remarkable interest, owing to its impact on the synthesis of enhanced-reliability control systems. This contribution consists of a geometric method which solves the unknown-state, unknown-input reconstruction problem in discrete-time systems with invariant zeros anywhere in the complex plane, except the unit circumference. The case of systems with the invariant zeros in the open set outside the unit disc is regarded as the basic one. The difficulties related to the presence of those invariant zeros are overcome by accepting a reconstruction delay commensurate to the invariant zero time constants and the accuracy required for reconstruction. The solution devised for that case also applies to systems without invariant zeros. However, in this case, reconstruction is exact and the delay depends on the number of iterations needed for a certain conditioned invariant algorithm to converge. Finally, the more general case of systems with invariant zeros lying anywhere in the complex plane, with the sole exception of the unit circumference, is reduced to the fundamental one through the synthesis of an appropriate filter.     

Further decomposition for singular systems and its properties on geometric subspace

This paper focuses on the relationship between the geometric subspaces and the structural decomposition of continuous-time singular systems. The original structural decomposition is not capable of revealing explicitly the invariant geometric subspaces for singular systems. As such, a further decomposition is necessary and is thus investigated in this paper. Under a new decomposition proposed, the supremal output-nulling (A,E, ImB)-invariant subspace of singular systems can be clearly expressed in an explicit form, and some of its applications are also addressed.     

New results on delay-dependent robust stability criteria of uncertain fuzzy systems with state and input delays

This paper focuses on the delay-dependent stability analysis and stabilization for uncertain T-S fuzzy control systems with state and input delays. The key features of the approach include the introduction of uncorrelated augmented matrix items into the Lyapunov functional and the use of a tighter bounding technology. In fact, these techniques lead to more general and less conservative stability conditions that guarantee a wide stability region. Our delay-dependent stability conditions thus obtained are given in terms of linear matrix inequalities (LMIs). Finally, some examples are included to show that the proposed criteria improve the existing results significantly.     

Some new results on the algebraic characterizations of an equality constrained optimization problem equivalent to the transportation problem

An equality constrained optimization problem equivalent to the transportation problem with m sources and n destinations is described. The optimality condition and some algebraic characterizations of the problem are investigated using its Hessian matrix. In addition, several algebraic characterizations of an equivalent case of the transportation problem are given using the spectral decomposition and generalized inverses of its coefficient matrix. It is shown that the transportation problem and its equivalent case have common algebraic characterizations.     

Further results on structural assignment of linear systems via sensor selection

The problem of assigning structural properties of a linear system through sensor selection is, for a given pair (A,B), to find an output pair (C,D) such that the resulting system (A,B,C,D) has the pre-specified structural properties, such as the finite and infinite zero structures and the invertibility properties. In this paper, by introducing the notion of infinite zero assignable sets for the pair (A,B), we establish necessary and sufficient conditions for the assignability of a given set of infinite zeros and a set of structural properties which includes the left invertibility property. In establishing these conditions, we develop a numerical algorithm for the construction of the required (C,D).     

Further results on the algebraic characterizations of a minimum cost network flow problem equivalent to the transportation problem

In this study a minimum cost network flow problem with m+n+2 nodes and mn arcs, which is equivalent to the transportation problem with m sources and n destinations, is described as an axial four-index transportation problem of order 1×m×n×1. Several algebraic characterizations of this problem of order 1×m×n×1 are investigated using the singular value decomposition and generalized inverses of its coefficient matrix. The results are compared with some results on the planar four-index transportation problem. It is shown that these problems have common algebraic characterizations and can be solved in terms of eigenvectors of the matrices J _m and J _n where J _m is an m×m matrix, all of whose entries are 1.     

On the problem of general structural assignments of linear systems through sensor/actuator selection

A systematic method is developed for determining an output matrix C for a given matrix pair (A,B) such that the resulting linear system characterized by the matrix triple (A,B,C) has the pre-specified system structural properties, such as the finite and infinite zero structure and the invertibility structures. Since the matrix C describes the locations of the sensors, the procedure of choosing C is often referred to as sensor selection. The method developed in this paper for sensor selection can be applied to the dual problem of actuator selection, where, for a given matrix pair (A,C), a matrix B is to be determined such that the resulting matrix triple (A,B,C) has the pre-specified structural properties.     

Discrete-time linear periodic systems: A note on the reachability and controllability interval length

Discrete-time linear systems with periodic coefficients of period T are considered in this paper. The reachability and controllability indices at time t for periodic systems, μ_rt and μ_ct, are defined. It is shown that the reachability [controllability] subspace at time t does coincide with the reachability [controllability] subspace over the interval (t − μ_rt,T, t) [(t, t + μ_ct.,T)] and properly includes the reachability [controllability] subspace over the interval (t −(μ_rt−1) T, t) [(t, t +(μ_cr−1)T)].     

The Nehari problem in systems with distributed input delays is inherently finite dimensional

The Nehari problem is considered in the context of linear systems with distributed input (or, equivalently, output) lags. It is shown that both the computation of the optimal value and the parametrization of suboptimal solutions reduce to matrix computations in an associated undelayed Nehari problem. The solution is then given in terms of a system of Neutral type.     

Further results on the observer design problem

In this paper the observer design problem for nonlinear systems is considered. Sufficient Lyapunov-like conditions are presented for the existence of a nonlinear observer. The theory we develop considerably improves and extends the results of our recent work [14].     

Disturbance decoupling of switched linear systems

In this paper, we consider disturbance decoupling problems for switched linear systems. We will provide necessary and sufficient conditions for three different versions of disturbance decoupling, which differ based on which signals are considered to be the disturbance. In the first version, the exogenous input is considered as the disturbance, in the second, the switching signal and in the third both of them are considered as disturbances. All three versions of disturbance decoupling have direct counterparts for linear parameter-varying (LPV) systems, while the latter instance of the problem is relevant for disturbance decoupling of piecewise linear systems, as we will show. The solutions of the three disturbance decoupling problems will be based on geometric control theory for switched linear systems and will entail both mode-dependent and mode-independent static state feedback.     

A note on the Hankel norm

It is shown that the triangle inequality for the Hankel norm of a stable, linear time-invariant system can be derived using properties of the system's controllability and observability Gramians.     

On the estimation problem of a class of continuous bioreactors with unknown input

The estimation problem of substrate–biomass reactors with isotonic or nonisotonic growth is addressed. The unmeasured reactor and time-varying feed substrate concentrations must be estimated from the dilution rate input and biomass measurements. First, the solvability conditions (nonlinear-global observability or detectability) are characterized, finding that: (i) the conditions depend on particular motions and growth mechanisms, and (ii) with isotonic (or nonisotonic) growth, the unknown input–state pair is observable in the classical (or nonstandard) single-valued (S) [or bivalued (B)] sense. Then, a robustly convergent S (or B)-observer is designed accordingly. The S-observer yields the estimate of the only unmeasured input–state trajectory pair associated with the measured input–output signal. The B-observer yields the estimates of the two possible unmeasured input–state trajectory pairs associated with the measured input–output signal, each one satisfying the reactor mass balances. The developments and findings are illustrated with representative (Monod and Haldane) examples through analytic assessment and numerical simulation.     

An addendum on the incremental assignment problem

In Toroslu and Üçoluk [I.H. Toroslu, G. Üçoluk, Incremental assignment problem, Information Sciences 177 (2007) 1523-1529] the incremental assignment problem is defined as follows. A new pair of vertices and their incident edges are added to a weighted bipartite graph whose maximum-weighted matching is already known, and the maximum-weighted matching of the extended graph is sought. An O(n²) algorithm for the problem has been derived, with n the size of a partition in the bipartite graph. We point out that such a result can be found in literature.     

Consensus problem of heterogeneous multi-agent systems with time delay under fixed and switching topologies

Consensus problem is investigated for heterogeneous multi-agent systems composed of first-order agents and second-order agents in this paper. Leader-following consensus protocol is adopted to solve consensus problem of heterogeneous multi-agent systems with time-varying communication and input delays. By constructing Lyapunov-Krasovkii functional, sufficient consensus conditions in linear matrix inequality(LMI) form are obtained for the system under fixed interconnection topology. Moreover, consensus conditions are also obtained for the heterogeneous systems under switching topologies with time delays. Simulation examples are given to illustrate effectiveness of the results.