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This paper investigates the disturbance decoupling problem by dynamic output feedback in the general case of systems with possible input-output feedthrough matrices. In particular, we aim to extend the geometric condition based on self-boundedness and self-hiddenness, which as is well-known enables to solve the decoupling problem without requiring eigenspace computations. We show that, exactly as in the case of zero feedthrough matrices, this solution maximizes the number of assignable eigenvalues of the closed-loop. Since in this framework we are allowing every feedthrough matrix to be non-zero, an issue of well-posedness of the feedback interconnection arises, which affects the way the solvability conditions are structured. We show, however, that the further solvability condition which originates from the problem of well-posedness is well-behaved in the case where we express such condition in terms of self bounded and self hidden subspaces.

The focus of this paper is the connection between two foundational areas of LTI systems theory: geometric control and eigenstructure assignment. In particular, we study the properties of the null-spaces of the reachability matrix pencil and of the Rosenbrock system matrix, which have been extensively used as two computational building blocks for the calculation of pole placing state feedback matrices and pole placing friends of output-nulling subspaces. Our objective is to show that the subspaces in the chains of kernels obtained in the construction of these feedback matrices interact with each other in ways that are entirely independent from the choice of eigenvalues. So far, these chains of subspaces have only been studied in the case of stationarity. In this case, it is known that these chains converge to the classic Kalman reachable subspace for the reachability matrix pencil and to the largest reachability subspace in the case of the Rosenbrock matrix, respectively. Here we are interested in showing that even before stationarity has been reached, the partial chains are linked to structural properties of the system, and are therefore independent of the closed-loop eigenvalues that we wish to assign. We further characterize these subspaces by investigating the notion of largest subspace on which it is possible to assign the closed-loop spectrum (possibly maintaining the output at zero) without resorting to non-trivial Jordan forms.

A new class of structured matrices is presented and a closed form formula for their determinant is established. This formula has strong connections with the one for Vandermonde matrices.

This paper deals with the computation of basis matrices of output-nulling and reachability subspaces for linear time-invariant (LTI) descriptor systems, which have been found to appear in the solution of many control and estimation problems. In particular, one of the classic approaches for the computation of these subspaces, which exploits the calculation of the null-space of the Rosenbrock system matrix pencil and which has found important applications in non-interacting and fault-detection problems, is generalized for descriptor systems.

In this paper we introduce a formula that parameterises the Pythagorean triples as elements of two series. With respect to the standard Euclidean formula, this parameterisation does not generate the Pythagorean triples where the elements of the triple are all divisible by 2. A necessary and sufficient condition is also proposed for a Pythagorean triple obtained from this formula to be primitive.

In this paper we lay the foundations of a not necessarily rational negative imaginary systems theory and its relations with positive real systems theory and, hence, with passivity. In analogy with the theory of positive real functions, in our general framework, negative imaginary systems are defined in terms of a domain of analyticity of the transfer function and of a sign condition that must be satisfied in such domain. In this way, on the one hand, our theory does not require to restrict the attention to systems with rational transfer function and, on the other hand | just by suitably selecting the domain of analyticity to be either the right half complex plane or the complement of the unit disc in the complex plane | we particularize our theory to both continuous-time and to discrete-time systems. Indeed, to the best of our knowledge, this is first time that discrete-time negative imaginary systems are studied in the literature. In this work, we also aim to provide a unitary view of the different notions that have appeared so far in the literature within the framework of positive real and in the more recent theory of negative imaginary systems, and to show how these notions are characterized and linked to each other. A stability analysis result for the interconnection of discrete-time systems is also derived.

In this paper we present a method for designing a linear time invariant (LTI) state-feedback controller to monotonically track a constant step reference at any desired rate of convergence for any initial condition. Necessary and sufficient constructive conditions are given to deliver a monotonic step response from all initial conditions. This method is developed for multi-input multi-output (MIMO) systems, and can be applied to square and non-square systems, strictly proper and non-strictly proper systems, and, importantly, also minimum and non-minimum phase systems. The framework proposed here shows that for MIMO LTI systems the objectives of achieving a rapid settling time, while at the same time avoiding overshoot and/or undershoot, are not necessarily competing objectives., arXiv admin note: text overlap with arXiv:1402.5181

This paper proposes a reduction technique for the generalised Riccati difference equation arising in optimal control and optimal filtering. This technique relies on a study on the generalised discrete algebraic Riccati equation. In particular, an analysis on the eigen- structure of the corresponding extended symplectic pencil enables to identify a subspace in which all the solutions of the generalised discrete algebraic Riccati equation are coin- cident. This subspace is the key to derive a decomposition technique for the generalised Riccati difference equation that isolates its nilpotent part, which becomes constant in a number of steps equal to the nilpotency index of the closed-loop, from another part that can be computed by iterating a reduced-order generalised Riccati difference equation.

This paper investigates the properties of the solutions of the generalised discrete algebraic Riccati equation arising from the solution of the classic infinite-horizon linear quadratic control problem. In particular, a geometric analysis is used to study the relationship existing between the solutions of the generalised Riccati equation and the output-nulling subspaces of the underlying system and the corresponding reachability subspaces. This analysis reveals the presence of a subspace that plays an important role in the solution of the related optimal control problem, which is reflected in the generalised eigenstructure of the corresponding extended symplectic pencil. In establishing themain results of this paper, several ancillay problems on the discrete Lyapunov equation and spectral factorisation are also addressed and solved.

The aim of this paper is to present a comprehensive range of design techniques for the synthesis of the standard compensators (Lead and Lag networks as well as PID controllers) that in the last twenty years have proved to be of great educational value in a vast number of undergraduate and postgraduate courses in Control throughout Italy, but that to-date remain mostly confined within this country. These techniques hinge upon a set of simple closed-form formulae for the computation of the parameters of the controller as functions of the typical specifications introduced in Control courses, i.e., the steady-state performance, the stability margins and the crossover frequencies.

This paper addresses the H2 optimal decoupling of previewed inputs. In particular, the synthesis of a decoupling filter exploiting the preview is carried out by recasting the problem as a single finite-horizon LQ regulator, with a generalized quadratic terminal cost. The present approach is based on a computationally attractive parametrization of the solutions of the associated Hamiltonian differential equation, and can also be applied for the solution of the dual problem, the H2 optimal fixed-lag smoothing problem.

This paper focuses on the implementation of a state feedback compensator for the solution of the linear quadratic control problem with an arbitrary initial state and with null end-point for discrete-time systems. The solution of the optimization problem is based on a finite horizon parametrization of the solutions of the associated extended symplectic difference equation.

The paper addresses a very important problem of control theory: the neutralization of the effect of measurable inputs on some significant output. The synthesis of the compensator is performed with a new algorithm embedded in the geometric control theory.

This paper addresses the cheap version of the classical linear quadratic (LQ) optimal control problem for continuous-time systems. The approach herein considered differs from those presented in literature, since it consists of applying the tools of the geometric control theory to the Hamiltonian system. In this way, it is possible to compute the stabilizing state-feedback gain achieving optimality by using standard geometric algorithms, whenever the initial state satisfies a suitable necessary and sufficient condition for solvability, also stated in geometric terms.

This paper presents a new direct digital design method for discrete proportional integral derivative PID + filter (PIDF) controllers employed in DC-DC buck converters. The considered controller structure results in a proper transfer function which has the advantage of being directly implementable by a microcontroller algorithm. Secondly, it can be written as an Infinite Impulse Response (IIR) digital filter. Thirdly, the further degree of freedom introduced by the low pass filter of the transfer function can be used to satisfy additional specifications. A new design procedure is proposed, which consists of the conjunction of the pole-zero cancellation method with an analytical design control methodology based on inversion formulae. These two methods are employed to reduce the negative effects introduced by the complex poles in the transfer function of the buck converter while exactly satisfying steady-state specifications on the tracking error and frequency domain requirements on the phase margin and on the gain crossover frequency. The proposed approach allows the designer to assign a closed-loop bandwidth without constraints imposed by the resonance frequency of the buck converter. The response under step variation of the reference value, and the disturbance rejection capability of the proposed control technique under load variations are also evaluated in real-time implementation by using the Arduino DUE board, and compared with other methods.