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In this paper we present and provide a proof for a set of non-recursive formulae arising in the computation of the largest output-nulling and the smallest input-containing subspaces which have been used in a variety of contexts in the framework of the geometric approach. These expressions have been used in the literature both in the strictly proper and in the non-strictly proper case, but, to the best of our knowledge, a proof is still missing. These formulae are established here in the general possibly non-strictly proper case. Some ancillary side results of independent interest are also proposed. [ABSTRACT FROM AUTHOR]

We develop a geometric approach for fractional linear time-invariant systems with Caputo-type derivatives. In particular, we generalize the fundamental notions of invariance and controlled invariance to the fractional setting. We then exploit this new geometric framework to address the disturbance decoupling problem via static pseudostate feedback, with and without stability. Our main contribution is a set of necessary and sufficient conditions for the disturbance decoupling problem that are related to the input-output properties of the closed-loop system, and hence they are applicable not just to Caputo-type derivatives but, more broadly, to any type of fractional system. These results show that, while the conditions for guaranteeing the existence of a decoupling pseudostate feedback remain essentially unchanged, the underlying theoretical framework is substantially different, because the fractional derivative is a nonlocal operator and this property plays a major role in the characterization of the evolution of the pseudostate trajectory. In particular, we show that, unlike the integer case, the infinite-dimensional nature of fractional systems means that feedback control is insufficient to maintain the pseudostate trajectory on a controlled invariant subspace, unless the entire past history of the pseudostate has evolved on that subspace. However, feedforward control can achieve this task under certain necessary and sufficient geometric conditions. [ABSTRACT FROM AUTHOR]

Abstract This paper addresses the tracking problem of constant references for multiple-input multiple-output linear time-invariant systems. We provide necessary and sufficient conditions for the solvability of the problem under the minimal set of assumptions that guarantee the well-posedness of every control problem requiring stability. Our approach allows to solve tracking problems for systems which are possibly nonright-invertible, simply (not asymptotically) stabilizable, and possibly with invariant zeros at the origin. Our methodology is constructive and results in the design of a stabilizing feedback matrix and a feedforward signal which solve the problem. Graphical abstract [ABSTRACT FROM AUTHOR]

The explicit reference governor is a constrained control scheme that was originally introduced for generic nonlinear systems. This paper presents two explicit reference governor strategies that are specifically tailored for the constrained control of linear time-invariant systems subject to linear constraints. Both strategies are based on the idea of maintaining the system states within an invariant set which is entirely contained in the constraints. This invariant set can be constructed by exploiting either the Lyapunov inequality or modal decomposition. To improve the performance, we show that the two strategies can be combined by choosing at each time instant the least restrictive set. Numerical simulations illustrate that the proposed scheme achieves performances that are comparable to optimisation-based reference governors. [ABSTRACT FROM AUTHOR]

In this study, the authors analyse some fundamental structural properties of linear time‐invariant multivariable systems in the controller canonical form and present a direct method for the computation of bases and associated friends for output‐nulling, input‐containing and reachability subspaces in terms of the parameters of the system and the invariant zero structure, both in the non‐defective and in the defective case. Using this analysis, it is possible to express the solvability conditions of important control and estimation problems in terms of easily checkable conditions on the system matrices. [ABSTRACT FROM AUTHOR]

In this paper, we investigate several theoretical and computational aspects of fundamental subspaces for linear time-invariant descriptor systems, which appear in the solution of many control and estimation problems. Different types of reachability and controllability for descriptor systems are described and discussed. The Rosenbrock system matrix pencil is employed for the computation of supremal output-nulling subspaces and supremal output-nulling reachability subspaces for descriptor systems. [ABSTRACT FROM AUTHOR]

We consider the classic problem of exact output regulation for a linear time invariant plant. Under the assumption that either a state feedback or measurement feedback output regulator exists, we give design methods to obtain a regulator that avoids overshoot and undershoot in the transient response. [ABSTRACT FROM AUTHOR]

We consider the classic problem of pole placement by state feedback. We offer an eigenstructure assignment algorithm to obtain a novel parametric form for the pole-placing feedback matrix that can deliver any set of desired closed-loop eigenvalues, with any desired multiplicities. This parametric formula is then exploited to introduce an unconstrained nonlinear optimisation algorithm to obtain a feedback matrix that delivers the desired pole placement with optimal robustness and minimum gain. Lastly we compare the performance of our method against several others from the recent literature. [ABSTRACT FROM AUTHOR]

Abstract: We consider the use of linear multivariable feedback control to achieve a nonovershooting and nonundershooting step response. Recently, Schmid and Ntogramatzidis (2010)  introduced a linear state feedback controller design method to avoid overshoot. In this paper, we describe conditions under which the design method may be modified to avoid undershoot. The method is applicable to square and nonsquare systems, minimum and nonminimum phase systems, and also strictly proper and bi-proper systems. [Copyright &y& Elsevier]

Abstract: We consider the use of linear multivariable feedback control to achieve a nonovershooting step response. A method is given for designing a linear time invariant state feedback controller to asymptotically track a constant step reference with zero overshoot and arbitrarily small rise time, under some mild assumptions. We present a unified design method that can be applied to continuous and discrete time systems, square and non-square systems, minimum and nonminimum phase systems, and also strictly proper and nonstrictly proper systems. [Copyright &y& Elsevier]

Abstract: This paper investigates several aspects of linear–quadratic (LQ) optimal control for Roesser models over a two-dimensional (2D) signal-index set of finite extent. First, we consider the characterisation and computation of open-loop control laws when constraints on the system semi-states are imposed at both the south-west and north-east boundaries of the frame (i.e. signal-index set) of interest; by virtue of the quarter-plane causal structure of the Roesser model, the south-west and north-east boundary conditions are analogous to initial conditions and terminal constraints, respectively. A necessary and sufficient characterisation of optimality is obtained and explicitly computable formulae are derived to characterise the corresponding control inputs and performance index under reasonable assumptions on the problem data. In the second part of the paper, the problem of optimal LQ control via semi-state feedback is considered. A 2D Riccati-like difference equation is introduced to characterise, in a sufficient sense, a solution to this problem. [Copyright &y& Elsevier]

Abstract: In this paper, the problem of reducing a given LTI system into a left or right invertible one is addressed and solved with the standard tools of the geometric control theory. First, it will be shown how an LTI system can be turned into a left invertible system, thus preserving key system properties like stabilizability, phase minimality, right invertibility, relative degree and infinite zero structure. Moreover, the additional invariant zeros introduced in the left invertible system thus obtained can be arbitrarily assigned in the complex plane. By duality, the scheme of a right inverter will be derived straightforwardly. Moreover, the squaring down problem will be addressed. In fact, when the left and right reduction procedures are applied together, a system with an unequal number of inputs and outputs is turned into a square and invertible system. Furthermore, as an example it will be shown how these techniques may be employed to weaken the standard assumption of left invertibility of the plant in many optimization problems. [Copyright &y& Elsevier]