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The disturbance decoupling problem is considered in the framework of switching linear systems assuming that no information on the actual value of the switching signal or on the current mode of the system is available. By introducing the novel notion of strong conditioned invariant subspace, the solvability of the problem is completely characterized by means of necessary and sufficient conditions. Constructive procedures to check the solvability conditions and to construct solutions, if any exists, are given.

This work concerns a new methodology to solve the ℋ2-optimal disturbance rejection problem by measurement feedback in the singular case: namely, when the plant has no feedthrough terms from the control input and the disturbance input to the controlled output and the measured output, respectively. A necessary and sufficient condition for problem solvability is expressed as the inclusion of two subspaces-a controlled-invariant subspace and a conditioned-invariant subspace. Such subspaces are directly derived from the Hamiltonian systems associated to the ℋ2-optimal control problem and, respectively, to the ℋ2-optimal filtering problem. The proof of sufficiency, which is constructive, provides the computational tools for the synthesis of the feedback regulator. A numerical example is worked out in order to illustrate how to implement the devised procedure.

A standard geometric-type environment, where only the very basic tools of the geometric approach are used (those supported by well-settled and well-tested computational aids) enables the development of algorithms for numerous control and estimation problems in the discrete-time case. These are: measurable or previewed signal localization problems, perfect or almost perfect tracking (right inversion), and, by duality, perfect or almost perfect unknown input estimation with possible post-knowledge and input reconstruction (left inversion). It is also shown that the devices obtained (compensator and observer), that may be noncausal when specific stability requirements are not met, can be implemented as dynamical systems including finite-horizon convolutors or finite impulse response systems.

The problem of compensating a given plant by means of a static compensator in such a way that, for any input, the output of the compensated system matches that of a given positive model, when both are initialized at 0, and its state evolves in the positive cone for positive initial conditions and inputs is considered. Under a mild structural assumption for the output-difference system between the plant and the model, a complete characterization of solvability of the problem in terms of necessary and sufficient conditions is obtained by means of structural geometric methods. Solvability conditions are practically checkable by algorithmic procedures and by solving a set of linear inequalities. The problem of asymptotic matching for any initial condition is then considered and solvability is characterized by necessary and sufficient conditions. A necessary condition that is practically checkable is given. Solvability by a dynamic compensator is also studied and a sufficient condition to characterize it is given.

International audience; This work deals with the problem of structural disturbance decoupling by state feedback for nonlinear impulsive systems. The dynamical systems addressed exhibit a hybrid behavior characterized by a nonlinear continuous-time state evolution interrupted by abrupt discontinuities at isolated time instants. The problem considered consists in finding a state feedback such that the system output is rendered totally insensitive to the disturbance. Both the case of static state feedback and that of dynamic state feedback are considered. A necessary and sufficient condition for the existence of a static state feedback that solves the problem in the multivariable case is proven by defining suitable tools in the context of the differential geometric approach. The situation concerning solvability by a dynamic state feedback is examined in the framework of the differntial algeraic approach. A necessary and sufficient solvaility condition is conjectured and discussed.

This work deals with the problem of annihilating the effect of a disturbance accessible for measurement on the output of an impulsive switching linear system: namely, a hybrid system whose state is subject to abrupt discontinuities at the same times when its dynamics are subject to switches. Jumping and switching are assumed to be instantaneously detectable and to satisfy a minimum dwell time requisite. The better exploitation of the information available on the disturbance is achieved through feedforward compensation. A necessary and sufficient condition for the solvability of the problem is proven. In particular, the proof of sufficiency is constructive, since it outlines the synthesis procedure for the sought compensator.

This work deals with the problem of noninteraction (and the dual problem of fault detection) for discrete-time linear time-invariant finite-dimensional state space systems. In particular, the standard, infinite horizon, noninteracting control requirement is replaced by a less demanding one, where noninteracting control is only sought on finite time horizons, suitably defined in connection with some structural properties of the system subblocks. A necessary and sufficient condition for solvability of the finite horizon problem thus stated is derived in terms of the almost controllability subspaces associated with the block structure of the system. The proof of the condition is constructive in the sense that it leads to a design procedure for the feedforward compensators that guarantee noninteracting control over the given time horizons. The underlying idea of the proof, i.e. the exploitation of the almost controllability subspace algorithm in the case of finite horizon noninteraction, is also compatible with a modified procedure for designing the compensators achieving infinite horizon noninteraction, which may be admissible for some specific subblocks of the system. In fact, in this latter case, it is the effective use of the controllability subspace algorithm which plays a key role. The dual counterpart in the context of fault detection introduces a structural means to identify and treat the cases where, due to the structural properties of the monitored system, some of the residuals which can be generated are only significant in a limited time. These concepts are also illustrated with a detailed numerical example.

An algebraic approach to the synthesis of a dynamic system that reconstructs the generic inaccessible input of a discrete-time linear multivariable system with unknown initial state is discussed. The method devised exploits geometric properties of key subspaces for the original system and algebraic properties of the Moore-Penrose inverse of Toeplitz matrices related to the algorithms for computing those subspaces. Nonminimum-phase invariant zeros are taken into account implicitly with the proposed techniques, while minimum-phase invariant zeros require that a filter be inserted between the original system and the reconstructor. The procedure applies to either strictly-proper or non-strictly-proper systems.

A geometric method for the design of a minimal-order dynamic feedforward compensator cancelling the minimum-phase invariant zeros of a linear multivariable system, while preserving the properties of controllability, observability, and right invertibility is discussed. The method is illustrated for continuous-time, non-strictly proper systems. Nonetheless, it also applies to strictly-proper systems.

This article deals with the optimization, expressed as the minimization of the l 2 norm of the tracking error, of the regulation transients caused by instantaneous, wide parameter variations occurring in the regulated systems. The parameter-varying regulated system is modeled by a set of discrete-time, linear, time-invariant systems and the regulated system switching law is assumed to be completely known a priori. For each linear time-invariant (LTI) system, a feedback regulator, including the exosystem internal model, is designed in order to guarantee closed-loop asymptotic stability and zero tracking error in the steady-state condition. The compensation scheme for the minimization of the regulation transients consists of feedforward actions on the regulation loop and a state switching policy for suitably setting the state of the feedback regulators at the switching times. Both the state switching policy and the feedforward actions are computed off-line: the former by exploiting some geometric properties of the multivariable autonomous regulator problem, the latter by resorting to a two-level, nested algorithm. The lower level includes a sequence of discrete-time, finite-horizon optimal control problems, each defined in the time interval between two consecutive switches. The upper level combines relevant data from the lower-level problems into a global, l 2 -control problem. A significant feature of the approach to the lower-level problems is the original procedure providing the solution of the finite-horizon, optimal control problem stated for discrete-time stabilizable systems through the structural invariant subspaces of the associated singular Hamiltonian system.

A dynamic feedforward scheme allows measurable signal decoupling to be solved independently of other problems simultaneously present in the design of a control system like, e.g., stabilization, robustness, insensitivity to disturbances. The synthesis procedure, based on the properties of self-bounded controlled invariant subspaces, ensures the minimal complexity of the feedforward unit, in terms of minimal unassignable dynamics and minimal dynamic order, in the case of left-invertible systems and, on some specific conditions, also in the case of non-left-invertible systems. The output dynamic feedback set up to guarantee stability (or, more generally, robustness or insensitivity properties) does not affect the complexity of the feedforward unit, since the peculiar layout where the feedback unit receives an additional input from the precompensator preserves, in the extended system, exactly the same set of unassignable internal eigenvalues of the minimal self-bounded controlled invariant subspace as that defined for the original system. The overall control structure turns out to be a two-degree-of-freedom controller completely devised in the geometric context.

The structural and geometric approach is a consolidated means of analysis and synthesis in multivariable linear systems. Introduced in the late 1960s, this approach has received a multiplicity of contributions throughout the following decades, thus giving rise to a complete theory, expounded in the authoritative books [Wonham 1985] and [Basile and Marro 1992]. The fundamental objects of this theory (e.g. controlled invariant subspaces, controllability subspaces, conditioned invariant subspaces, etc.) are also extensively treated in recent books for graduate level courses in system theory, like e.g. [Rugh 1996] and [Trentelman, Stoorvogel and Hautus 2001]. Nonetheless, the research domain of system structure and control is still very active (see e.g. [Malabre, Plenary lecture, 2nd IFAC Symposium on System, Structure and Control, Mexico, 2004] and the references cited therein). In fact, now that the ever increasing complexity of control systems shows as the most challenging research fields those concerning robustness, nonlinear systems, failure detection and fault tolerance, hybrid systems etc, the interest for the structural aspects of systems which are assumed to be exactly described by linear models, is still motivated by several reasons...

This paper proposes a valve stiction detection system which selects valve stiction detection algorithms based on characterizations of the data. For this purpose, novel data feature indexes are proposed, which quantify the presence of oscillations, meannonstationarity, noise and nonlinearities in a given data sequence. The selection is then performed according to the conditions on the index values in which each method can be applied successfully. Finally, the stiction detection decision is given by combining the detection decisions made by the selected methods. The paper ends demonstrating the effectiveness of the proposed valve stiction detection system with benchmark industrial data.

In this work, the problem of decoupling input signals accessible for measurement is formulated for continuous-time linear switching systems, with the requirement that the compensated system be quadratically stable under arbitrary switching. A dynamic feedforward switching compensator achieving the specifications is designed, on the assumption that the given plant be quadratically stable under arbitrary switching. The methodology devised exploits basic concepts of the geometric approach and linear matrix inequalities.

The aim of this paper is to show the applicability of geometric techniques to a regulation problem for linear, time-delay systems. Given a plant whose dynamics equations include delays and an exosystem that generates a reference signal, the problem we consider consists in finding a feedback regulator which guarantees asymptotic stability of the regulation loop and asymptotic command following of the reference signal, for any initial condition of the overall system in the presence of disturbances. By associating to the time-delay plant a corresponding abstract system with coefficients in a ring, it is possible to place our investigation in a finite dimensional algebraic context, where intuition and results obtained in the classical case, that is without delays, may be exploited. In particular, using tools and methods of the geometric approach to systems with coefficients in a ring, sufficient conditions for the solvability of the considered problem are found and a constructive procedure, which works under specific hypotheses, is given.

Prior results on input reconstruction for multi-input, multi-output discrete-time linear systems are extended by defining l-delay input and initial-state observability. This property provides the foundation for reconstructing both unknown inputs and unknown initial conditions, and thus is a stronger notion than l-delay left invertibility, which allows input reconstruction only when the initial state is known. These properties are linked by the main result (Theorem 4), which states that a MIMO discrete-time linear system with at least as many outputs as inputs is l-delay input and initial-state observable if and only if it is l-delay left invertible and has no invariant zeros. In addition, we prove that the minimal delay for input and state reconstruction is identical to the minimal delay for left invertibility. When transmission zeros are present, we numerically demonstrate l-delay input and state reconstruction to show how the input-reconstruction error depends on the locations of the zeros. Specifically, minimum-phase zeros give rise to decaying input reconstruction error, nonminimum-phase zeros give rise to growing reconstruction error, and zeros on the unit circle give rise to persistent reconstruction error.

In this work, we develop a geometric method for solving the problem of H2-optimal rejection of disturbance inputs in continuous-time linear systems without feedthrough terms from the control input and the disturbance input to the controlled output and the measured output. A necessary and sufficient condition for the solvability of the problem is expressed in terms of a pair of subspaces, a controlled-invariant subspace and a conditioned-invariant subspace, derived from the Hamiltonian systems associated with the problem. The if-part of the proof shows how to synthesize the feedback regulator, which is non-strictly-proper in general.

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.

Unknown-state, unknown-input reconstruction in systems with invariant zeros is intrinsically limited by the fact that, for any invariant zero, at least one initial state exists, such that when the mode of the invariant zero is suitably injected into the system, the output remains identically equal to zero. Nonetheless, the problem has recently attracted considerable interest, mainly due to its connections with fault diagnosis and fault tolerant issues. This paper discusses the synthesis of a system capable of reconstructing the generic initial state and the generic bounded input in discrete-time nonminimumphase linear systems with feedthrough terms. The procedure described is developed within the geometric approach.

The complete solution of the unknown-state, unknown-input reconstruction problem in systems with invariant zeros is intrinsically limited by the fact that for any invariant zero, at least one initial state exists, such that, when the mode associated to the invariant zero is suitably injected into the system, the corresponding output is zero. Although in the awareness of this restriction, the problem of reconstructing the initial state and the inaccessible inputs from the available measurements is the object of a fair amount of research activities because of its impact on a wide range of applications, specifically those dealing with the synthesis of enhanced-reliability control systems. In this context, the present paper contributes a geometric method aimed at solving the exact unknown-state, unknown-input reconstruction problem in discrete-time linear time-invariant multivariable systems with nonminimum-phase zeros. The case where all the system invariant zeros lie in the open set outside the unit disc of the complex plane is regarded as the basic one. The difficulties related to the presence of those invariant zeros are overcome by allowing a reconstruction delay commensurate to the invariant zero time constants. The same technique also applies to the case of systems without invariant zeros. In the latter circumstance, however, the reconstruction delay is related to the number of iteration required by the algorithm for the computation of a specific subspace to converge. Finally, the more general case where the problem is stated for a system whose invariant zeros lie both inside and outside the unit disc of the complex plane is reduced to the basic problem referred to a new system, derived from the original one through a procedure aimed at replacing the minimum-phase zeros with their mirror images with respect to the unit circle.

As an extension of existing results on input reconstruction, we define l-delay state and input reconstruction, and we characterize this property through necessary and sufficient conditions. This property is shown to be a stronger notion of left invertibility, in which the initial state is assumed to be known. We demonstrate l-delay state and input reconstruction on several numerical examples, which show how the input reconstruction error depends on the locations of the zeros. Specifically, minimum-phase zeros give rise to decaying input reconstruction error, nonminimum-phase zeros give rise to growing reconstruction error, and zeros on the unit circle give rise to persistent reconstruction error.

This paper deals with the problem of perfect elimination of the transients of the regulated output caused by parameter variations, in continuous-time multivariable linear systems. Parameter changes are assumed to be instantaneous and known in advance within a finite time horizon. The continuous-time case needs separate investigation from the discrete-time case, that has recently been solved both in the exact context and in the l 2 -optimal context. In fact, the continuous-time solution of the exact problem cannot be found through a plain conversion from continuous to discrete, since discretizations do not generically preserve the geometric properties of the set of switched systems playing a key role in the specific synthesis procedure. Moreover, the continuous-time exact solution cannot straightforwardly be retrieved as the zero-cost solution of the corresponding optimization problem, since the latter turns out to be a singular problem in general, due to the structure of the systems involved. In the light of these considerations, geometric conditions for the continuous-time problem solvability are proved and a complete design procedure is presented.

An approximate input reconstruction algorithm is used to reconstruct unknown inputs, which are then used for fault detection. The approximate input reconstruction algorithm is a least squares algorithm that estimates both the unknown initial state and input history. The estimated inputs are then compared to the commanded values and sensor values to assess the health of actuators and sensors. This approach is applied to the longitudinal and lateral dynamics of NASA's Generic Transport Model. The input reconstruction algorithm can be used for systems with minimum-phase or nonminimum-phase zeros; however, minimum-phase zeros entail an additional delay in reconstructing inputs, while zeros on the unit circle yield persistent estimation errors and thus poor input reconstruction regardless of the delay.

The object of this review presents a comprehensive study of filtering theory from an essentially structural perspective. The various classes of filtering problems treated in the literature are organized hierarchically on the basis of progressively less demanding performance requirements and different assumptions made on the noise characteristics.

The aim of this paper is to show the applicability of geometric techniques to a regulation problem for linear, time-delay systems. Given a plant whose dynamics equations include delays, the problem we consider consists in finding a feedback regulator which guarantees asymptotic stability of the regulation loop and asymptotic command following of the reference signal generated by an exosystem, for any initial condition of the overall system. By associating to the time-delay plant a corresponding abstract system with coefficients in a ring, it is possible to place our investigation in a finite dimensional algebraic context, where intuition and results obtained in the classical case, that is without delays, may be exploited.

This note introduces a geometric solution to the problem of perfect elimination of regulation transients in discrete-time, linear systems subject to swift and wide, a priori-known, parameter variations. The constructive proof of the conditions for problem solvability requires a preliminary, strictly geometric interpretation of the multivariable autonomous regulator problem, specifically aimed at discrete-time, linear systems. The novel concept of internal stabilizability of a robust controlled invariant subspace plays a key role in the formulation of those conditions as well as in the synthesis of the control scheme.

Abstract: H2preview control in the discrete-time domain is approached in a strict geometric perspective. The original formulation in the frequency domain is recast in the time domain. Then, it is shown how the problem in the time-domain can be reduced to the combination of elementary subproblems. This approach requires a structural analysis of the properties of the singular Hamiltonian system associated to the H2control problem.

This work introduces a methodology for the minimization, in terms of the l2 norm of the tracking error, of the regulation transients in discrete-time linear systems subject to instantaneous, wide parameter variations. A feedback regulator designed according to the well-known internal model principle guarantees closed-loop asymptotic stability and zero error in the steady-state condition. The compensation scheme for the minimization of transients includes a feedforward action on both the plant and the feedback regulator and a switching policy for the states of the exosystem and of the feedback regulator itself. The feedfoward action and the state switching policy are computed off-line, on the basis of the complete preview of the regulated-system switching law in a given time interval.

This contribution focuses on decoupling of previewed signals, i.e. the problem of making the output insensitive to input signals which are known a certain amount of time in advance. Inclusion of time delays in the problem formulation enables decoupling with preview to be turned into a causal problem. The solution of this latter problem is then completely set forth in the geometric approach context. Necessary and sufficient constructive conditions for decoupling of previewed signals are provided, where self-bounded controlled invariant subspaces play a key role in connection with internal stability of the devised system. If the minimal self-bounded controlled invariant subspace satisfying the structural constraint is internally stabilizable, decoupling just requires finite preview. Otherwise, infinite preview is demanded. In this latter case, resorting to finite impulse response systems yields a practically implementable solution which approximates the theoretical one with arbitrary accuracy. The procedure is illustrated by a benchmark example consisting of a flexible mechanical structure.

This work introduces an l2-optimal approach for minimizing the regulation transients in discrete-time, linear systems subject to instantaneous, wide, a-priori-known parameter variations. The theoretical bases are twofold. A geometric interpretation, specifically aimed at discrete-time linear systems, of the multivariable autonomous regulator problem is required to define the ideal state trajectories, corresponding to the zero-error, steady-state conditions. A geometric characterization of the structural invariant subspaces of the singular Hamiltonian system associated to the optimal control problem is used to derive the actual state trajectories, corresponding to the minimal l2-norm of the tracking error caused by parameter variations, given that the regulated system state cannot arbitrarily be imposed at the switching times. Since the proposed approach applies on the rather extensive conditions which guarantee solvability of a set of multivariable autonomous regulator problems as well as solvability of a set of optimal control problems, it is a valid option whenever the more restrictive conditions demanded to achieve perfect elimination of regulation transients are not satisfied.

This work encompasses the problem of the exact elimination of regulation transients for linear parameter varying systems in a straight geometric framework. Discrete-time, stabilizable systems are specifically addressed. Conditions for problem solvability are proved and the synthesis of the control scheme is illustrated in detail.

This work deals with the optimization, expressed as the minimization of the l2 norm of the tracking error, of regulation transients caused by instantaneous, wide parameter variations occurring in discrete-time linear systems. The regulated system switching law is assumed to be completely known a priori. A feedback regulator, designed according to the internal model principle, guarantees closed-loop asymptotic stability and zero error in the steady-state condition. The compensation scheme for the minimization of transients includes a feedforward action on both the plant and the feedback regulator and a switching policy for the states of the exosystem and the feedback regulator. The feedfoward action and the state switching policy are computed off-line, by means of a two-level nested algorithm. The lower level includes a sequence of finite-horizon optimal control problems (each one corresponding to a time interval between two subsequent switches), while the upper level combines relevant data from the lower-level problems into a global l2 optimization problem. A substantial feature of this contribution is that the approach to the lower-level problem relies on an original procedure which provides the solution of a discrete-time finite-horizon optimal control problem in closed form as a function of time. Thus, discrete-time intervals with a large number of samples can easily be handled.

A non-iterative solution for a class of discrete-time finite-horizon linear quadratic optimal control problems is obtained through the characterization of a pair of structural invariants of the singular Hamiltonian system associated to the H2 optimal control problem stated for the generic, discrete-time quadruple (A, B, C, D). On the assumption that the final state weighting function in the performance index is represented by a quadratic surface, it is shown that the optimal cost is a function of the initial state with the same structure. Optimal control laws and state trajectories are analytically expressed as functions of the initial state as well. The results hold under rather extensive conditions: those that guarantee the existence and uniqueness of the stabilizing solution of the corresponding discrete algebraic Riccati equation.

The non-iterative solution of the discrete-time finite-horizon linear quadratic optimal control problem with fixed terminal state derived by means of the Moore-Penrose inverse of an appropriately defined matrix is intrinsically subject to a constraint on the maximal length of the control time interval which can be considered. In fact, this is a consequence of the limitation on the computational power available for processing the matrix generalized inverse. This article introduces a computational framework where the dimensionality restriction is completely removed. The core of the proposed algorithm consists of a procedure where the time interval taken into account doubles at each step. This routine guarantees a fast convergence to the solution. Moreover, an arbitrarily accurate solution of the corresponding infinite-horizon problem can be retrieved by setting the final state to zero and welding a sufficient number of finite time intervals satisfying the original dimensionality constraint. It is worth noting that the procedure presented in this work returns an arbitrarily accurate solution of the infinite-horizon problem, with no additional complications, also when the to-be-controlled system is non-left-invertible.

This contribution is focused on a geometric methodology devised to achieve optimization, expressed as the minimization of the l2 norm of the tracking error, of regulation transients caused by instantaneous, wide parameter variations occurring in discrete-time, linear systems. The regulated system switching law is assumed to be completely known a priori in a given time interval. A set of feedback regulators, designed according to the internal model principle, guarantee closed-loop asymptotic stability and asymptotic tracking of the reference signal generated by an exosystem, for each regulated system (i.e., for each pair to-be-controlled system/exosystem). The compensation scheme for optimization of transients consists of feedforward actions on the regulation loop and switching policies for suitably setting the states of the feedback regulators and those of the exosystems at the switching times. The theoretical bases of this approach comprise (i) a geometric interpretation, specifically aimed at discrete-time stabilizable and detectable systems, of the multivariable autonomous regulator problem and (ii) a non-recursive solution, still aimed at discrete-time stabilizable systems, of the finite-horizon optimal control problem with final state wieghted by a generic quadratic function, based on a characterization of the structural invariant subspaces of the associated singular Hamiltonian system holding on the sole, fairly general, assumptions that guarantee the existence of the stabilizing solution of the corresponding discrete algebraic Riccati equation.

A multi-level computational framework which overcomes the dimensionality constraint intrinsic in the solution by pseudoinversion of the discrete-time finite horizon LQ optimal control problem with assigned final state is presented. Depending on design priorities, the algorithm can be based on either of two different nesting procedures: an additive procedure or a multiplicative procedure. In both cases, the solution of the infinite horizon problem can be retrieved if some rather extensive conditions are met. The algorithmic framework holds independently of the control weighting matrix being regular, singular, or zero. Moreover, the devised algorithm differs from those available in the literature in that it handles non-left-invertible systems with no further complications.

The non-iterative solution through Moore-Penrose inverse which applies to discrete-time finite-horizon LQ optimal control problems with fixed final state is subject to a constraint on the maximal length of the control time interval. This is a consequence of the limitation on the computational power available for processing the generalized inverse of properly constructed matrices. In this work, a computational framework where the dimensionality restriction is completely removed is presented. The core of the proposed algorithm consists in a procedure where the time interval taken into account doubles at each step. This routine guarantees a fast convergence to the solution. Moreover, the solution of the corresponding infinite-horizon problem is retrievable with arbitrary accuracy by setting the final state to zero and welding a sufficient number of arcs. The procedure returns an arbitrarily accurate solution of the infinite-horizon problem, with no additional complications, also when the to-be-controlled system is non-left-invertible.

The problem of minimizing, in the H2-norm sense, the effect on the output of an exogenous input signal known with finite preview is solved by means of a dynamic feedforward scheme designed on the basis of spectral factorization techniques. On standard assumptions, stability and robustness with respect to model uncertainties and unaccessible inputs are assumed to be guaranteed by an inner feedback, while the dynamic feedforward unit herein devised is utterly committed to the purpose of taking advantage of the preview available on the signal to be rejected. The design procedure is illustrated by a numerical example.

The problem of making the output of a discrete-time linear system totally insensitive to an exogenous input signal known with preview is tackled in the geometric approach context. A necessary and sufficient condition for exact decoupling with stability in the presence of finite preview is introduced, where the structural and the stabilizability aspects are considered separately. On the assumption that structural decoupling is feasible, internal stabilizability of the minimal self-bounded controlled invariant satisfying the structural constraint, namely Vm, guarantees stability of the dynamic feedforward compensator. However, if structural decoupling is feasible but Vm is not internally stabilizable, exact decoupling is nonetheless achievable with a stable feedforward compensator, on the sole assumption that Vm has no unassignable internal eigenvalues on the unit circle, provided that the signal to be rejected is known with infinite preview. An algorithmic framework based on steering along zeros techniques completely devised in the time domain shows how to compute the convolution profile of the feedforward compensator in each case.

Exact decoupling with preview, perfect tracking of previewed references, unknown-input state observation with fixed lag, and left inversion with fixed lag are considered from a unifying perspective where exact decoupling with preview is the basic problem. Necessary and sufficient constructive conditions for decoupling with finite preview are proved in the geometric framework. Structural and stabilizability conditions are considered separately and the use of self-bounded controlled invariant subspaces allows the dynamic compensator with the minimal unassignable dynamics to be straightforwardly derived. A steering along zeros technique is devised to guarantee decoupling with stability also in the presence of unstable unassignable dynamics of the minimal self-bounded controlled invariant.

Si è sviluppato un sistema di diagnosi predittiva per manovre elettriche di scambi ferroviari. Lo studio è stato motivato dal fatto che numerosi studi statistici, compiuti in vari Paesi Europei oltre che in Italia, sulle cause di guasto e disservizio nelle infrastrutture ferroviarie hanno messo in evidenza la cruciale importanza dell'efficienza dei sistemi di segnalamento, e, in particolare, dei dispositivi di scambio, che rientrano in questa categoria. L'implementazione di algoritmi efficienti per l'elaborazione di sequenze registrate degli ingressi accessibili e delle corrispondenti uscite misurabili del sistema monitorato risulta avere un impatto molto positivo sul programma di manutenzione della rete ferroviaria e sulle relative spese. Un primo importante risultato è quello di automatizzare un processo che, altrimenti, dipende fortemente dall'esperienza e dalla sensibilità dell'operatore umano. Contestualmente, il programma di manutenzione, normalmente costituito da sopralluoghi e interventi periodici, può essere modificato così da farlo dipendere essenzialmente da un sistema di monitoraggio fuori-linea. Questo permette un intervento immediato nel caso di guasti incipienti, indipendente dal programma di revisioni periodiche, la cui frequenza può, d'altra parte, essere ridotta.

The problem of making the output of a discrete-time linear system totally insensitive to an exogenous input signal known with preview is tackled in the geometric approach context. A necessary and sufficient condition for exact decoupling with stability in the presence of finite preview is introduced, where the structural aspects and the stabilizability aspects are considered separately. On the assumption that structural decoupling is feasible, internal stabilizability of the minimal self-bounded controlled invariant satisfying the structural constraint $$\mathcal{V}_m$$ guarantees the stability of the dynamic feedforward compensator. However, if the structural decoupling is feasible, but $$\mathcal{V}_m$$ is not internally stabilizable, exact decoupling is nonetheless achievable with a stable feedforward compensator on the sole assumption that $$\mathcal{V}_m$$ has no unassignable internal eigenvalues on the unit circle, provided that the signal to be rejected is known with infinite preview. An algorithmic framework based on steering-along-zeros techniques, completely devised in the time domain, shows how to compute the convolution profile of the feedforward compensator in each case.

Self-bounded controlled invariant subspaces are shown to play a key role in the synthesis of minimal-order dynamic regulators attaining model following by output feedback with stability. The approach, completely embedded in the geometric context, provides insight into the internal eigenstructure of the minimal self-bounded controlled invariant subspace, thus leading to an effective treatment of nonminimum-phase systems.

The problem of making the output insensitive to an exogenous input signal known with preview is tackled in the geometric approach context. Necessary and sufficient constructive conditions for decoupling with minimal preview are proved by means of simple geometric arguments. The structural and the stabilizability conditions are considered separately. The use of self-bounded controlled invariant subspaces enables the minimal order solution to be straightforwardly derived. A steering along zeros technique is devised to solve decoupling in the presence of unstable unassignable dynamics of the minimal self-bounded controlled invariant subspace satisfying the structural constraint. The procedure is illustrated by an example often considered in the literature.

The problem of detecting anomalous operating conditions in systems subject to deterioration is tackled by means of geometric tools, which provide sharp necessary and sufficient conditions for its exact solution. According to an approach often adopted, the observation problem is turned into the more congenial control problem by means of duality arguments. Hence, an extension of the well-known geometric conditions for noninteracting controls is introduced, which, when transferred to the observation context, enables detection of anomalies in the behavior of the monitored system by processing recorded sequences of the outputs. The algorithm herein proposed is particularly suitable and effective in condition monitoring of single-throw mechanical systems (railway switch machines, lifts, mechanical presses, automatic doors and barriers, etc.). A minimal-norm solution, which can be used when the geometric conditions are not satisfied is also presented.

The synthesis of a feedforward unit for H2-optimal rejection of previewed signals in continuous time-invariant linear systems is considered. The H2-optimal compensator herein devised consists of a finite impulse response system working in connection with a standard dynamic unit. The design strategy is based on a simple interpretation of the H2-optimal rejection problem withpre view as a compound optimal control problem, i.e. as a problem consisting of a finite-horizon LQ control problem with constrained final state, an infinite-horizon LQ control problem, and the problem of minimizing the global cost functional. An explicit description of the compensator is derived by exploiting some results on the parameterization of the solutions of the associated autonomous Hamiltonian system.

The structural properties of self-bounded controlled invariant subspaces are fundamental to the synthesis of a dynamic feedforward compensator achieving insensitivity of the controlled output to a disturbance input accessible for measurement, on the assumption that the system is stable or pre-stabilized by an inner feedback. The control system herein devised has several important features: i) minimum order of the feedforward compensator; ii) minimum number of unassignable dynamics internal to the feedforward compensator; iii) maximum number of dynamics, external to the feedforward compensator, arbitrarily assignable by a possible inner feedback. From the numerical point of view, the design method herein detailed does not involve any computation of eigenspaces, which may be critical for systems of high order. The procedure is first presented for left-invertible systems. Then, it is extended to non-left-invertible systems by means of a simple, original, squaring-down technique.

A dynamic feedforward scheme enables measurable signal decoupling to be solved independently of other problems simultaneously present in the design of an actual control system, like e.g. plant pre-stabilization, robustness with respect to uncertainties, insensitivity to unaccessible disturbances, etc. The synthesis procedure, based on the properties of self-bounded controlled invariant subspaces, ensures the minimal complexity of the dynamic feedforward unit, in terms of the minimal unassignable dynamics, in the case of left-invertible systems and, on certain conditions, also in the case of non-left-invertible systems. The dynamic output feedback in charge of pre-stabilization, or, more generally, ensuring some robustness or insensitivity properties, does not affect the complexity of the dynamic feedforward unit. In fact, the particular layout where the feedback unit receives an input directly from the precompensator, preserves the set of the internal unassignable eigenvalues of the minimal self-bounded controlled invariant. Hence, it maintains the unassignable dynamics of the precompensator.