70 results found
Shakib MF, Scarciotti G, Pogromsky AY, et al., 2023, Time-domain moment matching for multiple-input multiple-output linear time-invariant models, Automatica, Vol: 152, Pages: 1-8, ISSN: 0005-1098
Model reduction by moment matching for linear time-invariant (LTI) models is a reduction technique that has a clear interpretation in the Laplace domain. In particular, for the multiple-input multiple-output (MIMO) LTI case, Krylov subspace methods aim at matching the transfer-function matrix (and possibly its derivatives) of the reduced-order model to the transfer-function matrix of the full-order model along so-called tangential directions at desired interpolation points. A straightforward application of time-domain moment matching to MIMO LTI models does not result in such a match in the transfer-function matrix. In this paper, we derive a relation between the MIMO transfer-function matrices of the full- and the reduced-order models that follows from the application of time-domain moment matching on MIMO LTI models. This is subsequently exploited to formulate conditions on the parameters of time-domain moment matching under which the transfer-function matrix is matched along tangential directions, thus ensuring consistency with classical Krylov subspace methods.
Mellone A, Scarciotti G, 2023, Stochastic relative degree and path-wise control of nonlinear stochastic systems, IEEE Transactions on Automatic Control, Vol: 68, Pages: 1970-1985, ISSN: 0018-9286
We address the path-wise control of systems described by a set of nonlinear stochastic differential equations. For this class of systems, we introduce a notion of stochastic relative degree and a change of coordinates which transforms the dynamics to a stochastic normal form. The normal form is instrumental for the design of a state-feedback control which linearises and makes the dynamics deterministic. We observe that this control is idealistic, i.e. it is not practically implementable because it employs a feedback of the Brownian motion (which is never available) to cancel the noise. Using the idealistic control as a starting point, we introduce a hybrid control architecture which achieves practical path-wise control. This hybrid controller uses measurements of the state to perform periodic compensations for the noise contribution to the dynamics. We prove that the hybrid controller retrieves the idealistic performances in the limit as the compensating period approaches zero. We address the problem of asymptotic output tracking, solving it in the idealistic and in the practical framework. We finally validate the theory by means of a numerical example.
Gong Z, Scarciotti G, 2023, A hybrid observer for practical observability of linear stochastic systems, 61st IEEE Conference on Decision and Control, Publisher: IEEE
This paper focuses on the problem of observability and observer design for linear stochastic systems. To introduce our idea, we first construct an idealistic observer. This idealistic observer is not causal as it requires perfect knowledge of the Brownian motion. However, after introducing an a posteriori method to reconstruct the variations of the Brownian motion in discrete time, we propose a realistic hybrid observer which approximates the idealistic observer. The performance of this hybrid observer can be made arbitrarily close to that of the idealistic observer. Numerical simulations illustrate the results of the paper.
Mao J, Scarciotti G, 2022, Data-Driven Model Reduction by Two-Sided Moment Matching
In this brief paper, we propose a time-domain data-driven method for modelorder reduction by two-sided moment matching for linear systems. An algorithmthat asymptotically approximates the matrix product $\Upsilon \Pi$ fromtime-domain samples of the so-called two-sided interconnection is provided.Exploiting this estimated matrix, we determine the unique reduced-order modelof order $\nu$, which asymptotically matches the moments at $2 \nu$ distinctinterpolation points. Furthermore, we discuss the impact that certaindisturbances and data distortions may have on the algorithm. Finally, weillustrate the use of the proposed methodology by means of a benchmark model.
Mao J, Scarciotti G, 2022, Data-driven model reduction by moment matching for linear systems through a swapped interconnection, European Control Conference 2022, Publisher: IEEE, Pages: 1690-1695
In this work, we propose a time-domain data-driven technique for model reduction by moment matching oflinear systems. We propose an algorithm, based on the so-calledswapped interconnection, that (asymptotically) approximates anarbitrary number of moments of the system from a singletime-domain sample. A family of reduced-order models thatmatch the estimated moments is derived. Finally, the use of theproposed algorithm is demonstrated on the problem of modelreduction of a building model
Niu Z, Scarciotti G, 2022, Ranking the effectiveness of non-pharmaceutical interventions to counter COVID-19 in UK universities with vaccinated population, Scientific Reports, Vol: 12, ISSN: 2045-2322
Several universities around the world have resumed in-person teaching after successful vaccination campaigns have covered 70/80% of the population. In this study, we combine a new compartmental model with an optimal control formulation to discover, among different non-pharmaceutical interventions, the best prevention strategy to maximize on-campus activities while keeping spread under control. Composed of two interconnected Susceptible-Exposed-Infected-Quarantined-Recovered (SEIQR) structures, the model enables staff-to-staff infections, student-to-staff cross infections, student-to-student infections, and environment-to-individual infections. Then, we model input variables representing the implementation of different non-pharmaceutical interventions and formulate and solve optimal controlproblems for four desired scenarios: minimum number of cases, minimum intervention, minimum non-quarantine intervention, and minimum quarantine intervention. Our results reveal the particular significance of mask wearing and social distancing in universities with vaccinated population (with proportions according to UK data). The study also reveals that quarantining infected students has a higher importance than quarantining staff. In contrast, other measures such as environmental disinfection seems to be less important.
Mellone A, Scarciotti G, 2022, Output regulation of linear stochastic systems, IEEE Transactions on Automatic Control, Vol: 67, Pages: 1728-1743, ISSN: 0018-9286
We address the output regulation problem for a general class of linear stochastic systems. Specifically, we formulate and solve the ideal full-information and output-feedback problems, obtaining perfect, but non-causal, asymptotic regulation. A characterisation of the problem solvability is deduced. We point out that the ideal problems cannot be solved in practice because they unrealistically require that the Brownian motion affecting the system is available for feedback. Drawing from the ideal solution, we formulate and solve approximate versions of the full-information and output-feedback problems, which do not yield perfect asymptotic tracking but can be solved in a realistic scenario. These solutions rely on two key ideas: first we introduce a discrete-time a-posteriori estimator of the variations of the Brownian motion obtained causally by sampling the system state or output; second we introduce a hybrid state observer and a hybrid regulator scheme which employ the estimated Brownian variations. The approximate solution tends to the ideal as the sampling period tends to zero. The proposed theory is validated by the regulation of a circuit subject to electromagnetic noise.
Bai H, Mylvaganam T, Scarciotti G, 2022, Model reduction for quadratic-bilinear systems using nonlinear moments, European Control Conference 2022, Publisher: IEEE
We propose a steady-state based moment matchingmethod for model reduction of quadratic-bilinear systems.Considering a large-scale quadratic-bilinear system possessing astable equilibrium at the origin, the goal of this paper is to designa reduced order model which maintains certain properties of theoriginal system. More precisely, it is required that the reducedorder model also possesses a stable equilibrium at the originand, further, it may be desirable that its relative degree matchesthat of the original system. We use the notion of nonlinearmoments and exploit a formal power expansion to solve thisproblem. Two different families of reduced order models areprovided in this paper and their use is demonstrated on anillustrative numerical example.
Scarciotti G, Teel AR, 2022, On moment matching for stochastic systems, IEEE Transactions on Automatic Control, Vol: 67, Pages: 541-556, ISSN: 0018-9286
In this paper we study the problem of model reduction by moment matching for stochastic systems. We characterize the mathematical object which generalizes the notion of moment to stochastic differential equations and we find a class of models which achieve moment matching. However, differently from the deterministic case, these reduced order models cannot be considered “simpler” because of the high computational cost paid to determine the moment. To overcome this difficulty, we relax the moment matching problem in two different ways and we present two classes of reduced order models which, approximately matching the stochastic moment, are computationally tractable.
Shakib MF, Scarciotti G, Jungers M, et al., 2021, Optimal H∞ LMI-based model reduction by moment matching for linear time-invariant models, 2021 60th IEEE Conference on Decision and Control (CDC), Publisher: IEEE, Pages: 6914-6919
This paper proposes an approach to model order reduction of stable linear time-invariant (LTI) models. The proposed approach extends time-domain moment matching by the minimization of the H∞ norm of the error dynamics characterizing the difference between the full-order and reduced-order models given fixed interpolation points. The optimal H∞ moment matching problem is a constrained optimization problem with bilinear constraints. Introducing a novel numerical procedure, we minimize the approximation error, while respecting the constraints and, thereby, find a suboptimal H∞ reduced-order model. The effectiveness of the approach is illustrated in a numerical example.
Mellone A, Gong Z, Scarciotti G, 2021, Modelling, prediction and design of national COVID-19 lockdowns by stringency and duration, Scientific Reports, Vol: 11, Pages: 1-13, ISSN: 2045-2322
The implementation of lockdowns has been a key policy to curb the spread of COVID-19 and to keep under control the number of infections. However, quantitatively predicting in advance the effects of lockdowns based on their stringency and duration is a complex task, in turn making it difficult for governments to design effective strategies to stop the disease. Leveraging a novel mathematical “hybrid” approach, we propose a new epidemic model that is able to predict the future number of active cases and deaths when lockdowns with different stringency levels or durations are enforced. The key observation is that lockdown-induced modifications of social habits may not be captured by traditional mean-field compartmental models because these models assume uniformity of social interactions among the population, which fails during lockdown. Our model is able to capture the abrupt social habit changes caused by lockdowns. The results are validated on the data of Israel and Germany by predicting past lockdowns and providing predictions in alternative lockdown scenarios (different stringency and duration). The findings show that our model can effectively support the design of lockdown strategies by stringency and duration, and quantitatively forecast the course of the epidemic during lockdown.
Faedo N, Scarciotti G, Astolfi A, et al., 2021, On the approximation of moments for nonlinear systems, IEEE Transactions on Automatic Control, Vol: 66, Pages: 5538-5545, ISSN: 0018-9286
Model reduction by moment-matching relies upon the availability of the so-called moment. If the system is nonlinear, the computation of moments depends on an underlying specific invariance equation, which can be difficult or impossible to solve. This note presents four technical contributions related to the theory of moment matching: first, we identify a connection between moment-based theory and weighted residual methods. Second, we exploit this relation to provide an approximation technique for the computation of nonlinear moments. Third, we extend the definition of nonlinear moment to the case in which the generator is described in explicit form. Finally, we provide an approximation technique to compute the moments in this scenario. The results are illustrated by means of two examples.
Faedo N, Scarciotti G, Astolfi A, et al., 2021, Nonlinear energy-maximising optimal control of wave energy systems: A moment-based approach, IEEE Transactions on Control Systems Technology, Vol: 29, Pages: 2533-2547, ISSN: 1063-6536
Linear dynamics are virtually always assumed when designing optimal controllers for wave energy converters (WECs), motivated by both their simplicity and computational convenience. Nevertheless, unlike traditional tracking control applications, the assumptions under which the linearization of WEC models is performed are challenged by the energy-maximizing controller itself, which intrinsically enhances device motion to maximize power extraction from incoming ocean waves. \GSIn this article, we present a moment-based energy-maximizing control strategy for WECs subject to nonlinear dynamics. We develop a framework under which the objective function (and system variables) can be mapped to a finite-dimensional tractable nonlinear program, which can be efficiently solved using state-of-the-art nonlinear programming solvers. Moreover, we show that the objective function belongs to a class of generalized convex functions when mapped to the moment domain, guaranteeing the existence of a global energy-maximizing solution and giving explicit conditions for when a local solution is, effectively, a global maximizer. The performance of the strategy is demonstrated through a case study, where we consider (state and input-constrained) energy maximization for a state-of-the-art CorPower-like WEC, subject to different hydrodynamic nonlinearities.
Faedo N, Scarciotti G, Astolfi A, et al., 2021, Energy-maximising moment-based constrained optimal control of ocean wave energy farms, IET Renewable Power Generation
Shakib MF, Scarciotti G, Pogromsky AY, et al., 2021, Model reduction by moment matching for convergent Lur'e-type models, American Control Conference (ACC), Publisher: IEEE, Pages: 4459-4464, ISSN: 0743-1619
This paper proposes an approach to model order reduction of convergent Lur'e-type models, which consist of a linear time-invariant (LTI) block and a static nonlinear block that is placed in feedback with the LTI block. In the proposed approach, we match a finite number of moments of the LTI block and keep the static nonlinear block to approximate the moments of the Lur'e-type model. The benefits of this approach are that the Lur'e-type structure is preserved after reduction, that the reduction method has an interpretation in terms of the frequency response function of the LTI block and that global exponential stability properties of the full-order model are preserved. The effectiveness of the approach is illustrated in a numerical example.
Mellone A, Scarciotti G, 2021, Approximate feedback linearisation and stabilisation of nonlinear stochastic systems, 21st IFAC World Congress, Publisher: IFAC Secretariat, Pages: 4985-4990, ISSN: 2405-8963
This paper addresses the design of a practically sound control architecture to solvethe problem of feedback linearisation and stabilisation of single-input single-output nonlinearstochastic systems. We first present a causal method to obtain, from measurements of the state,a-posterioriestimates of the variations of the Brownian motion which affected the system. Thenwe employ these estimates to design a control law that approximately compensates for thediffusive dynamics of the system. We address the local stabilisation problem and we prove thatthe control law which performs the proposed stochastic compensations stabilises a broader classof systems with respect to feedback laws without compensation. We finally validate the theorythrough a numerical example.
Mellone A, Scarciotti G, 2021, The zero dynamics of nonlinear stochastic systems: stabilization and output tracking in the ideal case, 21st IFAC World Congress, Publisher: IFAC Secretariat, Pages: 4979-4984, ISSN: 2405-8963
This paper introduces the notion of zero dynamics and presents results of localstabilisation and output tracking for single-input single-output nonlinear stochastic systemsdescribed by stochastic differential equations. For this class of systems we define the zero dynamicswhen the stochastic relative degree is strictly smaller than the order of the system. We show that,under suitable conditions on the zero dynamics, the equilibrium at the origin can be stabilisedvia a coordinate change and a nonlinear state feedback. In an analogous way, we show that it ispossible to achieve local asymptotic output tracking of a reference signal. We validate the theorythrough a numerical example.
Mellone A, Scarciotti G, 2021, A note on the Ito and Stratonovich stochastic relative degree and normal form, 2020 IEEE Conference on Decision and Control (CDC), Publisher: IEEE
In this note we compare two notions of stochastic relative degree. Specifically, we consider nonlinear stochastic systems defined by the same stochastic differential equation and interpreted in either Ito’s or Stratonovich’s sense. We then recall the Ito stochastic relative degree and we introduce the concept of Stratonovich stochastic relative degree and Stratonovich normal form. We show, by means of examples, that the stochastic relative degrees arising from the two different interpretations of the same stochastic differential equations are, in general, different. We finally point out that this discrepancy can be eliminated through conversion formulas between Ito and Stratonovich integrals.
Astolfi A, Scarciotti G, Simard J, et al., 2021, Model reduction by moment matching: beyond linearity a review of the last 10 years, 2020 59th IEEE Conference on Decision and Control (CDC), Publisher: IEEE
We present a review of some recent contributions to the theory and application of nonlinear model order reduction by moment matching. The tutorial paper is organized in four parts: 1) Moments of Nonlinear Systems; 2) Playing with Moments: Time-Delay, Hybrid, Stochastic, Data-Driven and Beyond; 3) The Loewner Framework; 4) Applications to Optimal Control and Wave Energy Conversion.
Mellone A, Scarciotti G, 2020, Error-feedback output regulation of linear stochastic systems: a hybrid nonlinear approach, 11th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2019, jointly with MECHATRONICS 2019), Publisher: IFAC Secretariat, Pages: 520-525, ISSN: 2405-8963
The problem of output regulation for linear stochastic systems is addressed. Wefirst define and solve the ideal problem of output regulation via error feedback. We note thatits solution is not implementable in practice because the Brownian motion is not available formeasure. Therefore, we define an approximate problem for which we provide a practical solution.The implemented controller is hybrid, in that a continuous-time, deterministic control law issupplemented by a discrete-time, stochastic correction. This correction is performed using ana-posterioriapproximation of the variations of the Brownian motion provided by a nonlinearestimator. The resulting hybrid closed-loop system is nonlinear, as the scalars approximatingthe increments of the Brownian motion depend nonlinearly on the states and the inputs. Theerror between the solution of the approximate problem and the solution of the ideal problem ischaracterised. We show that the ideal solution is retrieved as the sampling time tends to zero.We illustrate the results by means of a numerical example.
Di Francor P, Scarciotti G, Astolfi A, 2020, Stability of nonlinear differential-algebraic systems via additive identity, IEEE/CAA Journal of Automatica Sinica, Vol: 7, Pages: 929-941, ISSN: 2329-9266
The stability analysis for nonlinear differential-algebraic systems is addressed using tools from classical control theory. Sufficient stability conditions relying on matrix inequalities are established via Lyapunov Direct Method. In addition, a novel interpretation of differential-algebraic systems as feedback interconnection of a purely differential system and an algebraic system allows reducing the stability analysis to a small-gain-like condition. The study of stability properties for constrained mechanical systems, for a class of Lipschitz differential-algebraic systems and for an academic example is used to illustrate the theory.
Scarciotti G, Jiang Z-P, Astolfi A, 2020, Data-driven constrained optimal model reduction, European Journal of Control, Vol: 53, Pages: 68-78, ISSN: 0947-3580
Model reduction by moment matching can be interpreted as the problem of finding a reduced-order model which possesses the same steady-state output response of a given full-order system for a prescribed class of input signals. Little information regarding the transient behavior of the system is systematically preserved, limiting the use of reduced-order models in control applications. In this paper we formulate and solve the problem of constrained optimal model reduction. Using a data-driven approach we determine an estimate of the moments and of the transient response of a possibly unknown system. Consequently we determine a reduced-order model which matches the estimated moments at the prescribed interpolation signals and, simultaneously, possesses the estimated transient. We show that the resulting system is a solution of the constrained optimal model reduction problem. Detailed results are obtained when the optimality criterion is formulated with the time-domain ℓ1, ℓ2, ℓ∞ norms and with the frequency-domain norm. The paper is illustrated by two examples: the reduction of the model of the vibrations of a building and the reduction of the Eady model (an atmospheric storm track model).
Di Franco P, Scarciotti G, Astolfi A, 2020, A globally stable algorithm for the integration of high-index differential-algebraic systems, IEEE Transactions on Automatic Control, Vol: 65, Pages: 2107-2122, ISSN: 0018-9286
The problem of constraint stabilization and numerical integration for differential-algebraic systems is addressed using Lyapunov theory. It is observed that the application of stabilization methods which rely on a linear feedback mechanism to nonlinear systems may result in trajectories with finite escape time. To overcome this problem we propose a method based on a nonlinear stabilization mechanism which guarantees the global existence and convergence of the solutions. Discretization schemes, which preserve the properties of the method, are also presented. The results are illustrated by means of the numerical integration of a slider-crank mechanism.
Mellone A, Scarciotti G, 2020, Normal form and exact feedback linearisation of nonlinear stochastic systems: the ideal case, 2019 IEEE Conference on Decision and Control (CDC), Publisher: IEEE
This paper introduces the concepts of stochastic relative degree, normal form and exact feedback linearisation for single-input single-output nonlinear stochastic systems. The systems are defined by stochastic differential equations in which both the drift and the diffusion terms are nonlinear functions of the states and the control input. First, we define new differential operators and the concept of stochastic relative degree. Then we introduce a suitable coordinate change and we show that the dynamics of the transformed state has a simplified structure, which we name normal form. Finally, we show that a condition on the stochastic relative degree of the system is sufficient for it to be rendered linear via a coordinate change and a nonlinear feedback. We provide an analytical example to illustrate the theory.
Faedo N, Garcia-Violini D, Scarciotti G, et al., 2019, Robust Moment-Based Energy-Maximising Optimal Control of Wave Energy Converters, 2019 IEEE 58th Conference on Decision and Control (CDC), Publisher: IEEE
Scarciotti G, Mellone A, 2019, ε-approximate output regulation of linear stochastic systems: a hybrid approach, 2019 European Control Conference, Publisher: IEEE, Pages: 287-292
The problem of output regulation for linear stochastic systems is addressed. The controlled system belongs to a general class of linear systems, namely the state, the control input and the exogenous input appear in both the drift and diffusion terms of the differential equations. Building upon the solution of the ideal, non-causal, stochastic regulator problem,we define an approximate full-information problem. By means of measurements of the state vector, we provide a way to compute a sequence of scalars approximating a posteriori the variations of the Brownian motion. Then, we propose a hybrid control architecture which solves the approximate problem.The continuous-time part of the controller is deterministic,whereas the discrete-time part has the function of “correcting”the control action by means of the approximate discrete-time Brownian motion. The solution of the ideal stochastic regulator problem is recovered as the sampling time tends to zero. We illustrate the results by means of a numerical example and conclude the paper with some final remarks:the proposed control architecture is the first causal solution of the full-information output regulation problem and is an essential intermediate step for the solution of the error-feedback problem.
Breschi V, Formentin S, Scarciotti G, et al., 2019, Simulation-driven fixed-order controller tuning via moment matching, 2019 European Control Conference, Publisher: IEEE, Pages: 2307-2312
We propose a controller tuning method basedon the data-driven model reduction by moment matchingtheory. By selecting a reference closed-loop transfer function,a moment matching data-driven model reduction algorithm isused to synthesize a fixed-order controller, via the identificationof a model of the inverse transfer function of the controller,i.e.the transfer function from the controlled input of thesystem to the mismatch error signal. The controller is finallyobtained by inverting this transfer function. The fixed-ordercontroller is guaranteed to match the steady-state behaviorof the ideal controller at certain pre-selected frequencies. Theeffectiveness of the resulting design method is assessed on acontrol problem for the48-th order model of the Los AngelesUniversity Hospital building.
Faedo N, Scarciotti G, Astolfi A, et al., 2019, Moment-based constrained optimal control of an array of wave energy converters, 2019 American Control Conference, Publisher: IEEE, Pages: 4797-4802, ISSN: 2378-5861
The roadmap to a successful commercialisation ofwave energy inherently incorporates the concept of anarrayorfarmof Wave Energy Converters (WECs). These interactinghydrodynamic structures require an optimised process that canensure the maximum extraction of time-averaged energy fromocean waves, while respecting the physical limitations of eachdevice and actuator characteristics. Recently, a novel optimalcontrol framework based on the concept ofmoment, for asingle WEC device, has been introduced in . Such a strategyoffers an energy-maximising computationally efficient solutionthat can systematically incorporate state and input constraints.This paper presents the mathematical extension of the optimalcontrol framework of  to the case where an array of WECsis considered, providing an efficient solution that exploits thehydrodynamic interaction between devices to maximise the totalabsorbed energy.
Mellone A, Scarciotti G, 2019, ϵ-approximate output regulation of linear stochastic systems: A hybrid approach, Pages: 287-292
The problem of output regulation for linear stochastic systems is addressed. The controlled system belongs to a general class of linear systems, namely the state, the control input and the exogenous input appear in both the drift and diffusion terms of the differential equations. Building upon the solution of the ideal, non-causal, stochastic regulator problem, we define an approximate full-information problem. By means of measurements of the state vector, we provide a way to compute a sequence of scalars approximating a posteriori the variations of the Brownian motion. Then, we propose a hybrid control architecture which solves the approximate problem. The continuous-time part of the controller is deterministic, whereas the discrete-time part has the function of 'correcting' the control action by means of the approximate discrete-time Brownian motion. The solution of the ideal stochastic regulator problem is recovered as the sampling time tends to zero. We illustrate the results by means of a numerical example and conclude the paper with some final remarks: the proposed control architecture is the first causal solution of the full-information output regulation problem and is an essential intermediate step for the solution of the error-feedback problem.
Scarciotti G, Mylvaganam T, 2019, Approximate infinite-horizon optimal control for stochastic systems, 57th IEEE Conference on Decision and Control (CDC), Publisher: IEEE
The policy of an optimal control problem fornonlinear stochastic systems can be characterized by a second-order partial differential equation for which solutions are notreadily available. In this paper we provide a systematic methodfor obtaining approximate solutions for the infinite-horizonoptimal control problem in the stochastic framework. Themethod is demonstrated on an illustrative numerical examplein which the control effort is not weighted, showing that thetechnique is able to deal with one of the most striking featuresof stochastic optimal control.
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