Papers by Alessandro Astolfi
2017 American Control Conference (ACC), 2017
This paper deals with the problem of model reduction by moment matching for linear differential i... more This paper deals with the problem of model reduction by moment matching for linear differential inclusions. The problem is formally formulated and the notions of momentset, perturbed moment trajectory, approximate reduced order model and robust reduced order model are introduced. Two sets of results are presented. The first part of the paper deals with robustness of the reduced order models with respect to input perturbations. Exploiting this result an enhanced model reduction scheme for linear differential equations is presented. In the second part of the paper we focus on the problem of model reduction by moment matching for time-varying systems driven by time-varying signal generators. Finally, these two sets of results are used to solve the problem of model reduction for linear differential inclusions. The results are illustrated by means of numerical examples.
2016 IEEE 55th Conference on Decision and Control (CDC), 2016
Model reduction by moment matching does not preserve, in a systematic way, the transient response... more Model reduction by moment matching does not preserve, in a systematic way, the transient response of the system to be reduced, thus limiting the use of this model reduction technique in control problems. With the final goal of designing reduced-order models which can effectively be used (not just for analysis but also) for control purposes, we determine, using a data-driven approach, an estimate of the moments and of the transient response of an unknown system. We compute the unique, up to a change of coordinates, reducedorder model which possesses the estimated transient and, simultaneously, achieves moment matching at the prescribed interpolation points. The error between the output of the system and the output of the reduced-order model is minimized and we show that the resulting system is a constrained optimal (in a sense to be specified) reduced-order model. The results of the paper are illustrated by means of a simple numerical example.
2021 60th IEEE Conference on Decision and Control (CDC), 2021
Infinite-horizon optimal control problems for nonlinear systems are studied and discussed. First,... more Infinite-horizon optimal control problems for nonlinear systems are studied and discussed. First, we thoroughly revisit the formulation of the underlying dynamic optimisation problem together with the classical results providing its solution. Then, we consider two alternative methods to construct solutions (or approximations thereof) of such problems, developed in recent years, that provide theoretical insights as well as computational benefits. While the considered methods are mostly based on tools borrowed from the theories of Dynamic Programming and Pontryagin's Minimum Principles, or a combination of the two, the proposed control design strategies yield innovative, systematic and constructive methods to provide exact or approximate solutions of nonlinear optimal control problems. Interestingly, similar ideas can be extended also to linear and nonlinear differential games, namely dynamic optimisation problems involving several decision-makers. Due their advantages in terms of computational complexity, the considered methods have found several applications. An example of this is provided, through the consideration of the multi-agent collision avoidance problem, for which both simulations and experimental results are provided.
2020 59th IEEE Conference on Decision and Control (CDC), 2020
We present a review of some recent contributions to the theory and application of nonlinear model... more 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.
2019 18th European Control Conference (ECC), 2019
We propose a controller tuning method based on the data-driven model reduction by moment matching... more We propose a controller tuning method based on the data-driven model reduction by moment matching theory. By selecting a reference closed-loop transfer function, a moment matching data-driven model reduction algorithm is used to synthesize a fixed-order controller, via the identification of a model of the inverse transfer function of the controller, i.e. the transfer function from the controlled input of the system to the mismatch error signal. The controller is finally obtained by inverting this transfer function. The fixed-order controller is guaranteed to match the steady-state behavior of the ideal controller at certain pre-selected frequencies. The effectiveness of the resulting design method is assessed on a control problem for the 48-th order model of the Los Angeles University Hospital building.
2019 IEEE 58th Conference on Decision and Control (CDC), 2019
We introduce a moment-based framework to design robust energy-maximising optimal controllers for ... more We introduce a moment-based framework to design robust energy-maximising optimal controllers for Wave Energy Converters (WECs). The technique explicitly allows model uncertainty in the computation of the optimal control input, by defining a suitable uncertainty polytope. The resulting robust optimisation is formulated as a minimax problem which has to be solved only at a small number of points of this uncertainty set. The objective function under the proposed strategy is shown to be of quadratic-type and the optimal solution is proven to be unique, providing a computationally efficient robust optimal control framework for WECs. The performance of the proposed controller is demonstrated by means of an application case, which considers a heaving point absorber WEC with imprecisely known model parameters.
2019 American Control Conference (ACC), 2019
The roadmap to a successful commercialisation of wave energy inherently incorporates the concept ... more The roadmap to a successful commercialisation of wave energy inherently incorporates the concept of an array or farm of Wave Energy Converters (WECs). These interacting hydrodynamic structures require an optimised process that can ensure the maximum extraction of time-averaged energy from ocean waves, while respecting the physical limitations of each device and actuator characteristics. Recently, a novel optimal control framework based on the concept of moment, for a single WEC device, has been introduced in [1]. Such a strategy offers an energy-maximising computationally efficient solution that can systematically incorporate state and input constraints. This paper presents the mathematical extension of the optimal control framework of [1] to the case where an array of WECs is considered, providing an efficient solution that exploits the hydrodynamic interaction between devices to maximise the total absorbed energy.
IEEE Transactions on Automatic Control, 2021
Model reduction by moment-matching relies upon the availability of the so-called moment. If the s... more 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 article 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.
IEEE Transactions on Automatic Control, 2019
The notions of eigenvalue, pole and moment at a pole of a continuous-time, nonlinear, time-invari... more The notions of eigenvalue, pole and moment at a pole of a continuous-time, nonlinear, time-invariant system are studied. Eigenvalues and poles are first characterized in terms of invariant subspaces. Tools from geometric control theory are used to define nonlinear enhancements of these notions and to study their relationship with the solution of certain partial differential equations, cascade decompositions and steady-state impulse responses. The theory is illustrated by means of worked-out examples and its significance is demonstrated by solving the model reduction problem by moment matching at poles for nonlinear systems.
European Journal of Control, 2019
Model reduction by moment matching can be interpreted as the problem of finding a reduced-order m... more 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 1 , 2 , ∞ norms and with the frequency-domain H 2 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).
IEEE Transactions on Automatic Control, 2019
Moments of continuous random variables admitting a probability density function are studied. We s... more Moments of continuous random variables admitting a probability density function are studied. We show that, under certain assumptions, the moments of a random variable can be characterised in terms of a Sylvester equation and of the steady-state output response of a specific interconnected system. This allows to interpret well-known notions and results of probability theory and statistics in the language of systems theory, including the sum of independent random variables, the notion of mixture distribution and results from renewal theory. The theory developed is based on tools from the center manifold theory, the theory of the steady-state response of nonlinear systems, and the theory of output regulation. Our formalism is illustrated by means of several examples and can be easily adapted to the case of discrete and of multivariate random variables.
Foundations and Trends® in Systems and Control, 2017
Mathematical models are at the core of modern science and technology. An accurate description of ... more Mathematical models are at the core of modern science and technology. An accurate description of behaviors, systems and processes often requires the use of complex models which are difficult to analyze and control. To facilitate analysis of and design for complex systems, model reduction theory and tools allow determining "simpler" models which preserve some of the features of the underlying complex description. A large variety of techniques, which can be distinguished depending on the features which are preserved in the reduction process, has been proposed to achieve this goal. One such a method is the moment matching approach. This monograph focuses on the problem of model reduction by moment matching for nonlinear systems. The central idea of the method is the preservation, for a prescribed class of inputs and under some technical assumptions, of the steady-state output response of the system to be reduced. We present the moment matching approach from this vantage point, covering the problems of model reduction for nonlinear systems, nonlinear time-delay systems, data-driven model reduction for nonlinear systems and model reduction for "discontinuous" input signals. Throughout the monograph linear systems, with their simple structure and strong properties, are used as a paradigm to facilitate understanding of the theory and provide foundation of the terminology and notation. The text is enriched by several numerical examples, physically motivated examples and with connections to well-established notions and tools, such as the phasor transform.
IFAC-PapersOnLine, 2016
In this paper we present a model reduction technique based on moment matching for a class of hybr... more In this paper we present a model reduction technique based on moment matching for a class of hybrid systems with state-dependent jumps. The problem of characterizing the steady-state for this class of systems is studied and a result which allows to described the steadystate response of hybrid systems through the use of a hybrid mapping is given. Then a family of hybrid reduced order models which achieve moment matching and are easily parameterizable is provided. The special case of periodic input signals is analyzed and conditions for applying the technique are given for this class. A numerical simulation illustrates the results.
IEEE Transactions on Automatic Control, 2017
The persistence of excitation of signals generated by time-invariant, autonomous, linear, and non... more The persistence of excitation of signals generated by time-invariant, autonomous, linear, and nonlinear systems is studied using a geometric approach. A rank condition is shown to be equivalent, under certain assumptions, to the persistence of excitation of the solutions of the class of systems considered, both in the discrete-time and in the continuous-time settings. The rank condition is geometric in nature and can be checked a priori, i.e. without knowing explicitly the solutions of the system, for almost periodic systems. The significance of the ideas and tools presented is illustrated by means of simple examples. Applications to model reduction from input-output data and stability analysis of skew-symmetric systems are also discussed.
Automatica, 2017
Theory and methods to obtain reduced order models by moment matching from input/output data are p... more Theory and methods to obtain reduced order models by moment matching from input/output data are presented. Algorithms for the estimation of the moments of linear and nonlinear systems are proposed. The estimates are exploited to construct families of reduced order models. These models asymptotically match the moments of the unknown system to be reduced. Conditions to enforce additional properties, e.g. matching with prescribed eigenvalues, upon the reduced order model are provided and discussed. The computational complexity of the algorithms is analyzed and their use is illustrated by two examples: we compute converging reduced order models for a linear system describing the model of a building and we provide, exploiting an approximation of the moment, a nonlinear planar reduced order model for a nonlinear DC-to-DC converter.
IEEE Transactions on Power Electronics, 2016
Power electronic devices are inherently discontinuous systems. Square waves, produced by intercon... more Power electronic devices are inherently discontinuous systems. Square waves, produced by interconnected transistors, are commonly used to control inverters. This paper proposes a novel phasor transform, based on the theory of moments, which allows to analyze the steady-state behavior of discontinuous power electronic devices in closed-form, i.e., without approximations. In the first part of this paper, it is shown that the phasors of an electric circuit are the moments on the imaginary axis of the linear system describing the circuit. Exploiting this observation, in the second part of this paper, we focus on the analysis of circuits powered by discontinuous sources. The new "discontinuous phasor transform" is defined and the v-i characteristics for inductors, capacitors, and resistors are described in terms of this new phasor transform. Since the new quantities maintain their physical meaning, the instantaneous power and average power can be computed in the phasor domain. The analytic potential of the new tool is illustrated studying the steady-state response of power inverters and of wireless power transfer systems with non-ideal switches.
2016 American Control Conference (ACC), 2016
In this short note the relation between the moments of a linear system and the phasors of an elec... more In this short note the relation between the moments of a linear system and the phasors of an electric circuit is discussed. We show that the phasors are a special case of moments and we prove that the components of the solution of a Sylvester equation are the phasors of the currents of the system. We point out several directions in which the phasor theory can be extended using recent generalizations of the moment theory that can benefit the analysis of circuits and power electronics.
2015 54th IEEE Conference on Decision and Control (CDC), 2015
An algorithm for the estimation of the moments of nonlinear systems and nonlinear time-delay syst... more An algorithm for the estimation of the moments of nonlinear systems and nonlinear time-delay systems from input/output data is proposed. The estimate is exploited to construct a family of reduced order models. Conditions to enforce additional properties, e.g. matching with asymptotic stability, matching with prescribed relative degree, matching with prescribed zero dynamics, upon the reduced order model are provided. The use of the technique is illustrated by a few examples based on the averaged model of the DC-to-DCĆuk converter.
IFAC Proceedings Volumes, 2014
The model reduction problem by moment matching for linear time-delay systems is addressed. A para... more The model reduction problem by moment matching for linear time-delay systems is addressed. A parameterized family of models achieving moment matching is characterized. The parameters can be exploited to derive a delay-free reduced order model or reduced order models with additional properties. The theory is illustrated by an example borrowed from the problem of automatic control of a platoon of vehicles.
IEEE Transactions on Automatic Control, 2016
The problem of model reduction by moment matching for linear and nonlinear differential time-dela... more The problem of model reduction by moment matching for linear and nonlinear differential time-delay systems is studied. The class of models considered includes neutral differential time-delay systems with discrete-delays and distributeddelays. The description of moment is revisited by means of a Sylvester-like equation for linear time-delay systems and by means of the center manifold theory for nonlinear time-delay systems. In addition the moments at infinity are characterized for both linear and nonlinear time-delay systems. Parameterized families of models achieving moment matching are given. The parameters can be exploited to derive delay-free reduced order models or time-delay reduced order models with additional properties, e.g. interpolation at an arbitrary large number of points. Finally, the problem of obtaining a reduced order model of an unstable system is discussed and solved.
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Papers by Alessandro Astolfi