2019 18th European Control Conference (ECC), Jun 1, 2019
A multi-agent continuous-time nonlinear model of social behaviour allowing for both competition a... more A multi-agent continuous-time nonlinear model of social behaviour allowing for both competition and cooperation is presented and analysed. The state of each agent is represented by its payoff, which the agent aims at maximising. The role of control variables is played by the model parameters, which account for the agents' decisions to either cooperate with or boycott the other agents and can vary in time within assigned intervals. Alliances and enmities can be established at any time, according to either a greedy or a longsighted criterion. The general nonlinear case is first considered. It is proved that, under realistic assumptions, the system evolution is bounded positive (no extinction) and there is a unique globallystable equilibrium point. As is somehow expected, the optimal decision for all agents corresponds to full cooperation (decision parameters kept at their positive maximum value) in the case of both shortsighted and farsighted criteria. This is not true if some parameters have negative upper bounds (meaning that some agents systematically boycott some others). Then, in the linear case, it is shown that the system is stable for arbitrarilyvarying decision parameters, provided that a Metzler matrix associated with full cooperation is Hurwitz. A characterisation of the long-term behaviour of the linear system is also provided. In particular, it is proved that, under stability conditions, a Nash equilibrium exists if a steady strategy is adopted. G.G. acknowledges support from the Aspasia grant and the DTF grant.
2010 IEEE 18th Signal Processing and Communications Applications Conference, 2010
This paper deals with the integer-order (finite-dimensional) approximation of a fractional-order ... more This paper deals with the integer-order (finite-dimensional) approximation of a fractional-order (infinite-dimensional) system using the interpolation approach in the Loewner framework. First given a set of interpolation frequencies and the corresponding values of the original noninteger-order transfer function, a Loewner matrix and a shifted Loewner matrix are created. Then, from these matrices an integer-order model in state-space descriptor form is constructed. Its transfer function matches the original system transfer function at the given frequencies. To avoid singularity problems related to data redundancy, a truncated singular value decomposition may finally be applied. Numerical simulations show that the suggested approach to fractional-order system approximation compares favourably with alternative techniques recently presented in the literature to the same purpose.
2019 24th International Conference on Methods and Models in Automation and Robotics (MMAR), 2019
A state-space integer-order approximation of a commensurate-order systems is obtained using a dat... more A state-space integer-order approximation of a commensurate-order systems is obtained using a data-driven interpolation approach based on Loewner matrices. Precisely, given the values of the original fractional-order transfer function at a number of generalised frequencies, a descriptor-form state-space model matching these frequency response values is constructed from a suitable Loewner matrix pencil, as already suggested for the reduction of high-dimensional integer-order systems. Even if the stability of the resulting integer-order system cannot be guaranteed, such an approach is particularly suitable for approximating (infinite-dimensional) fractional-order systems because: (i) the order of the approximation is bounded by half the number of interpolation points, (ii) the procedure is more robust and simple than alternative approximation methods, and (iii) the procedure is fairly flexible and often leads to satisfactory results, as shown by a pair of examples taken from the liter...
The frequency response of an evaporator for commercial refrigeration is investigated to develop a... more The frequency response of an evaporator for commercial refrigeration is investigated to develop a controloriented model which balances simplicity with accuracy. The dynamics of such devices is highly nonlinear; the response curve was experimentally identified around an operating point by recording the output of the system for several sinusoidal input signals. Among the possible linear approximating models, the collected data turn out to be best fitted with a fractional order transfer function. The proposed identification and process model allow for online tuning at different operating points for an efficient control design.
The paper presents and analyses some nonlinear continuous-time dynamic models of social systems w... more The paper presents and analyses some nonlinear continuous-time dynamic models of social systems whose members, groups or individuals, may change partners and/or opponents at any time, according to a greedy criterion. The main structural properties of these models, which belong to the class of positive switching systems, are investigated with particular regard to the existence of solutions, their positivity, boundedness and asymptotic behaviour. Simulations show how the cooperative or hostile attitudes of the participants affect their yield.
IEEE Transactions on Automatic Control, Jun 1, 1999
... of the elements in the old rows of order i + 1 and i): it turns out that, in this case too, t... more ... of the elements in the old rows of order i + 1 and i): it turns out that, in this case too, the new rows of ... As already stated in Section I, the classical proofs of the Routh criterion use notions not covered in most undergraduate courses and do not account for the rules for evaluating ...
IEEE Transactions on Automatic Control, May 1, 2004
This note deals with the problem of characterizing a class of second-order three-parameter contro... more This note deals with the problem of characterizing a class of second-order three-parameter controllers [including proportionalintegral-derivative (PID) and lead/lag compensators] satisfying given H closed-loop specifications. Design characterizations of similar form as in the recent work on PID control, are derived for a larger class of compensators using simple geometric considerations. Specifically it is shown that, given the value of one parameter: i) the region of the plane defined by the other two parameters where the considered H constraint is satisfied, consists of the union of disjoint convex sets whose number can be bounded by means of the pancake-cutting formula, and ii) the closed-loop pole distribution can be related to them. An example illustrates how the method can be applied to design a PID controller in the case of bounded sensitivity.
2018 23rd International Conference on Methods & Models in Automation & Robotics (MMAR), 2018
Simple conditions based on generalisations of the Routh-Hurwitz and Mikhailov criteria that ensur... more Simple conditions based on generalisations of the Routh-Hurwitz and Mikhailov criteria that ensure the absence of polynomial roots in an RHP sector straddling the positive real semi-axis ($\mathcal{S}$-stability) are presented. In particular, it is shown that $\mathcal{S}$-stability is ensured if the phase variation of a suitable power of the original $n$ th-degree characteristic polynomial is equal to $n\pi/2$, which implies that the zeros of the real and imaginary parts of this power must satisfy an interlacing property similar to the interlacing property satisfied by Hurwitz polynomials according to the classic Hermite-Biehler theorem. The condition can be checked by means of Sturm sequences. Examples show how the proposed methods operate.
2017 22nd International Conference on Methods and Models in Automation and Robotics (MMAR), 2017
The paper deals with the problem of reducing the order of an original high-order asymptotically s... more The paper deals with the problem of reducing the order of an original high-order asymptotically stable linear switching system by independently approximating the (stable) LTI systems corresponding to every fixed value of the switching signal. Precisely, each reduced-order model is obtained by minimising the L2 norm of a weighted equation error by means of an efficient algorithm that ensures model stability as well as the retention of a number of first- and second-order information indices, such as the Markov parameters and the impulse-response energies. Then, the stability of the switching system is guaranteed, irrespective of the switching law, by realising the aforementioned reduced models in such a way that they share a common Lyapunov function. To this purpose, a simple state-coordinate transformation amenable to online implementation is applied to the state models initially derived. To improve the approximation, the state after every switching is reset, with due care for stabil...
2021 25th International Conference on Methods and Models in Automation and Robotics (MMAR), 2021
This paper extends previous results of the same authors on the determination of the polynomial ro... more This paper extends previous results of the same authors on the determination of the polynomial root distribution with respect to a sector by means of elementary vector analysis. Specifically, it is shown how the overall phase variation of any real or complex polynomial along the radii of a sector accounts for the number of roots inside and outside the sector. The method applies to both symmetric and asymmetric sectors with respect to the real axis. Its practical application only requires plotting a Nyquist-like diagram (a hodograph). The procedure proves particularly useful in the stability analysis of fractional-order systems. A pair of examples is worked out to show how the method operates.
International Journal of Control, Automation and Systems, 2017
Popular model reduction methods can easily be adapted to retain the asymptotic response to inputs... more Popular model reduction methods can easily be adapted to retain the asymptotic response to inputs with rational transform. To this purpose, the forced response of the high-order system is decomposed into a transient and a steady-state component. Then, the reduced-0rder model is obtained by combining the unaltered steady-state component with an approximation of the transient component. Examples show that forcing the reduced-order model to retain the steady-state component does not compromise the transient accuracy.
The dynamics of ancient civilisations according to credited historians can be explained by means ... more The dynamics of ancient civilisations according to credited historians can be explained by means of a simple linear time-invariant (LTI) feedback model whose loop only consists of a first-order process and a pure time delay. It is shown that, despite its simplicity, this model can give rise to a variety of responses, either oscillatory or aperiodic, such as those envisaged by A. Toynbee. Since modern civilisations are characterised by fast parameter variations, their description calls instead for a time-variant model. Simulations with the linear parameter-varying model obtained from the aforementioned LTI model by letting its parameter vary are carried out to show the influence of different scenarios on the system response. Even if this model is elementary, it is believed that it may serve as a stimulating ''tool for thought'' in the ever-popular debate on the growth and decline of civilisations.
A major drawback hinders the application of Model Predictive Control (MPC) to the regulation of e... more A major drawback hinders the application of Model Predictive Control (MPC) to the regulation of electromechanical systems or, more generally, systems with fast dynamics: the time needed for the online computation of the control is often too long with respect to the sampling time. This paper shows how this problem can be overcome by suitably implementing the MPC technique. The main idea is to compute the control law using the discrete-time Euler Auxiliary System (EAS) associated with the continuous-time plant, and apply the control obtained for the discrete-time system to the continuous-time system. In this way the implementation sampling time can be much smaller than the EAS time parameter, which leads to significant savings in computation time. Theoretical results guarantee stabilisation, constraint satisfaction and robustness of such a control strategy, which is applied to the control of an electric drive and a cart-pendulum system.
2019 18th European Control Conference (ECC), Jun 1, 2019
A multi-agent continuous-time nonlinear model of social behaviour allowing for both competition a... more A multi-agent continuous-time nonlinear model of social behaviour allowing for both competition and cooperation is presented and analysed. The state of each agent is represented by its payoff, which the agent aims at maximising. The role of control variables is played by the model parameters, which account for the agents' decisions to either cooperate with or boycott the other agents and can vary in time within assigned intervals. Alliances and enmities can be established at any time, according to either a greedy or a longsighted criterion. The general nonlinear case is first considered. It is proved that, under realistic assumptions, the system evolution is bounded positive (no extinction) and there is a unique globallystable equilibrium point. As is somehow expected, the optimal decision for all agents corresponds to full cooperation (decision parameters kept at their positive maximum value) in the case of both shortsighted and farsighted criteria. This is not true if some parameters have negative upper bounds (meaning that some agents systematically boycott some others). Then, in the linear case, it is shown that the system is stable for arbitrarilyvarying decision parameters, provided that a Metzler matrix associated with full cooperation is Hurwitz. A characterisation of the long-term behaviour of the linear system is also provided. In particular, it is proved that, under stability conditions, a Nash equilibrium exists if a steady strategy is adopted. G.G. acknowledges support from the Aspasia grant and the DTF grant.
2010 IEEE 18th Signal Processing and Communications Applications Conference, 2010
This paper deals with the integer-order (finite-dimensional) approximation of a fractional-order ... more This paper deals with the integer-order (finite-dimensional) approximation of a fractional-order (infinite-dimensional) system using the interpolation approach in the Loewner framework. First given a set of interpolation frequencies and the corresponding values of the original noninteger-order transfer function, a Loewner matrix and a shifted Loewner matrix are created. Then, from these matrices an integer-order model in state-space descriptor form is constructed. Its transfer function matches the original system transfer function at the given frequencies. To avoid singularity problems related to data redundancy, a truncated singular value decomposition may finally be applied. Numerical simulations show that the suggested approach to fractional-order system approximation compares favourably with alternative techniques recently presented in the literature to the same purpose.
2019 24th International Conference on Methods and Models in Automation and Robotics (MMAR), 2019
A state-space integer-order approximation of a commensurate-order systems is obtained using a dat... more A state-space integer-order approximation of a commensurate-order systems is obtained using a data-driven interpolation approach based on Loewner matrices. Precisely, given the values of the original fractional-order transfer function at a number of generalised frequencies, a descriptor-form state-space model matching these frequency response values is constructed from a suitable Loewner matrix pencil, as already suggested for the reduction of high-dimensional integer-order systems. Even if the stability of the resulting integer-order system cannot be guaranteed, such an approach is particularly suitable for approximating (infinite-dimensional) fractional-order systems because: (i) the order of the approximation is bounded by half the number of interpolation points, (ii) the procedure is more robust and simple than alternative approximation methods, and (iii) the procedure is fairly flexible and often leads to satisfactory results, as shown by a pair of examples taken from the liter...
The frequency response of an evaporator for commercial refrigeration is investigated to develop a... more The frequency response of an evaporator for commercial refrigeration is investigated to develop a controloriented model which balances simplicity with accuracy. The dynamics of such devices is highly nonlinear; the response curve was experimentally identified around an operating point by recording the output of the system for several sinusoidal input signals. Among the possible linear approximating models, the collected data turn out to be best fitted with a fractional order transfer function. The proposed identification and process model allow for online tuning at different operating points for an efficient control design.
The paper presents and analyses some nonlinear continuous-time dynamic models of social systems w... more The paper presents and analyses some nonlinear continuous-time dynamic models of social systems whose members, groups or individuals, may change partners and/or opponents at any time, according to a greedy criterion. The main structural properties of these models, which belong to the class of positive switching systems, are investigated with particular regard to the existence of solutions, their positivity, boundedness and asymptotic behaviour. Simulations show how the cooperative or hostile attitudes of the participants affect their yield.
IEEE Transactions on Automatic Control, Jun 1, 1999
... of the elements in the old rows of order i + 1 and i): it turns out that, in this case too, t... more ... of the elements in the old rows of order i + 1 and i): it turns out that, in this case too, the new rows of ... As already stated in Section I, the classical proofs of the Routh criterion use notions not covered in most undergraduate courses and do not account for the rules for evaluating ...
IEEE Transactions on Automatic Control, May 1, 2004
This note deals with the problem of characterizing a class of second-order three-parameter contro... more This note deals with the problem of characterizing a class of second-order three-parameter controllers [including proportionalintegral-derivative (PID) and lead/lag compensators] satisfying given H closed-loop specifications. Design characterizations of similar form as in the recent work on PID control, are derived for a larger class of compensators using simple geometric considerations. Specifically it is shown that, given the value of one parameter: i) the region of the plane defined by the other two parameters where the considered H constraint is satisfied, consists of the union of disjoint convex sets whose number can be bounded by means of the pancake-cutting formula, and ii) the closed-loop pole distribution can be related to them. An example illustrates how the method can be applied to design a PID controller in the case of bounded sensitivity.
2018 23rd International Conference on Methods & Models in Automation & Robotics (MMAR), 2018
Simple conditions based on generalisations of the Routh-Hurwitz and Mikhailov criteria that ensur... more Simple conditions based on generalisations of the Routh-Hurwitz and Mikhailov criteria that ensure the absence of polynomial roots in an RHP sector straddling the positive real semi-axis ($\mathcal{S}$-stability) are presented. In particular, it is shown that $\mathcal{S}$-stability is ensured if the phase variation of a suitable power of the original $n$ th-degree characteristic polynomial is equal to $n\pi/2$, which implies that the zeros of the real and imaginary parts of this power must satisfy an interlacing property similar to the interlacing property satisfied by Hurwitz polynomials according to the classic Hermite-Biehler theorem. The condition can be checked by means of Sturm sequences. Examples show how the proposed methods operate.
2017 22nd International Conference on Methods and Models in Automation and Robotics (MMAR), 2017
The paper deals with the problem of reducing the order of an original high-order asymptotically s... more The paper deals with the problem of reducing the order of an original high-order asymptotically stable linear switching system by independently approximating the (stable) LTI systems corresponding to every fixed value of the switching signal. Precisely, each reduced-order model is obtained by minimising the L2 norm of a weighted equation error by means of an efficient algorithm that ensures model stability as well as the retention of a number of first- and second-order information indices, such as the Markov parameters and the impulse-response energies. Then, the stability of the switching system is guaranteed, irrespective of the switching law, by realising the aforementioned reduced models in such a way that they share a common Lyapunov function. To this purpose, a simple state-coordinate transformation amenable to online implementation is applied to the state models initially derived. To improve the approximation, the state after every switching is reset, with due care for stabil...
2021 25th International Conference on Methods and Models in Automation and Robotics (MMAR), 2021
This paper extends previous results of the same authors on the determination of the polynomial ro... more This paper extends previous results of the same authors on the determination of the polynomial root distribution with respect to a sector by means of elementary vector analysis. Specifically, it is shown how the overall phase variation of any real or complex polynomial along the radii of a sector accounts for the number of roots inside and outside the sector. The method applies to both symmetric and asymmetric sectors with respect to the real axis. Its practical application only requires plotting a Nyquist-like diagram (a hodograph). The procedure proves particularly useful in the stability analysis of fractional-order systems. A pair of examples is worked out to show how the method operates.
International Journal of Control, Automation and Systems, 2017
Popular model reduction methods can easily be adapted to retain the asymptotic response to inputs... more Popular model reduction methods can easily be adapted to retain the asymptotic response to inputs with rational transform. To this purpose, the forced response of the high-order system is decomposed into a transient and a steady-state component. Then, the reduced-0rder model is obtained by combining the unaltered steady-state component with an approximation of the transient component. Examples show that forcing the reduced-order model to retain the steady-state component does not compromise the transient accuracy.
The dynamics of ancient civilisations according to credited historians can be explained by means ... more The dynamics of ancient civilisations according to credited historians can be explained by means of a simple linear time-invariant (LTI) feedback model whose loop only consists of a first-order process and a pure time delay. It is shown that, despite its simplicity, this model can give rise to a variety of responses, either oscillatory or aperiodic, such as those envisaged by A. Toynbee. Since modern civilisations are characterised by fast parameter variations, their description calls instead for a time-variant model. Simulations with the linear parameter-varying model obtained from the aforementioned LTI model by letting its parameter vary are carried out to show the influence of different scenarios on the system response. Even if this model is elementary, it is believed that it may serve as a stimulating ''tool for thought'' in the ever-popular debate on the growth and decline of civilisations.
A major drawback hinders the application of Model Predictive Control (MPC) to the regulation of e... more A major drawback hinders the application of Model Predictive Control (MPC) to the regulation of electromechanical systems or, more generally, systems with fast dynamics: the time needed for the online computation of the control is often too long with respect to the sampling time. This paper shows how this problem can be overcome by suitably implementing the MPC technique. The main idea is to compute the control law using the discrete-time Euler Auxiliary System (EAS) associated with the continuous-time plant, and apply the control obtained for the discrete-time system to the continuous-time system. In this way the implementation sampling time can be much smaller than the EAS time parameter, which leads to significant savings in computation time. Theoretical results guarantee stabilisation, constraint satisfaction and robustness of such a control strategy, which is applied to the control of an electric drive and a cart-pendulum system.
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