Daniele Casagrande

Stable LPV Realization of Parametric Transfer Functions and Its Application to Gain-Scheduling Control Design

The paper deals with the stabilizability of linear plants whose parameters vary with time in a compact set. First, necessary and sufficient conditions for the existence of a linear gain-scheduled stabilizing compensator are given. Next, it is shown that, if these conditions are satisfied, any compensator transfer function depending on the plant parameters and internally stabilizing the closed-loop control system when the plant parameters are constant, can be realized in such a way that the closed-loop asymptotic stability is guaranteed under arbitrary parameter variations. To this purpose, it is preliminarily proved that any transfer function that is stable for all constant parameters values admits a realization that is stable under arbitrary parameter variations (linear parameter-varying (LPV) stability). Then, the Youla-Kucera parametrization of all stabilizing compensators is exploited; precisely, closed-loop LPV stability can be ensured by taking an LPV stable realization of the Youla-Kucera parameter. To find one such realization, a reasonably simple and general algorithm based on Lyapunov equations and Choleskys factorization is provided. These results can be exploited to apply linear time-invarient design to LPV systems, thus achieving both pointwise optimality (or pole placement) and LPV stability. Some potential applications in adaptive control and online tuning are pointed out.

Global asymptotic stabilization of the attitude and the angular rates of an underactuated non-symmetric rigid body

The paper deals with the global stabilization of both the attitude and the angular velocities of an underactuated rigid body. First a stability theorem is proven for a class of systems; subsequently, the equations describing the physics of the rigid body are presented, showing that the rigid body belongs to the considered class of systems, and a sufficient condition for the application of the theorem to the stability of the rigid body equilibrium is pointed out. Finally, some simulation results are reported showing the effectiveness of the proposed methodology.

Brief paper: Modal and transition dwell time computation in switching systems: A set-theoretic approach

We consider a plant switching among a set of dynamic systems each associated with a single stable equilibrium point. We assume that a constraint region for the state is assigned. The main problem is that of finding suitable limitations on the commutation speed in order to avoid constraints violations, even in the absence of state measurements. We introduce the concepts of modal and transition dwell times which lead to a dwell time vector and dwell time graph (represented by a proper matrix), respectively. The former imposes a minimum permanence on a discrete mode before commuting, the second imposes the minimum permanence on the current mode before switching to a specific new one. Both dwell time vector and dwell time graph, can be computed via set theoretic techniques. When systems share a single equilibrium stability can be assured as a special case. Finally, under the assumption of affine dynamics, non-conservative values are achieved.

Transfer Equivalence and Realization of Nonlinear Input-Output Delta-Differential Equations on Homogeneous Time Scales

Nonlinear control systems on homogeneous time scales are studied. First the concepts of reduction and irreducibility are extended to higher order delta-differential input-output equations. Subsequently, a definition of system equivalence is introduced which generalizes the notion of transfer equivalence in the linear case. Finally, the necessary and sufficient conditions are given for the existence of a state-space realization of a nonlinear input-output delta-differential equation.

Global stabilization of non-globally linearizable triangular systems: Application to transient stability of power systems

A general methodology to globally stabilize an equilibrium of a class of non-globally linearizable triangular systems is presented. The technique is applicable to all triangular systems described by analytic vector fields. The method is used to give an explicit solution to the challenging problem of transient stability of multimachine power systems with leaky transmission lines, for which only existence results are currently available.

Asymptotic stabilization of passive systems without damping injection: A sampled integral technique

Passivity in physical systems is a restatement of energy balancing, and therefore is a ubiquitous property in engineering applications. Under some weak conditions, the unique equilibrium state of passive systems is stable. However, to ensure asymptotic stability, strict output passivity and a detectability property are required. Although strict output passivity may be enforced via a damping injection that feeds back the passive output, this signal may be noisy or unmeasurable - the paradigmatic example being velocity in mechanical systems. In this paper a sampled integral stabilization (SIS) technique for the asymptotic regulation of passive systems, that requires only the knowledge of the time integral of the passive output - i.e. position in mechanical systems - is proposed. As a generalization of the previous result, it is shown that SIS is applicable to cascade connections of passive systems measuring only the storage function of the second one. Several examples, including a planar elbow manipulator and the rigid body dynamics are shown to satisfy the assumptions for the application of SIS.

Constant and switching gains in semi-active damping of vibrating structures

We consider the problem of optimal control of vibrating structures and we analyse the solution provided by collocated semiactive decentralised damping devices. We mainly consider the ℋ∞ criterion and we first study the case of constant dampers, showing that in the case of a single damper the performance is a quasi-convex function of friction so there is a single local minimum which is a global one. The case of multiple dampers does not exhibit this feature and time-expensive computations may be required. Secondly, we consider the case in which dampers may be tuned on line, and in particular the case in which they work in a switching on–off mode. We propose a state-switching feedback control strategy, which outperforms the constant damping approach with the optimal static gain performance as an upper bound. For large distributed flexible structures, state feedback is unrealistic and so we propose a stochastic strategy based on a Markov-jump criterion for which the transition probability are not assigned but designed to optimise average performance, with guaranteed asymptotic stability. Finally, we show that the same result provided for the ℋ∞ case holds for the ℋ2 and the l 1 criteria.

Hamiltonian-Based Clustering: Algorithms for Static and Dynamic Clustering in Data Mining and Image Processing

The large amount of data available for analysis and management raises the need for defining, determining, and extracting meaningful information from the data. Hence in scientific, engineering, and economics studies, the practice of clustering data arises naturally when sets of data have to be divided into subgroups with the aim of possibly deducting common features for data belonging to the same subgroup. For instance, the innovation scoreboard [1] (see Figure 1) allows for the classification of the countries into four main clusters corresponding to the level of innovation defining the “leaders,” the “followers,” the “trailing,” and the “catching up” countries. Many other disciplines may require or take advantage of a clustering of data, from market research [2] to gene expression analysis [3], from biology to image processing [4][7]. Therefore, several clustering techniques have been developed (for details see “Review of Clustering Algorithms”).

Switching-Driving Lyapunov Function and the Stabilization of the Ball-and-Plate System

This note introduces the notion of switching-driving Lyapunov function which can be used to provide a sufficient stabilizability condition for general nonlinear systems. The theory is illustrated by means of the ldquoball-and-platerdquo system, for which locally stabilizing control laws are explicitly derived.

Robust constrained Model Predictive Control of fast electromechanical systems

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.