Valentina Orsini

Efficient LMI-Based Quadratic Stabilization of Interval LPV Systems With Noisy Parameter Measures

The purpose of this note is to consider the quadratic stabilization of LPV systems in the realistic case where only Gaussian noisy parameter measures are available. Though neglected in the actual literature on LPV systems, this question is particular important because in all situations of a practical interest the parameter measurements (or estimates) are never exact. The assumed noisy nature of physical parameter readings requires a specifically developed approach consisting of mixed robust and LPV control methods. In the present case, an approach based on a vertex result on interval time varying (ITV) matrices is proposed. This allows the solvability conditions to be stated in terms of a set of LMIs, whose number is independent of the number of time-varying parameters.

Relaxed Conditions for the Exponential Stability of a Class of Linear Time-Varying Systems

This technical note states new sufficient conditions for the exponential stability of linear time-varying (LTV) systems of the form aeligmacr(ldr) =A(t)aelig(t). The approach proposed derives and uses the notion of perturbed frozen time (PFT) form that can be associated to any LTV system. Exploiting the Bellman-Gronwall lemma, relaxed stability conditions are then stated in terms of ldquoaveragerdquo parameter variations. Salient features of the approach are: pointwise stability of A(ldr) is not required, ||Amacr(ldr)|| may not be bounded, the stability conditions also apply to uncertain systems. The approach is illustrated by numerical examples.

Accurate output tracking for nonminimum phase nonhyperbolic and near nonhyperbolic systems

Abstract A new method to achieve an accurate output tracking for nonminimum phase linear systems with nonhyperbolic and near nonhyperbolic internal dynamics is presented. For the classical methods, developed in the framework of the preview based stable inversion, the nonhyperbolicity represents a rather serious inconvenience because the required preactuation time tends to be unacceptably large. Different stable inversion techniques, developed for SISO systems, are based on a proper redefinition of the desired output trajectory with the aim of canceling the undesired effects of unstable zeros. The main purpose of the new approach is to alleviate some theoretical and practical limitations inherent in the above methods. The desired output is partitioned into the transient and steady-state components. The transient input is “a priori” assumed to be given by a spline function. Once the desired output trajectory has been specified, this allows the computation of the unknown transient input as the approximate least-squares solution of Fredholm׳s integral equation corresponding to the explicit formula of the output forced response. The steady-state input is analytically computed exploiting the steady-state output response expressions for inputs belonging to the same set of the desired steady-state output. The main advantage of the resulting technique is that its generality does not require “ad hoc” procedures depending on the particular plant to be controlled. This allows the designer to freely specify the desired output trajectory without requiring it to depend on the unstable zeros of the plant or it to be null over an initial time interval.

An efficient LMI approach for the quadratic stabilisation of a class of linear, uncertain, time-varying systems

The purpose of this article is to provide a numerically efficient method for the quadratic stabilisation of a class of linear, discrete-time, uncertain, time-varying systems. The considered class of systems is characterised by an interval time-varying (ITV) matrix and constant sensor and actuator matrices. It is required to find a linear time-invariant (LTI) static output feedback controller yielding a quadratically stable closed-loop system independently of the parameter variation rate. The solvability conditions are stated in terms of linear matrix inequalities (LMIs). The set of LMIs includes the stability conditions for the feedback connection of a unique suitably defined extreme plant with an LTI output controller and the positivity of a closed-loop extremal matrix. A consequent noticeable feature of the article is that the total number of LMIs is independent of the number of uncertain parameters. This greatly enhances the numerical efficiency of the design procedure.

Almost perfect tracking through mixed numerical-analytical stable pseudo-inversion of non minimum phase plants

This paper considers the problem of computing the input u(t) of an internally asymptotically stable, possibly non minimum phase, linear, continuous-time system Σ yielding a very accurate tracking of a pre-specified desired output trajectory ỹ(t). The main purpose of the new approach proposed here is to alleviate some limitations inherent the classical methods developed in the framework of the preview based stable inversion, which represents an important reference context for this class of control problems. In particular the new method allows one to deal with arbitrary and possibly uncertain initial conditions and does not require a pre-actuation. The desired output ỹs(t) to be exactly tracked in steady-state is here assumed to belong to the set of polynomials, exponential and sinusoidal time functions. The desired transient response ỹt(t) is specified to obtain a fast and smooth transition towards the steady-state trajectory ỹs(t), without under and/or overshoot in the case of a set point reset. The transient control input ut(t) is “a priori” assumed to be given by a piecewise polynomial function. Once ỹ(t) has been specified, this allows the computation of the unknown ut(t) as the approximate least-squares solution of the Fredholms integral equation corresponding to the explicit formula of the output forced response. The steady-state input us(t) is analytically computed exploiting the steady-state output response expressions for inputs belonging to the same set of ỹs(t).

A Supervised Switching Control Policy for LPV Systems With Inaccurate Parameter Knowledge

This technical note deals with the switched supervised control of linear parameter varying systems whose parameters values are online acquirable only with an arbitrarily large degree of uncertainty. The purpose is to define a switching policy inside a family of predesigned controllers so that the switched closed-loop system result to be exponentially stable. The proposed switching logic is based on a performance-evaluation criterion which uses a measurable Lyapunov-like functional of the output. The exponential stability condition is derived imposing a sufficient long time interval over which the functional is decreasing. An interesting feature of the technical note is that no particular structure on the kind of uncertainty affecting the parameter values is assumed.

Event-triggered internally stabilizing sporadic control for MIMO plants with non measurable state

Abstract This paper proposes a new approach to the sporadic networked control problem: the output of a remotely controlled sampled plant is required to track an external reference with a reduced communication cost. The main novelty is that internal stability conditions are derived without assuming a measurable state of the plant. The new event driven Communication Logic (CL) is composed of a Sensor CL (SCL) and of a Controller CL (CCL). The SCL is based on the explicit computation of a Lyapunov-like functional of the tracking error and of a corresponding time-varying threshold: a network message from the sensor to the controller is triggered only if the functional exceeds the current value of the threshold. The CCL is directly driven by the SCL: the dynamic output controller sends a feedforward message to the plant only if it has received a message from the sensor at the previous time instant.

LMI Conditions for the Stability of Linear Uncertain Polynomially Time-Varying Systems

This technical note investigates the robust stability of uncertain linear time-varying systems with a mode-switch dynamics. Each mode is characterized by a dynamical matrix containing elements whose time behavior over bounded time intervals is described by interval polynomials of arbitrary degree. Using a quadratic Lyapunov function polynomially depending on time, stability conditions for each mode are stated in terms of linear matrix inequalities (LMIs). The stability conditions of the switching system are stated both in terms of minimum and average dwell time. A salient feature of the technical note is that the single-mode stability conditions given here allow the parameters and their derivatives to take values over arbitrarily large uncertainty sets.

A spline-based technique for optimal set point regulation through pseudo-inversion of nonminimum phase linear systems

This paper considers the optimal output set-point regulation for MIMO, non minimum phase sampled data systems. The usually proposed methods are based on stable model inversion whose exact solution is approximated through preview based implementation schemes. The new approach proposed here considers the meaningful practical situation of plants with a given, possibly uncertain, initial state, that can not be modified through pre-actuation. The structure of the optimal control input is ”a priori” assumed to be given by a smoothing spline function. In this way a twofold objective is achieved: a smooth behavior of the control input and its derivatives can be imposed, a very accurate tracking performance can be obtained by reducing the mesh size of spline [1]. Given the desired transient output response between two fixed set points, the spline coefficients are determined as the least-squares solution of the over determined system of linear equations obtained imposing that the spline function assumed as control input yields the specified output. In this way an optimal least square approximation of the desired output trajectory is obtained avoiding the stable explicit model inversion. Rather, this operation is implicitly approximately performed solving for the spline coefficients, the over-determined system of linear equations carrying the information on the model to be inverted and on the desired output. An interesting feature of this new method is that it also works for linear systems which are not required to be either square or right invertible.

Some remarks on the robust schur stability analysis of interval polynomials

The Schur stability analysis of an interval polynomial family can be quickly performed through a unique, suitably defined extreme polynomial. The purpose of this article is to provide some improvements with respect to the actually existing methods based on this approach.