Umberto Viaro

Characterization of PID and lead/lag compensators satisfying given H/sub /spl infin// specifications

This note deals with the problem of characterizing a class of second-order three-parameter controllers [including proportional-integral-derivative (PID) and lead/lag compensators] satisfying given H/sub /spl infin// 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/sub /spl infin// 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.

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.

An improvement in the Routh-Padé approximation techniques

A method for approximating high order stable systems via lower order stable models is presented. It is an improvement over previous Routh-Pade methods since the denominator polynomial of the model transfer function, even if based on the elements of a Routh array, contains two free parameters. A systematic procedure is illustrated for evaluating, besides the numerator coefficients, such denominator parameters in order to ensure a good fit, in the Pade sense, and a suitable stability margin.

Frequency-domain approach to robust PI control

The paper considers the problem of designing PI controllers for industrial processes approximated by a first-order time-delayed model. The suggested frequency-domain approach is based on a normalized open-loop transfer function and makes use of the loci of constant stability margins and other performance indices in the parameter space. In this way, it is possible to evaluate the effects of uncertainties in the process parameters and, thus, control system robustness. Some examples show how the procedure operates.

A method for the integer-order approximation of fractional-order systems

A procedure for approximating fractional-order systems by means of integer-order state-space models is presented. It is based on the rational approximation of fractional-order operators suggested by Oustaloup. First, a matrix differential equation is obtained from the original fractional-order representation. Then, this equation is realized in a state-space form that has a sparse block-companion structure. The dimension of the resulting integer-order model can be reduced using an efficient algorithm for rational L2 approximation. Two numerical examples are worked out to show the performance of the suggested technique.

Designing PI controllers for robust stability and performance

The design of proportional-integral (PI) controllers can profitably be carried out by referring to a normalized process model and to the loci of constant stability margins and crossover frequency in the parameter space. These loci are easily obtained from linear interpolation conditions. After checking the compatibility of the specifications with the controller structure, it is shown how the above-mentioned loci can be exploited for tuning purposes and robustness analysis. Criteria for choosing the controller parameters within the admissible region are discussed with the aid of examples.

Model reduction by matching Markov parameters, time moments, and impulse-response energies

This paper presents a simple procedure for constructing reduced-order models that match some Markov parameters and time moments of an original system as well as the energies of its impulse response and of other combinations of the system modes. In this way, the stability of the reduced model of a stable system is ensured. >

A program for solving the L2 reduced-order model problem with fixed denominator degree

A set of necessary conditions that must be satisfied by the L2 optimal rational transfer matrix approximating a given higher-order transfer matrix, is briefly described. On its basis, an efficient iterative numerical algorithm has been obtained and implemented using standard MATLAB functions. The purpose of this contribution is to make the related computer program available and to illustrate some significant applications.

Rational L/sub 2/ approximation: a non-gradient algorithm

The problem of determining the best rational approximant of a given rational transfer function of higher order according to the L/sub 2/-norm criterion is considered. An efficient algorithm is presented that makes it possible to find a (local) minimum without evaluating derivatives. It is based on a reformulation of the necessary conditions for optimality in terms of interpolation constraints. Examples show that the algorithm converges rapidly to a solution even if started from poor initial guesses.<<ETX>>

A stability test for continuous systems

Abstract A test for determining the zero distribution of a real polynomial with respect to the imaginary axis is presented. It is based on the construction of a sequence of polynomials of descending degree which draws on the Levinson algorithm. The proof of the stability criterion, which is based on intuitive geometrical considerations, is very simple, and the computational complexity is not greater than that for the Routh test. The critical situations that may arise are also examined and some examples are given.