Study On The Robust Stability Of Interval Systems Based On The Theory Of Real Lgebraic Geometry

The theoretical studies on robust stability of interval systems can date back to theend of the1970s. Since then, many researchers have devoted to this field, especially dur-ing the1980s and the1990s. Many important results have been derived, and the modelsof interval systems being studied have become more and more general. In practical en-gineering fields such as turbine generator, vibration control and flight control, modelingwithintervalsystemshasattractedmoreandmoreattentions. However, thepresentrobuststability criterions for interval systems can not be satisfactory. In view of this situation,wedevelopaseriesofcriterionsforrobuststabilityofintervalsystemsbasedonthetheoryof real algebraic geometry. The thesis is divided into seven chapters, mainly includingthe following four aspects:1. The robust stability of Hermitian interval systems is studied. Through analyseson the eigenvalues of Hermitian interval systems, we propose vertex criterions for robuststability of continuous and discrete systems, one of which is only sufficient, and the otheris necessary and sufficient. The necessary and sufficient criterion reduces the number ofvertex Hermitian matrices in the results of present literatures. Using the algorithm forconstructing these vertex Hermitian matrices, we make some simulations which illustratethe validity of our results.2. Based on the polynomial discriminant systems, we study2-D linear discrete sys-temsand2-Dlinearcontinuoussystemsdescribedbyrecursivemodels. Fromtheanalysisof the transfer functions of2-D systems, we propose necessary and sufficient criterionsfor stability of these systems. With small modifications of the deductions, the results canbe generalized to continuous-discrete systems, or2-D systems described by others mod-els. Theoretical analyses demonstrate that, the results are of low complexity, and can beapplied to2-D systems with a parameter as well, hence generalizing results from presentliteratures in some sense. Numerical examples and computer simulations illustrate thevalidity of the results.3. Study on robust stability of interval system is performed based on the methods fortesting positivity of homogeneous polynomials. First, we study a method(i.e., weighted difference substitition method) for testing positivity of homogeneous polynomials whichhas shown great efficiency in practice, and give estimates on the bounds for the steps re-quired when testing positivity of homogeneous polynomials through this method, henceproving the completeness of this method on positivity test of homogeneous polynomials.Second, we study three types of interval systems based on this method, that is, the inter-val systems described by polytopic matrices, the interval systems whose system matrixentries are rational functions of the uncertain parameters, and the2-D interval systemswhose coefficients of the characteristic polynomials are rational functions of the uncer-tain parameters. Necessary and sufficient criterions are developed for robust stability ofsuch systems. The deduction is performed for continuous systems. As a matter of fact,similar deduction can be made for discrete systems. Theoretic analyses show that, theresults in the thesis contain and develop the results of present literatures in some sense.Numerical simulations illustrate the validity of the results.4. A method for testing positivity of general functions is developed based on intervalanalysis. Through this method, we proposed sufficient criterions for robust stability of atype of interval systems which is more general then the usual models studied in presentliteratures, thatis, theentriesof systemmatricesaregeneral functionsofuncertainparam-eters varying in interval. Theoretic analyses and numerical examples demonstrate that,the method proposed in the thesis can make the result less conservative.