Sign Definite Condition Research Articles

Semidefinite Relaxation and New Conditions for Sign-Definiteness of the Quadratic Form under Quadratic Constraints

Use of the semidefinite relaxation in the problem of sign-definiteness of the quadratic form under quadratic constraints enables one to establish from the duality conditions an S-procedure. However, the S-procedure giving the necessary and sufficient conditions for signdefiniteness of the relaxed problem provides only the sufficient conditions for sign-definiteness for the original problem for the case of two and more quadratic constraints. This property is called the deficiency of S-procedure. A method was proposed enabling one in some cases to establish the conditional sign-definiteness in the case where the S-procedure provides a negative result. This method give the necessary and sufficient conditions for sign-definiteness in the two-dimensional case. An example was given.

Quasi-diagonalization of Hankel operators

We show that all Hankel operators $H$ realized as integral operators with kernels $h(t+s)$ in $L^2 ({\Bbb R}_{+}) $ can be quasi-diagonalized as $H= {\sf L}^* \Sigma {\sf L} $. Here ${\sf L}$ is the Laplace transform, $\Sigma$ is the operator of multiplication by a function (distribution) $\sigma(\lambda)$, $\lambda\in {\Bbb R}$. We find a scale of spaces of test functions where ${\sf L} $ acts as an isomorphism. Then ${\sf L}^*$ is an isomorphism of the corresponding spaces of distributions. We show that $h= {\sf L}^* \sigma$ which yields a one-to-one correspondence between kernels $h(t)$ and sigma-functions $\sigma(\lambda)$ of Hankel operators. The sigma-function of a self-adjoint Hankel operator $H$ contains substantial information about its spectral properties. Thus we show that the operators $H$ and $\Sigma$ have the same numbers of positive and negatives eigenvalues. In particular, we find necessary and sufficient conditions for sign-definiteness of Hankel operators. These results are illustrated at examples of quasi-Carleman operators generalizing the classical Carleman operator with kernel $h(t)=t^{-1}$ in various directions. The concept of the sigma-function directly leads to a criterion (equivalent of course to the classical Nehari theorem) for boundedness of Hankel operators. Our construction also shows that every Hankel operator is unitarily equivalent by the Mellin transform to a pseudo-differential operator with amplitude which is a product of functions of one variable only (of $x\in{\Bbb R}$ and of its dual variable).

Solving and visualizing nonlinear parametric constraints in control based on quantifier elimination

We present a new method with a software tool for parametric robust control synthesis by symbolic-numeric computation. The method is a parameter space approach and it is especially effective for analysis and design of fixed-structure controllers of rational type, which encompass PI and PID controllers. The real quantifier elimination (QE), which is one of the recent progresses in the symbolic computation, plays a key role in our development. The QE-based approach can uniformly deal with a lot of important design specifications for robust control such as frequency restricted H ∞ norm constraints, stability (gain/phase) margin and stability radius specifications, and pole location requirement by reducing such specifications to a particular type of formulae called a “sign definite condition (SDC)”. This is also useful for improving the efficiency of QE computations since we can utilize an efficient QE algorithm specialized to the SDC using the Sturm-Habicht sequence. We have developed a MATLAB toolbox for robust parametric control based on a parameter space approach accomplished by QE. The QE-based parameter space approach and numerical simulation of performances for specific controller parameter values taken from a controller parameter space are integrated conveniently in our toolbox with the assistance of a graphical user interface (GUI). With our toolbox the feasible regions of controller parameters are visualized in a parameter space for the controllers with three or two parameters. This enables control engineers to achieve multi-objective robust controller synthesis smoothly. We also discuss how to merge the numerical computation and the symbolic operation to make our new design methods more efficient in practical control design.

A parameter space approach to fixed-order robust controller synthesis by quantifier elimination

This paper proposes a new method of parameter space design for robust control synthesis which guarantees the real stability radius constraint, accomplished by using Quantifier Elimination (QE). We aim at practicality by employing the scheme to combine the Sign Definite Condition (SDC) and a special QE algorithm using the Sturm-Habicht sequence. The validity of our approach is confirmed by several design examples.

005 Robust control analysis considering real parametric perturbations based on sign definite condition

005 Robust control analysis considering real parametric perturbations based on sign definite condition

Robust Control Analysis Considering Real Parametric Perturbations Based on Sign Definite Conditions

Convenient tests for D-stability, H∞-norm constraints and gain/phase margin constrains of one-parameter systems are proposed in this paper. These tests are on the basis of Sign Definite Condition and the Euclidean algorithm in matrix form, which can be carried out in finite number of steps.