Stable polyhedra in parameter space

A typical uncertainty structure of a characteristic polynomial is P(s) = A(s) * Q(s) + B(s) with A(s) and B(s) fixed and Q(s) uncertain. In robust controller design Q(s) may be a controller numerator or denominator polynomial; an example is the PID controller with Q(s) = KI + KP(s) + KD (s*s) . In robustness analysis Q(s) may describe a plant uncertainty. For fixed imaginary part of Q(j*omega), it is shown that Hurwitz stability boundaries in the parameter space of the even part of Q(j*omega) are hyperplanes and the stability regions are convex polyhedra. A dual result holds for fixed real part of Q(j*omega) . Also sigma -stability with the real parts of all roots of P(s) smaller than sigma is treated

On Robust Stability of Polynomials with Polynomial Parameter Dependency

We consider uncertain polynomials whose coefficients depend polynomially on the elements of the parameter vector. The size of perturbation is characterized by the weighted norm of the perturbed parameter vector. The maximal perturbation defines the stability radius of the set of uncertain polynomials. It is shown that determining this radius is equivalent to solving a finite set of systems of algebraic equations and picking out the smallest solution. The number of systems depend crucially on the dimension of the parameter vector, whereas the complexity of systems increases mainly with the kind of polynomial dependency and the degree of the polynomial. This method also yields the critical parameter combination and the corresponding critical frequency. For a small number of parameters, this transfomed problem can be solved by symbolic computations. For a large number of parameters, numerical methods must be used

Discrete-Time Robust PID and Three-Term Control

It has been shown, that stability regions for PID-controllers in a (KD, KI)-plane for fixed Kp are convex polygons. This result allows a simple calculation of the set of all stabilizing PID controllers for a given plant. In the present paper this result is transfered to the case of discrete-time PID controllers or three-term controllers, where stability with respect to the unit circle or other circles in the z-plane must be checked. Since the orientation of the cross section planes for polygonial stability regions does not depend on the plant, it is easy to find the set of all simultaneous stabilizers for several representative plant parameters and to select a robust discrete-time controller from this set. A benchmark problem for digital robust control is used to illustrate the metho

On Robust Stability of Polynomials with Polynomial Parameter Dependency: Two/Three Parameter Cases

We consider real polynomials whose coefficients depend polynomially on the elements of an uncertain parameter vector. The size of perturbation is characterized by the weighted norm of the parameter vector. The smallest destabilizing perturbation defines the stability radius of the set of uncertain polynomials. It is shown that determining this radius is equivalent to solving a finite set of systems of algebraic equations and picking out the real solution with the smallest norm. The number of systems of equations increases mainly with the kind of polynomial dependency and the degree of the polynomial. This method also yields the smallest destabilizing parameter combination and the corresponding critical frequency. For two or three parameters this transformed problem can be solved using symbolic and numeric computations

Design of Robust PID Controllers

The family of characteristic polynomials of a SISO-PID loop with N representative plant operating conditions is Pi = Ai(s) (K1 + Kps+KDs2) + Bi(s), i=1,2..N. A basic task of robust control design is to find the set of all parameters K1, Kp, KD, that simultaneously place the roots of all Pi(s) into a specified region in the complex plane. For in form of the left half plane it is know that the simultaneously stabilizing region in the (KD,KI)-plane consists of one or more convex polygons. This fact simplifies the tomograhic rendering of the nonconvex set of all simultaneously stabilizing PID controllers by gridding of Kp. A similar result holds if is the shifted left half plane. In the present paper it is show that the nice geometric property also holds for circles with arbitrary real center and radius. It is further shown, that it cannot hold for any other region

Robust Gamma-Stability Analysis in Plant Parameter Space

Given a characteristic polynomial whose coefficients depend polynomially on l uncertain parameters, the following robustness problem arises: Determine whether all the roots of the polynomial are located in a prescribed region Gamma in the complex plane for all admissible parameter values. Mapping the boundary of Gamma into the parameter space. Testing this condition graphically etc. and applying the method to a track-guided bus with uncertain mass and velocity