The inverted pendulum is used as an ex- ample to illustrate controller fragility under minimum norm state feedback. The stabilization process gets trapped in a cusp. A parameter space representation shows a safe escape.
Linear systems with uncertain parameters q are analyzed for stability for all q in a box Q. An example with multilinear dependency of the characteristic polynomial P(s, q) on the parameters is used as a test case for the composition of four methods: (1) eigenvalues (s-plane), (2) zero exclusion from the complex value set (P-plane), (3) algebraic test of Hurwitz conditions, and (4) parameter space (stability boundary exclusion from Q in projected q-space). The methods are also discussed with respect to gridding, graphical warning for singular cases, extension to many parameters, extension to polynomic coefficient functions, extension to Gamma -stability, and insight beyond a yes-no answer to the analysis problem.< >
The family of characteristic polynomials of a SISO-PID loop with N representative plant operating conditions is P i = Ai(s)(K I ; + Kps + K D s 2 ) + B i (s), i = 1,2...N. A basic task of robust control design is to find the set of all parameters K I , K P , K D , that simultaneously place the roots of all P i (a) into a specified region Γ in the complex plane. For Γ in form of the left half plane it is known that the simultaneously stabilizing region in the (K D , K I )-plane consists of one or more convex polygons. This fact simplifies the tomographic rendering of the nonconvex set of all simultaneously stabilizing PID controllers by gridding of K P . A similar result holds if Γ is the shifted left half plane. In the present paper it is shown 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 Γ-region. A parameter space approach shows, that the roots of a polynomial P i (s) with fixed K P can cross the imaginary axis in three ways i) at zero, ii) at infinity, iii) at a finite number of singular frequencies ω k , k = 1,2,3... M. The singular frequencies ω k (K p ) are first determined as roots of a polynomial. For each ω k then a straight line with positive slope ω 2k is a boundary in the (K I , K D )-plane, where a pair of roots of P i (s) crosses the imaginary axis at s = ±ω k . Finally the resulting stable polygons are selected. Computationally the main task is the factorization of a polynomial for finding ω k (K P ). This step can be avoided by evaluating the inverse function K P (ω k ), which is explicitely given. The results are illustrated by the design of an additional PID controller for improved performance of a robustly decoupled car steering control system.
The stability test of polynomials whose coefficients depend multilinearly on interval parameters is considered. The authors describe and compare four brute-force solution approaches. These are eigenvalue calculation, zero exclusion from a specified value set, algebraic tests of real and complex Hurwitz roots, and the parameter space method. They are applied to a simple example with two parameters and third-order polynomial. An interesting feature of the example is that it can have an isolated unstable point. The example may be useful as a benchmark for future approaches to the multilinear problem. All four methods are shown to be feasible for the simple example, but they require effort.< >
The inverted pendulum is used as an example to illustrate controller fragility under minimum norm state feedback. The stabilization process gets trapped in a cusp. A parameter space representation shows a safe escape.
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.
