Parameter Space Approach
The parameter space approach can be utilized to determine the set of stabilizing parameters in a parameter space. This is accomplished by mapping the boundary of the admissible eigenvalue region via the characteristic polynomial into the parameter space. For practical applications a graphical representation is best suited. This means a restriction to two- or three-dimensional parameter spaces. Especially, a two-dimensional representation of stability boundaries is easy comprehensible. To determine the stability boundaries in a two-dimensional parameter plane, say t1 and t2, the characteristic polynomial p(s,t1,t2) is separated into real and imaginary part for s = sigma+jomega. For a fixed frequency s*, real and imaginary part are solved for t1 and t2. Doing this for a grid along the boundary of the admissible eigenvalue region yields the stability boundaries in the (t1,t2)-plane.
The following sketch illustrates the procedure: In the example the admissible eigenvalue region (indicated by Gamma) is bounded by a pair of lines of constant damping D, i.e. if the closed-loop eigenvalues are located in this region the system has at least the degree of damping D. Two general cases of stability boundaries can be distinguished: For a point on the real (complex) root boundary in the parameter plane the roots of the characteristic polynomial for this specific operating point will yield a real root (a complex pole pair) on the boundary of Gamma. The Gamma-stability boundaries separate the parameter plane into a finite number of regions. By checking Gamma-stability of arbitrary points of each region the set of Gamma-stabilizing parameters can be determined. In the example, the point t1 is assumed to be Gamma-stable and, hence, the entire region in which this point is located is Gamma-stable.
The approach can be applied to both design and analysis:
- For design of fixed gain controllers, t1 and t2 are controller parameters and the Gamma-stable set represents all controller parameters for which the plant is Gamma-stable. In the case of an uncertain plant these sets are calculated for a finite number of representatives of the plant, e.g. the vertices of the hyperrectangle. If the intersection of these sets is non-empty, the controllers of the intersection set simultaneously stabilize all representatives.
If the system has more than two controller parameters then the remaining m-2 controller parameters have either to be fixed or a suitable cross-section in the m-dimensional controller parameter space has to be selected.
- For design of gain scheduling controllers a mixed parameter plane of a controller parameter to be scheduled and a plant parameter which is used for scheduling is selected. The stability boundaries in this specific plane give immediate answer to the question of how to chose the gain scheduling law.
- For analysis, t1 and t2 are uncertain plant parameters. Then, the polynomial is robustly stable for the entire operating domain given by t1 and t2, if an arbitrary point of the operating domain is Gamma-stable and none of the Gamma-stability boundaries intersects the operating domain. If the system has more than two uncertain parameters then the remaining l-2 parameters have to be gridded. This limits the approach to a low number of uncertain parameters. For a graphical representation the stability boundaries are projected into the (t1,t2)-plane and the same procedure for examining robustness as given above is applied.
Invariance Plane Design
A goal of robust control design is low controller order. In most cases, low order controllers yield sufficient performance and the influence of controller parameters on the system dynamics is much easier comprehensible than for high order controllers. In the case of state-feedback controllers, however, the controller order is determined by the plant order. In this specific case, a systematic method is needed to obtain a robust controller. A possible approach is to shift only the most critical eigenvalues while the other eigenvalues remain at their location. For a nominal operating point this can be accomplished using an extension of Ackermann's formula. Hereby, an m-dimensional subspace of the n-dimensional controller parameter space is determined, such that n-m eigenvalues are made unobservable through this feedback. For practical applications m equals two and the stability boundaries can be visualized in this two-dimensional cross-section in controller parameter space using the parameter space approach.
In case of uncertain systems an invariance plane is fixed for a nominal operating point, e.g. the center of the operating domain. A finite number of representatives of the operating domain is selected, for example the vertices, and the set of stabilizing controller parameters for each representative is determined in the given invariance plane. If the intersection of these sets is non-empty, it indicates the set of suimultaneously stabilizing controller parameters.
These approaches are explained in detail in:
J. Ackermann, P. Blue, T. Bünte, L. Güvenc, D. Kaesbauer, M. Kordt, M. Muhler, and D.Odenthal, Robust Control: The Parameter Space Approach, Springer, London, 2002.
Preface and table of contents
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