Robust controller design using normalized coprime factor plant descriptions.

*(English)*Zbl 0688.93044
Lecture Notes in Control and Information Sciences, 138. Berlin etc.: Springer-Verlag. x, 206 p. DM 56.00 (1990).

This book focuses on the problem of robust stabilization for linear time invariant MIMO systems. The problem of robust stabilization is to find a single compensator which simultaneously stabilizes a whole family of plants around the nominal plant model. The uncertainty can be modeled as a free uncertainty parameter matrix \(\Delta\) being fed into a linear fractional map; the choice \(\Delta =0\) corresponds to the nominal plant and one seeks a compensator K which maintains internal stability with \(\Delta\) ’s having \(\| \Delta \|_{\infty}\leq \epsilon\). Special forms of the linear fractional map give rise to additive and multiplicative plant perturbations as special choices.

In early work on this problem it was usually assumed that the perturbed plant had the same number of right half plane poles; the problem of finding the largest \(\epsilon\) for which robust stabilization is possible leads to an H-infinity optimization problem for which good state space methods of solution are now available. However, except for some simple special cases, the associated H-infinity problem is of the “2-block” type for which no explicit closed form finite dimensional solution but rather only an iterative procedure (\(\gamma\)-iteration) is available. One of the main points of the book is that the 2-block H-infinity problem arising from the robust stabilization problem with respect to perturbations of the normalized coprime factors of the nominal plant can be reduced to a Nehari-Takagi problem of 1-block type, and thereby can be solved exactly in closed form via formulas involving only finite matrices. Also presented is an analysis of the generality of this last reduction, i.e., a characterization of which general 2-block H-infinity problems can be reduced to a Nehari-Takagi problem and thereby be solved exactly.

In practice one may want to perform a model reduction to get a smaller size model before doing the computations to find a robustly stabilizing compensator. The book includes a self-contained chapter on model reduction. One may choose to reduce the order of the coprime factors of the nominal plant initially or to reduce the order of the controller at the end. Also included is a physical discussion of various engineering tradeoffs and a recommendation of a particular general design strategy (“a loop shaping design procedure”) and several illustrative “real world” examples.

The book is well organized, well written and largely self-contained. Besides being an introduction to the topic of its main focus (robust stabilization), it also contains background information on inner-outer and coprime factorization, Riccati equations and the recent concise state space solution of the standard H-infinity problem. This makes the book a good introduction to the broad area of H-infinity control, a contribution beyond the specific focus of its title.

In early work on this problem it was usually assumed that the perturbed plant had the same number of right half plane poles; the problem of finding the largest \(\epsilon\) for which robust stabilization is possible leads to an H-infinity optimization problem for which good state space methods of solution are now available. However, except for some simple special cases, the associated H-infinity problem is of the “2-block” type for which no explicit closed form finite dimensional solution but rather only an iterative procedure (\(\gamma\)-iteration) is available. One of the main points of the book is that the 2-block H-infinity problem arising from the robust stabilization problem with respect to perturbations of the normalized coprime factors of the nominal plant can be reduced to a Nehari-Takagi problem of 1-block type, and thereby can be solved exactly in closed form via formulas involving only finite matrices. Also presented is an analysis of the generality of this last reduction, i.e., a characterization of which general 2-block H-infinity problems can be reduced to a Nehari-Takagi problem and thereby be solved exactly.

In practice one may want to perform a model reduction to get a smaller size model before doing the computations to find a robustly stabilizing compensator. The book includes a self-contained chapter on model reduction. One may choose to reduce the order of the coprime factors of the nominal plant initially or to reduce the order of the controller at the end. Also included is a physical discussion of various engineering tradeoffs and a recommendation of a particular general design strategy (“a loop shaping design procedure”) and several illustrative “real world” examples.

The book is well organized, well written and largely self-contained. Besides being an introduction to the topic of its main focus (robust stabilization), it also contains background information on inner-outer and coprime factorization, Riccati equations and the recent concise state space solution of the standard H-infinity problem. This makes the book a good introduction to the broad area of H-infinity control, a contribution beyond the specific focus of its title.

Reviewer: J.A.Ball

##### MSC:

93D15 | Stabilization of systems by feedback |

93B35 | Sensitivity (robustness) |

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

93C05 | Linear systems in control theory |

93C35 | Multivariable systems, multidimensional control systems |

93B50 | Synthesis problems |