[Paper Review] Constructive Stabilization and Pole Placement by Arbitrary Decentralized Architectures

Alborz Alavian, Michael Rotkowitz|arXiv (Cornell University)|Apr 6, 2017

Quantum chaos and dynamical systems8 references3 citations

TL;DR

This paper generalizes Wang and Davison's seminal 1973 results on decentralized control by rigorously proving that fixed modes—plant modes unmovable by static linear time-invariant (LTI) controllers with a given information structure—remain unmovable by dynamic LTI controllers under the same structure. It further establishes that non-fixed modes can be placed arbitrarily close to any desired locations via a constructive algorithm, thereby proving that stabilizability by a dynamic decentralized controller is equivalent to all fixed modes lying in the left half-plane.

ABSTRACT

A seminal result in decentralized control is the development of fixed modes by Wang and Davison in 1973 - that plant modes which cannot be moved with a static decentralized controller cannot be moved by a dynamic one either, and that the other modes which can be moved can be shifted to any chosen location with arbitrary precision. These results were developed for perfectly decentralized, or block diagonal, information structure, where each control input may only depend on a single corresponding measurement. Furthermore, the results were claimed after a preliminary step was demonstrated, omitting a rigorous induction for each of these results, and the remaining task is nontrivial. In this paper, we consider fixed modes for arbitrary information structures, where certain control inputs may depend on some measurements but not others. We provide a comprehensive proof that the modes which cannot be altered by a static controller with the given structure cannot be moved by a dynamic one either, and that the modes which can be altered by a static controller with the given structure can be moved by a dynamic one to any chosen location with arbitrary precision, thus generalizing and solidifying Wang and Davison's results. This shows that a system can be stabilized by a linear time-invariant controller with the given information structure as long as all of the modes which are fixed with respect to that structure are in the left half-plane; an algorithm for synthesizing such a stabilizing decentralized controller is then distilled from the proof.

Motivation & Objective

• To generalize Wang and Davison's 1973 results on fixed modes and pole placement from perfectly decentralized (block-diagonal) structures to arbitrary information structures.
• To provide a rigorous inductive proof that fixed modes under static LTI controllers remain unmovable under dynamic LTI controllers for any given information structure.
• To demonstrate that non-fixed modes can be placed arbitrarily close to any desired locations using a dynamic LTI controller with the prescribed structure.
• To derive an explicit, constructive algorithm for synthesizing a stabilizing decentralized controller based on the proof.

Proposed method

• Introduces a generalized definition of fixed modes that depends on both the information structure and controller type (static/dynamic, LTI/non-LTI).
• Uses an inductive argument to prove that the set of fixed modes is identical for static and dynamic LTI controllers under any given information structure.
• Applies a constructive approach to show that non-fixed modes can be moved to within an arbitrary distance of any desired pole locations in the complex plane.
• Develops Algorithm 1, which synthesizes a stabilizing dynamic LTI controller by iteratively assigning poles to non-fixed modes using state-space parameterization.
• Employs proximity-based conditions instead of exact eigenvalue matching to avoid unbounded controller gains.
• Extends results to arbitrary acceptable closed-loop eigenvalue regions by replacing the left half-plane with any open set of desired locations.

Experimental results

Research questions

• RQ1 Can the concept of fixed modes be generalized beyond perfectly decentralized (block-diagonal) information structures to arbitrary information structures in decentralized control?
• RQ2 Is it true that modes unmovable by static LTI controllers with a given information structure remain unmovable by dynamic LTI controllers with the same structure?
• RQ3 Can non-fixed modes be placed arbitrarily close to any desired locations in the complex plane using a dynamic LTI controller under a given information constraint?
• RQ4 What is a constructive, algorithmic procedure for synthesizing a stabilizing dynamic LTI controller under arbitrary information structures?
• RQ5 How can the results be extended to allow for arbitrary acceptable closed-loop eigenvalue regions beyond the left half-plane?

Key findings

• Fixed modes under static LTI controllers with a given information structure are identical to those under dynamic LTI controllers with the same structure, proving that dynamic controllers do not expand the set of movable modes.
• Non-fixed modes can be moved to within any desired accuracy of any chosen complex conjugate locations, ensuring arbitrary pole placement precision.
• A system is stabilizable by a dynamic LTI controller with a given information structure if and only if all fixed modes lie in the left half-plane.
• The proposed Algorithm 1 successfully synthesizes a stabilizing controller of order 3 for a 5th-order plant with 9 non-zero controller entries, outperforming alternative methods in controller order.
• The method avoids unbounded gains by using proximity-based eigenvalue placement instead of exact matching, enhancing numerical robustness.
• The results are extendable to any open set of acceptable closed-loop eigenvalues, not just the left half-plane, by replacing the complement of the acceptable region accordingly.