PhD Course on the Geometric Approach to System and Control Theory - 2011 (and 2010)
PhD Course in Control Systems and Operation Research
Introduction to the Geometric Approach to System and Control Theory – 2011 (and 2010)
Course Contents
Algebraic structures.
Vector spaces, subspaces, inner product, orthoghonal vectors, partial ordering relations, lattices (Handwritten Notes 1B-3B, [Ref. 1, pp. 370-371]).
Subspace algebra.
Operations on subspaces. Properties of operations on subspaces (Handwritten Notes 4B, [Ref. 1, pp. 126-129]).
Invariant subspaces.
Definition and characterization (necessary and sufficient condition) of invariant subspace (Handwritten Notes 5B, [Ref. 1, p. 129]). Invariant subspaces and changes of basis: change of basis (or similarity transformation); subspace invariance and change of basis; restriction of a linear map to an invariant subspace (Handwritten Notes 6B-7B, [Ref. 1, pp. 129-130]). Lattices of invariant subspaces and related algorithms: algorithm for the minimal A-invariant subspace containing imB (statement and proof); algorithm for the maximal A-invariant subspace contained in kerC (statement and proof - by duality) (Handwritten Notes 8B-9B, [Ref. 1, pp. 131-132]). Complementability of an A-invariant subspace: definition and characterization (necessary and sufficient condition) (Handwritten Notes 10B, [Ref. 1, pp. 133-134]). Invariant subspaces and state trajectories: the fundamental lemma of the geometric approach (Handwritten Notes 11B). Internal and external stability of an A-invariant subspace (Handwritten Notes 12B, [Ref. 1, pp. 136-138]). The Kalman canonical decomposition (Handwritten Notes 12C, [Ref. 1, pp. 143-144]). State feedback, output injection, and dynamic measurement feedback (Handwritten Notes 13C, [Ref. 1, pp. 160-175]).
Controlled and conditioned invariant subspaces.
Controlled invariant subspace - definition. Conditioned invariant subspace - definition. Sum of controlled invariant subspaces. Intersection of conditioned invariant subspaces. Upper semilattice of (A,B)-controlled invariant subspaces contained in kerE. Lower semilattice of (A,C)-conditioned invariant subspaces containing imH. Algorithm for the minimal(A,C)-conditioned invariant subspaces containing imH. Duality between the maximal (A,B)-controlled invariant subspace contained in kerE and the minimal (A,C)-conditioned invariant subspaces containing imH. Algorithm for the maximal (A,B)-controlled invariant subspace contained in kerE (Handwritten Notes 1D-2D, [Ref. 1, pp. 204-210]). Orthogonal complement of an (A,L)-controlled invariant subspace. Orthogonal complement of an (A,L)-conditioned invariant subspace. Theorem: controlled invariant subspaces and controlled state trajectories (statement and proof) (Handwritten Notes 2D’, [Ref. 1, p. 207]). Controlled invariant subspace (necessary and sufficient condition). Controlled invariance is a coordinate-free notion (Handwritten Notes 3D, , [Ref. 1, p. 207]). Controlled invariance and state-feedback invariance (Handwritten Notes 4D, [Ref. 1, pp. 208]). Conditioned invariance and output-injection invariance (Handwritten Notes 5D, [Ref. 1, pp. 208]).
Self-bounded controlled invariant subspaces and self-hidden conditioned invariant subspaces.
Property: controlled invariant subspaces and invariant subspaces by the same state-feedback. Property: conditioned invariant subspaces and invariant subspaces by the same output-injection (Handwritten Notes 6D, [Ref. 1, pp. 208]). Self-bounded controlled invariant subspace (definition and characterization). Intersection of self-bounded controlled invariant subspaces. Lattice of self-bounded controlled invariant subspaces (Handwritten Notes 7D-8D, [Ref. 1, pp. 210-214]). Self-hidden conditioned invariant subspace (definition and characterization). Sum of self-hidden conditioned invariant subspaces. Lattice of self-hidden conditioned invariant subspaces (Handwritten Notes 9D, [Ref. 1, p. 215]). Internal and external stabilizability of a controlled invariant subspace (Handwritten Notes 10D, [Ref. 1, pp. 217-220]). The minimal internally stabilizable (A,B)-controlled invariant subspace contained in kerE and containing imH. Basile and Marro’s conjecture (Handwritten Notes 11D-11D’, [Ref. 1, pp. 223-226]). Disturbance decoupling with stability by state feedback (Handwritten Notes 12D-12D’, [Ref. 1, pp. 223-230]). Measurable signal decoupling. The routine hud.m of the Geometric Approach Toolbox and an illustrative example (Handwritten Notes 1G-5G, [Ref. 1, pp. 223-230], [Ref. 2]).
Computational Support
Some Basic Matlab Commands
A.1 Vectors, matrices and polynomials. A.2 Interaction with the “Command Window”. A.3 Cell arrays. A.4 Binary logic. A.6 Conditional execution of a block of commands. A.7 Further commands ([Ref. 3, pp. 75-79]). A.7 Linear time-invariant systems ([Ref. 3, pp. 79-80]). A.8 M-files and functions ([Ref. 3, p. 81]). A.9 Some system connections ([Ref. 3, pp. 82-84]).
The Geometric Approach Toolbox
([Ref. 1, pp. 444-453]).
Examination papers
PhD Course in Control Systems and Operation Research: Midterm Test (November 5, 2010)
PhD Course in Control Systems and Operation Research: Midterm Test (November 11, 2010)
References
[1] G. Basile and G. Marro, “Controlled and Conditioned Invariants in Linear System Theory”, Prentice-Hall, Englewood Cliffs, NJ, 1992.
[2] E. Zattoni, “Decoupling of measurable signals via self-bounded controlled invariant subspaces: minimal unassignable dynamics of feedforward units for prestabilized systems,” IEEE Transactions on Automatic Control , vol. 52, no. 1, pp. 140–143, January 2007 [DOI: 10.1109/TAC.2006.886499].
[3] G. Marro, “Some Basic Matlab Commands”, Department of Electronics, Computer Science and Systems, University of Bologna, Italy, 2010.
Recommended Readings
[4] W. M. Wonham, Linear Multivariable Control: A Geometric Approach, 3rd ed. New York: Springer-Verlag, 1985.
[5] H. Trentelman, A. Stoorvogel, and M. Hautus, Control theory for linear systems, ser. Communications and Control Engineering. Great Britain: Springer, 2001.
