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Book of Open Problems Presented at MTNS 2002 2002 MTNS Problem Book Open Problems on the Mathematical Theory of Systems August 12-16, 2002 Editors Vincent D. Blondel Alexander Megretski Associate Editors Roger Brockett Jean-Michel Coron Miroslav Krstic Anders Rantzer Joachim Rosenthal Eduardo D. Sontag M. Vidyasagar Jan C. Willems Some of the problems appearing in this booklet will appear in a more extensive forthcoming book on open problems in systems theory. For more information about this future book, please consult the website http://www.inma.ucl.ac.be/~blondel/op/ We wish the reader much enjoyment and stimulation in reading the problems in this booklet. The editors ii Program and table of contents Monday August 12. Co-chairs: Vincent Blondel, Roger Brockett. 8:00 PM Linear Systems 75 State and first order representations 1 Jan C. Willems 1 59 The elusive iff test for time-controllability of behaviours 4 Amol J. Sasane 4 58 Schur Extremal Problems 7 Lev Sakhnovich 7 69 Regular feedback implementability of linear differential behaviors 9 H.L. Trentelman 9 36 What is the characteristic polynomial of a signal flow graph? 13 Andrew D. Lewis 13 35 Bases for Lie algebras and a continuous CBH formula 16 Matthias Kawski 16 10 Vector-valued quadratic forms in control theory 20 Francesco Bullo, Jorge Cort´es,Andrew D. Lewis, Sonia Mart´ınez 20 8:48 PM Stochastic Systems 4 On error of estimation and minimum of cost for wide band noise driven systems 24 Agamirza E. Bashirov 24 25 Aspects of Fisher geometry for stochastic linear systems 27 Bernard Hanzon, Ralf Peeters 27 iii 9:00 PM Nonlinear Systems 81 Dynamics of Principal and Minor Component Flows 31 U. Helmke, S. Yoshizawa, R. Evans, J.H. Manton, and I.M.Y. Mareels 31 55 Delta-Sigma Modulator Synthesis 35 Anders Rantzer 35 44 Locomotive Drive Control by Using Haar Wavelet Packets 37 P. Mercorelli, P. Terwiesch, D. Prattichizzo 37 61 Determining of various asymptotics of solutions of nonlinear time-optimal problems via right ideals in the moment algebra 41 G. M. Sklyar, S. Yu. Ignatovich 41 9:24 PM Hybrid Systems 28 L2-Induced Gains of Switched Linear Systems 45 Jo~aoP. Hespanha 45 48 Is Monopoli's Model Reference Adaptive Controller Correct? 48 A. S. Morse 48 82 Stability of a Nonlinear Adaptive System for Filtering and Parameter Estimation 53 Masoud Karimi-Ghartemani, Alireza K. Ziarani 53 70 Decentralized Control with Communication between Controllers 56 Jan H. van Schuppen 56 9:48 PM Infinite dimensional systems 80 Some control problems in electromagnetics and fluid dynamics 60 Lorella Fatone, Maria Cristina Recchioni, Francesco Zirilli 60 9:54 PM Identification, Signal Processing 19 A conjecture on Lyapunov equations and principal angles in subspace identification 64 Katrien De Cock, Bart De Moor 64 84 On the Computational Complexity of Non-Linear Projections and its Relevance to Signal Processing 69 iv J. Manton and R. Mahony 69 10:06 PM Identification, Signal Processing 2 H -norm approximation 73 1 A.C. Antoulas, A. Astolfi 73 14 Exact Computation of Optimal Value in H Control 77 1 Ben M. Chen 77 Tuesday August 13. Co-chairs: Co-chairs: Anders Rantzer, Eduardo Sontag, Jan Willems. 8:00 PM Algorithms and Computation 40 The Computational Complexity of Markov Decision Processes 80 Omid Madani 80 Tuesday August 13 80 Co-chairs: Anders Rantzer, Eduardo Sontag, Jan Willems 80 7 When is a pair of matrices stable? 83 Vincent D. Blondel, Jacques Theys, John N. Tsitsiklis 83 8:12 PM Stability, stabilization 12 On the Stability of Random Matrices 86 Giuseppe C. Calafiore, Fabrizio Dabbene 86 37 Lie algebras and stability of switched nonlinear systems 90 Daniel Liberzon 90 21 The Strong Stabilization Problem for Linear Time-Varying Systems 93 Avraham Feintuch 93 86 Smooth Lyapunov Characterization of Measurement to Error Stability 95 Brian P. Ingalls, Eduardo D. Sontag 95 13 Copositive Lyapunov functions 98 M.K. C¸amlıbel, J.M. Schumacher 98 v 56 Delay independent and delay dependent Aizerman problem 102 Vladimir R˘asvan 102 6 Root location of multivariable polynomials and stability analysis 108 Pierre-Alexandre Bliman 108 18 Determining the least upper bound on the achievable delay margin 112 Daniel E. Davison, Daniel E. Miller 112 53 Robust stability test for interval fractional order linear systems 115 Ivo Petr´aˇs,YangQuan Chen, Blas M. Vinagre 115 73 Stability of Fractional-order Model Reference Adaptive Control 118 Blas M. Vinagre, Ivo Petr´aˇs,Igor Podlubny, YangQuan Chen 118 29 Robustness of transient behavior 122 Diederich Hinrichsen, Elmar Plischke, Fabian Wirth 122 85 Generalized Lyapunov Theory and its Omega-Transformable Regions 126 Sheng-Guo Wang 126 9:24 PM Controllability, Observability 20 Linearization of linearly controllable systems 130 R. Devanathan 130 5 The dynamical Lame system with the boundary control: on the structure of the reachable sets 133 M.I. Belishev 133 30 A Hautus test for infinite-dimensional systems 136 Birgit Jacob, Hans Zwart 136 34 On the convergence of normal forms for analytic control systems 139 Wei Kang, Arthur J. Krener 139 27 Nilpotent bases of distributions 143 Henry G. Hermes, Matthias Kawski 143 9 Optimal Synthesis for the Dubins' Car with Angular Acceleration Control 147 Ugo Boscain, Benedetto Piccoli 147 vi Problem 75 State and first order representations Jan C. Willems Department of Electrical Engineering - SCD (SISTA) University of Leuven Kasteelpark Arenberg 10 B-3001 Leuven-Heverlee Belgium [email protected] 75.1 Abstract We conjecture that the solution set of a system of linear constant coefficient PDE's is Markovian if and only if it is the solution set of a system of first order PDE's. An analogous conjecture regarding state systems is also made. Keywords: Linear differential systems, Markovian systems, state systems, kernel representations. 75.2 Description of the problem 75.2.1 Notation First, we introduce our notation for the solution sets of linear PDE's in the n real independent variables n w x = (x1; : : : ; xn). Let Dn0 denote, as usual, the set of real distributions on R , and Ln the linear w subspaces of (Dn0 ) consisting of the solutions of a system of linear constant coefficient PDE's in the w w real-valued dependent variables w = col(w1; : : : ; ww). More precisely, each element B Ln is defined w 2 by a polynomial matrix R R [ξ1; ξ2; : : : ; ξn]; with w columns, but any number of rows, such that 2 •× w @ @ @ B = w (Dn0 ) R( ; ;:::; )w = 0 : f 2 j @x1 @x2 @xn g w We refer to elements of Ln as linear differential n-D systems. The above PDE is called a kernel w w representation of B Ln. Note that each B Ln has many kernel representations. For an in depth w 2 2 study of Ln, see [1] and [2]. Next, we introduce a class of special three-way partitions of Rn. Denote by P the following set of partitions of Rn: n [(S ;S0;S+) P]: [(S ;S0;S+ are disjoint subsets of R ) − 2 , − n (S S0 S+ = R ) (S and S+ are open, and S0 is closed)]: ^ − [ [ ^ − n n Finally, we define concatenation of maps on R . Let f ; f+ : R F, and let π = (S ;S0;S+) P. − ! − 2 1 n Define the map f π f+ : R F, called the concatenation of (f ; f+) along π, by − ^ ! − f (x) for x S (f π f+)(x) := − − f+(x) for x 2 S0 S+ − ^ 2 [ 75.2.2 Markovian systems Define B Lw to be Markovian : 2 n , n w [(w ; w+ B C1(R ; R )) (π = (S ;S0;S+) P) − 2 \ ^ − 2 (w S0 = w+ S0 )] [(w π w+ B]: ^ −j j ) − ^ 2 Think of S as the `past', S0 as the `present', and S+ as the `future'. Markovian means that if two solutions of− the PDE agree on the present, then their pasts and futures are compatible, in the sense that the past (and present) of one, concatenated with the (present and) future of the other, is also a solution. In the language of probability: the past and the future are independent given the present. We come to our first conjecture: w B Ln is Markovian 2if and only if it has a kernel representation that is first order. I.e., it is conjectured that a Markovian system admits a kernel representation of the form @ @ @ R0w + R1 w + R2 w + Rn w = 0: @x1 @x2 ··· @xn w Oberst [2] has proven that there is a one-to-one relation between Ln and the submodules of w w R [ξ1; ξ2; : : : ; ξn], each B L being identifiable with its set of annihilators 2 n w @ @ @ NB := n R [ξ1; ξ2; : : : ; ξn] n>( ; ;:::; )B = 0 : f 2 j @x1 @x2 @xn g Markovianity is hence conjectured to correspond exactly to those B Lw for which the submodule 2 n NB has a set of first order generators. 75.2.3 State systems In this section we consider systems with two kind of variables: w real-valued manifest variables, w = col(w1; : : : ; ww), and z real-valued state variables, z = col(z1; : : : ; zz). Their joint behavior is again assumed to be the solution set of a system of linear constant coefficient PDE's. Thus we w+z w+z consider behaviors in Ln , whence each element B Ln is described in terms of two polynomial (w+z) 2 matrices (R; M) R [ξ1; ξ2; : : : ; ξn] by 2 •× w z B = (w; z) (D0 ) (D0 ) f 2 n × n j @ @ @ @ @ @ R( ; ;:::; )w + M( ; ;:::; )z = 0 : @x1 @x2 @xn @x1 @x2 @xn g Define B Lw+z to be a state system with state z : 2 n , n w+z [((w ; z ); (w+; z+) B C1(R ; R )) (π = (S ;S0;S+) P) − − 2 \ ^ − 2 (z S0 = z+ S0 )] [(w ; z ) π (w+; z+) B]: ^ −j j ) − − ^ 2 2 Think again of S as the `past', S0 as the `present', S + as the `future'.
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