Lecture 3 The ELIMINATION Problem Jan C. Willems University of Leuven, Belgium Minicourse ECC 2003 Cambridge, UK, September 2, 2003 Lecture 3 The ELIMINATION Problem p.1/22
Problematique Develop a theory and algorithms for eliminating latent variables Lecture 3 The ELIMINATION Problem p.2/22
Introduction First principles model auxiliary, latent variables e.g. interconnection variables theoretical latent variables: momenta, potentials, driving noise, state variables Lecture 3 The ELIMINATION Problem p.3/22
Given the behavioral eq ns for the components, how do those of the interconnected system look like? Lecture 3 The ELIMINATION Problem p.4/22
Recall the definitions: A dynamical system with latent variables =, the time-axis, the signal space, the latent variable space : the full behavior Lecture 3 The ELIMINATION Problem p.5/22
Recall the definitions: A dynamical system with latent variables =, the time-axis (= the set of relevant time instances)., the signal space (= the variables that the model aims at)., the latent variable space (= auxiliary modeling variables). : the full behavior (= the pairs that the model declares possible). Lecture 3 The ELIMINATION Problem p.5/22
The manifest behavior Call the elements of manifest variables, those of latent variables. The latent variable system the manifest system with manifest behavior induces such that Lecture 3 The ELIMINATION Problem p.6/22
The manifest behavior Call the elements of manifest variables, those of latent variables. The latent variable system the manifest system with manifest behavior induces such that In convenient equations for, the latent variables are eliminated. But how do these equations look like, and how are they obtained? Lecture 3 The ELIMINATION Problem p.6/22
L B full W B Given mathematical structure for, what mathematical structure for emerges? Lecture 3 The ELIMINATION Problem p.7/22
Examples 1. The projection of a linear subspace is again a linear subspace. Lecture 3 The ELIMINATION Problem p.8/22
! Examples 1. The projection of a linear subspace is again a linear subspace. 2. The projection of an algebraic variety is, in general, not an algebraic variety: # " Lecture 3 The ELIMINATION Problem p.8/22
$ Examples 1. The projection of a linear subspace is again a linear subspace. 2. The projection of an algebraic variety is, in general, not an algebraic variety: 3. How about the projection of (the sol n set of) a smooth differential equation? # &(&'( & &' &(&'( & &' % Again a differential eq n?? Lecture 3 The ELIMINATION Problem p.8/22
) Examples 1. The projection of a linear subspace is again a linear subspace. 2. The projection of an algebraic variety is, in general, not an algebraic variety: 3. How about the projection of (the sol n set of) a smooth differential equation? 4. How about the projection of a constant coefficient linear differential equation? & &' & &' Again a constant coefficient linear differential eq n?? Lecture 3 The ELIMINATION Problem p.8/22
) + * * ) Elimination First principle models latent variables. For systems described by linear constant coefficient ODE s: & &' & &' with..,- Lecture 3 The ELIMINATION Problem p.9/22
) + * * ) 2 / 4 3 Elimination First principle models latent variables. For systems described by linear constant coefficient ODE s: & &' & &' with..,- This is the natural model class to start a study of finite dimensional linear time-invariant systems! Much more so than 1 2 / 0 / & &' Lecture 3 The ELIMINATION Problem p.9/22
) Is it(s manifest behavior) also a differential system?? Consider 5 56 5 56 Lecture 3 The ELIMINATION Problem p.10/22
) : 9 8 7 ;: 9> 9 8 7 < < < Is it(s manifest behavior) also a differential system?? Consider 5 56 5 56 Full behavior: ; =< < < ( belongs to (, by definition. Its manifest behavior equals ( such that Lecture 3 The ELIMINATION Problem p.10/22
9> Does belong to? Lecture 3 The ELIMINATION Problem p.11/22
9> Does belong to? Theorem: It does! Proof: The fundamental principle. Lecture 3 The ELIMINATION Problem p.11/22
A A K E FFFE I @ M E UG T L B I M @ M E UG T LK M E FFF A A I? Y J W M LW E FFFE E FFFE 9>? Does belong to Theorem: It does! Proof: The fundamental principle. The fundamental principle (for PDE s) states that I BJ @? ACBHG ACBD iff GP is solvable for GN S E R G E QR D OGP GN?L Z \ K J Y@ W [ Z SX R G QR D GN ACBCG ACBD Lecture 3 The ELIMINATION Problem p.11/22
^ ] Example: Consider once again our electrical RLC - circuit: I + R C L environment C R L system!! Model the relation between manifest and!! Lecture 3 The ELIMINATION Problem p.12/22
I + R C L environment C R L system Lecture 3 The ELIMINATION Problem p.13/22
I + R C L environment C R L system Lecture 3 The ELIMINATION Problem p.13/22
I + I C I R C + R C C + + L R L I L + I R L The circuit graph Introduce the latent variables: the voltage across and the current in each branch: ^ ] ^ _ e ]`_fe ^ c ]dc ^ _ba ]`_ba Lecture 3 The ELIMINATION Problem p.13/22
& & 4 h ] ] ^ ^ ] ^ ^ c &' ] _je ] ]gc ^ _ke ^ c Lecture 3 The ELIMINATION Problem p.14/22 System equations Constitutive equations (CE): ]gc ^ _fe ) ] _fe ^ _ba ) c ] _ba &' Kirchhoff s voltage laws (KL): ] _ a ] _je ] ]gc ] _ia Kirchhoff s current laws (KCL): ^ _ke ^ ^ c ^ _ja ^ ^ _ja
^ ] n m J l a e sq n q a a @ a a e e e q n q I a J a e After elimination, we obtain the following explicit differential equation the between and :. Case 1: It l o l I o @p o s qr qr o I@p l o @p u \ qr qr Lecture 3 The ELIMINATION Problem p.15/22
^ ] n m J l a e sq n q a a @ a a e e e q n q I a J a e n e q q a u I a @ a a e ^ ] After elimination, we obtain the following explicit differential equation the between and :. Case 1: It l o l I o @p o s qr qr o I@p l o @p u \ qr qr. l J a Case 2: l o @p It J l o qr qr! and These are the exact relations between Lecture 3 The ELIMINATION Problem p.15/22
First principles modeling ( CE s, KL, & KCL) 15 behavioral equations. Include both the port and the branch voltages and currents. v Lecture 3 The ELIMINATION Problem p.16/22
First principles modeling ( CE s, KL, & KCL) 15 behavioral equations. Include both the port and the branch voltages and currents. v Why can the port behavior be described by a system of linear constant coefficient differential equations? Lecture 3 The ELIMINATION Problem p.16/22
w x y First principles modeling ( CE s, KL, & KCL) 15 behavioral equations. Include both the port and the branch voltages and currents. v Why can the port behavior be described by a system of linear constant coefficient differential equations? Because: 1. The CE s, KL, & KCL are all linear constant coefficient differential equations. 2. The elimination theorem. capacitor D y a{z, inductor, series, parallel, may give erroneous results n} Lecture 3 The ELIMINATION Problem p.16/22
w First principles modeling ( CE s, KL, & KCL) 15 behavioral equations. Include both the port and the branch voltages and currents. v Why can the port behavior be described by a system of linear constant coefficient differential equations? Because: 1. The CE s, KL, & KCL are all linear constant coefficient differential equations. 2. The elimination theorem Why is there exactly one equation? Passivity!. Lecture 3 The ELIMINATION Problem p.16/22
~ Remarks: Number of equations after elimination (constant coeff. lin. ODE s) number of variables. Elimination fewer, higher order equations. Lecture 3 The ELIMINATION Problem p.17/22
~ ~ ) ) Remarks: Number of equations after elimination (constant coeff. lin. ODE s) number of variables. effective computer algebra/gröbner bases type algorithms for elimination Lecture 3 The ELIMINATION Problem p.17/22
~ ~ ~ 8 7 : $ Remarks: Number of equations after elimination (constant coeff. lin. ODE s) number of variables. effective computer algebra/gröbner bases type algorithms for elimination Depends on sol n smoothness! elimination theorem on with compact support): # &' ' 8 8 & &' Lecture 3 The ELIMINATION Problem p.17/22
~ ~ ~ ~ Remarks: Number of equations after elimination (constant coeff. lin. ODE s) number of variables. effective computer algebra/gröbner bases type algorithms for elimination Depends on sol n smoothness! Not generalizable to smooth nonlinear systems. Why are differential equations models so prevalent? Lecture 3 The ELIMINATION Problem p.17/22
~ ~ ~ ~ ~ Remarks: Number of equations after elimination (constant coeff. lin. ODE s) number of variables. effective computer algebra/gröbner bases type algorithms for elimination Depends on sol n smoothness! Not generalizable to smooth nonlinear systems. Generalizable to linear constant coefficient PDE s. Lecture 3 The ELIMINATION Problem p.17/22
ƒ F A Œ ƒ F A s Example: Maxwell s equations E p ˆ J Ž E Ar J ƒš Z E J \ Ar o p ˆ J ƒš Lecture 3 The ELIMINATION Problem p.18/22
ƒ F A Œ ƒ F A s Example: Maxwell s equations E p ˆ J Ž E Ar J ƒš Z E Ž J \ Ar o p ˆ Ž J ƒš (time and space), 1 (electric field, magnetic field, current density, charge density),, set of solutions to these PDE s. Lecture 3 The ELIMINATION Problem p.18/22
ƒ F Œ ƒ F s Example: Maxwell s equations E p ˆ J Ž E A Ar J ƒš Z E Ž J \ A Ar o p ˆ Ž J ƒš (time and space), 1 (electric field, magnetic field, current density, charge density),, set of solutions to these PDE s. Note: 10 variables, 8 equations! free variables. Lecture 3 The ELIMINATION Problem p.18/22
Which PDE s describe ( ) in Maxwell s equations? Lecture 3 The ELIMINATION Problem p.19/22
Which PDE s describe ( ) in Maxwell s equations? Eliminate 1 from Maxwell s equations Lecture 3 The ELIMINATION Problem p.19/22
< # < ) in Maxwell s equations? Which PDE s describe ( from Maxwell s equations 1 Eliminate š < ' š # ' œ š š ' Lecture 3 The ELIMINATION Problem p.19/22
< # < Which PDE s describe ( ) in Maxwell s equations? Eliminate 1 from Maxwell s equations š < ' š # ' œ š š ' Elimination theorem this exercise is exact & successful (+ gives algorithm). Lecture 3 The ELIMINATION Problem p.19/22
9> 9> 9> D 9> D. + 9 D. + 9 D 9> D has very nice properties. It is closed * > It follows from all this that under:. 9> Intersection:. 9> Addition:. ž : 9s Ÿ D Projection: Action of a linear differential operator:,- 9s 9> D 9s > 5 56 Inverse image of a linear differential operator:,- 9s 9s > 5 56 Lecture 3 The ELIMINATION Problem p.20/22
The elimination theorem (and the related algorithms) is one of the nice, important, new problems that have emerged from the behavioral theory. Lecture 3 The ELIMINATION Problem p.21/22
< < < The elimination theorem (and the related algorithms) is one of the nice, important, new problems that have emerged from the behavioral theory. Equally important as elimination is introducing convenient latent variables: state models, first order equations image representations and controllability Lecture 3 The ELIMINATION Problem p.21/22
End of Lecture 3 Lecture 3 The ELIMINATION Problem p.22/22