Purdue Oldenburger Lecture 10-09 - University of Michiganulsoy/pdf/Oldenburger_Lecture.pdf · Rufus Oldenburger Lecture ... Application to Automotive Powertrain Control,” 2009 DSCC

A. Galip Ulsoy C.D. Mote, Jr. Distinguished University Professor of Mechanical Engineering and the William Clay Ford Professor of Manufacturing Dept. of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2125 USA

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Mechanical Engineering The University of Michigan, Ann Arbor

Department of Mechanical Engineering, College of Engineering University of Michigan 2266 GG Brown Building, 2350 Hayward Street, Ann Arbor MI 48109-2125 USA

Analysis and Control of Time Delay Systems via the Lambert W

Function

A. Galip Ulsoy C.D. Mote, Jr. Distinguished University Professor of Mechanical Engineering

and the William Clay Ford Professor of Manufacturing

Rufus Oldenburger Lecture Mechanical Engineering Department,

Purdue University, October 1, 2009


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Rufus T. Oldenburger •  Contributions to Time Delay Systems

•  The solution of PDEs, for modeling fluid flows through pipes, is given in terms of transcendental functions (e.g., sinh and tanh). Then, the transfer function has a similar form to systems with delay. Rational approximations were used [Oldenburger & Goodson, J. Basic Eng., 1964]

•  Wiener-Hopf equation, which represents physical systems (e.g., fluid dynamics), can be approximated with delay line models. Using delay equations, a method to estimate the transfer function with input and output signals was developed. [Desai & Oldenburger, IEEE Trans. Aut. Cont., 1969]

•  Delay line approx. of the Wiener-Hopf equation: -  Estimate g(t) using r(t) and c1(t). [Desai &

Oldenburger, IEEE Trans Aut Cont, 1969] -  Select best value of the delay, T, to better

approximate Wiener-Hopf equation. [Oldenburger & Hoberock, Technical Report, 1964]

*Delay line model [Oldenburger and Hoberock, Technical Report, 1964]


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Acknowledgements and References •  Collaborators:

-  Graduate students: Sun Yi and Farshid M. Asl -  Math colleague: Prof. Patrick W. Nelson

•  Pleased to acknowledge financial support from NSF via Grant # 0555765 and ERC/RMS

•  Selected references: -  F.M. Asl and A.G. Ulsoy, “Analysis of a System of Linear Delay Differential Equations,” ASME J. Dyn. Syst., Meas.

Cont., vol. 125, no. 2, pp 215-223, June 2003. -  S. Yi, P.W. Nelson and A.G. Ulsoy, "Delay differential equations via the matrix Lambert W function and bifurcation

analysis: Application to machine tool chatter," Math. Biosci. Eng., vol. 4, pp. 355-368, April 2007. -  S. Yi, P.W. Nelson and A.G. Ulsoy, "Survey on analysis of time delayed systems via the Lambert W function,"

Dynamics of Continuous, Discrete and Impulsive Systems (Series A), vol.14 (S2), pp. 296-301, 2007. -  S. Yi, P.W. Nelson and A.G. Ulsoy, “Controllability and observability of systems of linear delay differential equations

via the matrix Lambert W function," IEEE Trans. Auto. Cont, April 2008. -  S. Yi, P.W. Nelson and A.G. Ulsoy, ”Eigenvalue and Sensitivity Analysis for a Model of HIV-1 Pathogenesis with an

Intracellular delay," 2008 DSCC. -  S. Yi, P.W. Nelson and A.G. Ulsoy, " Eigenvalue assignment via the Lambert W function for control of time delayed

systems," J. Vib. Control, 2009. (in press, invited) -  S. Yi, P.W. Nelson and A.G. Ulsoy, "Robust Control and Time-Domain Specifications for Systems for Delay

Differential Equations via Eigenvalue Assignment," 2008 ACC and 2009 JDSMC (accepted) -  S. Yi, A.G. Ulsoy, and P.W. Nelson, ““Design of Observer-Based Feedback Control for Time-Delay Systems with

Application to Automotive Powertrain Control,” 2009 DSCC and Journal of the Franklin Institute, 2009 (in press, invited).

-  Time Delay Systems: Analysis and Control Using the Lambert W Function, S. Yi, P.W. Nelson, A.G. Ulsoy, Imperial College Press, London, 2010 (submitted).


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Outline •  Introduction

-  Background on time delay systems -  LTI system of DDEs with single delay, h -  Motivational example – capacity management

•  Solution of DDEs via the Lambert W Function -  Lambert W function and scalar DDEs -  Systems of DDEs and the matrix Lambert W function

•  Stability and Rightmost Eigenvalues -  Conjecture: principal branch determines stability -  Analogy with ODEs

•  Observability, Controllability and Eigenvalue Assignment

-  Controllability and observability Gramians -  Controller design via eigenvalue assignment

•  Concluding Remarks


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Introduction •  Delays are inherent in many physical and

engineering systems, e.g. -  Vibration and noise control, transportation, transmission,

conveyors, tele-operation, etc. •  Control of time-delay systems is challenging:

-  Delay operator leads to infinite spectrum due to

-  Difficulty in stability analysis and controller design •  Delay differential equations (DDEs)

-  18th century: Laplace and Condorcet [Gorecki et al 1989] -  Bellman [1963], Hale and Lunel [1993], etc. -  Approximate, numerical, graphical methods: Padé

approx, Lyapunov, LMIs, Nyquist, etc. [Richard 2003]

Imag(s)

Re(s)


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Approaches for Time-delay Systems  Representative current approaches:

-  Approximation, e.g., Padé approximation of the delay:

-  Bifurcation analysis, e.g., [Olgac et al, 1997]

-  Prediction-based methods, e.g., Smith predictor [Smith, 1958], finite spectrum assignment (FSA) [Manitius and Olbrot, 1979]

-  Numerical solutions, e.g., dde23 in Matlab (Runge-Kutta)

-  Graphical methods, e.g., Nyquist [Desoer and Wu, 1968]

-  Lyapunov methods, e.g., LMI, ARE [Niculescu, 2001]

-  Numerous monographs have been devoted to this field of active research: [Richard, 2003; Yi et al, 2010]


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Background •  We will consider linear time invariant (LTI) dynamic systems with a

single delay, h, represented as a system of delay differential equations (DDEs):

•  This formulation can represent various systems, e.g.: -  Manufacturing Plant Capacity Management [Asl and Ulsoy, 2003] -  Machine Tool Chatter Stability [Yi, Nelson and Ulsoy, 2007] -  HIV Pathogenesis Dynamic Model [Yi, Nelson and Ulsoy, 2008] -  Automotive Powertrain Systems Control [Yi, Nelson and Ulsoy 2009],

Hepatitis Viral Dynamics, Teleoperation, etc.

t

x(t)

-h 0

g(t)

x0

h 2h 3h 4h


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Motivational Example:

Capacity Management •  Manufacturing plant capacity management [Asl & Ulsoy, CIRP

Annals, 2003]

– via stochastic dynamic programming 1.  Stochastic model of market demand 2.  Cost model for optimal capacity

Example of optimal solution Example of effect of delay


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Outline •  Introduction •  Solution of DDEs via the Lambert W

Function -  Lambert W function and scalar DDEs -  Systems of DDEs and the matrix Lambert W

function •  Stability and Rightmost Eigenvalues

-  Conjecture: principal branch determines stability -  Analogy with ODEs





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The Lambert W function •  Definition:

•  Infinite number of branches

•  Lambert [1758], Euler [1779], and recently Corless [1996] •  Already embedded in Maple, Mathematica, Matlab, e.g.,

(k = 0)

( 0)k ≠ “Principal branch”

�

Wk (x)eWk (x ) = x

�

w = lambertw(k,x)


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Solution of DDEs via the Lambert W Function Method [Asl & Ulsoy, JDSMC, 2003]

Roots in terms of the parameters, a, ad, h

•  Obtain a transcendental characteristic equation:

•  Consider the scalar (first-order) delay differential equation (DDE)

Using the definition of the Lambert W function


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Example: Free Scalar DDEs

[ )0

with preshape function ( ) 1, for -1,01, for 0

g t tx t

= ∈= =

( ) ks t Ik

kx t e C

∞

=−∞

= ∑

[Asl & Ulsoy, ASME JDSMC, 2003]


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Example: Forced Scalar DDEs

Forced Soln. Form [Bellman et al. 1963]

Free solution [Asl & Ulsoy, JDSMC 2003]

*[Yi et. al., 2006]


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Systems of DDEs [Yi, Ulsoy & Nelson, 2006]

Substitution into the above equation yields

As in scalar case, postmultiply by heSheAh

Asl and Ulsoy (2003) was correct only for free response of scalar DDE and special case of systems of DDEs when A and Ad commute.

This was extended to free & forced solution of general systems of DDEs in [Yi, Ulsoy and Nelson 2006, 2007] by assuming a solution form:

where S is an nxn matrix.

( ) 0he −+ + =SdS A A

( ) h h hh e e he+ = −S A AdS A A

�

x(t) = eStC

�

SeStC +AeStC +AdsS(t−h )C = 0

�

(S +A +AdeS(−h ))eStC = 0


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Systems of DDEʼs (cont.)

Thus, we introduce an unknown matrix Q that satisfies

Substitute (2) into (1) to obtain the condition for the unknown matrix Q

However, unless S and A commute (or equivalently A & Ad commute) ( )( ) ( )h h hh e e h e ++ ≠ +S A S AS A S A

( )( ) hh e h++ = −S AdS A A Q 1 ( )h

h= − −dS W A Q A

( )( ) h hh e h− −− = −dW A Q Ad dW A Q A


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Matrix Lambert W Function •  Define the argument as Hk = AdhQk, •  Use the Jordan Canonical Form Jk to get Hk = ZkJkZk

−1. •  Then, the matrix Lambert W function is defined as:

•  Finally,


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Example: Chatter in Turning


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Example: Chatter Equation [Yi et al, MBE, 2007]

linearization


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Outline •  Introduction •  Solution of DDEs via the Lambert W

Function •  Stability and Rightmost Eigenvalues

-  Conjecture: principal branch determines stability -  Analogy with ODEs





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Rightmost Eigenvalues from k=0 branch

Unstable Stable

DDEs have an infinite eigenspectrum: Stability is determined by the rightmost eigenvalues; our conjecture:

Has been proven only for scalar case, and for systems of DDEs when A and Ad commute [Shinozaki and Mori, 2006]


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Chatter in Machining •  Linearized Chatter Equation for Turning

Spindle Speed Increase Productivity!

Incr

ease

Pro

duct

ivity

!

[Yi et al, MBE, 2007]


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HIV Pathogenesis Dynamics

Infected White Blood Cell

Infectious Virus

Non-infectious Virus

death rate

clearance rate

[Yi et al, DSCC, 2008]


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Analogy to Systems of ODEs •  The state transition

matrix: 1.  ODEs: the matrix

exponential function 2.  DDEs: infinite series

using matrix exponentials of the matrix Lambert W function

•  Can be utilized to develop free and forced solutions in the time and Laplace domains

•  Rightmost eigenvalues, and stability, determined by principal (k=0) branch

•  Similar to ODEs, one can derive observability and controllability Gramians for systems of DDEs, and assign eigenvalues


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Outline

•  Introduction •  Solution of DDEs via the Lambert W

Function •  Stability and Rightmost Eigenvalues •  Observability, Controllability and

Eigenvalue Assignment -  Controllability and observability Gramians -  Controller design via eigenvalue assignment



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Controllability & Observability Conditions for controllability & observability of DDEs [Yi et al., IEEE TAC, 2008]

•  Uses our solution form, and previous results by [Weiss, 1967] •  Gramians: balanced realization and sensitivity w.r.t. parameters

full rank:

linearly independent rows:

full rank:

linearly independent columns:


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Example: Controllability Gramian Point-wise controllability:

Control u(t) such that x(t1)=0

Approximating Controllability Gramian:

[Yi et al., IEEE TAC, 2008]


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Controller/Observer Design via Eigenvalue Assignment

•  Select desired eigenvalues

•  “Linear” feedback controller

•  With the new coefficients,

•  Equate the selected eigenvalues to those of the matrix S0 as (i.e., k = 0 branch only)

( ) ( ) ( ) ( )t t t h t= + − +dx Ax A x Bu unknown unknown

( ) ( ) ( )t t t h= + −du K x K x

ˆ ˆ

( ) ( ) ( ) ( ) ( )t t t h= + + + −d

d dA A

x A BK x A BK x

, , for 1, ,i desired i nλ = …


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Diesel Engine Example

observer

feedback

[Jankovic and Kolmanovsky, 2009]


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Diesel Engine: Feedback Control •  Diesel Engine: linearized system w/ transport time delay at N=1500 RPM

•  This system has an unstable eigenvalue at 0.9225 (> 0) without feedback •  With feedback control: •  Desired location: -10

*eigenvalues by k=0


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Diesel Engine: State Observer - Observer (pole at -15):

Unstable (one positive real

eigenvalue)

Goal: Find stabilizing “L”

eigenvalues for k=0


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Response of Controlled System   Feedback Control with Observer:

*[Yi et al., JFI 2010]

↑ was unstable


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Lambert W Function Approach Given system of LTI delay differential equations with single delay h

1. Derive the solution (free and forced): [DCDIS 2007]

3. Check conditions for controllability and observability: [IEEE TAC 2008]

4. Design feedback controller & Observer via eigenvalue assignment: [JVC 2008 & DSCC 2009 & JFI]

* For systems of ODEs, the above approach is standard. * Using the Lambert W function-based approach we have extended this to DDEs.

2. Determine stability of the system: [MBE 2007]

5. Robust stability and time-domain specifications: [ACC 2008 & JDSMC]


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Concluding Remarks •  Matrix Lambert W Function Method

-  Free and forced solution to systems of DDEs 1. Analytical expression in terms of the parameters of a system 2. Each eigenvalue is distinguished by the branches of W

-  Stability depends on the rightmost eigenvalues (k=0 branch) -  Observability, controllability and eigenvalue assignment -  Analogous to methods for ODEs

•  Open Research Questions -  General proof that stability is determined by principal branch, k = 0 -  Existence/uniqueness conditions for Qk -  Connection between controllability and eigenvalue assignment

•  Our Current and Future Research -  HIV/HBV dynamics for optimal therapy -  Automotive powertrain control (e.g., A/F ratio, ISC, etc.) -  Extensions to multiple delays; to nonlinear or time-varying systems;

to PWAS, etc.

Documents

Purdue Oldenburger Lecture 10-09 - University of Michiganulsoy/pdf/Oldenburger_Lecture.pdf · Rufus Oldenburger Lecture ... Application to Automotive Powertrain Control,” 2009 DSCC