Reacting flows and control theory

Harvey LamPrinceton University

http://www.princeton.edu/~lam

Numerical Combustion 08Monterey, CA

Model reduction for reacting flows

Start with an initial value problem of N nonlinear ODEs.

Goal is to find a “slow manifold” which provides M algebraic relations between N unknowns after the transients die.

Mathematical tools: QSSA (quasi-steady state approximation) and PE (partial equilibrium). Time scale separation!

Control Theory

Start with a dynamical system with N state variables governed by N nonlinear ODEs which contain M unknown control forces.

Real time sensor measurements are available. It is desired that the sensor measurements

honor the M given (user-specified) control objectives after some initial transient.

Goal: find those M control forces (using feedback) to honor the M control objectives!

Control theory mathematics

System to be controlled:

where u is unknown and to be determined.* Sensor measurements Y=C(X;t) are available!* Want Y(t) to honor M user-specfied control

objectives (after the transients die):€

dX

dt= A(X;t) + Bu

€

f m (Y,dY

dt) ≈ 0, m =1, .., M.

The control problem

The desired result is u(Y;t)--- the control force as some function of the current and past values of the sensor measurements Y(t).

The conventional wisdom is that one can only control the system if a good model A(X;t) of the system is known.

Question: can the system be controlled if we don’t know A(X;t)?

Generic control objectives

Consider the generic user-specifiedcontrol objectives on Y:

Ym=Cm(X;t). Thus, we want:€

dY m

dt= E m (Y;t), m =1,...,M.

€

f m (Y,dY

dt) =

dY m

dt − E m (Y;t)

=∂Cm

∂t+

∂Cm

∂X

dX

dt

⎛

⎝ ⎜

⎞

⎠ ⎟− E m (Y;t) ≈ 0, m =1,..., M .

Dynamics of the sensor measurements

Since Ym=Cm(X;t), we have:

where

has clear physical meanings.

€

dY m

dt=

∂Cm

∂t+

∂Cm

∂X

dX

dt

=∂Cm

∂t+

∂Cm

∂XA(X;t) + Bnun

n=1

M

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟,

=∂Cm

∂t+

∂Cm

∂XA(X;t) + Dn

m

n=1

M

∑ un , m =1,..., M .

€

Dnm ≡

∂Cm

∂XBn, m,n =1,..., M .

Exact control law…

The exact actual ODE for Y:

The desired ODE for Y:

Equating dY/dt, we obtain the exact control law:

€

dY m

dt=

∂Cm

∂t+

∂Cm

∂XA(X;t) + Dk

m

k=1

M

∑ uk, m =1,...,M .

€

dY m

dt= E m (Y;t), m =1,...,M.

€

uexactm ≡ [Dm

k ]−1

k=1

M

∑ E k (Y;t) −∂C k

∂t−

∂C k

∂XA(X;t)

⎧ ⎨ ⎩

⎫ ⎬ ⎭

, m =1,..., M .

Conventional wisdom:knowledge of A(X;t) is needed for control!

The exact control law is a “manifold” in [u,X] space:

Look! Knowledge of A(X;t) is needed! Is it possible to control the system without detailed knowledge of the

A(X;t) of the system? It is assumed that the “time scale” of the actual system is O(1).

We assume the control system is microprocessor-based (with CPU clock speed of xx giga-hertzs).

€

uexactm ≡ [Dm

k ]−1

k=1

M

∑ E k (Y;t) −∂C k

∂t−

∂C k

∂XA(X;t)

⎧ ⎨ ⎩

⎫ ⎬ ⎭

, m =1..., M.

The reacting flows idea…

Imagine u to be chemical radicals which are involved in some fast reactions …

The f k(Y,dY/dt)’s are given and known control objectives…

• Apply QSSA to these radicals in the small limit…

• Question: what should K be? €

dum

dt=

1

εKk

m

k

M

∑ f k Y,dY

dt

⎛

⎝ ⎜

⎞

⎠ ⎟=

R(u,Y)

ε, m =1,...,M.

How to make QSSA legitimate…

We need the Jacobian of R(u,Y) with respect to u to be negative definite.

is at our disposal. We can make the “chemical reaction rate” sufficiently fast by using very small values…

€

dum

dt=

1

εKk

m

k

M

∑ dY k

dt− E k (Y;t)

⎛

⎝ ⎜

⎞

⎠ ⎟=

Rm (u,Y)

ε, m =1,..., M .

€

Rm (u,Y) ≡ KkmDn

k

k=1

M

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

n=1

M

∑ un − uexactn (Y;t)( ), m =1,...,M .

Some details … The u dependence of R(u,Y):

Condition on K: to make J negative definite!

€

f k (Y,dY

dt) =

dY k

dt− E k (Y;t),

=∂C k

∂t+

∂C k

∂X

dX

dt

⎛

⎝ ⎜

⎞

⎠ ⎟− E k (Y;t),

=∂C k

∂t+

∂C k

∂XA(X;t) + Bnun

n=1

M

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟− E k (Y;t), k =1,...,M;

Rm (u,Y) = Kkm

k=1

M

∑ ∂C k

∂t+

∂C k

∂XA(X;t) − E k (Y;t) + Dn

kun

n=1

M

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟, m =1,...,M .

€

Jnm ≡ Kk

mDnk

k=1

M

∑ = Kkm ∂C k

∂Xk=1

M

∑ Bn, m,n =1,...,M .

Universal Dynamic Control Law

System to be controlled (integrated by nature):

Desired dynamics of Ym=Cm(X):

The UDCL (integrated by the black box):

No knowledge of A(X;t) is needed!(Need to pick K)

€

dX n

dt= An (X;t) + Bk

n

k=1

M

∑ uk, n =1,...,N.

€

dY m

dt= E m (Y;t), m =1,..., M.

€

dum

dt=

1

εKk

m

k=1

M

∑ dY k

dt− E k (Y;t)

⎛

⎝ ⎜

⎞

⎠ ⎟, m =1,..., M .

How to pick K The actual Y dynamics:

Thus Dm

k is the Ym response to a unit pulse of uk. (easy to determine)… D must not be singular!

… K being the inverse of -D would work (sufficient but not necessary).

€

dY m

dt=

∂Cm

∂t+

∂Cm

∂XA(X;t) + Bku

k

k=1

M

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟,

=∂Cm

∂t+

∂Cm

∂XA(X;t) + Dk

m

k=1

M

∑ uk, m =1..., M.

Summary… for N=M=1 case

Dynamics of system to be controlled:

Desired Y dynamics:

The real time UDCL (for any A(X;t)):

€

dX

dt= A(X;t) + Bu; Y = CX .

€

dY

dt= E(Y;t).

€

du

dt=

K

εD−1 dY

dt− E(Y;t)

⎧ ⎨ ⎩

⎫ ⎬ ⎭

, D = CB.

Numerical example:joy-stick control!

Desired Y(t) dynamics:

The red line is any Ytarget(t) joy-stick trajectory.

The black line is the UDCL controlled trajectory for any A(X;t).

0

10

20

30

40

50

60

70

80

90

1stQtr

2ndQtr

3rdQtr

4thQtr

EastWestNorth

QuickTime™ and a decompressor

are needed to see this picture.

€

dY

dt=

1

τYtarget (t) −Y( )

Time scale separation? The physical system’s time scale is O(1). The controller black box’s

hardware/software turn-around time is O().

The UDCL exploits <<1. What happens to those components

of X not involved in the M control objectives? (cross our fingers and pray!)

Concluding remarks Linearity offers no advantage…

A(X;t) can include unknown disturbances… It is highly preferred that sensor

measurements of both Y(t) and dY(t)/dt are available. Numerical differentiation of Y(t) is not recommended.

Controllability, observability, and “relative degree” are relevant concepts.

CSP can be helpful to two-point boundary value problems encountered in optimal controls.

http://www.princeton.edu/~lam