Reacting flows and control theory Harvey Lam Princeton University Numerical Combustion 08 Monterey, 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):

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-specified control objectives on Y:  Y m =C m (X;t). Thus, we want:

Dynamics of the sensor measurements  Since Y m =C m (X;t), we have: where has clear physical meanings.

Exact control law…  The exact actual ODE for Y:  The desired ODE for Y: Equating dY/dt, we obtain the exact control law:

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).

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?

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…

Some details …  The u dependence of R(u,Y):  Condition on K: to make J negative definite!

Universal Dynamic Control Law  System to be controlled (integrated by nature):  Desired dynamics of Y m =C m (X):  The UDCL (integrated by the black box): No knowledge of A(X;t) is needed! (Need to pick K)

How to pick K  The actual Y dynamics:  Thus D m k is the  Y m response to a unit pulse of u k. (easy to determine) … D must not be singular!  … K being the inverse of -D would work (sufficient but not necessary).

Summary… for N=M=1 case  Dynamics of system to be controlled:  Desired Y dynamics:  The real time UDCL (for any A(X;t)):

Numerical example: joy-stick control!  Desired Y(t) dynamics: The red line is any Y target (t) joy-stick trajectory.  The black line is the UDCL controlled trajectory for any A(X;t).

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. 