Education:
Ph.D., | Texas A&M University, Texas | (2006) |
M.S., | Tsinghua University, Beijing, China | (2002) |
B.S., | Tsinghua University, Beijing, China | (1999) |
| |
Research Interests
Analysis and Reduction of Nonlinear Systems via Covariance
Matrices
The objective of process control is to
maintain a desired level of product quality and safety while making the process
more economical. Because these goals apply to a variety of industries, process
control systems are used in facilities for the production of chemicals, pulp
and paper, metals, food, pharmaceuticals, and microelectronics. While the
methods of production vary from industry to industry, the principles of process
control are generic in nature and can be universally applied, regardless of the
size of the plant.
Process controllers that are based explicitly on dynamic mathematical models (i.e.,
sets of differential equations) have become increasingly popular in the
chemical process industry. This is due to several reasons, the most important
being that highly accurate models can now be solved with modern dynamic
simulators and powerful optimization algorithms. However, computational
requirements grow with the complexity of the models. Many rigorous dynamic
models require too much computation time to be useful for real-time model based
controllers. Real-time control is important, however, because the action of the
controller must be computed in a time span that is faster than the dominant
process time constant. Because control calculations for techniques such as
linear model predictive control scale according to n^{3} (n = number
of state variables), model reduction is a promising avenue for real-time
implementation.
We have developed a novel strategy to nonlinear model reduction that includes
system analysis, nonlinearity
quantification, and model reduction. The advantages of this new approach are
that the procedure is applicable to any kind of nonlinear system
and that several system properties can be determined within the same framework
resulting in an efficient strategy for the analysis of nonlinear systems. In
the first step, covariance matrices, which are an extension of linear
gramians to any type of system, are computed. These
covariance matrices are then balanced by a novel algorithm that can be applied
to nonlinear systems, with no assumption about rank deficiency of the computed
covariance matrices being required. Two new types of nonlinearity measures were
introduced that are based upon the balanced covariance matrices. If the
measures indicate that the model exhibits only a mild degree of nonlinearity
then it is appropriate to work with the linearization of the model, which
greatly simplifies on-line application of the model. However, if the model is
found to be strongly nonlinear over the required operating region, the
information contained in the covariance matrices can be further used for
reduction of the nonlinear model. Two new model reduction procedures for
nonlinear systems were introduced in our research and are given by equations
(2) and (3), where the
T and
P_{1} refer to the
transformation and projection matrices, respectively.
The main task of the model reduction procedure is to
determine the invertible state transformation
T such that the states are ordered according to their importance for
the input-output behavior of the system. The approaches are based upon a strong
theoretical foundation, in that they reduce to well known facts from linear
systems theory when applied to linear systems, while they allow for the
flexibility required for nonlinear systems.
These covariance matrix based methods
have been applied in simulation to a series of reactors, a distillation column,
and a polymerization reactor. The procedures reduced the number of differential
equations in the model as well as the computation time required for the
solution of these models. We have also demonstrated that several model
reduction procedures, which have been developed over the last 20 years and were
assumed to be independent from one another, form special cases of the procedure
developed in my dissertation.
While these procedures were developed with the control
of nonlinear systems in mind, they are applicable to any nonlinear system of
the form of equation (1) regardless of its use and can elucidate several of the
properties exhibited by nonlinear systems.
Current research focuses on extending this framework
to systems described by partial differential equations (PDEs)
as well as differential algebraic equations (DAEs).