Page 18 - Industrial Plants 2014

Space state techology
For a long time the control literature has described
modern control algorithms based on a flexible type
of multivariable model. The model was based on
linear differential equations that mapped the
relationships between process inputs and process
outputs through use of intermediate variables, called
the state vector. This type of model was called a
state space model. MPC algorithms came along
Fig. 4 - Standard MPC the 1980’s when several independent sources all after state space models were introduced, but did
structure began converging on a basic architecture. The key not use this type of model.
to this architecture is the use of an internal linear State space models became linked to optimal
dynamic model in the controller calculation. The control theory for aerospace applications and did
algorithm computes an estimate of process not include many of the practical control objectives
disturbances acting on the process variables being that were part of the design basis of MPC. The
controlled. The disturbance estimate, the process result was that state space models were ignored for
variable setpoints and feedforward signal levels a long time by the process industries, but recent
become inputs to the controller calculation. With enhancements in new algorithms have changed
these inputs and the process model, the controller that.
is able to calculate the required values for the The equations that represent a discrete-time state
independent, manipulated variables. This structure space model are presented in the equation:
is illustrated in figure 4.
In the controller error minimization calculation, the x(k + 1) = Ax(k) + Bu[u(k) + w(k)] + Bt d(k)
model helps predict future values of z(k) = Cx(k)
the process variables. This led to the y(k) = Cx(k) + v(k)
MPC algorithms have names Internal Model Control (IMC)
become the dominant and Model Predictive Control (MPC). where:
method for dealing The MPC algorithms became the first • x is the state vector
with interactive process large-scale deployment of computer • u is the process input or control effort vector
control problem and based multivariable process • d is a vector of measured disturbance variables,
have proven to be very controllers (also called Advanced also known as feedforwards
fexible in expanding to Process Control or APC). To make • w, v are noise vectors
large systems and in the calculations efficient and • z is the vector of process variables
handling complicated convenient, the algorithms use • y is the vector of process variables with
constraint scenarios discrete impulse response models. measurement noise

These models can predict the values • Ax, Bu, Bd and C are process matrices
of future process outputs through the • k is the time in number of sampling intervals
discrete convolution equation. The equation is fairly
simple to program and lends itself to incorporation In this case, the MPC controller (figure 5) uses an
in the optimization algorithms needed to calculate explicit estimation of state vector X to compute the
the values of future manipulated variables, while future moves on manipulated variables.
minimizing process variable deviations.
MPC algorithms have become the dominant method
for dealing with interactive process control problem
and have proven to be very flexible in expanding to
large systems and in handling complicated
constraint scenarios. An enhancement that first
appeared in the early 1990’s posed the controller
optimization problem as a multi-objective
optimization, where each stage of the optimization
problem added a new constraint while adhering to
the optimal solution for previously solved higher
ranked constraints. This innovation made tuning the
controllers with varying sets of active constraints
much easier. Fig. 5 - State Space MPC

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