Page 46 - Distributed model predictive control for plant-wide systems

20 Distributed Model Predictive Control for Plant-Wide Systems

• It allows operation closer to constraints (compared with conventional control), which fre-
quently leads to more profitable operation.
• Control update rates are relatively low in these applications, so there is plenty of time for
the necessary online computations.

As pointed out in [69], early versions of MPC and generalized predictive control did not
automatically ensure stability, thus requiring tuning. This is the reason why that research in
the 1990s devoted considerable attention to this topic. Research on the stability of model pre-
dictive controlled systems has now reached a relatively mature stage. The important factors
for stability have been isolated and employed to develop a range of model predictive con-
trollers that are stabilizing and differ only in their choice of the three ingredients (terminal
cost, terminal constraint set, and terminal local controller) that are common to most forms
of MPC [69]. These conditions are merely sufficient and several researchers [70] are seeking
relaxation. Among the stabilized MPC design methods, a dual mode MPC was proposed in
[71] and developed in [72, 73]. In the dual mode version, a terminal set is chosen. It solves an
open loop optimal control problem solved online until all states enter the terminal set. When
all the states are in their terminal set, a local feedback control law is employed to steer the
states to the origin.
In this chapter, the dynamic matrix control, state space model-based predictive control, and
the dual mode predictive control are introduced, since these will be used as fundamentals in
the following chapters.

2.2 Dynamic Matrix Control
DMC is an algorithm based on the step response model. It applies incremental algorithms,
which are very effective in removing the steady-state error. Up to now, DMC is the most widely
accepted in the process industry. This section mainly refers to [2, 74].

2.2.1 Step Response Model
Suppose the system is at rest. For a linear time-invariant single-input single-output (SISO)
system, let the output change for a unit input change Δu be given by
{0, s , s , … , s , s N+1 , …}
2
N
1
Here, we suppose that the system settles exactly after N steps. The step response
{s , s , … , s } constitutes a complete model of the system, which allows us to compute the
1 2 N
system output for any input sequence,
N
∑
y(k)= s Δu(k − l)+ s u(k − N − 1) (2.1)
l N+1
l=1
where Δu(k − l) = u(k − l) − u(k − l − 1) Note that when s = s , (2.1) is equivalent to
. N N − 1
N
∑
y(k)= s Δu(k − l)+ s u(k − N) (2.2)
l
N
l=1
Step response model (2.1) can only be used in stable processes. For a multiple-inputs and
multiple-outputs (MIMO) process with m inputs and r outputs, one obtains a series of step