Page 61 - Distributed model predictive control for plant-wide systems

Model Predictive Control 35

control for stabilization. Such switching from an MPC law to a terminal controller, once
the state reaches a suitable neighborhood of the origin, is referred to as the dual mode
MPC [71, 78].
In what follows, we formulate the optimization problem for the dual mode MPC.

Problem 2.3 Consider system (2.35). Let > 0 be as specified in Lemma 2.1. Let the update
time be k ≥ 1. Given x(k), find the control sequence u(k + l − 1|k)∶{1, 2, … , N} → U that
minimizes
N−1
∑ 2 2
2
J(k)= ‖x(k + N|k)‖ + (‖x(k + l|k)‖ + ‖u(k + l|k)‖ ) (2.41)
P Q R
l=0
subject to the following constraints:
u(k + l − 1|k)∈ U, l = 1, … , N (2.42)
x(k + N|k)∈Ω( ) (2.43)

2.4.3 Algorithms
Before stating the dual mode MPC algorithm, we make the following assumption to facilitate
the initialization phase.
Assumption 2.2 At initial time k , there exists a feasible control u(k + l − 1|k )∈ U,
0
0
0
l = 1, 2, … , N, for system (2.35), such that the solution to the system x(k + l|k ) = Ax(k +
0
0
0
l − 1|k ) + Bu(k + l − 1|k ), l = 1, 2, … , N, satisfies x(N + k ) ∈Ω ( ) and results in a bounded
0 0 0 0 i
cost J(k ).
0
Assumption 2.2 bypasses the difficult task of actually constructing an initially feasible solu-
tion. In fact, finding an initially feasible solution for many optimization problems is often a
primary obstacle, whether or not such problems are used in a control setting. As such, many
centralized implementations of MPC also assume that an initially feasible solution is available
[50, 51].
Algorithm 2.3 Dual Mode MPC Algorithm
The dual mode MPC law at every time instant k is constructed as follows:

Step 1 If x(k) ∈Ω( ), then apply the terminal controller u(k) = Kx(k); else go to Step 2.
Step 2 Solve Problem 2.3 for u(k|k), and apply u(k|k);
Step 3 Let k + 1 → k; repeat Step 1.
Algorithm 2.3 gives a method to solve the dual mode predictive control. In the next
subsection, it will be shown that the dual mode predictive control policy drives the state
x(k + l)to Ω( ) in a finite number of updates. As a result, if Ω( ) is chosen sufficiently small,
then MPC can be employed for all time without switching to a terminal controller. Of course,
in this case, instead of asymptotic stability at the origin, we can only drive the state toward
the small set Ω( ).
The analysis in the next subsection shows that the dual mode predictive control algorithm
is feasible at every update and is stabilizing.