Page 59 - Distributed model predictive control for plant-wide systems

Model Predictive Control 33

2.4 Dual Mode Predictive Control
From the above, it is obvious that, compared with other control technologies, MPC has
powerful abilities to deal with the constraints. However, the early version of MPC for process
control did not address stability theoretically so that they are not automatically stabilizing.
A common method to obtain the stability of the early version of MPC is to increase the
length of the control horizon, which indicates the stability property of an infinite horizon
predictive control scheme. Until the 1990s, the stability theory of MPC was established [69],
and the theoretical framework of that guarantees feasibility and stability, and is now
well understood.
A method to obtain a stable MPC was proposed in [71]. The proposed strategy, which is
denominated as dual mode control, is based on the inclusion of an inequality constraint that
forces the terminal state to lie in a terminal positive-invariant region Ω at the end of the pre-
diction horizon. When the state is outside the terminal region, the control action is provided
by the conventional MPC with the above constraint. Once the state is in Ω, the controller is
switched to a linear previously defined controller [71]. In this section, we will introduce the
dual mode predictive control.

2.4.1 Invariant Region

Consider the following linear time-invariant system:
{
x(k + 1)= Ax(k)+ Bu(k)
(2.35)
y(k)= Cx(k)
where x ∈ ℝ , u ∈ U ⊂ ℝ , and y ∈ ℝ n y are, respectively, the state, input, and output vec-
n x
n u
tors, and U is the feasible set of the input u, which is used to bound the input according to
the physical constraints on the actuators, the control requirements, or the characteristics of the
plant.

Assumption 2.1 For system (2.35), there exists a state feedback u(k) = Kx(k) such that the
closed-loop system x(k + 1) = A x(k) is asymptotically stable, where A = A + BK.
c
c
This assumption presumes that each subsystem is able to be stabilized by a feedback
control Kx. The control gain K can be obtained by the linear matrix inequality (LMI) or
linear-quadratic regulator (LQR) technology.

Lemma 2.1 Under Assumption 2.1, for any positive scalar c and P ≻ 0, the set
Ω(c)={x ∈ ℝ n x ∶ ‖x‖ ≤ c}
P
is a positive invariant region of attraction for the closed-loop system x(k + 1) = A x(k). Addi-
c
tionally, there exists a small enough positive scalar such that Ω( ) is in the feasible input set
U ∈ ℝ n u for all x ∈Ω( ).