Page 105 - Distributed model predictive control for plant-wide systems

Local Cost Optimization-based Distributed Model Predictive Control 79

5.2.4 Stability Analysis

In this section, the stability condition of the entire closed-loop system is deduced by analyzing
the entire closed-loop system’s dynamic matrix which could be specified on the basis of the
closed-loop solution stated by Theorem 5.1.
For a simple reason, we directly give the stability condition, Theorem 5.2. If the readers
want to study the detail of the deducing procedure, they can refer to Appendix C at the end of
this chapter or [46].

Theorem 5.2 (Distributed stability). The closed-loop system given by the system S with feed-
back distributed control laws given by (5.30) is asymptotically stable, if and only if,

| ⎧ A B ⎫ |
| ⎡ 0 ⎤ |
| ⎪ ⎪|
̃
| ⎢ S A SA S B ̃ ⎥ |
| ⎪ ( ) SB ⎪ |
| ⎢ ⎥ ⎬ | < 1
j ⎨ ̃ ̃
| ⎢ A + S A SA ( + B + S B) SB⎥ |
| ⎪ ⎪|
| ⎢ I ⎥ |
⎪
| ⎪ ⎣ Mn u 0 ⎦ |
| ⎩ ⎭|
∀j = 1, … , m N (5.40)
where m = Pn + n + 2Mn of the global closed-loop system.
x
u
x
N
Remark 5.3 It can be seen from Equation (5.40) that the first two block rows of the dynamic
matrix A depend on the system matrix A (in the first two block columns) and the system matrix
N
B (in the last two block columns), the third block row depends on process matrices A, B, and C,
weight matrices Q and R , and horizons P and M. Therefore, the stability of unconstrained
i
i
LCO-DMPC introduced in this chapter could be designed and tuned by adjusting the weight
matrices Q , R and horizons P and M which introduce significant modifications on the third
i
i
block row of matrix A .
N
5.2.5 Simulation Results
In this section, the following two input and two output nonminimum phase plant S is taken
as a control objective to investigate the introduced unconstrained LCO-DMPC; the transfer
function of system S is
⎡ −0.024 (z − 1.492) (z + 0.810) 0.018(z + 0.935) ⎤
[ ] [ ]
y (z) ⎢ 2 2 ⎥ u (z)
1 (z − 0.819)(z − 1.922z + 0.961) (z − 1.676z + 0.819) 1
= ⎢ ⎥
y (z) ⎢ 0.126 0.147(z − 0.668) ⎥ u (z)
2 2
⎢ (z − 0.368) 2 ⎥
⎣ (z − 1.572z + 0.670) ⎦
A state-space realization for S has the form
{
x (k + 1) = Ax(k)+ Bu(k)
(5.41)
y(k)= Cx(k)