4. Results

ii. If the closed loop system is asymptotically stable (AS) near the region of the equilibrium x ¼ 0, then the nonlinear process P is said to be locally multi-loop integral controllable

In Figure 2, we assume the state equation of the general process P~ (which includes original process P and the two scalar non-integral controllers c<sup>1</sup> andc2) is modeled as follows (with the

> <sup>P</sup><sup>~</sup> : <sup>x</sup>\_ <sup>¼</sup> f xð Þ ; <sup>u</sup><sup>~</sup> y ¼ g xð ; u~Þ

> > ¼ η

Consider the closed loop system in Figure 1, and assume that the general process P~ and the linear part of the controller Cl are described by Eqs. (2) and (3), respectively. If the following

i. The equation 0 ¼ f xð ; u~Þ obtained by setting x\_ ¼ 0 in Eq. (2) implicitly defines a unique

iii. If two <sup>C</sup><sup>2</sup> functions can be found <sup>f</sup>1ð Þ� and <sup>f</sup>2ð Þ� such that that the steady-state input output function g h<sup>ð</sup> ð Þ <sup>u</sup>~Þ; <sup>u</sup><sup>~</sup> of the general process <sup>P</sup><sup>~</sup> satisfies the following requirements:

.

asymptotically stable (GAS) and locally exponentially stable (LES).

k1 k2

e ¼ �η

k1 k2

, the equilibrium x ¼ hðu~Þ of the system x\_ ¼ f xð ; u~Þ is globally

" # y

" #

(2)

(3)

�

1 ξ\_ 2

" #

The state equation for the linear integral controller is expressed as:

Cl : <sup>ξ</sup>\_ <sup>¼</sup> <sup>ξ</sup>\_

The following theorem presented a sufficient condition for MIC:

8 ><

>:

(Steady-state MIC conditions for nonlinear 2ISO processes).

<sup>C</sup><sup>2</sup> function <sup>x</sup> <sup>¼</sup> <sup>h</sup>ðu~<sup>Þ</sup> for <sup>u</sup><sup>~</sup> <sup>∈</sup> <sup>U</sup><sup>~</sup> <sup>⊂</sup>R<sup>2</sup>

u~ ¼ ξ

around the equilibrium x ¼ 0 [19, 20].

Figure 2. Multi-loop integral controllability for 2ISO system.

280 Adaptive Robust Control Systems

same assumptions for Eq. (1) of process P):

3.1.2. Theorem 1

assumptions are satisfied:

ii. For any fixed u~ ∈ U~ ⊂ R<sup>2</sup>
