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Page 382 Zhang et al. Intell Robot 2022;2(4):371­90 I http://dx.doi.org/10.20517/ir.2022.26

Therefore, the controller design can be summarized as the following:

min
(41)
>0, >0,
s. t. (26), (34), (38), (40),and 0 < < 1.

The T-S fuzzy state feedback controller is derived by solving the optimization problem in Equation (40) under
parameter uncertainties and external persistent perturbations. The inequality Equation (34) ensures that the
cost function J ∞ ( ) is upper-bounded by the Lyapunov function ( ( )), the inequality Equation (38) guar-

antees that the input constraints are satisfied, and the inequality Equation (40) leads to ( ) ∈ Ω , where Ω
is an RPI set.

Remark3.1: Notethat isminimizedinsteadof intheabovementionedoptimizationproblem. Thisapproach
is used because both and need to be minimized, and a smaller implies a higher system performance. We
adopt the approach given in Ref. [23] to simultaneously optimize and by defining = in Equation (34).
The introduction of the variable makes the constraint Equation (40) a bilinear matrix inequality, which can
be handled with existing solvers, e.g., PENBMI. The computational load is reduced further by predefining a
suboptimal value of by trial and error.

Theorem3.1: TheoptimizationproblemEquation(40)hasthepropertyofrecursivefeasibility,thatis,asolution
will always exist once the problem is initially solvable.

Proof: Implementationofthepredictivecontrolstrategy basedon theT-Sfuzzymodel requiresthe constrained
optimization problem Equation (40) to be solved at each time instant. Therefore, it is important to guarantee
the recursive feasibility of the optimization problem. As an external disturbance is considered, the recursive
feasibility is no longer a natural characteristic of the proposed controller. In this study, only the constraint
Equation (26) depends on the time instant , which involves ( ). Therefore, we only need to ensure the
feasibility of the constraint Equation (26).

Note that the inequality Equation (26) is equal to ( ) ∈ Ω . The inequality Equation (40) ensures that the
set Ω is an RPI set, which implies that the inequality Equation (26) is still feasible at the ( + 1) time instant.
That is, ( + 1) ∈ Ω +1 ∈ Ω is still satisfied. Hence, recursive feasibility is guaranteed. Thus, the proof is
completed. □

Theorem 3.2: The closed-loop system in this paper has ISS based on the proposed MPC strategy under an
external disturbance.

Proof: It has been proven that the optimization problem Equation (41), once solvable, will always be solvable.

∑ 2
∗
∗
∗
∗
∗
Definetheoptimalsolutionattimeinstant as { , , , , , },and ( ( )) = =1 ( ) ( ) ( ),
∗
∗
∗

1
2
¯
where = ∗−1 . We need to prove that the ( ( )) is an ISS-Lyapunov function. Define as the upper
∗
∗
∗

bound of eigenvalue of , and as the lower bound of the eigenvalue of . We can obtain:
∗
∗

2
¯
2
∗
k ( )k ≤ ( ( )) ≤ k ( )k . (42)
Furthermore, from inequality Equation (24), we can derive the following inequality:

∗

( ( + 1)) − ( ( )) ≤ − ( ) ( ) − ( ) ( ) + ( ) ( ), (43)
∗
∗
∗
∑ 2 −1
∗
where = ( ) . Therefore, we get
=1

( ( + 1)) − ( ( )) ≤ − ( ) ( ) + ( ) ( ). (44)
∗
∗