Decentralized Fuzzy MPC Framework

• Decentralized MPC is an approach that assigns local control to subsystems while ensuring robust overall stability and low computational cost.
• It employs interval type‐2 fuzzy Takagi–Sugeno models to capture membership uncertainty and dynamic coupling among nonlinear subsystems.
• The framework decomposes global optimization into parallel quadratic programs, significantly reducing online computational effort and enhancing disturbance rejection.

A decentralized Model Predictive Control (MPC) framework allocates control authority to local controllers associated with subsystems of a large‐scale system, while ensuring robust stability, recursive feasibility, low online computational cost, and effective handling of coupling and uncertainty. In the decentralized robust interval type‐2 fuzzy MPC (IT2‐Fuzzy‐MPC) for Takagi–Sugeno (T–S) large‐scale systems (Sarbaz et al., 2021), each nonlinear subsystem is modeled by an interval type‐2 fuzzy rule base capturing membership uncertainty, with affine dynamic couplings incorporated as additive terms, and the global online optimization is decomposed into parallel local quadratic programs (QPs).

1. Interval Type-2 Fuzzy Takagi–Sugeno Large-Scale System Modeling

Each subsystem $S_i$, for $i=1,\ldots,N$, is described by an interval type‐2 (IT2) fuzzy T–S rule base with $r_i$ rules:

• Rule $l=1,\ldots,r_i$:

$$\text{IF } z_{i1} \text{ is } F_{i1} \text{ and } \ldots \text{ and } z_{ig} \text{ is } F_{ig}$$

$$\text{THEN } x_i(k+1) = A_i^l x_i(k) + B_i^l u_i(k) + E_i^l d_i(k) + \sum_{j=1}^N G_{ij}^l x_j(k)$$

• The aggregated update over all rules and interval firing strengths $$w_i^l(z_i(k)) \in [\,\underline w_i^l(z_i(k)),\,\overline w_i^l(z_i(k))\,]$$ becomes:

$$x_i(k+1) = \sum_{l=1}^{r_i} w_i^l(z_i(k)) [A_i^l x_i(k) + B_i^l u_i(k) + E_i^l d_i(k)] + \sum_{l=1}^{r_i} \sum_{j=1}^N w_i^l(z_i(k)) G_{ij}^l x_j(k)$$

• The interval weights are defined with state‐dependent splitting:

$$\underline w_i^l(z_i) = p_i^l(x_i)\,\underline{\mu}_{i^l}(z_i),\quad \overline w_i^l(z_i) = p_i^l(x_i)\,\overline{\mu}_{i^l}(z_i),\quad p_i^l(x_i) + q_i^l(x_i)=1$$

• Footprint of uncertainty (FOU) is encoded in each membership function via $$\,\mu(z_i)=\frac{1}{1+e^{-(x_i-c)}} \pm \Delta(x_i),\,\Delta(x_i)\in[-\delta,\delta]$$.

2. Formulation of the Decentralized Fuzzy-MPC Optimization Problem

Prediction Model:

For a receding horizon $N$, predictions are computed for each subsystem by applying the IT2‐T–S update (with weights from current state measurements and predicted neighbor states): $$x_i(k+m+1|k) = \sum_{l=1}^{r_i} w_i^l(z_i(k)) [A_i^l x_i(k+m|k) + B_i^l u_i(k+m|k)] + \overline{E}_i d_i(k+m|k) + \sum_{l=1}^{r_i} \sum_{j=1}^N w_i^l(z_i(k)) G_{ij}^l x_j(k+m|k)$$

Cost Function and Constraints:

Each local MPC solves a min–max Ho‐type cost: $$J_i(k) = \max_{d_i\in D_i} \min_{u_i\in U_i} \left\{ \sum_{m=0}^{N-1} [x_i^T Q_i x_i + u_i^T R_i u_i - T_i d_i^T d_i] + x_i(N)^T P_i x_i(N) \right\}$$

• Terminal set: $x_i(N) \in \mathcal{X}_i^f$ (robust positively invariant, RPI).
• State, input, disturbance constraints: $x_i\in\mathcal{R}_i$, $u_i\in\mathcal{U}_i$, $d_i\in D_i = \{ d_i: \|d_i\| \leq n_i \}$
• Coupling: Only appears affinely via predicted subsystem states, no hard cross-subsystem inequalities.

Robust Tightening: Disturbance handling is embedded via the negative-definite Ho term and enforced through selection of RPI terminal sets.

3. Decentralized Parallel Implementation Strategy

• The global MPC decomposes into $N$ parallel QPs, one per subsystem, as neighbor coupling terms $G_{ij}^l x_j$ require only immediate state information $x_j(k)$ at the current sample.
• Controllers exchange only current neighbor states, with no multi-step look-ahead or future trajectory communication.
• All local QPs are solved in parallel, leveraging measured or communicated neighbor states.

Pseudocode for Decentralized IT2-Fuzzy MPC:

1. Measure $x_i(k)$ and neighbor states $\{x_j(k)\}_{j \in neigh}$
2. Evaluate firing intervals $h_i^l(z_i(k))$
3. Formulate the local LMI‐QP: minimize $\epsilon_i$ subject to LMIs (17),(18),(23),(26) for gains $\{K_i^l\}$, matrices $P_i$, constants $\epsilon_i$
4. Apply $u_i(k)=\sum_l h_i^l(z_i(k)) K_i^l x_i(k)$
5. Send $x_i(k)$ to neighbors

4. Robustness and Stability Analysis

RPI Terminal Set and Feasibility:

For each subsystem, construct $$\mathcal{X}_i^f = \{ x_i : x_i^T P_i x_i \leq \epsilon_i \}$$ and seek state-feedback gains $K_i^l$ guaranteeing, for all $l, j$: $$\begin{bmatrix} \Xi_i + \Xi_i^T - 2\epsilon_i I & X_i E_i^l & \dots & g_{ij}^l X_j G_{ij}^l \ \vdots & \ddots & & \vdots \ (E_i^l)^T X_i & \dots & G_{ij}^l X_j G_{ij}^l & - (2\epsilon_i - 1) X_i^f \end{bmatrix} < 0$$ with $X_i = \epsilon_i P_i$, $X_i^f = \epsilon_i P_i^f$, and $\Xi_i = A_i^l + B_i^l K_i^l$.

ISS Lyapunov Argument:

Defining $V_i(x_i)=x_i^T P_i x_i$, under the closed-loop policy: $$V_i(x_i(k+1)) - V_i(x_i(k)) \leq -x_i^T Q_i x_i + T_i d_i^T d_i$$ guarantees input-to-state stability (ISS) under bounded disturbances.

5. Computational Efficiency and Numerical Performance

• Online cost is minimized by precomputing and fixing $(P_i, Q_i, R_i, T_i)$ offline so that the online QP (subject to the required LMIs) is low-dimensional for each subsystem.
• Parallel computation: All local QPs are solved independently and in parallel.
• Solve time: Examples report $\sim$2 ms per local QP on standard PC hardware.

Numerical Studies:

• Chain of three subsystems ($r_i=2$): $A_1^1=[0.55,0.05;0.05,0.55]$, $B_1^1=[1;0]$, $E_1^1=[0.1;0.05]$, $G_{12}^1=[0.08,0.05]$. Horizon $N=10$, quadratic weights, disturbance bound $n=0.5$.
• Double-inverted pendulum ($l=3$ rules/subsystem): Closed-loop tracking error within $5$ s, disturbance rejection tested.
• Compared to decentralized PI, the decentralized fuzzy-MPC reduces settling time by $>30\%$ and control effort by $>40\%$.

6. Advantages, Limitations, and Scalability

Advantages:

• True decentralization: Only current states are exchanged—no higher-order coms or trajectory sharing.
• Interval type-2 fuzzy modeling: Models membership uncertainty directly, greatly reducing conservatism while retaining robust performance.
• Min–max Ho-MPC: Guarantees robust ISS and effective disturbance rejection by explicit handling in cost/terminal set.
• Low online computational cost: Off-line LMI pre-tuning and parallel QP solution for each subsystem enable real-time capabilities even in large networks.

Limitations:

• Model structure is tailored to IT2 fuzzy T–S systems—the benefits of membership uncertainty handling are specific to this formulation.
• Performance is contingent on the linearity of coupling and measurement/communication accessibility of neighbor states at every sample.

7. Context within Modern Decentralized MPC Research

This decentralized IT2-Fuzzy-MPC (Sarbaz et al., 2021) advances the scalable and robust control of nonlinear large-scale networked systems by combining interval type-2 fuzzy modeling with min–max MPC for each subsystem, relying solely on neighbor state exchange. It directly addresses key limitations of existing fuzzy-MPC approaches (excessive computational burden and conservatism) by decomposing the global optimization and exploiting parallel structure. The method is demonstrably more effective and computationally efficient than contemporary decentralized PI schemes in disturbance rejection and stabilization metrics.