Model Predictive Control (MPC) System

• Model Predictive Control (MPC) system is a feedback control strategy that optimizes control moves over a finite horizon while directly handling constraints.
• It integrates state-space modeling, linearization, and augmented prediction to enable accurate tracking and offset elimination in coupled multivariable processes.
• Simulations demonstrate MPC's capability for smooth setpoint regulation and effective disturbance rejection in challenging liquid-level regulation scenarios.

Model Predictive Control (MPC) systems are a class of feedback controllers that compute optimal control actions through the repeated solution of constrained optimization problems, leveraging an explicit predictive model of future plant behavior. MPC's receding horizon principle and capacity for direct constraint handling have resulted in broad adoption for complex multi-input multi-output (MIMO) process systems, such as those encountered in chemical or process industry liquid-level regulation. The core methodology involves iterative model-based prediction, optimization over a finite horizon, and real-time feedback action, enabling both reference tracking and disturbance rejection in dynamically coupled systems.

1. State-Space Modeling and Linearization of Coupled Systems

A canonical MIMO application of MPC is the coupled tank liquid level system—critically relevant in process plant operations. The physical system is described by nonlinear differential equations that capture both the inter-tank hydraulic coupling and flow resistance nonlinearity: $$\begin{align*} A_1 \frac{dL_1}{dt} &= F_{i1} - a_1\sqrt{L_1 - L_2} \ A_2 \frac{dL_2}{dt} &= F_{i2} - a_2\sqrt{L_2} + a_1\sqrt{L_1 - L_2} \end{align*}$$ where $L_1, L_2$ are tank levels, $F_{i1}, F_{i2}$ are the controlled inflows, and $a_1, a_2$ are outflow coefficients. For controller synthesis, these equations are linearized about an equilibrium point—application of Taylor series expansion yields a discrete-time linear state-space model for small-signal deviations: $$\begin{align*} x(k+1) &= A_m x(k) + B_m u(k) \ y(k+1) &= C_m x(k) + D_m u(k) \end{align*}$$ with state $x = [h_1\ h_2]^T$ (level deviations), control $u = [fi_1\ fi_2]^T$ (inlet flows), and output $y$ (tank level measurements).

To further align the plant model with the incremental/offset-free prediction structure of MPC, an augmentation is performed: $$\Delta x_m(k+1) = A_m \Delta x_m(k) + B_m \Delta u(k)$$ where $\Delta x_m$, $\Delta u$ represent incremental changes, resulting in an augmented state-space model tailored for accurate open-loop prediction and closed-loop disturbance rejection.

2. Design and Implementation of the MPC Controller

The controller design proceeds by discretizing the augmented state-space model and framing the finite-horizon output prediction as: $Y = \Psi x(k_i) + \Phi \Delta U$ Here, $Y$ is the predicted stacked output vector over the horizon, $\Delta U$ is the sequence of future control increments to be optimized, and $\Psi, \Phi$ are constructed from the system matrices ($A,B,C$). The cost function penalizes both predicted output errors and control effort: $J = (R_s - Y)^T (R_s - Y) + \Delta U^T R \Delta U$ with reference trajectory $R_s$ and weighting matrix $R$. The unconstrained quadratic program yields a closed-form solution: $$\Delta U^* = (\Phi^T\Phi + R)^{-1} \Phi^T (R_s - \Psi x(k_i))$$ In standard form, the algorithm computes a sequence of optimal future input increments based on the current state, but applies only the first increment—iteratively repeating the process as new plant measurements become available.

The concept of prediction horizon ($N_p$) and control horizon ($N_c$) is explicit: $N_p$ defines the number of steps over which future plant outputs are predicted, while $N_c$ restricts the number of control moves being optimized, adding a regularization effect and reducing computational complexity.

3. Disturbance Modeling, Rejection, and Robustness

The coupled tank system is subjected to additive input disturbances (formally, pulse-like perturbations on inlet flows). Disturbances are introduced as exogenous input components ($u_3$), and the controller's disturbance rejection performance is analyzed by imposing a 10% inlet flow pulse. The key property of receding horizon MPC is its ability to reject measured disturbances: following detection (via plant measurement at each sampling instance), the output predictor is recalibrated and the optimal input sequence re-optimized. Simulation demonstrates that, post-disturbance, tank levels rapidly and smoothly return to their setpoints without oscillation; this property is attributed to the incremental model structure and state feedback inherent to the MPC algorithm.

The approach's robustness extends to linearization/model mismatch: even when the real plant deviates from the linearized model used in controller synthesis, the update-and-reoptimize cycle maintains effective setpoint tracking and disturbance tolerance.

4. Performance Evaluation: Setpoint Tracking and Output Regulation

Simulation studies demonstrate that MPC achieves tight tracking of both step and pulse reference changes for both tank levels ($h_1, h_2$). The MIMO structure of the controller naturally accounts for inter-tank coupling: control actions coordinated across flow valves maintain both tank levels at setpoints, without crosswise overshoot. Under pulse disturbance conditions, output deviations are promptly corrected, returning to the desired value with no observable oscillatory artifacts—indicative of sound closed-loop pole placement and well-designed cost function weights.

Smoothness of recovery (no overshoot/oscillation), tracking accuracy, and the rapid suppression of disturbance transients collectively establish the practical efficacy of the MPC workflow in multivariable process contexts.

5. Methodological Advances and Application-Specific Features

Several design characteristics distinguish this implementation:

• Explicit MIMO (Multiple Input Multiple Output) Formulation: The controller design and simulation address a two-input, two-output system, representative of many process control scenarios. The use of an augmented, incremental state-space predictor is both necessary for correct setpoint regulation (eliminating steady-state error) and generic across process control applications with interacting variables.
• Comprehensive Design Workflow: The sequence of derivations—nonlinear modeling, linearization, state-space discretization, incremental/augmented modeling, MPC optimization law development—is detailed and methodical, making the methodology reproducible for similar systems.
• Disturbance Analysis: Explicit scenario-based testing quantifies the disturbance rejection properties.
• Extensibility: While constraints are not activated in the simulated optimization, the quadratic programming structure enables direct extension to constrained MPC with bounds on both flow input and tank level outputs.

6. Mathematical Summary and Key Equations

The essential equations comprising the MPC application are as follows:

Step	Mathematical Expression
Linearized state-space	$x(k+1) = A_m x(k) + B_m u(k)$
Output equation	$y(k+1) = C_m x(k) + D_m u(k)$
Augmented prediction	$Y = \Psi x(k_i) + \Phi \Delta U$
MPC quadratic cost	$J = (R_s - Y)^T (R_s - Y) + \Delta U^T R \Delta U$
Optimal control law	$$\Delta U^* = (\Phi^T\Phi + R)^{-1} \Phi^T (R_s - \Psi x(k_i))$$

The system operates in discrete time, with incremental modeling and augmented state/output representations ensuring flexibility for practical plant constraints and disturbance rejection.

7. Conclusion and Relevance

This MPC system exemplifies contemporary model-based control strategies for coupled multivariable process plants. The theoretical workflow—from physical plant modeling to online optimization and receding horizon feedback—promotes robust, accurate tracking, and exceptional disturbance recovery for nontrivial MIMO liquid-level systems. The incremental/augmented model formulation is essential for eliminating offset errors and achieving reliable constraint management (if extensions are required). The methodology, as demonstrated for two coupled tanks, is extensible to more complex industrial process systems with higher-dimensional coupling and disturbance environments.