###### Abstract  
This work presents the design of a control algorithm based on an augmented incremental state-space model, emphasizing its compatibility with Takagi-Sugeno (T-S) fuzzy models for nonlinear systems. The methodology integrates key components such as incremental modeling, fuzzy system identification, discrete Linear Quadratic Regulator (LQR) design, and state observer implementation. To optimize controller performance, the Extremum Seeking Control (ESC) technique is employed for the automatic tuning of LQR gains, minimizing a predefined cost function. The control strategy is formulated within a generalized framework that evolves from conventional discrete fuzzy models to a higher-order incremental-N state-space representation. The simulation results on a nonlinear multivariable thermal mixing tank system validate the effectiveness of the proposed approach under reference tracking and various disturbance scenarios, including ramp, parabolic, and higher-order polynomial signals. The main contribution of this work is that the proposed scheme achieves zero steady-state error for reference inputs and disturbances up to order N\(-\)1 by employing the incremental-N formulation. Furthermore, the system exhibits robustness against input and load disturbances, as well as measurement noise. Remarkably, the ESC algorithm maintains its effectiveness even when noise is present in the system output. Additionally, the proposed incremental-N model is applicable to fast dynamic systems, provided that the system dynamics are accurately identified and the model is discretized using a suitable sampling rate. This makes the approach particularly relevant for control applications in electrical systems, where handling high-order reference signals and disturbances is critical. The incremental formulation, thus, offers a practical and effective framework for achieving high-performance control in both slow and fast nonlinear multivariable processes.  
incremental model; Takagi-Sugeno model; Extremum Seeking Control  
17 June 2025  
7 July 2025  
###### Abstract  
This work presents the design of a control algorithm based on an augmented incremental state-space model, emphasizing its compatibility with Takagi-Sugeno (T-S) fuzzy models for nonlinear systems. The methodology integrates key components such as incremental modeling, fuzzy system identification, discrete Linear Quadratic Regulator (LQR) design, and state observer implementation. To optimize controller performance, the Extremum Seeking Control (ESC) technique is employed for the automatic tuning of LQR gains, minimizing a predefined cost function. The control strategy is formulated within a generalized framework that evolves from conventional discrete fuzzy models to a higher-order incremental-N state-space representation. The simulation results on a nonlinear multivariable thermal mixing tank system validate the effectiveness of the proposed approach under reference tracking and various disturbance scenarios, including ramp, parabolic, and higher-order polynomial signals. The main contribution of this work is that the proposed scheme achieves zero steady-state error for reference inputs and disturbances up to order N\(-\)1 by employing the incremental-N formulation. Furthermore, the system exhibits robustness against input and load disturbances, as well as measurement noise. Remarkably, the ESC algorithm maintains its effectiveness even when noise is present in the system output. Additionally, the proposed incremental-N model is applicable to fast dynamic systems, provided that the system dynamics are accurately identified and the model is discretized using a suitable sampling rate. This makes the approach particularly relevant for control applications in electrical systems, where handling high-order reference signals and disturbances is critical. The incremental formulation, thus, offers a practical and effective framework for achieving high-performance control in both slow and fast nonlinear multivariable processes.