## 1 Introduction  
New designs and operations of gas turbine engines (GTEs) are increasingly complex where several constraints and control modes should be satisfied simultaneously to achieve a safe and optimal performance for the engine. To satisfy these requirements, current engine control systems are still mainly working with a min-max control philosophy. This strategy is known as a practical algorithm to satisfy all engine control modes simultaneously without any error and malfunction [1,2]. Results of the design and implementation of the min-max controller were presented within the Basic Research in Industrial Technologies for Europe-Europe/America (BRITE-EURAM) project OBIDICOTE (On Board Identification, Diagnosis and Control of gas Turbine Engines) and confirmed by all OBIDICOTE partners [3,4]. Therefore, this strategy is the most practical control method for gas turbine engines [5, 6, 7]. However, given the ambitious targets and severe limitations set by governments and organizations (e.g., the advisory council for aviation research and innovation in Europe has a target of a 75% reduction in CO\({}_{2}\) emissions and a 90% reduction of NOx emissions by 2050 [8,9]), more robustness and flexibility is required for the next generation of GTEs control systems to adapt GTEs and aircraft performance to these standards and regulations. So, the application of optimal control strategies for GTEs control modes satisfaction has been investigated extensively in recent years. Generally, the main feature of the optimal control theory is to use quadratic functions to solve the optimal control problems in complex systems. The linear quadratic regulator (LQR) is a typical case for solving general nonlinear optimal control problems that indicates the prospective design for the task of performance optimization [10]. This approach is used extensively for gas turbine engines controller design and optimization [11,12]. So, its basics and methodology are well understood, straightforward, and easy to implement. However, the shortcomings would be the iterative process required to deal with the problem, tuning the weighting matrix, and parameters handling [13, 14, 15]. An LQR is based on the receding horizon concept such that future outputs are predicted at every time step in order to minimize a global criterion/cost function. By estimating future outputs based on past outputs, the user is able to better regulate offset in tracking. Usually, the output horizon is infinite; otherwise, the user ends up with a model predictive controller. H_\(\infty\) and neural networks (NN) are also used for GTEs optimal control problems [16,17]. However, the main shortcomings of these approaches are the level of mathematical understanding needed to apply them successfully and the need for a reasonably good model of the system to be controlled. Another optimal control theory is model predictive control (MPC). The main advantage of MPC is that it allows the current timeslot to be optimized while keeping future timeslots into account. MPC is working based on the idea of optimizing a finite time horizon, but only implementing the current timeslot and then optimizing again, repeatedly, thus differing from LQR [18]. This approach has also the ability to anticipate future events and can take control actions accordingly (PID (Proportional-Integral-Derivative) controllers do not have this predictive ability) [19]. On one hand, the prediction of future dynamic response in the MPC eliminates the iterative process as well as tuning the weighting matrix of LQR [20,21]. On the other hand, constraints to the engine parameters can be easily implemented to the MPC, which overcomes the problem of handling the parameters in LQR [22]. In comparison with H_\(\infty\) and NN, the MPC is simpler, which does not require an accurate dynamic model to initialize transient operation, and the engine constraints can be directly embedded in the control process [23,24].