# Dynamic Energy Efficient Control of Induction Machines Using Anticipative Flux Templates
## 1 Introduction
An increase in the efficiency of induction machines in dynamic operation will also increase the attractiveness of induction motors for high dynamic servo applications. In this case, the solution of the optimal control problem is more complex for several reasons: In contrast to the optimal control problem for steady state operation, no closed-form solution can be given. As a result, the optimal control problem must be solved online in the inverter at low sampling times. For energy efficient control at varying speed and torque values, the power losses are not constant, but depend on the time varying stator phasor components and the flux linkage dynamics. This dynamic optimization problem is more complex, which requires more computation time. Numerical offline optimization using dynamic programming was considered in [12; 13; 14]. A numerical online solution to the problem is challenging due to the high computational complexity of the problem and the requirement of a low sample time. To cope with these challenges, a model predictive control approach using a parameterized template was proposed in [15], thus simplifying the problem. This approach assumes a particular behavior of the process variables and rewrites the optimal control problem in terms of a parameter optimization problem. A simple suboptimal online method based on the steady state solution was discussed in [16; 17]. An alternative solution based on a filtered flux linkage setpoint in the transient regime was discussed in [18]. In [19; 20], polynomial flux linkage prototypes obtained from a numerical solution of the optimization problem were used to reduce losses. In [21], a closed-form solution to the optimization problem based on [PERSON]'s principle was proposed. A model predictive control based on state control was developed in [22]. The approaches discussed above do not consider current and voltage limits. These strategies can therefore only be applied in a restricted operating range. Considering these limits requires knowing the behavior of the process quantities in dynamic operation, which can only be determined numerically, e.g., through offline optimization. A strategy for considering these limits was developed in [23]. The method uses general predictive control to solve the optimization problem. In the cited publications addressing dynamic optimization, iron losses were considered in [12; 15; 19; 20; 23], whereas in [16; 21], only copper losses were considered. In all cited publications, the optimization interval during dynamic operation of the motor started at the beginning of the torque or speed change. This paper integrates [24; 25; 26], where in contrast to all other approaches, the speed ramp does not start right at the beginning of the optimization interval. The optimization problem does consider the voltage and current limits, as well as the main inductance saturation effect. The obtained optimal offline solution shows that the flux should be increased in an anticipatory manner to the flux in order to minimize losses. The results from the offline optimization procedure are used as a reference trajectory, which now must be achieved with lower computation times, to allow for an online implementation. This is achieved by deriving a template from the results obtained by the offline optimization procedure. The template therefore also considers voltage and current limits, as well as the main inductance saturation effect. This paper further extends the template based solution by adding a template adaptation algorithm. Using this algorithm, template based flux anticipation is possible. The implementation in an online system becomes feasible, because the offline generated template only needs to be adapted to the current driving situation in the online implementation, which results in significantly lower computation effort compared to the optimization procedure in [24]. This approach is validated, tested, and compared, which is done by simulations and measurements on a test bench.