Complex Controllers Applied to Space Vectors: A Survey on Characteristics and Advantages

Abstract

This work presents the characteristics and advantages of using complex controllers applied to space vectors. A mathematical background is presented to show that the harmonic spectrum of a space vector indicates the phase-sequence of each harmonic that composes this vector. As consequence, a reference space vector can be used for applications in which the control system has three-phase reference signals. This approach allows the implementation of complex controllers that present lower order, require less memory elements and have better dynamic response than real controllers. A literature review is made to enable a structural comparison (in terms of computational cost) between several real and complex control schemes based on the internal model principle. Two experimental applications are used to evaluate the performance of complex controllers.

Unified Approach to Evaluation of Real and Complex Repetitive Controllers

“…Since nk ± m RCs are real controllers [6], all their poles and zeros must be real or pairs of complex conjugates. However, when evaluating the term 1 of (4), and considering the Euler's formula, it can be seen that…”

“…Regarding the implementation of this class of controllers in digital signal processors, it is done using its scalar representation [6]. This means that a complex single-input singleoutput controller, whose input and output signals are space vectors, can be represented using a real multi-input multioutput representation, where the input and output signals are the real and imaginary components of the space vector.…”

“…10. The magnitudes of the positive and negative frequency components correspond to the magnitudes of the positive-and negative-sequence threephase components, respectively [6]. As a matter of fact, this space vector is represented by…”

“…Since these linear solutions incorporate the mathematical model of the reference signal in the closedloop control system, according to the internal model principle [5], they ensure zero steady-state error for tracking periodical reference signals whose all harmonic components are in the selected set. The control strategies based on the internal model principle that are used in three-phase inverters can still be classified into three categories [6]: real controllers, which are used to separately control each phase signal or each component of the space-vector that represents the three-phase signal in a stationary reference-frame, such as in [4]; complex controllers in stationary reference frame, which are used to control the space-vector formed by a three-phase signal in the αβ reference frame, such as in [7] [8]; and real controllers in synchronous reference frame, which are used to control d and q signals in a synchronous reference frame, such as in [9]- [11], and have frequency spectrum characteristics that are similar to those obtained for complex controllers in stationary reference frame.…”

“…• Presenting a detailed analysis on how real and complex RC can be represented and evaluated through PRCs (the concept of complex RCs is covered in [6]); • Proposing a generic nk ± m RC (Fig. 5), which is a real (a) nk ± m RC proposed in [13].…”