Abstract
In this chapter, fundamental definitions and terminology are given to the reader regarding the closed-loop control system. The analysis of the control loop takes place in the frequency domain and, therefore all necessary transfer functions of the control loop are presented in Sect. 2.2. The important aspect of internal stability of a control loop is presented in Sect. 2.3, whereas in Sect. 2.4 the property of robustness in a control loop is analyzed. In Sect. 2.5, a clear definition of the type of the control loop is given, since in Part II, the proposed theory is dedicated to the design of type-I, type-II, and type-III, ... type-p control loops. Last but not least, in Sect. 2.6, the definitions of sensitivity and complementary sensitivity functions are presented so that the tradeoff feature in terms of controller performance that these two functions introduce is made clear to the reader. Finally, in Sect. 2.7, the principle of the Magnitude Optimum criterion is presented and certain optimization conditions are proved that comprise the basic tool for all control laws’ proof throughout this book. These optimization conditions serve to maintain the magnitude of the closed-loop frequency response equal to the unity in the widest possible frequency range as the Magnitude Optimum criterion implies. In the same section, the Magnitude Optimum criterion is proved to be considered as a practical aspect of the \(H_\infty \) design control principle.
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Notes
- 1.
I.e., rise of temperature during a motor’s operation, aging of materials after a certain time (for example, copper conductors in a squirrel-cage induction motor).
- 2.
Negative feedback is present to the water clock invented by Ktesibios (Greek inventor and mathematician in Alexandreia, 285–222 BC) and in the steam engine governor patented by James Watt in 1788.
- 3.
In the case of electric motor drives, for example, \(k_\mathrm{p}\) stands for the proportional gain introduced by the power electronics circuit, which finally applies the command signal \(u(t)\) to the plant which in this case is the electric motor. For voltage source inverters, the command signal \(u(t)\) is voltage. In the sequel it is explained that the gain introduced by the actuator has to be linear and proportional so that the command signal of the controller remains unaltered. In the specific case of electric motor drives, the power part introduces also a time delay with time constant \(T_\mathrm{d}\) which corresponds to the time the controller decides the command \(u(t)\), until the time it is finally applied by the power electronic circuit. Therefore in this case, the model of the actuator is given as \(k_\mathrm{p}\mathrm{e}^{-sT_\mathrm{d}}\)
- 4.
\( {S}(s)\) stands for the sensitivity of the closed-loop control system and is defined as \( {S}( s ) = \frac{{y( s )}}{{{d_\mathrm{o}}( s )}}\) when \(r(s)=n_\mathrm{r}(s)=d_\mathrm{i}(s)=n_\mathrm{r}(s)=0\).
- 5.
In grid-connected power converters and when vector control is followed for regulating the DC link voltage to be utilized by the motor connected converter, there is one inner loop for regulating the current of the power converter and one outer loop for regulating its DC link voltage. In this case, the inner current control loop is of type-I, since in its open-loop transfer function there exists only one integrator coming from the current PI control action, whereas the outer control loop is of type-II, since the open-loop transfer function introduces two integrators, one coming from the DC link voltage PI control action and another coming from the capacitor bank path (\(\frac{1}{sC}\)). A case of type-II control loop in the field of electric motor drives is the speed control loop in vector-controlled or direct torque-controlled drives. In this case, one integrator comes from the speed PI control action and another integrator comes from the inertia (\(\frac{1}{sJ}\)) of the shaft of the motor the speed of which is controlled.
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Papadopoulos, K.G. (2015). Background and Preliminaries. In: PID Controller Tuning Using the Magnitude Optimum Criterion. Springer, Cham. https://doi.org/10.1007/978-3-319-07263-0_2
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