Input–output current regulation of Zeta converter using an optimized dual-loop current controller

Abstract

In this paper, an optimized dual-loop current controller for current balancing of a Zeta converter is presented and analysed in continuous current mode. The proposed strategy has an inner loop which is defined based on the input inductor current control. The reference signal of the sliding manifold is changed through an outer loop which works to regulate the output current. The stability analysis of the two-loop controllers is established by means of Routh–Hurwitz criterion and the equivalent control method. Then, the gains of the outer loop compensator are optimized using the integral gain maximization method to guarantee the robustness and disturbance rejection of the closed loop system in the presence of model uncertainties. The controller performance is investigated in depth taking into account the parametric variations associated with the converter operation in various equilibrium points. Moreover, a laboratory set-up of the suggested controller has been implemented by using analogue component devices. The experimental results demonstrate the effective performance of the controller.

Appendix A

When the inner loop is nearly constant (i.e. $\frac{{{\text{d}}I_{E} (t)}}{{{\text{d}}t}} < < \frac{{{\text{d}}i_{O} }}{{{\text{d}}t}}$ and also $I_{E} (t) = I_{E}$), the ideal equation of the closed loop system in (16) can be expressed as:

$$ \begin{aligned} C\frac{{{\text{d}}v_{C} }}{{{\text{d}}t}} & = - i_{o} \left( {\frac{{v_{C} }}{{E + v_{C} }}} \right) + I_{E} \left( {\frac{E}{{E + v_{C} }}} \right) \\ L_{2} \frac{{{\text{d}}i_{o} }}{{{\text{d}}t}} & = v_{C} - v_{O} \\ \end{aligned} $$

(a.1)

where $v_{o} = Ri_{O}$. The equilibrium point of the system given by (17) can be rewritten as follows:

$$ I_{L} = I_{E} = \frac{{RI_{O}^{2} }}{E},_{{}} V_{C} = RI_{O}^{2} ,_{{}} I_{O} = I_{d} $$

(a.2)

By using (a.2) and with some manipulations, (a.1) can be expressed as follows:

$$ \begin{aligned} C\frac{{{\text{d}}v_{C} }}{{{\text{d}}t}} & = - i_{o} \left( {\frac{{v_{C} - V_{C} }}{{E + v_{C} }}} \right) - RI_{d} \left( {\frac{{i_{O} - I_{d} }}{{E + v_{C} }}} \right) \\ L_{2} \frac{{{\text{d}}i_{o} }}{{{\text{d}}t}} & = - R(i_{O} - I_{d} ) + (v_{C} - V_{C} ) \\ \end{aligned} $$

(a.3)

Now, select the following positive definite Lyapunov function:

$$ V(v_{C} ,i_{O} ) = \frac{1}{2}C(v_{C} - V_{C} )^{2} + \frac{1}{2}L_{2} (i_{O} - I_{d} )^{2} \ge 0 $$

(a.4)

The time derivative of the defined Lyapanov function is given by:

$$ \mathop V\limits^{.} (v_{C} ,i_{O} ) = - \frac{{i_{O} }}{{E + v_{C} }}(v_{C} - V_{C} )^{2} - \frac{{RI_{d} }}{{E + v_{C} }}(v_{C} - V_{C} )(i_{O} - I_{d} ) - R(i_{O} - I_{d} )^{2} + (v_{C} - V_{C} )(i_{O} - I_{d} ) $$

(a.5)

Expressing (a.5) in form of $\mathop V\limits^{.} = - XQX^{T}$, leads to:

$$ \dot{V}(v_{C} ,i_{O} ) = - XQX^{T} = - [v_{C} i_{O} ]\left[ {\begin{array}{*{20}c} {\frac{{i_{O} }}{{E + v_{C} }}} & {\frac{{RI_{d} }}{{E + v_{C} }}} \\ { - 1} & R \\ \end{array} } \right][v_{C} i_{O} ]^{T} $$

(a.6)

If $Q > 0$; then, $\mathop V\limits^{.} (v_{C} ,i_{O} )$ will be negative definite. For the Zeta converter, it is valid: $R > 0$ and $I_{d} > 0$. If the conditions $i_{O} > - I_{d}$ and $v_{C} > - E$ are fulfilled, then $\mathop V\limits^{.} (v_{C} ,i_{O} ) < 0$. Therefore, the proposed system is asymptotically stable based on the LaSalle’s stability principles [43].