Higher-Order Sliding Mode Control Scheme with an Adaptation Low for Uncertain Power DC–DC Converters

Abstract

In this paper, a higher-order sliding mode controller for a power DC–DC converter is proposed. The uncertainty and disturbance are implicit to be unknown. A detailed analysis to explore the local and global stability of a Buck DC–DC converter was presented in this paper. Different control schemes were implemented to regulate the output voltage and to eliminate the high current ripples for the proposed converter, beginning with a proportional-integral-derivative compensation scheme, then a sliding mode controller, a higher-order sliding mode controller and finally an adaptation low with higher-order sliding mode controller (AHOSMC). Stability and robustness of the AHOSMC are proved by using the classical Lyapunov criterion. The sensitivity of parameters variation is analyzed, and a detailed bifurcation analysis is undertaken.

Appendix

1.1 PID and SMC Controllers

We have introduced firstly a PID feedback controller which has the following transfer function:

$$\begin{aligned} G_\mathrm{contr} (p)=k_p \frac{1+p\cdot T_i +p^{2}\cdot T_i \cdot T_d }{p\cdot T_i } \end{aligned}$$

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Then, we have presented the SMC controller. In SMC, the trajectory of the system is constrained to move or slide along a predetermined hyper plane in the state space. Such mode is completely robust and independent of parametric variations and disturbances (Utkin 1992; Song and Sun 2014). By eliminating the parasitic effect of the capacitor ($r_{C} = 0$), The system is described by the following state-space equations (Mondal and Mahanta 2013):

$$\begin{aligned} \left\{ {\begin{array}{l} \dot{x} =F\left( x \right) +G\left( {x,V_{e}} \right) T_{R} \\ y=S\left( x \right) \\ \end{array}} \right. \end{aligned}$$

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where the matrices F and G are given by:

$$\begin{aligned} F\left( x \right) = \left[ {\begin{array}{l} -\frac{r_{L} }{L}i_L -\frac{1}{L}V_\mathrm{C} \\ \frac{1}{C}i_L -\frac{1}{\hbox {RC}}V_\mathrm{C} \\ \end{array}} \right] , \quad G\left( {x,V_{e} } \right) = \left[ {\begin{array}{l} \frac{V_{e} }{L} \\ 0 \\ \end{array}} \right] \end{aligned}$$

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And S(x) is the measured output function known as the sliding variable. The general form of S(x) is given as follows (Utkin 1992):

$$\begin{aligned} S\left( x \right) =\left( {\frac{d}{dt}+\uplambda _x } \right) ^{r-1}e\left( x \right) \end{aligned}$$

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Table 1 Parameters of the buck DC–DC converter (Hadri Hamida 2011)

It is the first convergence condition which permits dynamic system to converge toward the sliding surfaces. It is a question of formulating a positive scalar function $\hbox {V(x) }> 0$ for the system states variables which are defined by the following Lyapunov function (Utkin et al. 1999; Filippov 1964):

$$\begin{aligned}&V\left( x \right) =\frac{1}{2}S\left( x \right) ^{T}S\left( x \right) \end{aligned}$$

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$$\begin{aligned}&\dot{V} \left( x \right) <0\Rightarrow S\left( x \right) ^{T}\dot{S} \left( x \right) <0 \end{aligned}$$

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Now, we define

$$\begin{aligned} T_{R} \left( t \right) =\frac{1}{2}\left( {1+\hbox {sign}\left( S \right) } \right) =T_{R_{eq} } \left( t \right) +T_{R_n } \left( t \right) \end{aligned}$$

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where $T_\mathrm{Req}(t)$ and $T_{Rn}(t)$ represent the equivalent control (Utkin 1992) and the nonlinear switching control and:

$$\begin{aligned} \hbox {sign} \left( S \right) = \left\{ {\begin{array}{ll} 1 &{}S\left( x \right) >0 \\ -1 &{}S\left( x \right) <0 \\ \end{array}} \right. \end{aligned}$$

The sliding surfaces are given by the following expression (Table 1):

$$\begin{aligned} S=e\left( {V_s } \right) +e\left( {i_L } \right) =V_{r\acute{e}f} -H_v V_s +i_{r\acute{e}f} -H_i i_L \end{aligned}$$

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and consequently, their derivatives are given by:

$$\begin{aligned} \dot{S} =E\left( x \right) +QT_{R} \end{aligned}$$

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where $x = [i_{L} \quad V_\mathrm{C}]^{ T}$,

$$\begin{aligned} E=\left( {\frac{H_{i} r_{L} }{L}-\frac{H_{\mathrm{v}}}{\mathrm{C}}} \right) i_{L} +\frac{H_{i}}{L}V_{C} +\frac{H_{\mathrm{v}}}{C}I_{s}, \end{aligned}$$

and

$$\begin{aligned} Q=-\frac{H_{i}}{L}V_{e}. \end{aligned}$$

Finally, the control law is given by:

$$\begin{aligned} T_R =-Q^{-1} E\left( x \right) +K \hbox {sign} \left( S \right) \end{aligned}$$

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1.2 Parameters of the Buck DC–DC Converter

1.3 How to Plot a Bifurcation Diagram?

Let us consider the state equation:

$$\begin{aligned} \left\{ {\begin{array}{l} \dot{x} =Ax+BV_e \\ x_0 \\ \end{array}} \right. \end{aligned}$$

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The diagram of bifurcation of this system is defined in the following way: One represents $V_{e}$ in X-coordinate ($V_{e}$ ranging between 15 and 50 V), and we present in ordinate, the values obtained by the solution of the state equation, after a certain iteration count (400 for example). For each value of $V_{e}$, the operation is started again a great number of times by choosing each time a random value of the first term $x_{0}$ (equilibrium values). One thus obtains the bifurcation diagram of our system.