Output feedback control of multicellular converters

Abstract

This paper presents an output feedback control for the multicellular converter using hybrid systems theory. First, we study the observability of the continuous state of the converter under predetermined finite switching sequence. Using some dynamical properties of the converter, a time independent necessary and sufficient condition is given for the continuous state observability. This interesting condition allows the avoidance of the usual matrix exponential computations for the observability analysis and leads to major notes on the multicellular converter observability. Second, as an application of the above results, we consider the case of the 2-cell converter where we establish a new switching control scheme that guarantees the existence and the finite-time stability of a limit cycle. The corresponding repetitive switching sequence of the limit cycle is used to prove the continuous state observability. We next design a super twisting sliding mode observer that guarantees the finite continuous state convergence. Simulation results confirm the effectiveness and the robustness of the output control scheme under different perturbations: variations of the input voltage and load resistance.

Appendix A

Proof: Firstly, we should demonstrate the asymptotic stability of the region R (Fig. 4). This is done in [20]. Next to that, we will proof the existence of isolated closed trajectory (limit cycle), which will be followed by its stability analysis.

As we mentioned, we have two cases depending on the reference current value. The idea is to analyze the function that transforms the point A on the surface $v_{c} = 0.5E - \Delta v$ to the point E (under sequence $\sigma_{1}$ or $\sigma_{2}$) on the same surface (Figs. 9, 10).

Fig. 9

Ponctual transformation under the proposed control scheme (case 1) in two possible situations

Fig. 10

Ponctual transformation under the proposed control scheme (case 2) in two possible situations

To obtain this transformation, we will define the augmented system as follows,

$$ \dot{X} = A_{augi} X,\;\;{\text{with}}\;\;A_{augi} = \left[ {\begin{array}{*{20}c} {A_{{q_{i} }} } & {B_{{q_{i} }} } \\ 0 & 0 \\ \end{array} } \right] $$

(A.1)

The initial condition of this system is $X_{0}^{\text{T}} = \left[ {\begin{array}{*{20}c} {x_{0}^{\text{T}} } & 1 \\ \end{array} } \right]$.

1.1 A.1 Case 01($I_{ref} < 0.5I_{\rm{max} }$)

Figure 9 represents the steady state in the phase plan under the proposed switching surfaces (case 1). To analyze the punctual transformation, we distinguish two situations:

Situation a ($i_{a} < I_{ref} - \Delta i$)

We have:

Point A to B:

$$ \begin{aligned} X_{B} & = e^{{A_{aug2} t_{1} }} X_{A} \\ C_{aug1} X_{B} & = 0,C_{aug1} = \left[ {\begin{array}{*{20}c} 1 & 0 & { - E/2 - \Delta v} \\ \end{array} } \right] \\ \end{aligned} $$

Point B to C:

$$ \begin{aligned} X_{C} & = e^{{A_{aug1} t_{2} }} X_{B} = e^{{A_{aug1} t_{2} }} e^{{A_{aug2} t_{1} }} X_{A} \\ C_{aug2} X_{c} & = 0,C_{aug2} = \left[ {\begin{array}{*{20}c} 0 & 1 & { - I_{ref} + \Delta i} \\ \end{array} } \right] \\ \end{aligned} $$

Point C to D

$$ \begin{aligned} X_{D} & = e^{{A_{aug3} t_{3} }} X_{C} = e^{{A_{aug3} t_{3} }} e^{{A_{aug1} t_{2} }} e^{{A_{aug2} t_{1} }} X_{A} \\ C_{aug3} X_{D} & = 0,C_{aug3} = \left[ {\begin{array}{*{20}c} 1 & 0 & { - E/2 + \Delta v} \\ \end{array} } \right] \\ \end{aligned} $$

Point D to E:

$$ \begin{aligned} & X_{E} = e^{{A_{aug1} t_{4} }} X_{D} = e^{{A_{aug1} t_{4} }} e^{{A_{aug3} t_{3} }} e^{{A_{aug1} t_{2} }} e^{{A_{aug2} t_{1} }} X_{A} \\ & C_{aug2} X_{E} = 0, \\ \end{aligned} $$

Thus, the punctual transformation is given by,

$$ X_{E} = e^{{A_{aug1} t_{4} }} e^{{A_{aug3} t_{3} }} e^{{A_{aug1} t_{2} }} e^{{A_{aug2} t_{1} }} X_{A} = f_{1} (X_{A} ) $$

(A.2)

With the following constraints:

$$ \begin{aligned} & C_{aug1} e^{{A_{aug2} t_{1} }} X_{A} = 0 \\ & C_{aug2} e^{{A_{aug1} t_{2} }} e^{{A_{aug2} t_{1} }} X_{A} = 0 \\ & C_{aug3} e^{{A_{aug3} t_{3} }} e^{{A_{aug1} t_{2} }} e^{{A_{aug2} t_{1} }} X_{A} = 0 \\ & C_{aug2} e^{{A_{aug1} t_{4} }} e^{{A_{aug3} t_{3} }} e^{{A_{aug1} t_{2} }} e^{{A_{aug2} t_{1} }} X_{A} = 0 \\ \end{aligned} $$

(A.3)

This transformation can be limited to one of the continuous states (Capacitor voltage or load current). This choice is justified by the intersection of the trajectory with the switching surfaces. For that, we will analyze only the load current under the punctual transformation. We have:

$$ i_{E} = C_{aug} f_{1} (i_{A} ) = C_{aug} e^{{A_{aug1} t_{4} }} e^{{A_{aug3} t_{3} }} e^{{A_{aug1} t_{2} }} e^{{A_{aug2} t_{1} }} X_{A} $$

(A.4)

With $C_{aug} = \left[ {\begin{array}{*{20}c} 0 & 1 & 0 \\ \end{array} } \right]$ is the augmented output matrix. We denote $M = e^{{A_{aug1} t_{4} }} e^{{A_{aug3} t_{3} }} e^{{A_{aug1} t_{2} }} e^{{A_{aug2} t_{1} }}$

One can simply calculate exponential of the state matrix of mode 1.

$$ e^{{A_{aug1} t_{4} }} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & {e^{{ - \frac{R}{L}t_{4} }} } & 0 \\ 0 & 0 & 1 \\ \end{array} } \right] $$

(A.5)

The form of the matrix M(or any exponential matrix of the augmented system) is given by

$$ M = \left[ {\begin{array}{*{20}c} * & * & * \\ * & * & * \\ 0 & 0 & 1 \\ \end{array} } \right] $$

(A.6)

Thus, one can obtain the following results

$$ \begin{aligned} C_{aug2} e^{{A_{aug1} t_{4} }} & = \left[ {\begin{array}{*{20}c} 0 & {e^{{ - \frac{R}{L}t_{4} }} } & { - I_{ref} + \Delta i} \\ \end{array} } \right] \\ C_{aug} e^{{A_{aug1} t_{4} }} & = \left[ {\begin{array}{*{20}c} 0 & {e^{{ - \frac{R}{L}t_{4} }} } & 0 \\ \end{array} } \right] \\ \end{aligned} $$

By comparison, one can find,

$$ C_{aug} e^{{A_{aug1} t_{4} }} = C_{aug2} e^{{A_{aug1} t_{4} }} + \left[ {\begin{array}{*{20}c} 0 & 0 & {I_{ref} - \Delta i} \\ \end{array} } \right] $$

(A.7)

Using (A.7) in (A.4), we obtain:

$$ i_{E} = C_{aug2} MX_{A} + \left[ {\begin{array}{*{20}c} 0 & 0 & {I_{ref} - \Delta i} \\ \end{array} } \right]e^{{A_{aug3} t_{3} }} e^{{A_{aug1} t_{2} }} e^{{A_{aug2} t_{1} }} X_{A} $$

The term $C_{aug2} MX_{A}$ is null due to the constraints on the switching surfaces (A.3). Therefore, the transformation of the load current from A to E is given by:

$$ i_{E} = f_{1} (i_{A} ) = I_{ref} - \Delta i $$

(A.8)

Situation b ($i_{a} > I_{ref} - \Delta i$)

We proved that for any initial current $i_{a} < I_{ref} - \Delta i$, the punctual transformation leads to $i_{E} = I_{ref} - \Delta i$ and in this situation, the dynamics of mode 1 leads to the same current $i_{E} = I_{ref} - \Delta i$

Thus, the obtained results can be resumed as follows:

• This transformation has a unique fixed point $x_{p}^{T} = \left[ {\begin{array}{*{20}c} {E/2} & {I_{ref} - \Delta i} \\ \end{array} } \right]$; This proofs the existence of the limit cycle under the proposed switching surfaces in case1;

• The derivative of the transformation is null, which means that $x_{p}$ is a super attractive fixed point. This leads to the conclusion that the limit cycle is more than asymptotic stable, it is finite time stable under this control scheme;

1.2 A. 2 Case 02($I_{ref} < 0.5I_{\rm{max} }$)

Figure 10 represents the steady state in the phase plan under the proposed switching surfaces (case 2). To analyze the punctual transformation, we distinguish two situations:

• Situation a ($i_{a} > I_{ref} + \Delta i$)

• Situation b ($i_{a} < I_{ref} + \Delta i$)

With same manner, one can found the punctual transformation of the load current in case 2.

$$ i_{E} = f_{2} (i_{A} ) = I_{ref} + \Delta i $$

(A.9)

Henceforth, the same remarks can be concluded in case 2 where the fixed point of the punctual transformation is $x_{p}^{T} = \left[ {\begin{array}{*{20}c} {E/2} & {I_{ref} + \Delta i} \\ \end{array} } \right]$.

This ends the proof.□