On the switching control of the DC–DC zeta converter operating in continuous conduction mode

Here, a switching control mechanism for the stabilization of a DC–DC zeta converter operating in continuous conduction mode is proposed. The switching control algorithm is based on a control Lyapunov function and extends the method proposed for a two-dimensional boost converter model presented in the literature to a four-dimensional zeta converter model. The local asymptotical stability of the operating point is established using LaSalle’s invariance principle for differential inclusions. By applying spatial regularization, a modiﬁed switching control algorithm reduces the switching frequency and keeps the state-trajectory around a neighbourhood of the operating point. The method works well even if the operation point changes signiﬁcantly and it is valid for both step-up and step-down operations. Furthermore, by approximating the state-trajectory near the operating point, an explicit relation between the modiﬁed switching algorithm and the switching frequency is obtained, which allows to choose systematically the desired switching frequency for the converter to operate. The effectiveness of the proposed method is illustrated with simulation results.


INTRODUCTION
In energy harvesting systems, DC-DC converters are part of the power management system. Because of the uncertain nature of the ambient energy, for example, low or high irradiance of sun and fluctuation of the wind speed, the voltage generated by the energy harvester, which is connected to the input of the DC-DC converter, can be higher or lower than the output voltage. For this reason, a fourth-order DC-DC converter is a good candidate to be deployed, since it has step-up and step-down capability. There are a few topologies available, and the zeta topology is selected for our research due to two reasons: (1) positive output voltage, and low output voltage ripple [3], (2) natural DC input-to-output voltage isolation [4].
To control a DC-DC converter, the conventional fixedfrequency, average-based system control methods are commonly deployed such as proportional integral (PI) [5][6][7][8][9], optimal [11][12][13][14][15][16], sliding mode [17,18], fuzzy [19,20], model predictive [21,22], adaptive [23], and fuzzy neural [24], to name a few. The PI control produces fast output voltage regulation, however, it suffers from high control duty-ratio effort [9] that can lead to PWM circuitry problem [12]. The conventional optimal linear quadratic regulator (LQR) control produces optimal compensation with minimal control effort, but lack of robustness if the parameter is uncertain [14]. While LMI-LQR control is robust, its control duty-ratio signal has quite a large ripple [15], which may produce non-linear effect if the ripple exceeded 20% [10]. As for the sliding mode, fuzzy, model predictive, adaptive, and fuzzy neural, because they use the so-called small-signal average model, when the duty ratio largely deviates from the nominal one, the small-signal average model is not a good approximation. As a result, the controller design is no longer valid, which in turn jeopardizes the system performance. On the other hand, non-average-based system control, typically known as hybrid control, is a variable switching frequency type of control where the switching frequency is initially low and it becomes arbitrarily fast at operating point. The hybrid control is more robust than average-based system control [2], due to the former's ability to execute the switching mechanism online. Hybrid control has been implemented for the stabilization of the DC-DC converter [25][26][27][28][29][30][31][32]. In [25][26][27][28], the authors propose a switching algorithm by approximating the state-trajectory and restricting the state-trajectory within a limits specified by the guard conditions. Even though the output voltage regulation is achieved and demonstrated, no theoretical work is presented to prove the stability of the system. In [29][30][31][32], the authors propose a Lyapunov-based hybrid control to stabilize the DC-DC converters. The study [29] basically proposes the switching rule that assigns the mode decreasing the value of the Lyapunov function most. When the trajectory reaches the switching boundary, it evolves as a sliding mode solution, which means that the switching interval becomes infinitesimally small. In [30,31], the authors use sampled-data control to avoid sliding mode solutions. Though switching frequency is controlled by the sampling period, it tends to be small because the method is based on sufficient conditions. Hence the trajectory is close to the sliding solution. Though our paper uses the same switching mechanism as in [32], the stability analysis is fundamentally different. A second-order boost converter model in [32] can be analysed by the standard Lyapunov approach by showing the decrease of the Lyapunov function along the trajectory. The derivative along the trajectory of the fourth-order zeta converter model becomes zero even if the point is not the operating point. Hence LaSalle's invariance principle proved for this class of differential inclusion should be established. The preliminary and much shorter versions of our work were presented in the conference proceedings [1,2].
The remainder of the paper is organized as follows. In Section 2, we establish a switching control mechanism for a twomode system. The two-mode system is instrumental to model the DC-DC zeta converter operating in continuous conduction mode (CCM). In Section 3, we analyse the stability of the switching system in Section 2, and apply it to the zeta converter. In Section 4, the switching control mechanism discussed in Section 2 is modified to limit the switching frequency of the zeta converter. In addition, by using linear-line approximation of the trajectory, we show how to decide the switching frequency. Simulation results to show the effectiveness of our proposed method are presented in Section 5. Lastly, in Section 6, we conclude our work and state the plan for future work.
Notation: ℝ denote the set of real numbers. For ∈ ℝ, ℝ > , ℝ < , ℝ ≥ , and ℝ ≤ denote the set of real numbers larger, smaller, larger than or equal to, and smaller than or equal to , respectively. The notation conv denotes the convex hull of a set. For a function: ℝ n → ℝ, −1 denotes the inverse image of . For a singleton {c}, c ∈ ℝ, we use the simplified notation, −1 (c)

SWITCHING CONTROL OF TWO-MODE SYSTEM
Because transistors and diodes exhibit on-off behaviours, many converters can be modelled as multi-mode systems, where each mode is described as a linear state-space model. A two-mode system can be used to model the CCM of any DC-DC converters. In this section, we introduce a Lyapunov-function-based switching control strategy and motivate its stability analysis.
Two linear systems of the same state dimension are given by where x(t ) ∈ ℝ n is the state and u(t ) ∈ ℝ is the input. Fix u 0 ∈ ℝ and ∈ ℝ. Assume that Note that we do not assume the stability nor the nonsingularity of the matrices A 1 and A 2 . Let  i (i = 1, 2) denote the set of stationary points of the systems (1) and (2); namely If A i is non-singular,  i is a singleton; otherwise it may be empty or infinite. The following proposition is easy to derive, but useful in the subsequent discussions.

Proposition 1.
The point x * is given by (3) if and only if it satisfies the following equation: Furthermore, Proof. Since the equivalence of (3) and (5) which implies x # = x * by the non-singularity of We are interested in a switching control law that drives the state of the switching system with the modes (1) and (2) to x * under u(t ) ≡ u 0 . For this, define a candidate Lyapunov function where P is a positive definite matrix. Because P > 0, there exist c 1 > 0 and c 2 > 0 such that holds. The derivatives of V (x) along the trajectories of (1) and (2) are respectively.

Proposition 2. Suppose that PA
−1 The proof is based on the following observation.

Proof of Proposition 2. Define
Notice that the assumptions of Lemma 1 are satisfied since (5) holds. Then the proof is immediate from Lemma 1.
□ Based on Proposition 2, we propose the following switching control mechanism.
Switching Mechanism A • If the system is operating in mode 1 and reaches −1 1 (0), then it switches to mode 2. • If the system is operating in mode 2 and reaches −1 2 (0), then it switches to mode 1.
Remark 1. Switching Mechanism A is initially proposed for a boost converter model in [32]. Proposition 2 clarifies a condition that ensures that Switching Mechanism A is well defined.
To analyse the stability of the switching control law, we consider the differential inclusion The set-valued map F : ℝ n →ℝ n is upper semi-continuous, and its values are bounded closed convex sets. Solutions of (12) include solutions of (1), (2) with the switching control mechanism. We shall analyse the stability of the operating point x * of the differential inclusion (12).
Remark 2. The switching mechanism that selects the mode defined by is in line with the method used in [29], except that [29] assumes that for some λ ∈ (0, 1). This mechanism does not stabilize the operating point asymptotically when we only assume PA 1 + A T 1 P ≤ 0 and PA 2 + A T 2 P ≤ 0. The following simple example If [ 0 1 ] x 0 = 0, then the differential inclusion has the unique solution x(t ) ≡ x 0 , which shows that it is not asymptotically stable. Note that this is not a counterexample of [29,Theorem 2]. Nevertheless, the example shows that the distinction of positive definiteness and positive semi-definiteness is meaningful.

STABILITY OF SWITCHING SYSTEM
In this section, we analyse the stability of the switching mechanism proposed in the previous section and apply the method to a DC-DC zeta converter operating in the CCM.

Stability analysis
Consider the differential inclusion (12) and the function V (x) in (6). DefineV (x) : ℝ n →ℝ bẏ It is easy to verify thaṫ The inverse imageV −1 (S ), where S ⊂ ℝ, is defined bẏ Proposition 3. Consider the differential inclusion (12) and the Lyapunov function (6). Then, . Let x * ,  1 , and  2 be defined by (3) and (4). Then By Proposition 5, the operation point x * is not globally asymptotically stable if  1 ∪  2 ≠ ∅. We shall study the local asymptotic stability of x * . The next result shows that x * is stable in this sense.
The proof of Theorem 1 hinges on a couple of Lemmas. The first one states that the operating point x * is stable.
Proof. Note that the set-valued function F (x) in (12) is defined for all x ∈ ℝ n from (10) in Proposition 2. Let > 0, and choose = by (13), and hence V ( (t, x 0 )) ≤ V (x 0 ) holds. This implies that ‖ (t, x 0 ) − x * ‖ 2 ≤ , and therefore x * is stable. □ One of the important observations is the property of the limiting set of a solution of a differential inclusion with an upper semi-continuous set-valued map. Let be a differential inclusion where F : ℝ n →ℝ n is upper semicontinuous and its values are bounded closed convex sets. Let (t, x 0 ) be a solution of (14). A point ∈ ℝ n is called a limit point of (t, The set of all limit points of (t, x 0 ) is called the limit set of (t, x 0 ) and is denoted as Ω.
Lemma 3. Consider the differential inclusion (14). Suppose that a solution (t, x 0 ) is bounded. Then the limit set Ω is non-empty, closed, and bounded. Furthermore, if ∈ Ω, then there exists a solution of (t, ) of (14) with initial condition x (0) = satisfying (t, ) ∈ Ω for all t ≥ 0.
Proof. The first half is elementary, see for example [33,Lemma 5.30]. To prove the second half, let {t k } be a sequence in [0, ∞) such that t k → ∞ and k := (t k , x 0 ) tends to ∈ Ω. Let T > 0 be fixed, and define k (t ) : Note that k (t ) is a solution of (14) with the initial condition x (0) = k . Using a similar argument as in [34,p.13,Theorem 4] and [34, p.104, Theorem 1], one can prove there exists and a convergent subsequence of { k } where limit is , and (t ) = (t, ) is a solution of the differential inclusion (14) with the initial condition x (0) = . Then any point on (t ) is a limit point of (t, x 0 ) and hence (t, ) ∈ Ω for 0 ≤ t ≤ T . Since this is true for any T > 0, this concludes the proof. □ Lemma 4. There exists r > 0 such that for every in the set and recall from (9) that It follows thaṫ1(x) is a continuous function; moreover, for almost all t . Let  = {t : 1 ( ( , )) > 0}. Note that  is an open set. If t ∈  , then by (12) )) is twice continuously differentiable, and This implies that ( (t, )) > ( (0, )) = 0 for some t . But  = ∅, and this is not possible. Hence, V ( ( , )) ≤ V ( (t, )) < V ( ) for some t ∈ [0, ]. The proof for ∈ ( −1 2 (0) ⧵ −1 1 (0)) ∩ N is similar. □ Proof of Theorem 1. Since (t, x 0 ) is bounded, its limit set Ω is an invariant set by Lemma 3. Since V (x) is bounded from below and V ( (t, x 0 )) is monotonically non-increasing for every sequence {t k } such that t k → ∞ as k → ∞, c := lim k→∞ V ( (t k , x 0 )) exists. If ∈ Ω, then there exists a sequence Remark 3. Theorem 1 is a consequence of LaSalle's invariance principle proved for the differential inclusion (12). This is a useful tool to prove the stability of the switching control applied to a DC-DC zeta converter in Section 3.2.

DC-DC zeta converter operating in continuous conduction mode
Consider the DC-DC zeta converter circuit shown in Figure 1. The circuit consists of two inductors L 1 and L 2 , two capacitors C 1 and C 2 , an ideal diode d and C 2 , a DC voltage source v g , a resistive load R, and an ideal switch S . Denote the currents of L 1 and L 2 as i L1 and i L2 , the voltages of C 1 and C 2 as v C 1 and v C 2 , respectively.
The converter is in CCM if the diode d is open when the switch S is on and it is shorted when the switch is off. When the switch is closed (mode 1), the converter is equivalent to the circuit shown in Figure 2, and when the switch is open (mode 2), the converter is equivalent to the circuit shown in Figure 3. With the state vector x = [i L1 i L2 v C 1 v C 2 ] T and the input u = v g , the matrices for the two modes are given by

FIGURE 3 Equivalent circuit when the switch is open
If u 0 > 0, then mode 1 has no stationary solution, and mode 2 has a unique stationary solution x * 2 = [ 0 0 0 0 ] T .
Remark 5. The switching mechanisms proposed in [29,30,31] basically pick up the mode which nearly decreases the Lyapunov function most while our method retains the mode as long as it decreases the Lyapunov function. The trajectory of [29] evolves as a sliding mode solution when it approaches the sliding boundary, which means that the switching interval becomes infinitesimally small. The sampled-data control approach in [30] and [31] can reduce the switching frequency by adjusting the sampling period. However, the period depends on the feasibility of matrix inequality, which is a sufficient condition and hence incurs conservativeness. Our method approaches a sliding mode solution only when the trajectory is near the operating point. Further reduction of switching frequency to the predetermined level is possible by using the modified switching mechanism stated in the next section.

LIMITING THE SWITCHING FREQUENCY
The switching control proposed in Section 2 requires unbounded number of switching as a solution approaches the operating point x * . In this section, the switching control mechanism discussed in Section 2 is modified to limit the switching frequency of a zeta converter.

Modified switching mechanism
The switching mechanism considered in Section 2 is based on the signs of the derivatives along the trajectories (8) and (9). Let 1 , 2 > 0 and̃1 and̃2 be modified switching functions which satisfy the following: Notice that (27) is equivalent tõ Based on (26) and (27), we propose the following modified switching control mechanism: Switching Mechanism B • If the system is operating at mode 1 and reaches̃− 1 1 ( 1 ), then it switches to mode 2. • If the system is operating at mode 2 and reaches̃− 1 2 ( 2 ), then it switches to mode 1.
The differential inclusion (12) is modified accordingly.

Example for the DC-DC zeta converter
From (19), Define Then (20) is a basis of ℝ 4 , and thus any x ∈ ℝ 4 can be written as The modified functions̃1(x) and̃2(x) can be defined as where ‖ 1 , 2 ‖ is any norm in ℝ 2 , and : ℝ ≥0 → ℝ ≥0 is a monotone non-decreasing function satisfying where Δi L2b Δv C 1b 0] T are the difference of the approximated statetrajectory from the operating point at their respective switching instants as shown in Figure 4. Observing Figure 4 and from (21) and (22), the gradient of the state-trajectory at the operating point is given by  (17)) the above expressions can be rewritten as Define penalty functions 1 := k 1 2 4 + (c 1 1 , 2 ) and 2 := k 2 2 4 + (c 2 1 , 2 ) and assume the state-trajectory near the operating point. Therefore, the penalty functions are close to 0 such that 1 ≈ 0 and 2 ≈ 0, consequently, 1 ≈ 1 (x * + Δx 1 ) and 2 ≈ 2 (x * + Δx 2 ). Nevertheless, the effect of 1 > 0 and 2 > 0 will be investigated and illustrated graphically later in Section 5. Therefore, with (17) and (36), and from (29) and (30), we have From (37) and (38), we observe how the desired switching frequency f is related to the thresholds 1 and 2 . Therefore, the DC-DC zeta converter will operate at the prescribed switching frequency under the modified switching rule. Though the expressions of 1 and 2 look complex, they are straightforwardly processed beforehand (offline). Nowadays, considering the capability of the high-speed processors like in the DSP, FPGA, or even (maybe) microcontroller, there should be no performance issue in executing the switching mechanism.

SIMULATION RESULTS
The simulations are carried out using the circuit simulation software PSIM® with the parameters shown in Table 1. With the input voltage v g , the capacitor voltage v C 2 , and the load current i o are the variables that are sensed in the circuit in Figure 1, and fixed number computation instead of floating-point computation is used practically to reduce computational burden; 1 and 2 in (37) and (38), respectively, can be rewritten as .
In Figure 5, the simulation results for 1 ≈ 0 and 2 ≈ 0 are shown. As can be seen, with the nominal v g = 18 V and i o = 2 A, no overshoot for the output voltage v o is observed at the start-up, and the settling time is approximately 10 ms. At t = 20 ms, v g drops to 9 V and i o reduces to 1 A. Despite the large input voltage drops, the overshoot at the output voltage is considerably small with some oscillations can be seen before it settles down at approximately t = 30 ms. Afterwards, at t = 40 ms, the input voltage drops further to 3 V and i o = 0.33 A. Similarly, although more oscillation and longer setting time are observed, nonetheless the output voltage is able to return to its operating point. Moreover, the converter is now operating in step-up mode (instead of step-down mode for the first two perturbations), thus proving the effectiveness of the switching control in regulating the output voltage at both operation modes. Finally, at t = 80 ms, the input voltage returns to its nominal value of 18 V. Although the increment is very significant (+500%), the switching control can regulate the output voltage well with minimum overshoot (approximately 10%) and considerably fast settling time (approximately 8 ms). On the other hand, the steady-state switching waveforms in close view for the three different input voltage perturbations are illustrated in Figure 6. As shown, the switching control algorithm is able to produce the desired switching frequency of appoximately 100 kHz for all three instances.
In the next simulation, the effect of introducing the penalty functions 1 > 0 and 2 > 0, defined in Section 4, are shown in Figure 7. As can be observed, the introduction of 1 and 2 does not have much effect on the response of the output voltage. Increasing 1 and 2 , hovewer, increases the switching frequency as shown in Figure 8. These observations are expected: (37) and (38) are no longer valid, since 1 and 2 are not approximately zero. As 1 and 2 reache 1 and 2 , respectively, the number of switching becomes unbounded, which is identical for the case of the switching control mechanism in Section 2.

CONCLUSION AND FUTURE WORK
Here, we have presented a switching control mechanism which induces closed-loop stability of the DC-DC zeta converter. A two-mode system is use to model the DC-DC zeta converter operating in CCM. A switching control algorithm is derived from a Lyapunov functional candidate which is basically the energy storage function of the zeta converter. Instrumental in our work is the establishment of an analysis of the fourth-order zeta converter that is simple but sufficient to prove the stability of the control. Moreover, our switching control mechanism is not only able to reduce the switching frequency, but most essentially one can systematically choose the desired switching frequency for the converter to operate. Important to highlight here is how we use two different thresholds 1 and 2 (for mode 1 and mode 2, respectively) to define the spatial regularization, as opposed to a single common threshold for the two modes as adapted in [32]. As a result, we were able to solve the switching control flaw in [32] by eliminating the output voltage steady-state error. Although the approximate state waveforms are used to find 1 and 2 , the close agreement between the theoretical and simulation results of the desired switching frequency shows that the approximation is indeed justified. In future, we plan to add the internal resistances in the zeta converter model, consider the effect of interference, and most importantly validate the findings with experimental results.