**1. Introduction**

Pulse width-modulated (PWM) DC-DC converters are typical piecewise-smooth dynamical systems [1], and an external control loop is required for them to produce a precise and stable output voltage. Because of some desirable features, such as fast dynamic response, automatic overload protection, good current sharing, current limiting, and so on [2], the peak-current-mode (PCM) control has been widely applied [3].

However, it is commonly accepted that converters with PCM control are confronted with instability issues [4,5]. A wide variety of nonlinear dynamic phenomena, such as bifurcations and chaos, have been observed in these converters [6–13], which could deteriorate the performances of the converters, and are undesired in practice. As a result, the active duty ratio of PCM control in DC-DC converters is usually restricted in the range of (0, 1/2) in continuous-conduction-mode (CCM) and (0, 2/3) in discontinuous-conduction-mode (DCM) [14,15]. Hence, these limitations in turn lead to a restrained application of DC-DC converters. In renewable energy grid-connected power systems, for example, DC-DC converters with high step-up gain are needed, ye<sup>t</sup> the classical boost topology

cannot service the request in these scenarios [16]. An approach to overcoming this drawback is to build new topologies [17–21]. Unfortunately, some unavoidable features of these new proposed topologies, especially a complicated structure, have brought di fficulties in design and burdened costs in scalable applications.

Building new topologies is not the only solution that can be found; in fact, improving control strategies is another option. For example, hybrid predictive control is designed to improve the features of a boost converter operating in both CCM and DCM operation [22,23]. Additionally, slope compensation is the conventional strategy preferred by engineers [24]. According to analysis works on PCM-controlled DC-DC converters with CCM operation, when the active duty ratio is beyond the range of (0, 1/2), the quality factor, Q, of the control-to-inductor transfer function is negative [24,25]. It means that the eigenvalues of the converters distribute on the right half of the *S* plane, and the converters are in an unstable state. However, by adding a su fficient compensation ramp signal to the current loop, the factor can turn into a positive value, which causes the converters to return to a stable state.

Unfortunately, for the conventional ramp compensation technique, there is an over-compensation (low Q) or under-compensation (high Q) problem that would limit the control bandwidth. Therefore, a dynamic ramp compensation scheme was put forward in [26]. Another drawback associated with the conventional ramp compensation technique is that the use of constant slope compensation reduces some of the benefits of PCM control. For example, the peak value of the inductor current deviates from the desired reference, which is not desirable in applications where accurate tracking of the reference signal is needed [27]. Hence, a self-compensation technique was proposed in [27], which can provide a more accurate current limiting capacity and does not require an external slope generator. In [28], a time-varying ramp compensation technique was presented to eliminate the fast-scale instability of a PCM-controlled power factor correction (PFC) boost converter. In [29], adaptive ramp compensation was proposed to improve the robustness of the conventional ramp compensation technique. Yet, the inherent over-compensation or under-compensation problem has still not been solved well.

In this paper, a new technique based on the parameter-perturbation method [30] is proposed, which can make PCM control work in a wider duty ratio range without using ramp compensation. The parameter-perturbation method is a recently proposed chaos control strategy, which aims to stabilize a chaotic system to a desired unstable period orbit (UPO). This method applies only very small perturbations to a carefully chosen system parameter once per control period. Compared with some classical chaos control methods [31], the parameter-perturbation method does not require the unstable fixed point to be a saddle node, and thus there is no limitation on the types of targeting UPOs. Hence, this method is applied to enhance the performance of a PCM-controlled DC-DC converter. In some works on nonlinear dynamic analysis of PCM-controlled DC-DC converters [8,10,13], the duty cycle of PCM control can be simply approximated by a function of the reference current. Therefore, the reference current of PCM control is used as a parameter to be perturbed in this paper. Then, the UPO-1 of the converters can be calculated by the real-time sampling data of inductor current and capacitor voltage, and the stabilizer can be designed.

The rest of the paper begins with a quick glimpse of the typical limitations of a boost converter with classical PCM control in Section 2. The principles of the parameter-perturbation method and the detailed procedures of implementing this method are presented in Section 3. Section 4 analyzes the performances of the boost converter with the proposed control scheme. In Section 5, experiments are performed for a further verification of the proposed scheme. Finally, the conclusion is outlined in Section 6.

#### **2. Typical Limitation of Boost Converter with Classical PCM Control Scheme**

The circuit diagram of the boost converter is shown in Figure 1a, where *r*L, *r*C, and *r*T denote the parasitic resistances of the inductor, *L*, the capacitor, *C*, and the switch, *ST*, respectively. The control loop consists of a comparator, and an R-S flip-flop. The operation can be briefly described as follows. The flip-flop is set periodically by the clock signal, turning on the switch, *ST*. Then, the inductor current, *iL*, goes up linearly, and is compared with the reference level, *I*ref. When the peak value of *iL* reaches the level, *I*ref, the output of the comparator resets the flip-flop, thereby turning off *ST*. When *ST* is <sup>o</sup>ff, the inductor current falls almost linearly.

**Figure 1.** Boost converter with a classical PCM control scheme: (**a**) Basic circuit diagram; (**b**) typical steady-state waveforms.

Assuming that the converter operates in CCM, there are two operating modes depending on whether or not *ST* is on, and typical waveforms of *i*L and the clock signal are shown in Figure 1b. The PCM-controlled boost converter can be described by:

$$\begin{cases}
\dot{\mathbf{x}} = \mathbf{A}\_1 \mathbf{x} + \mathbf{B}\_1 E, \quad nT\_s \le t < (n + d\_n)T\_s \\
\dot{\mathbf{x}} = \mathbf{A}\_2 \mathbf{x} + \mathbf{B}\_2 E, \quad (n + d\_n)T\_s \le t < (n + 1)T\_s
\end{cases} \tag{1}$$

where *E*, *Ts*, and *dn* denote the input voltage, switching period, and duty ratio in the *n*th cycle, respectively. The state vector was set to be **x** = [*i*L, *u*O]Tr, where the superscript 'Tr' means the transposition of a matrix. The system matrices, **A***i* (*i* = 1, 2) and **B***i* (*i* = 1, 2), are given by:

$$\mathbf{A}\_{1} = \begin{bmatrix} -\frac{r\_{\mathrm{T}} + r\_{\mathrm{L}}}{L} & 0\\ 0 & -\frac{1}{C(R + r\mathrm{C})} \end{bmatrix}, \mathbf{A}\_{2} = \begin{bmatrix} -\frac{r\_{\mathrm{L}}}{L} & -\frac{1}{L} \\\ \frac{R(L - r\_{\mathrm{C}}r\_{\mathrm{L}}\mathbb{C})}{C(R + r\mathrm{C})} & -\frac{L + R\mathrm{C}r\mathrm{C}}{LC(R + r\mathrm{C})} \end{bmatrix} \tag{2}$$

and:

$$\mathbf{B}\_1 = \begin{bmatrix} \frac{1}{L} \\ 0 \end{bmatrix}, \mathbf{B}\_2 = \begin{bmatrix} \frac{1}{L} \\ \frac{Rr\_\mathbf{C}}{L(R+r\_\mathbf{C})} \end{bmatrix}. \tag{3}$$

A stroboscopic map is a discrete map or iterative map obtained by sampling a continuous system periodically, which describes the dynamics of a discrete variable in terms of a difference equation. According to (1), the stroboscopic map can be obtained as:

$$\begin{aligned} \mathbf{x}\_{n+1} &= \mathbf{F}(\mathbf{x}\_{\text{tr}}, I\_{\text{ref}})\\ &= \Phi\_2((1 - d\_n)T\_s) \left[ \Phi\_1(d\_n T\_s) \mathbf{x}\_n + \Psi\_1(d\_n T\_s) E \right] + \Psi\_2((1 - d\_n)T\_s) E, \end{aligned} \tag{4}$$

in which **x***n* and **x***n*+1 denote the state vector at the instant of *t* = *nTs* and *t*= (*n* + 1)*Ts*, respectively. **Φ***i*(ξ) and **Ψ***i*(ξ) are calculated by the following equations as:

$$\Phi\_i(\boldsymbol{\xi}) = e^{\mathbf{A}\_i(\boldsymbol{\xi})} = \mathbf{I} + \sum\_{k=1}^{\infty} \frac{1}{k!} \mathbf{A}\_i^k \boldsymbol{\xi}^k, \quad \mathbf{\varPsi}\_i(\boldsymbol{\xi}) = \int\_0^{\boldsymbol{\xi}} \boldsymbol{\Phi}\_i(\boldsymbol{\tau}) \mathbf{B}\_i d\boldsymbol{\tau}, \tag{5}$$

where 'I' is a symbol of the unit matrix, and the subscript *i* = 1, 2. The switching function is defined as:

$$s(\mathbf{x}\_{n\prime}, d\_n) = d\_n - (I\_{\text{ref}} - i\_{\text{Ln}}) / m\_1 T\_{s\prime} \tag{6}$$

in which *m*1 = *E*/*L* is the rising slope of *i*L, and *d*n can be determined by setting the switching function to be zero, i.e., *<sup>s</sup>*(**<sup>x</sup>**n, *dn*) = 0.

Parameters of the converter are listed in Table 1, which were chosen to be the same as those in [32]. When *I*ref is changed from 0.5 to 5.5 A, typical bifurcation diagrams of *i*L and *u*C versus *I*ref can be obtained by numerical simulation. Bifurcation is the sudden change of the qualitative behavior of a system when one or more parameters are varied. Bifurcation literally means splitting into two parts. In nonlinear dynamics, the term has been used to mean the splitting of the behavior of a system at a threshold parameter value into two qualitatively different behaviors, corresponding to parameter values below and above the threshold. So, a bifurcation diagram is a summary chart of the behavioral changes as some selected parameters are varied. DC-DC converters are typically nonlinear, and the bifurcation diagram has become a very common tool for analysis of the dynamic behaviors of converters. It can be seen in Figure 2 that the period doubling bifurcation occurs at *I*ref = 1.52 A with *d* = 0.39 and the voltage set-up ratio *M* = 1.64. Along with the increase of *I*ref, the converter undergoes several period doubling bifurcations, and evolves into a chaotic state eventually. Obviously, when the system operates in a chaotic state, there are usually different kinds of values of *u*C corresponding to the same *I*ref. Then, it is desirable to stabilize the system from a chaotic state to the target period-1 orbit with larger *u*C, by some kind of chaotic control method, such as the method based on parameter perturbation.

**Table 1.** Circuit parameters of the PCM-controlled boost converter.


**Figure 2.** Bifurcation diagram of a classical PCM-controlled boost converter: (**a**) For the inductor current *i*L versus *I*ref; (**b**) For the output capacitor voltage *u*C versus *I*ref.

#### **3. An Improving Scheme Based on the Target UPO Tracking Method**

#### *3.1. Principle of the Parameter-Perturbation Method*

The parameter-perturbation method is designed to stabilize a discrete chaotic system on a desired unstable period-1 orbit (UPO-1) [30]. For a two-dimensional discrete system, it can be described by the following equation as:

$$\mathbf{x}\_{n+1} = \mathbf{F}(\mathbf{x}\_{n}, \mathbf{p}\_{n}),\tag{7}$$

where **F**(·) is a smooth vector function, **x** ∈ *R*<sup>2</sup> is state vector of the system, *P* is a parameter that can be changed in a neighborhood of a nominal value of *P* [30], and the subscript *n* denotes the *n*th iteration. Assuming that the system is in chaotic state, and has an unstable fixed point, **X***P*, the following equation can be obtained:

$$\mathbf{X}\_{\mathcal{P}} = \mathbf{F}(\mathbf{X}\_{\mathcal{P}}, P). \tag{8}$$

Then, in a sufficiently small neighborhood of **X***P*, the system (Equation (7)) can be approximated by a linear map as:

$$\mathbf{x}\_{n+1} = \mathbf{J}\_x(\mathbf{x}\_n - \mathbf{X}\_P) + \mathbf{J}\_p(p\_n - P) + \mathbf{X}\_P \tag{9}$$

where the coefficient matrices, **J***x* and **J***<sup>p</sup>*, are defined as:

$$\mathbf{J}\_{\mathbf{x}} = \left. \frac{\partial \mathbf{F}(\mathbf{x}\_{n\prime}, p\_n)}{\partial \mathbf{x}} \right|\_{(\mathbf{X}\_{p\prime}, P)} , \mathbf{J}\_p = \left. \frac{\partial \mathbf{F}(\mathbf{x}\_{n\prime}, p\_n)}{\partial p} \right|\_{(\mathbf{X}\_{p\prime}, P)} . \tag{10}$$

According to Equation (9), two steps of perturbations are needed for the selected parameter, the iterative function for **x***n*+2 can be obtained, that is:

$$\mathbf{x}\_{\rm ll+2} = \mathbf{J}\_{\mathbf{x}}^2 \mathbf{(x}\_{\rm ll} - \mathbf{X}\_{\rm p}) + \begin{bmatrix} \mathbf{J}\_{\mathbf{x}} \mathbf{J}\_{\mathbf{p}} & \mathbf{J}\_{\mathbf{p}} \end{bmatrix} \begin{bmatrix} p\_n - P \\ p\_{n+1} - P \end{bmatrix} + \mathbf{X}\_{\rm p} \tag{11}$$

and it is the expression for the UPO-1 that corresponds to **X***P*.

When the system is stabilized to the desired UPO-1, then **x***n*+2 = **x***n*+1 = **X***P*, and the following perturbation increments for *Pn* can be acquired by:

$$
\begin{bmatrix}
\Delta p\_n\\\Delta p\_{n+1}
\end{bmatrix} = \begin{bmatrix}
p\_n - P\\p\_{n+1} - P
\end{bmatrix} = \begin{bmatrix}
\mathbf{J}\_\mathbf{x} \mathbf{J}\_p & \mathbf{J}\_p
\end{bmatrix}^{-1} \mathbf{J}\_\mathbf{x}^2 \begin{pmatrix}
\mathbf{X}\_p - \mathbf{x}\_n
\end{pmatrix} = \mathbf{M} \begin{pmatrix}
\mathbf{X}\_p - \mathbf{x}\_n
\end{pmatrix}.
\tag{12}
$$

## *3.2. Implementation Scenario*

This subsection provides a scenario to illustrate the common functionality that can be implemented in a PCM-controlled boost converter. Based on dynamic analysis of the boost converter with a classical PCM control scheme, one ensures that the converter operates in a chaotic state when the reference level is set to be *I*ref = 3 A, and the other parameters are shown in Table 1. According to the basic principle of the parameter-perturbation method above, let **x***n*+1 = **x***n* in Equation (4), one can obtain the unstable fixed point as:

$$\mathbf{X}\_{\mathfrak{p}} = \begin{bmatrix} I\_{\mathbf{L}\mathfrak{p}} \ \mathcal{U}\_{\mathbf{O}\mathfrak{p}} \end{bmatrix}^{\mathbf{Tr}} = \begin{bmatrix} 2.4344, \ 26.2895 \end{bmatrix}^{\mathbf{Tr}}.\tag{13}$$

The perturbing values for the reference current can be calculated according to Equation (12), and the reference current becomes:

$$I\_{\text{refn}}^{(p)} = \Delta I\_{\text{refn}} + I\_{\text{ref}}.\tag{14}$$

By using the above computed perturbations, the UPO-1 of the converter can be found, which will be in the form of Equation (11). The schematic diagram of the stabilizer can be designed. As shown in Figure 3, in addition to the conventional peak current mode control scheme, a perturbation module is added to generate the disturbance increments for the controlled parameter. Additionally, the reference

level in each period is the sum of the nominal value, *I*ref, and the disturbance increments, Δ*I*refn, which is calculated by the perturbation module at the beginning of the *n*th period according to Equation (14). The additional switch, *S*, in the control loop is used to enable and disable this target orbit control scheme.

**Figure 3.** Circuit diagram of a PCM-controlled boost converter with a parameter-perturbation module.

Figure 4a,b respectively show the simulated time-domain waveforms and the phase trajectories of the state variables at *I*ref = 3 A. Obviously, it can be seen from Figure 4a that before *t* = 0.02 s, the conventional PCM-controlled boost converter operates in a chaotic state, with larger ripples accompanying the inductor current and output voltage. When the switch, *S*, is enabled at the instant of *t* = 0.02 s, the proposed control scheme comes into force. Additionally, the system can be stabilized quickly from the chaotic state to the desired target period-1 orbit (UPO-1), as shown in Figure 4b. Moreover, the peak-peak values of the inductor current and output voltage are reduced to 0.57 A and 5.5 V from 1.45 A and 15 V, respectively.

**Figure 4.** Simulations of the boost converter with the proposed control scheme: (**a**) Time-domain waveforms of state variables; (**b**) chaotic attractor and the stabilized UPO-1.

## *3.3. Performance Assessment*

The performances of the converter can be evaluated by checking the movement of the eigenvalues when some chosen circuit parameters are varied. Any crossing from the interior of the unit circle to the exterior indicates a bifurcation. Particularly, if a real eigenvalue goes through −1 as it moves out of the unit circle, a period-doubling occurs.

According to the stroboscopic map of the boost converter, the Jacobian matrix evaluated in the neighborhood of **X***p* (the equilibrium point) is defined as:

$$\mathbf{J}\left(\mathbf{X}\_{\mathcal{P}}\right) = \left.\frac{\partial \mathbf{F}}{\partial \mathbf{x}\_n} - \frac{\partial \mathbf{F}}{\partial d\_n} \left(\frac{\partial \mathbf{s}}{\partial d\_n}\right)^{-1} \frac{\partial \mathbf{s}}{\partial d\_n}\right|\_{\mathbf{x}\_n = \mathbf{X}\_{\mathcal{P}}}\tag{15}$$

one can ge<sup>t</sup> the following derivatives by using Equations (4) and (6):

$$\frac{\partial \mathbf{F}}{\partial \mathbf{x}\_{\rm nl}} = \Phi\_2((1 - d\_{\rm n})T\_s)\Phi\_1(d\_{\rm n}T\_s),\tag{16}$$

$$\begin{array}{ll}\frac{\partial \mathbf{F}}{\partial d\_{n}} &= T\_{s} \Phi\_{2}((1-d\_{n})T\_{s})(\mathbf{A}\_{1}-\mathbf{A}\_{2})\Phi\_{1}(d\_{n}T\_{s})\mathbf{x}\_{n} \\ &- T\_{s} \mathbf{A}\_{2} \Phi\_{2}((1-d\_{n})T\_{s})\mathbf{A}\_{2}^{-1}\mathbf{B}\_{2}E+\Phi\_{2}((1-d\_{n})T\_{s})\mathbf{A}\_{1}^{-1}\mathbf{B}\_{1} \\ &\cdot E\left[-T\_{s}\mathbf{A}\_{2}\Phi\_{2}((1-d\_{n})T\_{s})+T\_{s}\mathbf{A}\_{1}\Phi\_{1}(d\_{n}T\_{s})+T\_{s}\mathbf{A}\_{2}\right] \\ \end{array} \tag{17}$$

$$\frac{\partial \mathbf{s}}{\partial d\_n} = \frac{(r\mathbf{L} + r\mathbf{r})T\_s}{L} \Big(\dot{\mathbf{u}}\_{\mathbf{L}} - \frac{E}{r\_{\mathbf{L}} + r\_{\mathbf{T}}}\Big) \mathbf{e}^{-\frac{r\_{\mathbf{L}} + r\mathbf{T}}{L}d\_nT\_s} \Big. \tag{18}$$

$$\frac{\partial \mathbf{s}}{\partial d\_{\mathrm{ll}}} = \left[ 0.5042 - e^{-\frac{\eta\_{\mathrm{L}} + \tau\_{\mathrm{T}}}{L} d\_{\mathrm{u}} T\_{\mathrm{s}}}, \ -0.0002 \right]. \tag{19}$$

Additionally, by introducing the above derivatives into Equation (15), it leads to the following characteristic equation as:

$$\det\left(\lambda\mathbf{I} - f(\mathbf{X}\_{\mathbb{P}})\right) = 0.\tag{20}$$

Then, eigenvalues can be obtained by solving Equation (20).

When the reference level is set to be *I*ref = 3 A, the loci of eigenvalues of the converter with the proposed control technique is provided in Figure 5, in which the load and source disturbances are considered. For the variation of the load resistance, *R*, from 1 to 90 Ω, the loci of the eigenvalues can be obtained as shown in Figure 5a. From Figure 5a, the boost converter is stable when the load resistance is in the range of 1 to 57.4 Ω. The first point of period doubling bifurcation occurs at *R* = 57.4 Ω when one of the eigenvalues equals −1. After that, an eigenvalue moves out of the unit circle with the increase of the load resistance, *R.* For the variation of the input voltage, *E*, from 1 to 20 V, the loci of eigenvalues is depicted in Figure 5b. One can see that the converter is stable when the input voltage, *E*, is in the range of 4.95 to 20 V. Otherwise, the converter will be in an unstable state.

**Figure 5.** Loci of eigenvalues of a PCM-controlled boost converter with the proposed technique: (**a**) Arrows indicate the direction of movement of the eigenvalues with *R* increasing, (**b**) Arrows indicate the direction of movement of the eigenvalues with *E* increasing.

As a comparison, the loci of eigenvalues of the converter with the conventional PCM control technique is provided in Figure 6. It can be seen from Figure 6 that with the conventional PCM control scheme, the boost converter is stable for the load resistance being in the range of 1 to 3.06 Ω, and the source voltage, *E*, being in the range of 18.78 to 20 V.

**Figure 6.** Loci of eigenvalues of the classical PCM-controlled boost converter: (**a**) Arrows indicate the direction of movement of the eigenvalues with *R* increasing; (**b**) Arrows indicate the direction of movement of the eigenvalues with *E* increasing.

From Figures 5 and 6, the stable operating range for input disturbance is expanded remarkably from 18.78 to 20 V to 4.95 to 20 V, and similarly, the stable range for the load variation is also enlarged from 1 to 3.06 Ω to 1 to 57.4 Ω. Obviously, the performances of the classical PCM control scheme can be improved with the proposed stabilizer.

#### **4. Comparison and Discussion**

The general remedy to avoid the chaotic operation state of PCM-controlled converters is to introduce a compensating ramp to *I*ref. The basic circuit diagram of a PCM-controlled boost converter with ramp compensation and the typical steady-state waveforms are shown in Figure 7, where *m*c denotes the slope of the compensating signal.

**Figure 7.** PCM-controlled boost converter with ramp compensation: (**a**) Basic circuit diagram; (**b**) typical steady-state waveforms.

According to the ramp compensation technique as shown in Figure 7, the reference current is given by the following equation:

$$I\_{\rm ref}^{(\varepsilon)} = I\_{\rm ref} - m\_{\rm c}(\rm tmod \, T), \tag{21}$$

where the term '*t* mod *T*' means a modulo operation, which is equivalent with '*t*-*kT*', and *k* is the integer quotient of *t*/*T*. Additionally, if the self-compensation technique in [25] is adopted, the reference can be described by:

$$I\_{\rm ref}^{(s)} = I\_{\rm ref} - \frac{1}{T} \int\_0^t [I\_{\rm ref} - \frac{m\_1 DT}{2} - i\iota\_\cdot(\tau)] d\tau. \tag{22}$$

In Equations (21) and (22), *m*1 is the rising slope of the Boost converter. The reference current of the converter is set to be *I*ref = 3 A, and all the other circuit parameters are chosen as the same as those listed in Table 1. Additionally, to confirm that the converter with a constant slope of *mc* operates in a stable state, bifurcation analysis is performed. According to the results in Figure 8, period-doubling bifurcation occurs when *mc* = 3220 V/s. Then, in the following simulations, the value of *mc* is set to be 3250 V/s, and the converter can attain a higher voltage output than that in the case of *mc* > 3250 V/s.

**Figure 8.** Bifurcation diagram of a PCM-controlled boost converter with constant ramp compensation.

To perform a comparison, both the conventional ramp compensation technique and the selfcompensation technique are adopted in simulations. The behaviors and characteristics of the converter are put together with the proposed technique. The waveforms of applying different techniques to stabilize the PCM-controlled boost converter can be seen in Figure 9, where a transition from a chaotic state to a stable state is depicted.

**Figure 9.** Stabilizing PCM-controlled boost converter by different techniques.

In Figure 9, the red and black solid lines represent the reference current and the inductor current of the converter, respectively. It can be seen when stabilizers come into e ffect at *t* = 0.02 s, the converter with both the self-compensation technique and the proposed technique can be stabilized quickly. Yet, the converter with the conventional ramp compensation technique takes more than 20 switching cycles to enter a stable state.

Furthermore, the simulated stable current and voltage waveforms of the boost converter with di fferent control strategies are provided in Figure 10. The black solid lines represent the results with the conventional ramp compensation technique, the blue dotted lines are obtained by using the self-compensation technique, and the red solid lines result from the proposed technique. It is seen that the three control techniques can be used to stabilize the converter from a chaotic state. Additionally, by setting UPO-1 as the control objective, the proposed technique can achieve a performance very close to that of using the self-compensation technique. However, the output voltage of the converter with the conventional ramp compensation technique is a little lower than those with the self-compensation technique and the proposed technique. Moreover, both the self-compensation technique and the proposed technique can ensure that the peak value of the inductor current does not deviate from the desired reference, whereas the conventional slope compensation technique cannot.

**Figure 10.** Stable waveforms of the PCM-controlled boost converter with di fferent techniques.

Additionally, with variations of the load from 2 to 80 Ω, and the input voltage from 0.5 to 20 V, respectively, the performances of the boost converter with three di fferent kinds of control technique are compared, as listed in in Table 2. According to the simulation results summarized in Table 2, the boost converter has almost the same voltage gain under the proposed control technique and the self-compensation method. Both methods provide an accurate current limiting capacity, and their robustness is better than the conventional ramp compensation technique.


