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This paper proposes a novel discrete-time sliding mode (DTSM) control approach to address the robust stability problem of buck converters with multiple disturbances. The contributions lie in the “unified” modelling and controller design. In modelling, all the possible model uncertainties and external disturbances are considered and further classified into two cases. It can also be extended to the situations with individual/several disturbances. While for the controller design, differing from the traditional DTSM based on the nominal model, the disturbances are directly introduced in the process, giving the robust stability condition and four new regulation subranges. It is suitable for both nominal and perturbed systems. Finally, the influences of the sampling time and disturbances on the control performance are investigated. Simulations and experiments confirm the benefits of the unified approach with greater accuracy and wider applications.

The issue of model uncertainty and external disturbance is widespread and challenging for the control of power converters. It might be related to the internal circuit elements, input power supply, measurement device, external environment interference, and so on [

As a kind of favourite DC/DC power converters, buck converters have been widely used in renewable energy systems, distributed generation systems, marine power plants, and so on [

For the control of buck converters, the pulse width modulation (PWM) is a conventional approach but suffers from the bottleneck of model precision, especially when facing the possible disturbances [

With the rapid development of programmed microprocessor chips, the discretization is another issue not to be avoided for SM controlled converters [

Based on the above analysis, a unified approach is proposed to address the robust stability problem of DTSM controlled buck converters with multiple disturbances. How to regulate the controller parameter and sampling time? What are the disturbances that affect the quality of the output voltage? They are two key questions. As a result, a unified approach of modelling and design of DTSM controller is proposed. To be specific, the contributions of this paper can be concluded as follows: (1) all possible model uncertainties and external disturbances of buck converters are considered in modelling and control, including the variations of input and reference output voltage, parameter uncertainties of load resistor, inductor, and capacitor, and time-varying external disturbances; (2) differing from the traditional DTSM approach [

The structure of this paper is organized as follows. In Section

^{n} denotes the

Figure _{L} and _{C} are the currents flowing through _{ref} is the DC reference voltage of _{1} and _{2} are defined as the output voltage error and its derivative as follows:

Diagram of a SM controlled buck converter system.

To realize the ON/OFF control of the power switch

Here, we assume that the buck converter works in continuous conduction mode. Based on the average state-space modelling approach and Kirchhoff’s circuit law, the ON/OFF operations of the buck converter in Figure

By combining (

In this paper, we consider all the possible model uncertainties and external disturbances. Therefore, (_{1} (_{2} (

Here, we define Δ_{ref} as the variation of the reference output voltage _{ref}. By combining (_{1}(_{2}(

Rewrite (_{1}, _{2}]^{T} and the nominal matrix _{1}, ∆_{2}(_{3}(

Observed from (_{ref}, ∆_{1}(_{2}(

In the following, a unified modelling approach is proposed to cover all the possible disturbances as well as the possible situations with individual/several disturbances. Therefore, two cases are discussed.

where _{1} + Δ_{2}(_{3}(

_{ref}, _{1}(_{2}(_{i}(_{j}, Δ_{1i}, and Δ_{2j},

Average state-space models of buck converters with individual disturbances.

Disturbing factors | Lumped disturbances | Average state-space model | Unified expression | |||
---|---|---|---|---|---|---|

Δ | Δ_{1} | Δ_{2}( | Δ_{3}( | |||

ΔE | √ | |||||

ΔC | √ | √ | √ | |||

ΔL | √ | √ | √ | |||

ΔR | √ | |||||

ΔV_{ref} | √ | |||||

_{1}(_{2}( | √ |

By combining (_{j} could hinder the controller design so that is categorized separately where

In the following, we adopt the zero-order holder (ZOH) to realize the discretization of the buck converter system [

By adopting ZOH, the state value can be held constant during the sampling period ^{2} and ^{3}) is the item with orders higher than ^{3} but ignored in this paper.

Furthermore, for (_{2}(_{2}(_{k} =

By combining (

Observed from (_{1}, Δ_{2}(_{3}(_{1} + Δ_{2}(_{3}(_{4}(_{3}(^{T}, where

It is worth noticing that the discrete model in (_{j}_{i}(_{i}(

From (_{i}(_{3i}(

In the following, a unified DTSM control approach is proposed based on the above unified modelling and analysis of the robust stability for buck converters. Still taking the worse situation with all the disturbances in (

Due to the advantage of easy implementation, the discrete signals _{1}(_{2}(

(robust stability analysis). For the discrete model of buck converter in (_{1}(_{2} (

Then, the whole closed-loop DTSM control system can be guaranteed stable.

In this paper, differing from the traditional DTSM approach based on the nominal model [

By substituting (

From (

Based on the existing condition of DTSM in (

If _{1}, _{2}), i.e., ((_{1}(_{2}(

If

Furthermore, by substituting (

For the items ^{2}^{2}/2) and ^{2}/2) in (

Therefore, by substituting (

From the guaranteed stability condition in Theorem _{3}(_{1}, _{2}).

(relationship of the sampling time and DTSM parameter). In order to guarantee the robust stability of DTSM controlled buck converter system in (_{1}; (2) _{1} < _{2}; (3) _{2} < _{3}; and (4) _{3}, where

Based on the guaranteed stability condition in (_{1}(_{2} (

For the comparative study imposed by the lumped disturbance _{3}(

Here, we define two variables _{1}(_{2}(

According to the ON/OFF operation of the power switch, as well as the two critical variables _{1}(_{2}(_{1}(_{2} (_{1}(_{2}(_{1}(_{2}(_{3}(_{1}, _{2}), four cases will be discussed in the following.

^{2}C^{2} + _{2}(_{1}(^{2}^{2} + _{1}(_{2}(^{2}^{2} +

_{0} − 1/^{2}^{2} + ^{2}^{2} + _{1}(_{2}(

_{1}(_{2}(

_{1}(_{2}(

Based on the robust stability analysis of the above four cases, the relationship of the sampling time

Robust stability regions: (a) ^{2}^{2} +

Differing from the traditional DTSM based on the nominal model [

Difference of the traditional and proposed DTSM parameter.

For the recommended choice of DTSM parameter _{1}(_{2} (

Considering the limitation of the hardware circuit, special attention should be paid to the subrange 0_{1}, where the sampling time is restricted with _{s} is expected to be less than half of sampling frequency based on the Shannon sampling theorem [_{s} < 1/4_{s} < 1/4_{1} in (_{2}, (2) _{2}_{3}, and (3) _{3} can be used to replace the four subranges in Theorem

For the lumped disturbance _{3}(_{3}(

(influence of the lumped disturbance on the steady-state error). For the DTSM controlled buck converter in (_{3}(_{3}(_{3}(

For the DTSM controlled buck converter in (_{1}, _{2}) can be calculated as (0, 0). From (_{1}(_{ref} and _{2}(

In Region I and Region II with lumped disturbances _{3}(_{1}(_{2}(_{1} axis and _{2} axis, i.e., _{1}(_{2}(_{1}(_{2}(

While in Region III and Region IV without lumped disturbances (nominal system), the four points of _{1} axis and _{2} can be obtained from (

By comparing the four points _{1}(_{2}(_{ref} varies within [0, _{3}(_{3}(

If we further let _{1}(_{2}(

By comparing (_{3}(_{3}(_{3}(

In order to validate the unified approach of modelling and controller design for the buck converter in Figure _{ref} = 9 V.

Comparisons of the output voltage with nine pairs of (

Output voltages affected by disturbances: (a) output voltages with different pairs of (

Performance comparisons with nine pairs of (

Λ | Label | Response time (ms) | Steady error (mV) | Relative steady error | |
---|---|---|---|---|---|

1 ms | 15 | 4 | 3.902 | 0.434‰ | |

60 | 4 | 3.902 | 0.434‰ | ||

250 | 4 | 3.902 | 0.434‰ | ||

0.5 ms | 15 | 9.5 | 0.465 | 0.052‰ | |

60 | 8 | 0.465 | 0.052‰ | ||

250 | 6.5 | 0.465 | 0.052‰ | ||

0.25 ms | 15 | 20.5 | 0.057 | 0.006‰ | |

60 | 15.8 | 0.057 | 0.006‰ | ||

250 | 7.5 | 0.057 | 0.006‰ | ||

0.5 ms (disturbance) | 15 | 9.5 | 112 | 1.244% | |

60 | 8 | 84 | 0.933% | ||

250 | 6.5 | 67 | 0.744% |

Based on the results of Theorems

Based on the relationship of the sampling time _{2} = 1/_{1} and _{3} are determined by the sampling time _{s} is expected to be less than half of sampling frequency, i.e., _{s} < 1/4_{3}, and (3) _{3}. In this paper, we choose the sampling time _{3} = 189.98, 109.99, and 70.47, respectively. Correspondingly from (

In Figure

Step 2: For the bounded lumped disturbance _{3}(^{T}, Δ_{1} = [0, 0.0025_{ref} sin(2^{T}, Δ_{2}(_{ref} sin(2^{T}, and Δ_{3}(_{ref} sin(2_{1}_{2})/^{T}. The reason of choosing the same amplitude 0.0025_{ref} for Δ_{1}, Δ_{2}(_{3}(

In Figure _{3}(

In order to validate this paper, Figure

DSpace experiment platform of the buck converter system: (a) structure diagram of the experiment system; (b) experiment flow.

In order to prove the proposed unified approach to be suitable for diverse disturbances, four working modes of buck converters are selected as examples: (1) time-varying noise 0.05sin(2_{ref} = 0.2 V; (3) load resistor jumps from 50 Ω to 100 Ω and then back to 50 Ω; and (4) input voltage changes from 16 V to 25 V and then back to 16 V.

In experiment, the sampling time

Experiment results of output voltages.

In this paper, the robust stability problem of DTSM controlled buck converters is investigated and a unified approach of modelling and controller design is proposed for all the possible model uncertainties and external disturbances. In modelling, two unified expressions are proposed to describe the possible individual/several/all disturbances. In the design of DTSM controller, the disturbances are directly included in the process, instead of ignoring them in the traditional approach. Innovatively, four more accurate subranges arise for the controller regulation. Based on the robust stability analysis, the influences of the sampling time and disturbances on the control system have also been investigated. Simulations and experiments validate the effectiveness and wider applications of the proposed approach.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China under Grants 5130703 and 61673132, and the authors would like to thank the experimental support from the 716 Research Institute, China Shipbuilding Industry Corporation.