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\begin{frontmatter}

\title{Fixed time control based on finite time observer for Buck converter system with load disturbance}
% Title, preferably not more than 10 words.

\thanks[footnoteinfo]{Sponsor and financial support acknowledgment
goes here. Paper titles should be written in uppercase and lowercase
letters, not all uppercase.}

\author[First]{Xiaolong Tian}
\author[First]{Fazhan Tao}
\author[First]{Zhumu Fu}
\author[First]{Nan Wang}
\author[First]{Shuzhong Song}

\address[First]{College of Information Engineering, Henan University of Science
and Technology, Kaiyuan Avenue, Luoyang, 471023, Henan,
China(e-mail: xintianlinglong@163.com, taofazhan@haust.edu.cn, fuzhumu@haust.edu.cn, wswn2019@163.com, sszhong@haust.edu.cn).}

\begin{abstract}                % Abstract of not more than 250 words.
To improve transient response and reduce current ripple, a fixed time voltage regulation control strategy based on finite time observer for Buck converter is proposed in this paper. Different from the previous methods of adding additional circuits, the proposed fixed time controller combined with load finite time observer can be directly applied in integrated circuits. Meantime it avoids the chatter and vibration of sliding mode control strategy, relaxing constraint on initial condition of finite time control strategy. Firstly, a finite time observer is designed to estimate the uncertain load resistance. Secondly, combined with the backstepping method, a fixed time voltage regulation control strategy is designed by fixed time control theory, so that the output voltage has a credible ability to converge to the reference voltage in a fixed time. Finally, the proposed control strategy is verified by comparing with the finite time controller\cite{WB:8} and sliding mode controller\cite{WB:13}, where the obtained simulation and experiment results show the proposed strategy can make better performance in terms of convergence speed and ripple reduction when the reference voltage and the load change.
\end{abstract}

\begin{keyword}
Buck converter, finite time observer, fixed time control, fast response, backstepping method.
\end{keyword}

\end{frontmatter}
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\section{Introduction}
Switching DC-DC converter can be divided into Buck converter, Boost converter and Buck-Boost converter according to the topological structure. Buck converter is a simple converter, being widely used in various intelligent devices. For occasions with high real-time requirements, it is urged to consider the transient response in addition to  power conversion efficiency. However, due to the characteristics of nonlinear and time-varying of Buck converter, the ability of response is not sufficient and satisfied, which will cause instability of whole system. Moreover, the load changes slowly over time, making the model information not accurate and leading to the failure of the control strategy. It is necessary to investigate the approaches to improve the transient response of Buck converter.

In order to improve the transient response of Buck converter, researchers have focused on two areas to improve performance: (1) adding additional circuits; (2) adopting nonlinear control methods.

To realize fast transient response, some additional auxiliary switches are added to change the equivalent output inductance \cite{WB:1}. To track the output voltage of Buck converter, an additional auxiliary converter is designed as the load changes \cite{WB:2}. This method adds inductance and power transistors, it is not suitable for applying to integrated circuits. The authors of \cite{WB:3} adds an error amplifier to the pulse width modulation (PWM) circuit to achieve transient response by accelerating the output level drift of the error amplifier. However, the driving ability is related to the size of the external transistor. Therefore, to obtain sufficient driving ability, a certain scale of area of chips has to be occupied forcibly. To acquire fast transient response, limiting error amplifier and fast error signal control are utilized \cite{WB:4}. Thus, limited by size and cost, more and more researchers are focusing on improving the transient response of Buck converter with nonlinear control methods.

In the aspect of adopting nonlinear control methods, finite time control \cite{WB:5}-\cite{WB:9}and sliding mode control \cite{WB:10}-\cite{WB:14} are mainly used. Based on finite time control technique, a fast control strategy for Buck converter is proposed to realize the output voltage converges to the reference voltage in finite time \cite{WB:5}-\cite{WB:8}. But the upper bound of convergence time is related to the change of initial state. As the initial state of the system changes, the upper bound of convergence time also changes correspondingly. In \cite{WB:9}, the problem of the common current controller of the parallel Buck converter system is studied, and a new finite time current sharing control strategy is designed. The output voltage of the converter system can reach the desired reference voltage in a limited time. With the application of sliding mode control methods, many researchers have started to use sliding mode control to improve the response speed of Buck converter. A digital fast terminal sliding mode control method is studied for Buck converter with mismatched interference, which has high accuracy for tracking voltage and great dynamic characteristics under different working conditions \cite{WB:10}. In \cite{WB:11}, a sliding mode adaptive Buck control method is proposed for Buck converter system. The sliding mode surface and controller are established to ensure that the output voltage error converges to the neighborhood near the equilibrium point. In \cite{WB:12}, to improve the performances of fast transient response, a sliding mode control combined with fixed time control is proposed. However, in the actual sliding mode control \cite{WB:10}-\cite{WB:14}, due to the uncertainty of the model and the existence of interference, chattering phenomenon and steady-state error always exist and bring many bad effects.

It is worth pointing out that there are three problems existed on the two aspects mentioned above to solve the transient response of Buck converter.

Firstly, the aforementioned increased circuit structure generally has the problem of increasing cost or volume, which is difficult to be applied in integrated circuits. Finite time control is related to the initial state of the system, and the convergence speed can not meet the real-time requirements. Sliding mode control has a frightening problem of chattering, which makes the stability requirement of electric vehicles and other occasions difficult to be satisfied. Furthermore, it is difficult to design and construct the sliding mode surface.

Secondly, in the above researches, less attention is paid to the ripple in the circuit. The transient response of the load is affected by the equivalent output inductance. The smaller the inductance, the faster the transient response. However, reducing the equivalent inductance will increase the current ripple and reduce the efficiency of the converter \cite{WB:15}. The larger the ripple current is, the greater the heat will be. It will shorten the service life of the capacitor and lead to breakdown in serious cases. There are many researches on ripple suppression of Buck converter \cite{WB:15}-\cite{WB:17}, but most of them are realized by changing the circuit structure, being not easy to realize.

Thirdly, most of the above researches need to establish accurate converter state space model. In the actual system, the external load often changes with time, and this information is often impossible to know precisely, because of the inability to obtain accurate parameter knowledge, resulting in stable but not ideal transient response. Therefore, some scholars have carried out researches on finite time observer. For the Buck converters without current signal, the finite time output feedback controller based on the observer is designed by using finite time control and finite time convergent observer technology, which ensures that the system converges to the equilibrium state in finite time\cite{WB:21}. A non-smooth current constrained control algorithm is proposed for buck converters with system parameter uncertainties and disturbances. This control algorithm prevents transient current from being too large, but sacrifices the transient performance of the system\cite{WB:22}. The methods with and without disturbance observer are analyzed and compared, which results show that the control strategy with observer has more advantages in anti-interference\cite{WB:23}.

However, above analysis does not take into account the finite time observer combined with fixed time controller. On the one hand, finite time observer fast tracks load disturbance, on the other hand, because of the fixed time controller has nothing to do with the system initial state parameters, which further to improve the convergence speed and anti-interference ability of the system.


In order to solve the above problems, a composite control strategy is proposed by combining fixed time technique with finite time observer. A fixed time controller is designed to improve the transient response of the system, and a finite time observer observer is constructed to further improve the  performance dynamic performance and anti-interference of the closed-loop system. The main innovations of the
paper are as follows:


(1)The combination of finite time observer and fixed time controller can improve the fast tracking performance of the system when the load changes. The finite time observer improves the speed of load identification and is more suitable for practical circuits. The fixed time controller improves the transient response and anti-interference of the system because the parameters are independent of the initial state of the system.

(2) The proposed strategy has no increase in circuit size and hardware cost. The fixed time controller is designed and applied to Buck circuit by combining backstepping method with fixed time control theory. Although the combination of fixed time and sliding mode control has been studied in \cite{WB:12}, the sliding mode control has chattering.

(3) Different from finite time controllers, which are influenced by the system state, the fixed time controller is independent of the initial state of the system and the convergence rate is determined by the parameters of the controller. So the transient response of the system can be further improved.

(4) Due to the advantages of fixed time theory, the designed controller can provide faster convergence speed and stronger anti-interference  performance for the Buck converter system. The fixed time voltage regulation control strategy can effectively reduce the ripple of capacitor current and improve the power quality.

(5) To verify the convergence rate, a finite time controller is designed. In the
comparison with sliding mode controller\cite{WB:13} and finite time controller\cite{WB:8}, the effectiveness of the fixed time controller in response speed and reducing current ripple is verified.

The rest of this paper is organized as follows. System description and preliminary knowledge are shown in Section 2. Design of the fixed time controller with load finite time observer is given in Section 3. A finite time controller for comparison is designed in Section 4. Simulation and experiment results are presented in Section 5. And the conclusions are drawn in Section 6.

\section{ System description and preliminary knowledge}\label{sec2}

\subsection{Model description of Buck converter}\label{subsec2}

The model of Buck DC-DC converter based on fixed time controller is shown in Fig. 1, where $V_{o}$ is the output voltage, $V_{in}$ is the input voltage,   $V_{ref}$ is the reference voltage;    $i_{C}$ is the output capacitor current, $i_{L}$ is the inductance
current;  $S_{w}$ is the switching element of MOSFET ; ${D}$  is a freewheeling diode; ${C}$ is the output capacitance; ${L}$ is the input inductance; ${R}$ is the load resistance,  ${\hat R}$ is the estimate of R;  ${u}$  is the controller output;  $x_{1}$ is the output voltage error, $x_{2}$ is the $i_{C}$.

\begin{figure}[H]
	\begin{center}
		\includegraphics[width=8.4cm]{fig1.eps}    % The printed column width is 8.4 cm.
		\caption{Buck converter based on fixed-time controller.}
		\label{fig:1}
	\end{center}
\end{figure}

When the switch turns on, the state space equation of the system is given as\cite{WB:8}:

\begin{equation}
\frac{{d{i_C}}}{{dt}} = \frac{{{V_{in}} - {V_o}}}{L} - \frac{{{{\dot V}_o}}}{R}
\end{equation}
\begin{equation}
\frac{{d{V_o}}}{{dt}} = \frac{{{i_C}}}{C}{\rm{     }}
\end{equation}

When the switch turns off, the state space equation of the system is given as\cite{WB:8}:

\begin{equation}
\frac{{d{i_C}}}{{dt}} =  - \frac{{{V_o}}}{L} - \frac{{{{\dot V}_o}}}{R}{\rm{ }}
\end{equation}
\begin{equation}
\frac{{d{V_o}}}{{dt}} = \frac{{{i_C}}}{C}{\rm{      }}
\end{equation}

According to equations (1), (2), (3) and (4), the average model of Buck converter in continuous conduction mode can be expressed as:



\begin{equation}
{{\dot V}_o} = \frac{1}{C}{i_C}
\end{equation}
\begin{equation}
{{\dot i}_C} = \frac{{u{V_{in}} - {V_o}}}{L} - \frac{{{{\dot V}_o}}}{R}
\end{equation}

Assuming that ${V_{ref}}$ is the expected reference output voltage, the output voltage error is $ {x_1} = {V_o} - {V_{ref}}$ , the output capacitance current is ${x_2} = {i_C}$ , and the equations (5) and (6) can be converted into:

\begin{equation}
{{\dot x}_1} = \frac{{{x_2}}}{C}
\end{equation}
\begin{equation}
{{\dot x}_2} =  - \frac{{{x_1}}}{L} - \frac{{{x_2}}}{{RC}} - \frac{{{V_{ref}}}}{L} + \frac{{{V_{in}}}}{L}u
\end{equation}


Our purpose  is to design a controller to ensure that the output voltage error of the system consisting of equations (7) and (8) converges in a fixed time.

\subsection{Preliminary knowledge}\label{subsec2}
\textbf{Lemma 1}\cite{WB:18} : For the system consisting of equations (7) and (8) , if there is positive definite in the domain and the continuous function $V(x)$ satisfies:
\begin{equation}
\dot V(x) \le  - \eta  V{(x)^{\bar p}}{\rm{ , }}x \in U\backslash \{ 0\} {\rm{ }}
\end{equation}
where $ \eta $ and $ \bar p $ are normal numbers and satisfy  $\bar p \in (0,1){\rm{ }}$ , then the system is stable in a finite time.

\textbf{Lemma 2}\cite{WB:19} : Considers nonlinear the system consisting of equations (7) and (8) , assuming that there exists a Lyapunov function  $V(x)$ with constants  $\sigma  $ and $\delta > 0$,  $0 < {\upsilon _1} < 1 $, ${\upsilon _2} > 1$, the following equation can be obtained:
\begin{equation}
\dot V(x) \le  - \sigma V{(x)^{{\upsilon _1}}} -\delta  V{(x)^{{\upsilon _2}}}{\rm{    }}
\end{equation}
the system is stable in a fixed time.The settling time T meets the following conditions(in this paper, ${\upsilon _1}=1/2$, $ {\upsilon _2}=2$):
\begin{equation}
T \le {T_{\max }}: = \frac{1}{{\sigma (1 - {\upsilon _1})}} + \frac{1}{{\delta ({\upsilon _2})}}
\end{equation}

\textbf{Lemma 3}\cite{WB:20} (Cauchy-Schwartz inequality) : For any positive number  ${q_t}$, the following relationship holds:
\begin{equation}
{(\sum\limits_{t = 1}^n {{q_t}} )^2} \le n{\rm{ }}\sum\limits_{t = 1}^n {q_t^2} {\rm{  }}
\end{equation}

\textbf{Lemma 4}\cite{WB:19} (Young`s inequality) : For any variable $g$ and $f$ , the following inequality holds:

\begin{equation}
\left| g \right|\left| f \right| \le \frac{{{\kappa ^p}}}{p}{\left| g \right|^p} + \frac{1}{{q{\kappa ^q}}}{\left| f \right|^q}
\end{equation}

where $\kappa $ is positive number, $p > 1$, $q > 1$, and $\frac{1}{p} + \frac{1}{q} = 1$ .

\textbf{Lemma 5}\cite{WB:24}: For the following systems:

\begin{equation}
\left\{ \begin{array}{l}
{{\dot \phi }_1} = \varphi {\phi _2} - \varphi {r_1}{\left| {{\phi _1}} \right|^{{\rho _1}}}sign({\phi _1})\\
{{\dot \phi }_2} =  - \varphi {r_2}{\left| {{\phi _1}} \right|^{{\rho _2}}}sign({\phi _1})
\end{array} \right.
\end{equation}

where $\varphi  > 0$, $0.5 < {\rho _1} < 1$, and ${\rho _2} = 2{\rho _1} - 1$. By choosing appropriate parameters ${r_1}$ and ${r_2}$, system(14) will be stabilized in a finite time.


\section{ Design of the fixed time controller with load finite time observer}\label{sec3}

\subsection{Design of the load finite time observer}\label{subsec3}
For the Buck converter system, this paper does not consider the influence of the changes of inductance and capacitance, and assumes that the main disturbance in the system is the load disturbance. In order to eliminate the influence of load disturbance on the system, an observer is used to estimate the load resistance, and the estimated value of load resistance is fed back to the controller in real time. Set $\theta  = {{ - 1} \mathord{\left/
 {\vphantom {{ - 1} R}} \right.
 \kern-\nulldelimiterspace} R}$,${\hat \theta }$ is the estimate of $\theta $, ${{\hat V}_0}$ is the estimate of ${V_0}$. The load observer is designed as:

\begin{equation}
{{\hat V}_0} = \frac{1}{C}({i_L} + \hat \theta {V_0}) + {\upsilon _1}{V_0}{\left| {{V_0} - {{\hat V}_0}} \right|^{{\eta _1}}}sign({V_0} - {{\hat V}_0})
\end{equation}
\begin{equation}
\dot {\hat \theta}  = {\upsilon _2}{V_0}{\left| {{V_0} - {{\hat V}_0}} \right|^{{\eta _2}}}sign({V_0} - {{\hat V}_0})
\end{equation}
where $\upsilon _1, \upsilon _2  > 0$, $0.5 < {\eta _1} < 1$, and ${\eta _2} = 2{\eta _1} - 1$.

\textbf{Proof:} In order to analyze the stability of the designed observer, the error of the observer is defined as:

\begin{equation}
{\phi _1} = {V_0} - {{\hat V}_0}
\end{equation}
\begin{equation}
{\phi _2} = \theta  - \hat \theta
\end{equation}


Since the load resistance R is a constant, $\theta$  is also a constant. According to equations (15)(16) and (17)(18), the error state equation is:

\begin{equation}
{{\dot \phi }_1} = \frac{{{V_0}}}{C}{\phi _2} - {\upsilon _1}{V_0}{\left| {{\phi _1}} \right|^{{\eta _1}}}sign({\phi _1})
\end{equation}
\begin{equation}
{{\dot \phi }_2} =  - {\upsilon _2}{V_0}{\left| {{\phi _1}} \right|^{{\eta _2}}}sign({\phi _1})
\end{equation}

Set $\varphi  = \frac{{{V_0}}}{C}$, equation (19)(20) can be converted to:

\begin{equation}
{{\dot \phi }_1} = \varphi {\phi _2} - {\upsilon _1}C\varphi {\left| {{\phi _1}} \right|^{{\eta _1}}}sign({\phi _1})
\end{equation}
\begin{equation}
{{\dot \phi }_2} =  - {\upsilon _2}C\varphi {\left| {{\phi _1}} \right|^{{\eta _2}}}sign({\phi _1})
\end{equation}

According to \textbf{Lemma 5}, equation (15)(16) is stable in  finite time, so the error of the load observer can converge to 0 in finite time, which means that the estimated value can converge to the true value in finite time.

\subsection{Design of the fixed time controller}\label{subsec3}

According to the error dynamic system consisting of equations (7) and (8), the original equation can be expressed as:
\begin{equation}
{{\dot x}_1} = a{x_2}
\end{equation}
\begin{equation}
{{\dot x}_2} =  - b{x_1} - c{x_2} - d + ku
\end{equation}
where  $b = \frac{{1}}{L} $ , $c = \frac{{1}}{{RC}}$ , $d = \frac{{ {V_{ref}}}}{L}$ , $k = \frac{{{V_{in}}}}{L}$ .

The system consisting of equations (23) and (24) will be transformed by coordinating transformation, mainly to facilitate the parameter design of the controller. Introduce coordinate transformation to equation (23) and (24), where  $\beta $ is the intermediate control law:

\begin{equation}
{z_1} = {x_1}
\end{equation}
\begin{equation}
{z_2} = {x_2} - \beta
\end{equation}

For the system after coordinating transformation , we will adopt the fixed time theory to design the fixed time controller. According to the fixed time theory, the controller of Buck converter with fixed time is designed. Firstly, the Lyapunov function is constructed. Then, the intermediate control rate  is designed by the backstepping method. Finally, the suitable $u $ is found to satisfy the fixed time control theory.

\textbf{Theorem 1}: For the transformed error system consisting of equations (25) and (26) , if the controller $u$ is designed to
\begin{equation}
u = \frac{{b{x_1} + c{x_2} + d + \dot \beta  - {m_3} - {m_4}z_2^3 - \frac{a}{2}{z_2}}}{k}
\end{equation}
then ,the state of the system  will converge to the origin in a fixed time.(Note:${m_i}$ is positive number, $i$ is positive integer, $i \in (1,12)$.)

\textbf{Proof:}

\textbf{Step 1:} When $\beta  =  - \frac{1}{2}{z_1} - \frac{{{m_1}}}{a} - \frac{{{m_2}}}{a}z_1^3$ ,  $V_1$ is satisfied to ${\dot V_1} \le \frac{a}{2}z_2^2 - {m_1}V_1^{\frac{1}{2}} - {m_2}V_1^2$.

Constructing Lyapunov function:

\begin{equation}
V = {V_1} + {V_2}{\rm{     }}
\end{equation}
\begin{equation}
{V_1} = \frac{1}{2}z_1^2{\rm{      }}
\end{equation}
\begin{equation}
{V_2} = \frac{1}{2}z_2^2{\rm{     }}
\end{equation}

Derivative of equation (28):
\begin{equation}
\dot V = {\dot V_1}{\rm{ + }}{\dot V_2}{\rm{      }}
\end{equation}


Derivative of equation (29):
\begin{equation}
{\dot V_1} = {z_1}{\dot z_1} = {z_1}{z_2}a + {z_1}a\beta {\rm{       }}
\end{equation}

By \textbf{Lemma 4} inequality scaling:
\begin{equation}
{\dot V_1} \le \frac{a}{2}z_1^2 + \frac{a}{2}z_2^2 + {z_1}a\beta {\rm{ }}
\end{equation}

Therefore, it is designed that $\beta  =  - \frac{1}{2}{z_1} - \frac{{{m_1}}}{a} - \frac{{{m_2}}}{a}z_1^3$ . When $\dot \beta  =  - \frac{1}{2}{\dot z_1} - \frac{{{m_2}}}{a}z_1^2{\dot z_1}$ ,  $\beta$ is brought into equation (32), and $ {\dot V_1}$ satisfies:

\begin{eqnarray}
%\begin{align}
{{\dot V}_1} &\le \frac{a}{2}z_2^2 - {m_1}{z_1} - {m_2}z_1^4 \nonumber\\
{\rm{ }}&=   \frac{a}{2}z_2^2 - {m_1}V_1^{\frac{1}{2}} - {m_2}V_1^2
%\end{align}
\end{eqnarray}

\textbf{Step 2:} Prove that when $ u$ is satisfied to  $ku = b{x_1} + c{x_2} + d + \dot \beta  - {m_3} - {m_4}z_2^3 - \frac{a}{2}{z_2} $ , equation (31) satisfies the fixed time control theorem.

Substituting $ {\dot V_1}$ obtained in step 1 into equation (31) to get:

\begin{equation}
\begin{split}
\dot V &= {{\dot V}_1} + {{\dot V}_2}\\
 &\le  - {m_1}V_1^{\frac{1}{2}} - {m_2}V_1^2 + \frac{a}{2}z_2^2 + {z_2}\dot z_2^{}\\
 &=  - {m_1}V_1^{\frac{1}{2}} - {m_2}V_1^2 + \frac{a}{2}z_2^2\\ &\quad+ {z_2}( - b{x_1} - c{x_2} - d + ku - \dot \beta )
\end{split}
\end{equation}

Substituting $ku = b{x_1} + c{x_2} + d + \dot \beta  - {m_3} - {m_4}z_2^3 - \frac{a}{2}{z_2} $ into the above equation (35) to get:

\begin{equation}
\dot V = {\dot V_1} + {\dot V_2} \le  - {m_1}V_1^{\frac{1}{2}} - {m_2}V_1^2 - {m_3}V_2^{\frac{1}{2}} - {m_4}V_2^2
\end{equation}

From \textbf{Lemma 3}:

\begin{equation}
{(V_1^{\frac{1}{2}} + V_2^{\frac{1}{2}})^2} = {V_1} + {V_2} + 2{({V_1}{V_2})^{\frac{1}{2}}} \ge ({V_1} + {V_2}){\rm{     }}
\end{equation}

Substituting (37) into equation (36) to get:

\begin{equation}
\begin{split}
\dot V &= {{\dot V}_1} + {{\dot V}_2}\\
 &\le  - {m_1}V_1^{\frac{1}{2}} - {m_2}V_1^2 - {m_3}V_2^{\frac{1}{2}} - {m_4}V_2^2\\
 &\le  - {m_5}(V_1^{\frac{1}{2}} + V_2^{\frac{1}{2}}) - {m_6}(V_1^2 + V_2^2)\\
 &\le  - {m_5}{(V_1^{} + V_2^{})^{\frac{1}{2}}} - {m_6}{(V_1^{} + V_2^{})^2}\\
&({m_5} = \min ({m_1},{m_3}),{m_6} = \min ({m_2},{m_4}))\\
\end{split}
\end{equation}

To sum up, under the action of the controller (27) which satisfies Lemma 2, the system output voltage error converges to the equilibrium point in a fixed time.

\section{Design of the finite time controller}\label{sec4}

In this section, according to the finite time theory, the controller of Buck converter with finite time is designed as comparison. Firstly, the Lyapunov function is constructed. Then, the intermediate control rate  is designed by the backstepping method. Finally, the suitable $u$ is found to satisfy the finite time control theory.

\textbf{Theorem 2.} For the transformed error system consisting of equations (25) and (26), if the controller design $u$ meets the following conditions:

\begin{equation}
u = \frac{b}{k}{x_1} + \frac{c}{k}{x_2} + \frac{1}{k}\dot {\bar {\beta }}  + \frac{d}{k} - \frac{a}{{2k}}{z_2} - \frac{{{m_8}}}{k}
\end{equation}

Then ,the state of the system will converge to the origin in a finite time.

\textbf{Proof: }

\textbf{ Step 1:} When $\bar \beta  =  - \frac{1}{2}{z_1} - {m_7}{\rm{ }}$ ,  $\bar V$ is satisfied to $\dot{\bar V} = {{\dot {\bar V}}_1} + {{\dot {\bar V}}_2} \le  - {m_8}\bar V_1^{\frac{1}{2}} + \frac{a}{2}z_2^2 + {z_2}\dot z_2^{}$.


Constructing Lyapunov function:

\begin{equation}
\bar V = {\bar V_1} + {\bar V_2}{\rm{     }}
\end{equation}
\begin{equation}
{\bar V_1} = \frac{1}{2}z_1^2{\rm{         }}
\end{equation}
\begin{equation}
{\bar V_2} = \frac{1}{2}z_2^2{\rm{         }}
\end{equation}

Derivative of equation (40):
\begin{equation}
{\dot{\bar V} = {{\dot{\bar V}}_1} + {{\dot{\bar V}}_2}}
\end{equation}

Derivative of equation (41):
\begin{equation}
{\dot{\bar {V_1}}} = {z_1}{\dot z_1} = {z_1}{z_2}a + {z_1}a\bar \beta  {\rm{       }}
\end{equation}



By \textbf{Lemma 4} inequality scaling:
\begin{equation}
{\dot{\bar {V_1}}} \le \frac{a}{2}z_1^2 + \frac{a}{2}z_2^2 + {z_1}a\bar \beta  {\rm{ }}
\end{equation}

Therefore, it is designed that $ \bar \beta   =  - \frac{1}{2}{z_1} - {m_7}{\rm{   }}$ . When $\dot {\bar \beta}   =  - \frac{1}{2}{\dot z_1}$ ,  $\bar \beta $ is brought into equation (45), and $ {\dot {\bar V}}$ satisfies:

\begin{equation}
\begin{split}
\dot{\bar V} &= {{\dot{\bar V}}_1} + {{\dot{\bar V}}_2} \\
&\le  - {m_8}{z_1} + \frac{a}{2}z_2^2 + {z_2}\dot z_2^{}\\
 &=  - {m_8}{\bar {V_1}}^{\frac{1}{2}} + \frac{a}{2}z_2^2 \\&\quad+ {z_2}( - b{x_1} - c{x_2} - d + ku - \dot {\bar {\beta}}  )
\end{split}
\end{equation}

\textbf{Step 2: }Prove that when $ u$ is satisfied to  $ku = b{x_1} + c{x_2} + d + \dot {\bar \beta}   - \frac{a}{2}{z_2} - {m_9}{\rm{    }} $ , equation (43) satisfies the finite time control theorem.


Substituting $ku = b{x_1} + c{x_2} + d + \dot{\bar \beta}   - \frac{a}{2}{z_2} - {m_9}{\rm{  }} $ into the above equation (43) to get:

\begin{equation}
\dot{\bar V} = {{\dot{\bar V}}_1} +{{\dot{\bar V}}_2} \le  - {m_{10}}\bar V_1^{\frac{1}{2}} - {m_{11}}\bar V_2^{\frac{1}{2}}
\end{equation}

From \textbf{Lemma 3}:

\begin{equation}
{(\bar V_1^{\frac{1}{2}} + \bar V_2^{\frac{1}{2}})^2} = {\bar V_1} + {\bar V_2} + 2{({\bar V_1}{\bar V_2})^{\frac{1}{2}}} \ge ({\bar V_1} + {\bar V_2}){\rm{     }}
\end{equation}

Substituting (48) into equation (47) to get:

\begin{equation}
\begin{split}
\dot{\bar V} &= {{\dot{\bar V}}_1} +{{\dot{\bar V}}_2}\\
& \le  - {m_{12}}({\bar V_1}^{\frac{1}{2}} + {\bar V_2}^{\frac{1}{2}})\\
 &\le  - {m_{12}}{({\bar V_1} + {\bar V_2})^{\frac{1}{2}}}\\
({m_{12}}& = \min ({m_{10}},{m_{11}}))
\end{split}
\end{equation}

Therefore, it can be proved that under the action of the controller (39) which satisfies Lemma 1, the state of the system  can converge to the origin in a finite time.

\section{Simulation and Experiment analysis}\label{sec5}

In this simulation, the control strategy is shown in Fig 2. The switching frequency of PWM generator is 100kHz.

\begin{figure}[H]
	\begin{center}
		\includegraphics[width=8.4cm]{fig2.eps}    % The printed column width is 8.4 cm.
		\caption{Simulation circuit based on fixed-time control.}
		\label{fig:2}
	\end{center}
\end{figure}

\subsection{Selection of simulation circuit parameters}\label{subsec5}


The parameters of three controllers(the parameters of the sliding mode controller is designed according to the method in \cite{WB:13} ) and system models are given, as shown in Table 1 and Table 2 below. In order to get the better control effect, the control parameters are obtained by trial and error method.
\begin{table}[h]
	\centering
	\caption{System model parameters}
	\begin{tabular}{llll}
		\toprule
		&Description           & Parameters                 & Value        \\
		\midrule
		&Input voltage          &  ${{V_{in}}} $               & 50 V         \\
		& Reference voltage       & $ {{V_{ref}}}$             & 25 V         \\
		
		&Inductance                & L       & 150 $\mu H $     \\
		&Capacity               &   C    & 1000 $\mu F $     \\
		&Rsistance            &  R    & 1 $\Omega $      \\
		
		\bottomrule
	\end{tabular}
\end{table}

\begin{table}[h]
\centering
\caption{Controller model parameters}
\begin{tabular}{lllllll}
\toprule
&Fixed-time controller              & b/k    &c/k          &(d-m3)/k      &m4/k   \\
\midrule
  & Value    & -0.01	&-6000	&-100	&-1000  & \\
\midrule
\midrule
&Finite-time controller              & b/k    &c/k          &(d-m8)/k      & --  \\
\midrule
  & Value    &-0.01	 &-6000	 &-100 & --\\
\midrule
\end{tabular}
\end{table}



\subsection{Transient response analysis}\label{subsec5}

The load is kept constant at 1 $\Omega$ , and the reference voltage is set at 25 V. In Fig. 3, the tracking results show that the fixed time controller is obviously superior to the finite time controller and sliding mode controller in response speed and steady-state error, and the fixed time controller is stable at about 0.3 $ms$, while the finite time controller is greater than 8 $ ms $.The sliding mode controller has an obvious chattering.

\begin{figure}[H]
	\begin{center}
		\includegraphics[width=8.4cm]{fig3.eps}    % The printed column width is 8.4 cm.
		\caption{Comparison of three controllers tracking voltage speed.}
		\label{fig:3}
	\end{center}
\end{figure}

We set the reference voltage to change from 25V to 30V in 2 $ms$ , and other parameters remain unchanged. In Fig. 4 , for the dynamic response waveform of the system. it can be seen that the proposed fast fixed time voltage control strategy is obviously superior to the response speed of finite time and sliding mode strategy, in which the stable time of the fixed time control system is at about 2.1 $ms$ and that of sliding mode controller is greater than 2.2 $ms$ . When the reference voltage changes from 25V to 30V, the finite time control system can reach stability only at after 2.6 $ms$ at the beginning of startup. There is a certain gap between the steady-state errors of the three controllers. The steady-state error of the fixed time controller is the smallest of the three.
\begin{figure}[H]
	\begin{center}
		\includegraphics[width=8.4cm]{fig4.eps}    % The printed column width is 8.4 cm.
		\caption{Tracking speed of the three controllers when the reference voltage changes.}
		\label{fig:4}
	\end{center}
\end{figure}
When the load becomes 1.2 $\Omega$ at 1 $ms$, it can be seen from the Fig. 5 that the adjustment time of the proposed fixed time controller is faster than that of the finite time controller and the sliding mode controller. The finite time controller reaches stability at 1.4 $ms$, while the fixed time controller and the sliding mode controller reach stability before 1.1 $ms$. Moreover, the error and convergence time of fixed time control are smaller.
\begin{figure}
	\begin{center}
		\includegraphics[width=8.4cm]{fig5.eps}    % The printed column width is 8.4 cm.
		\caption{Comparison of tracking voltages of three controllers under load changes.}
		\label{fig:5}
	\end{center}
\end{figure}

\subsection{ Current ripple analysis}\label{subsec5}

It can be clearly seen from Fig. 6 that there is a big gap between sliding mode controller and the other two controllers. Compared with finite time controller and fixed time controller, sliding mode controller has a larger ripple current (the amplitude is about 0.4A). In Fig. 7, it can be seen that effect of ripple suppression of fixed time controller is the best, and the ripple amplitude is the smallest. The current ripple of the circuit with the finite time controller is close to that of the fixed time controller. Therefore, the proposed fixed time Buck control method is effective against suppressing current ripple and has smaller current fluctuation.

\begin{figure}[H]
	\begin{center}
		\includegraphics[width=8.4cm]{fig6.eps}    % The printed column width is 8.4 cm.
		\caption{Comparison of current ripple under three controllers.}
		\label{fig:6}
	\end{center}
\end{figure}

\begin{figure}[H]
	\begin{center}
		\includegraphics[width=8.4cm]{fig7.eps}    % The printed column width is 8.4 cm.
		\caption{Comparison of current ripple details between fixed-time controller and finite-time controller.}
		\label{fig:7}
	\end{center}
\end{figure}

\subsection{Experiments Results with the Buck circuit system}\label{subsec5}

Hardware circuit (as shown in Fig. 9) consists of Buck circuit, voltage and current sampling circuit and control circuit composed of DSP28335 main control board. The input voltage is 40V and the resistance is 25 $\Omega$, the set voltage is 25V. In order to facilitate the early debugging, the control software platform, Buck circuit control interface (as shown in Fig. 8), is designed in PC, and communicates with DSP through 485 serial port. The whole design framework (as shown in Fig. 10) and the control strategy are downloaded to the DSP board. The DSP outputs PWM wave to the driving circuit, and the driving circuit to the Buck circuit. The current and voltage information of the main circuit can be collected through the sampling circuit.



\begin{figure}[H]

	\begin{center}
		\includegraphics[width=8.4cm]{fig8.eps}    % The printed column width is 8.4 cm.
		\caption{Buck circuit control system in PC}
		\label{fig:8}
	\end{center}
\end{figure}
\begin{figure}
	\begin{center}
		\includegraphics[width=8.4cm]{fig9.eps}    % The printed column width is 8.4 cm.
		\caption{Hardware test platform}
		\label{fig:9}
	\end{center}
\end{figure}

\begin{figure}
	\begin{center}
		\includegraphics[width=8.4cm]{fig10.eps}    % The printed column width is 8.4 cm.
		\caption{Hardware block diagram}
		\label{fig:10}
	\end{center}
\end{figure}

As shown in Fig. 11 and Table 3, the overshoot of the proposed strategy is much smaller than that of the finite time controller and the sliding mode controller. The settling time with the proposed strategy is shortened by 14\% compared with that of the finite time control and 24\% compared with that of the sliding mode control.

\begin{figure}[H]
	\begin{center}
		\includegraphics[width=8.4cm]{fig11.eps}    % The printed column width is 8.4 cm.
		\caption{Comparison of three controllers at set voltage 25V}
		\label{fig:11}
	\end{center}
\end{figure}


\begin{table}[h]
	\centering
	\caption{Comparisons under Sliding mode controller, Finite time controller and Fixed time controller}
	\begin{tabular}{llll}
		\toprule
		&Controller           & Overshoot(V)                 & Settling Time(ms)        \\
		\midrule
		&Sliding mode  controller        &  2.1               & 365.6         \\
		& Finite time controller       & 0.5             & 323.1         \\
		
		&Fixed time controller                & 0.2       & 276.8 \\
	
		\bottomrule
	\end{tabular}
\end{table}

The simulation and experiment results show that, the composite controller combining a fixed time controller with a finite time observer has advantages in convergence speed and interference resistance, the reason of which is the convergence time of fixed time controller does not depend on the initial state. Meanwhile, the load observer can accurately and quickly track the load changes. This can be well reflected on Buck converter when the load and reference voltage changes. Hence, the fixed time controller based on load finite time observer can track reference voltage quickly and reduce current ripple, which has been verified in convergence speed and anti-interference.

\section{Conclusions}\label{sec5}

This paper proposes a fixed time voltage regulation control strategy based on load finite time observer to improve the transient response of the Buck converter and also to reduce the current ripple in the circuit. The fixed time theory is used instead of the traditional method of adding additional circuits and sliding mode control to improve the stability and fast response of the system. By constructing a Lyapunov function, the fixed time stability has been proved. Simlation results have shown that the output voltage can track the reference volatge in a fixed time. Compared with the sliding mode controller and finite time controller, the Buck converter with fixed time controller has faster transient and smaller current ripple when the load and reference voltage changes.



%\bibliography{ifacconf}             % bib file to produce the bibliography
                                                     % with bibtex (preferred)

\begin{thebibliography}{xx}  % you can also add the bibliography by hand

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