Sliding Mode Control in Power Converters and Drives: A Review

Nomenclature

Vo$V_o$ Output voltage of the dc/dc power converter

Vin$V_{in}$ Input voltage of the dc/dc power converter

iL$i_L$ Inductor current of the dc/dc power converter

vdc$v_{dc}$ Output voltage of grid-connected power converter

vabc$v_{abc}$ AC voltage vector in abc$abc$frame

vαβ$v_{\alpha \beta }$ AC voltage vector in αβ$\alpha \beta$ frame

vdq$v_{dq}$ AC voltage vector in dq$dq$ frame

iabc$i_{abc}$ AC current vector in abc$abc$ frame

iαβ$i_{\alpha \beta }$ AC current vector in αβ$\alpha \beta$ frame

idq$i_{dq}$ AC current vector in dq$dq$ synchronous frame

idq$i_{dq}$ AC current vector in dq$dq$ synchronous frame

p,q$p,q$ Active power and reactive power

C DC-link capacitor

L Line inductor

RL$R_L$ Load resistance

ω Rotor speed

T Electromagnetic torque

Ψ Rotor flux

I. Introduction

SLIDING mode control (SMC) is a special kind of nonlinear control which has proven to be an effective robust control strategy for incompletely modeled or nonlinear systems since its first appearance in the 1950s [1]-[4]. One of the most distinguished properties of SMC is that it utilizes a discontinuous control action which switches between two distinctively different system structures such that a new type of system motion, called sliding mode, exists in a specified manifold. This peculiar characteristic of the motion in the manifold provides insensitivity to the matched disturbances. In order to deal with the unmatched uncertainties, SMC with other approaches such as adaptive approach [5], LMI-based approach [6] and observer based approach [7], etc., have been proposed in many works [8], [9].

SMC has been applied primarily to the control of variable structure systems, its analysis and design are well presented in books, survey and tutorial papers [10]-[17], both from a theoretical and implementation perspective. In general, SMC suffers from the so-called chattering phenomena, which is undesirable because it often causes control inaccuracy, high heat loss in electric circuitry, and high wear of moving mechanical parts [18]-[20]. In addition, the chattering action may excite the unmodeled high-order dynamics, which could damage actuators, systems and even leads to unforeseen instability. The chattering in SMC systems is usually caused by

• Utilization of digital controllers with the finite sampling rate, which causes the so-called discretization chattering. Theoretically, the ideal sliding mode implies infinite switching frequency. Since the conventional SMC action is constant within a sampling interval, the switching frequency can not exceed that of half the sampling frequency, which leads to chattering.

• The unmodeled dynamics with small time constants, which are often neglected in the ideal model.

Whatever the case may be, a high switching frequency is not feasible or undesirable for practical power electronics applications due to limitations of switching devices, such as losses, time delay, response time constant, the presence of dead zone, hysteresis and saturation of device switching frequency [21].

Despite the chattering phenomena, the inherent switching nature of SMC is quite suitable and even advantageous for power converters due to its switched operation. Therefore, SMC is an interesting solution to deal with power electronics systems, such as switching dc/dc power converters, grid-connected power converters, and motor drives.

During the last few years, power electronics has undergone an intense technological evolution through the advancements of the power semiconductor industry. For example, the new generation of semiconductor switches operates with faster switching frequency and handles higher powers than the previous one. The widely used real-time computer controllers make the implementation of advanced and complex control algorithms a reality. These factors together have led to the development of cost-effective and grid-friendly converters, which play the fundamental roles in applications such as renewable energy sources and their integration into the electrical grid, motor drives, etc [22]-[25].

This paper is focused on the application of SMC for power converters and drives. The basic SMC theory is revisited and the particular problems and solutions, when applied to power electronics systems, are discussed and analyzed. Methods for solving major challenges of conventional SMC such as chattering phenomenon and variable switching frequency are addressed, and new trends and challenges for its application to power electronic systems are examined.

This paper is organized as follows: Section II briefs the fundamental theory and methodologies of SMC. The use of SMC for different types of power electronics systems are presented in Sections III to V. Future challenges to adopt SMC as an industry solution to power converters are addressed in Section VI. Finally, Section VII concludes the paper.

II. SMC Fundamental Theory and Methodologies

SMC has been recognized as an efficient tool to design robust controllers for complex high order nonlinear dynamic plants operating under various uncertainty conditions since its first appearance in the 1950s. The major advantage of the sliding mode is the low sensitivity to plant parameter variations and external disturbances which relaxes the necessity of exact modeling. SMC enables the decoupling of the overall system motion into independent partial components of lower dimension, which reduces the complexity of feedback design. SMC has been developed as a new control design method for a wide spectrum of systems including nonlinear, time-varying, discrete, large-scale, infinite-dimensional, stochastic, and distributed systems [26]. Also, in the past two decades, SMC has successfully been applied to a wide variety of practical systems such as robot manipulators, aircrafts, underwater vehicles, spacecrafts, flexible space structures, power electronics, control of electric drives, doubly fed induction generator, robotics, and automotive engines [10], [27]-[29].

In this section, the basic notion of SMC, the controller design principle and their distinguishing features are presented.

A Fundamental Theory of SMC

Let us consider the following nonlinear system:

where x(t)∈Rn$x(t)\in {\mathbb{R}}^n$ is the state variable vector, u(t)∈Rm$u(t)\in {\mathbb{R}}^m$ is the control input, f(⋅,⋅)$f(\cdot ,\cdot )$ and g(⋅,⋅)$g(\cdot ,\cdot )$ are continuous functions in x and t vector fields [10], [30]. Note that the n is the dimension of the state variable vector x(t)$x(t)$, and the m is the dimension of the control input vector u(t)$u(t)$.

The sliding mode controller

is designed as

where u+i(t)≠u−i(t)$u_i^{+}(t)\ne u_i^{-}(t)$ and s(x)∈Rm$s(x)\in {\mathbb{R}}^m$ is the switching vector function s(x)=[s1(x)s2(x)⋯sm(x)]T$s(x)=[s_1(x)s_2(x)\cdots s_m(x){]}^{\mathrm{T}}$. It undergoes discontinuities on the surface s(x)=0$s(x)=0$.

Note that SMC law (3) is designed to ensure that the sliding surface (s(x)=0)$(s(x)=\mathbf{0})$ is reached and then motion on the sliding surface is maintained. This means that the so-called ‘reachability condition’ should be satisfied by manipulating the control law u(t)$u(t)$. The sufficient condition for the system (1) to satisfy the reachability condition is expressed as

Condition (4) guarantees that the trajectory of the system states always points towards the sliding surface. A more strict reachability condition called ‘η-condition’ is given as follows

where η is a positive scalar. Condition (5) ensures that the sliding surface is reached in finite time.

B SMC Design Methods

Several SMC design methods have been proposed in literature which mainly consist of two steps [10], [30]:

Step 1: Design a sliding manifold s(x)$s(x)$ which provides desired performance in the sliding mode, such as stability, disturbance rejection capability and tracking;

Step 2: Design a discontinuous feedback control u(t)$u(t)$ which will force the system states to reach the sliding manifold in finite time, thus the desired performance is attained and maintained.

For ease of implementation, the sliding variable si(x), i=1,2,…,m$s_i(x),i=1,2,\dots ,m$ is chosen as a linear combination of the state variables, expressed as

where αji${\alpha }_{ji}$ denotes the sliding coefficients and xj(t)∈x(t)$x_j(t)\in x(t)$. The main objective of the sliding mode controller is to drive the system state trajectories onto the specified sliding surface in a finite time and maintained there for all subsequent time. Typical SMC strategies will be introduced in the following.

1) Equivalent control-based design: For the system (1), assuming that the term ∂s∂xg(x,t)$\frac{\mathrm{\partial }s}{\mathrm{\partial }x}g(x,t)$ is non-singular, the control law u(t)$u(t)$ is designed as follows,

where ueq(t)$u_{\mathrm{e}\mathrm{q}}(t)$ represents a continuous component and uN(t)$u_N(t)$ represents a discontinuous component.

The equivalent control ueq(t)$u_{\mathrm{e}\mathrm{q}}(t)$ is derived from the so-called equivalent control method, i.e., in the case when s(x)=s˙(x)=0$s(x)=\dot{s}(x)=0$. Thus, ueq(t)$u_{\mathrm{e}\mathrm{q}}(t)$ is calculated as

Substituting the above equivalent control (8) into the original system (1), it follows that the motion of sliding mode is determined by

where (9) is considered as the equation of the sliding mode in the manifold s(x)=0$s(x)=0$.

The high frequency switching action uN(t)$u_N(t)$ is designed as

such that the derivative of the Lyapunov function V=12s(x)Ts(x)$V=\frac{1}{2}s(x{)}^{\mathrm{T}}s(x)$ is negative, that is

Remark 1: The physical meaning of the equivalent control can be interpreted as the low-frequency component of the discontinuous control law u(t)$u(t)$, because the high-frequency uN(t)$u_N(t)$ can be filtered out by a low pass filter of the system

which means z≃ueq$\mathit{\text{z}}\simeq u_{\mathrm{e}\mathrm{q}}$.

2) Reaching law approach: The reaching law specifies the dynamics of a switching function, which can be described by the following differential equation:

where

and Υ=diag{ε1,ε2,…,εm}$\mathrm{{\rm Y}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{{\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_m\}$, εi>0${\epsilon }_i>0$, K=diag{k1,k2,…,km}$K=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{k_1,k_2,\dots ,k_m\}$, ki>0$k_i>0$, gi(0)=0${\mathfrak{g}}_i(0)=0$, si(x)gi(si(x))>0$s_i(x){\mathfrak{g}}_i(s_i(x))>0$,  i=1,…,m$i=1,\dots ,m$.

Equation (13) is only a general form of reaching law. In fact, there are many reaching laws and some special cases are

1)$1)$ The constant rate reaching law:

2)$2)$ The constant plus proportional rate reaching law:

3)$3)$ The power rate reaching law:

The reaching law approach not only guarantees the reaching condition but also specifies the dynamic characteristics of the motion during the reaching phase.

C Chattering Phenomenon

Chattering problem is one of the main obstacles for applying SMC to real applications. It is caused by unmodeled dynamics or discrete time implementation. Chattering leads to undesirable results, such as low control accuracy, high heat loss in electric circuitry, and high wear of moving mechanical parts [31]. In addition, it may excite the unmodeled high-order dynamics, which probably leads to unforeseen instability. Therefore, various methods have been proposed in literature to reduce or soften the chattering action [32]-[36]. Among others, the main approaches to avoid or limit the chattering problems are shown in Table I. It should be noted that in addition to these methods mentioned in Table I, fractional-order SMC [37] and disturbance observer based SMC approaches [7] are also common and typical approaches to soften the chattering problem.

Table  I.  Main Approaches to Alleviate or Limit the Chattering Problems

Approaches	Operation principles
Boundary layer approach	To insert a boundary layer near the sliding surface so that a continuous control action replaces the discontinuous one when the system is inside the boundary layer [38]. For this purpose, the discontinuous component of the controller:
	uN(t)=−Kssign(s(x))$u_N(t)=-K_s\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s(x))$,
	is often replaced by the saturation control:
	uN(t)≈−Kss(x)∥s(x)∥+δ,$u_N(t)\approx -K_s{\displaystyle \frac{s(x)}{\Vert s(x)\Vert +\delta }},$
	for some, preferably small, δ>0$\delta >0$. The boundary layer approach has been utilized extensively to practical applications. However, this method has some disadvantages such as:
	• It may give a chattering-free system but a finite steady-state error must exist;
	• The boundary layer thickness has the trade-off between control performance of SMC and chattering mitigation;
	• Within the boundary layer, the characteristics of robustness and the accuracy of the system are no longer assured.
Reaching law approach	Since the amplitude of chattering depends on the magnitude of control, the intuitive way of chattering reduction is to decrease the amplitude of the discontinuous control [39]. This technique affects the robustness property of the controller and degrades the transient response of the system. Therefore, exists a trade-off between the chattering reduction and the system performance. A compromised approach is to decrease the amplitude of the discontinuous control, uN(t)$u_N(t)$, when the system state trajectories are near to sliding surface (to reduce the chattering), and to increase the amplitude when the system states are far from the sliding surface.
Dynamic SMC approach	The main idea of the dynamic SMC approach is to insert an integrator (or any other strictly proper low-pass filter) between the SMC and the controlled plant [40]. The concept is illustrated here:
	The time derivative of the control input, ω, is treated as the new control input for the augmented system. Since the low-pass integrator filters out the high frequency chattering in ω, the control input to the real plant, u, becomes continuous which offers a possibility to reduce chattering. Such a method can eliminate chattering and ensure zero steady-state error, however, it should be noted that the system order is increased by one and the transient responses may be degraded [41].
SOSMC approach	Second order sliding mode control (SOSMC) approach is one of the most popular methods to alleviate the chattering problems, which not only effectively eliminates the disadvantages of the conventional SMC, but also maintains its all major attributes such as strong robustness and finite time convergence. The common SOSMC algorithms include twisting algorithm, super-twisting algorithm, drift algorithm and prescribed convergence law algorithm, etc. All these algorithms guarantee that system state can converge s=s˙$s=\dot{s}$ in finite time [42]–[45]. Among these second order sliding mode control algorithms, super-twisting algorithm viewed as nonlinear proportional-integral control is very effective to deal with the system in presence of strong nonlinearity effects [46]. The super-twisting algorithm can be expressed as,
	usta(t)=usta1(t)+usta2(t),usta1(t)=−λ1sign(s(x)),usta2(t)=−λ2|s(x)|0.5sign(s(x)),$\begin{array}{c}{u}_{sta}(t)=u_{sta1}(t)+u_{sta2}(t),\\ u_{sta1}(t)=-{\lambda }_1\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s(x)),u_{sta2}(t)=-{\lambda }_2|s(x){|}^{0.5}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s(x)),\end{array}$
	where λ1${\lambda }_1$ and λ2${\lambda }_2$ are positive design parameters.
Intelligent SMC approach	The main idea of the intelligent SMC approach is integration of SMC with intelligent control technologies, such as neural networks (NNs), fuzzy logic technologies, etc., to make it smarter [47]–[49]. Depending on the technology different goals can be defined as follows.
	• NNs allows one to approximate smooth nonlinear functions to arbitrary accuracy. Therefore, they can be used to approximate and compensate external disturbances and model uncertainties to soften the chattering. On the other hand, NNs in SMC can be also aimed to estimate the switching control term in which the discontinuous control signal is converted to a continuous control signal providing an efficient method to reduce the chattering.
	• The purpose of adding fuzzy logic systems to SMC is similar to NNs. It allows one to approximate smooth nonlinear functions to arbitrary accuracy. Then it is used to estimate the external disturbance and modeling uncertainties to alleviate chattering.
	• Evolutionary computation is an alternative to seek optimal control parameters. Thus it can be used to alleviate chattering.

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III. SMC Strategies of DC/DC Power Converters

SMC is naturally well suited for the control of variable structure systems. Since power converters inherently include switching devices, they belong to variable structure systems. Therefore, it is straightforward to apply SMC that yields a discontinuous control law [10]. Moreover, given that power converters are usually modeled using the state space averaging method, SMC forms an efficient analysis and design tool for the control of switched mode power converters because it offers excellent large-signal handling capability.

Conventional linear control is small signal based. It only allows one to optimally operate the converters for a specific range of operating conditions and often fails to achieve satisfactory performance under large parameter/load variations, i.e., large-signal operating condition. SMC as a kind of nonlinear control method. It is suitable for controlling the power converters, which is able to achieve better regulation and dynamical performance for a wider range of operating conditions. The main reason is that there’s no need to have a linear model of the power converter for nonlinear controller design. However, the main obstacle associated with the application of SMC is its variable frequency nature, which makes the design of output filter difficult. Nonetheless, if this problem is properly handled, SMC is a powerful control design method for power converters and has a huge potential in industrial applications [21].

Many sliding-based controllers have been proposed for dc/dc converters, such as SMC based on Hysteresis-Modulation (HM) technique or fixed-frequency SMC [52]-[54]. As it is well known, the direct implementation of SMC will result in some high and uncontrolled switching frequency which makes them unsuitable for industrial applications. HM-based SMC is adopted to limit the operating frequency via tuning the hysteresis parameter. It should be noted that the operating frequency is only limited but it is still variable [21].

Generally, there are three approaches to make the switching frequency constant. The first approach is to incorporate a constant ramp or timing function directly into the controller [55]. The second approach is to apply the adaptive hysteresis concept, with a hysteresis level which can be varied with both input and output voltages to force the switching frequency to remain a constant under all load conditions [50]. The third approach is to achieve constant switching frequency by employing PWM technique. The PWM-based SMC can be obtained by translating the equivalent control ueq$u_{\mathrm{e}\mathrm{q}}$ to the duty ratio of the PWM, d. The equivalent control signal ueq$u_{\mathrm{e}\mathrm{q}}$ is calculated by setting the derivative of the sliding variable to zero, i.e, s˙=0$\dot{s}=0$. The control signal u is compared to the PWM carrier to generate a discrete gate pulse signal.

Mazumder et al. proposed a first fixed-frequency PWM-based integral variable structure sliding mode controller for parallel buck converters [53]. This technique was subsequently extended for control of the voltage regulator module application in [56]. Later, a unified fixed-frequency PWM-based direct sliding mode voltage control design was proposed for the buck, boost and buck-boost converters [57]. Its main disadvantage is that it lacks robustness against system parameters, i.e., load resistance and capacitors. Pointed out by [12] that direct sliding mode voltage control for boost and buck/boost converters may result in the instability of the system. Cascade control structure which consists of an inner current loop and outer voltage loop is introduced to solve this problem. This control method increases the overall system’s stability (phase) margin, and hence simplifies the design of the outer voltage loop. It can be concluded from [58] that sliding mode current controller may be a good alternative over conventional current-mode controllers for fast-response boost-converter applications but at a higher implementation cost and circuit complexity.

As an illustrative example, three SMC strategies for buck converter are presented in the discussion. The mathematical model of the typical buck converter can be expressed as,

where L, C and RL$R_L$ represent inductor, storage capacitor and load resistance, respectively; iL$i_L$, Vin$V_{in}$ and Vo$V_o$ are inductor current, input voltage and output voltage, respectively.

However, these strategies are generally applicable to any dc/dc converter types. Basic structures of typical HM-based SMC, fixed-frequency PWM based SMC and cascade control structure for a dc/dc converter system are shown in Table II.

Table  II.  SMC Strategies for Buck Converter

Techniques	Structures	Sliding surfaces	Control laws
HM-Based SMC [50]		s=Kp(Vref−Vo)+io−∫uVi−VoLdt,$s=K_{\mathrm{p}}\left(V_{\mathrm{r}\mathrm{e}\mathrm{f}}-V_{\mathrm{o}}\right)+i_o-{\displaystyle \int {\displaystyle \frac{u{V}_{\mathrm{i}}-V_{\mathrm{o}}}{L}}dt,}$ where Kp$K_p$ is a positive constant and io$i_o$ is the output current.	u = ⎧⎩⎨1,0,unchanged,ifs>κ,ifs<−κ,otherwise.$\begin{array}{r}u=\{\begin{array}{ll}1,& \mathrm{i}\mathrm{f}s>\kappa ,\\ 0,& \mathrm{i}\mathrm{f}s<-\kappa ,\\ \mathrm{u}\mathrm{n}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{d},& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}.\end{array}\end{array}$
PWM-Based Direct Voltage SMC [21]		s=α1(Vref−V0)+α2d(Vref−V0)dt$s={\alpha }_1(V_{ref}-V_0)+{\alpha }_2{\displaystyle \frac{d(V_{ref}-V_0)}{dt}}$ +α3∫(Vref−V0)dt,$+{\alpha }_3{\displaystyle \int (V_{ref}-V_0)dt,}$ where α1,α2${\alpha }_1,{\alpha }_2$ and α3${\alpha }_3$ are the sliding coefficients.	u=ueq+Kp3sign(s),$u=u_{eq}+K_{p3}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s),$ ueq=Kp1iC+Kp2(Vref−V0)+V0,$u_{eq}=K_{p1}{i}_C+K_{p2}(V_{ref}-V_0)+V_0,$ where Kp2=α3α2LC$K_{p2}={\displaystyle \frac{{\alpha }_3}{{\alpha }_2}}LC$, Kp3>0$K_{p3}>0$ and Kp1=L(α1α2−1RLC)$K_{p1}=L\left({\displaystyle \frac{{\alpha }_1}{{\alpha }_2}}-{\displaystyle \frac{1}{R_{L}C}}\right)$.
PWM-Based Cascaded SMC [51]		External loop s1=V∗ref−Vo$s_1=V_{ref}^{\ast }-V_o$ and internal loop s2=i∗L−iL$s_2=i_L^{\ast }-i_L$.	i∗L=λ1|s1|12sign(s1)+α1∫tt0sign(s1)ds,$i_L^{\ast }={\lambda }_1{\left|s_1\right|}^{\frac{1}{2}}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s_1)+{\alpha }_1{\displaystyle {\int }_{t_0}^t\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s_1)ds,}$ u=LE(u2(s2)+VoL+RLLiL),$u={\displaystyle \frac{L}{E}}\left(u_2(s_2)+{\displaystyle \frac{V_o}{L}}+{\displaystyle \frac{R_L}{L}}{i}_L\right),$ u2(s2)=μ2|σ(s2)|12sign(s2)+α2∫tt0sign(s2)ds,$u_2(s_2)={\mu }_2{\left|\sigma (s_2)\right|}^{\frac{1}{2}}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s_2)+{\alpha }_2{\displaystyle {\int }_{t_0}^t\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(s_2)ds,}$ where μ1${\mu }_1$, μ2${\mu }_2$, α1${\alpha }_1$ and α2${\alpha }_2$ are positive constants.

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