Novel Power-Rate Reaching Law for Quasi-Sliding Mode Control

Abstract

This study elaborates on the quasi-sliding mode control design for discrete time dynamical systems subject to matched external disturbances and modeling uncertainties. In order to provide finite time convergence to the sliding surface and at the same time restrict the control effort, we propose a novel power-rate reaching law utilizing a hyperbolic tangent function. The construction of the reaching law ensures that when the distance between the representative point of the system and the sliding surface is significant then the convergence pace is limited, which results in a reduced control effort. However, as the representative point of the system approaches the sliding surface, the convergence pace increases. Moreover, the study adopts a non-switching-type definition of the sliding motion, which eliminates undesirable chattering effects in the sliding phase. In order to reduce the impact of external disturbances on the system, the model following approach is taken, which allows for the rejection of all but the last disturbance value.

1. Introduction

Research on variable structure control began in the mid-20th century [1,2,3]. Among various designs, sliding-mode control has become the most popular and widely applied control within the control engineering community [4,5,6,7,8,9,10,11]. It is well-known to provide extraordinarily beneficial properties of the closed-loop system, such as stability, reduction of the dimensions of the dynamical problem and robustness, or in continuous time cases even insensitivity [3] to matched external disturbances and modeling uncertainties.

The key to the sliding mode controller is the design of an appropriate sliding surface [12]. As the method restricts the dynamics of the system to a preselected hyperplane, the behavior of the system becomes fully dependent on the choice of this hyperplane. Therefore, the dimensions of the problem are reduced by one and stability on the switching surface may be ensured, for example, with a simple pole-placement method. Moreover, the control law is altered each time the representative point tries to deviate from the sliding surface, which results in the alleviation of disturbances in the sliding phase. For an ideal sliding mode to emerge, the control signal must be switched continuously so that the system’s dynamics belong to the sliding surface. However, when the actuator’s frequency is limited or for the case of discrete time systems, delays between subsequent control updates exist. Consequently, the representative point of the system remains within a certain band around the sliding hyperplane. Such motion was defined by Milosavljevic in the late 20th century as a quasi-sliding mode [13].

Numerous authors have studied the design and stability of the quasi-sliding motion, defining the convergence and stability requirements [14,15,16]. Utkin, in [17], proposed to drive the system from any initial position, represented with the sliding variable s(0), to the switching surface, s = 0, in one sampling period, by assigning s(k + 1) = 0. His design formed a basic equivalent control method, which was further developed in [18,19]. However, due to the presence of unknown disturbances, the method results in bounding the system’s dynamics to the vicinity of the switching hyperplane only. The radius of the achieved band is a direct result of the disturbances and uncertainties of the model and therefore, it represents the measure of the system’s robustness. Moreover, forcing the system to reach the sliding surface in one control step may often require an extensively large control magnitude, which is possibly unobtainable in practice. A method to significantly reduce the control effort was proposed by Bartoszewicz [20], who introduced a demand trajectory s_d(k), which defined the evolution of the sliding variable profile before it reaches the sliding surface. Therefore, the sliding mode could be obtained gradually, in a selected number of steps k₀. Moreover, the author of [20] proposed a method to partially compensate for disturbances, which reduced the width of the quasi-sliding mode band. A similar approach was presented in [21], where a reaching function was introduced, and in [22], where each phase of the movement was separately defined.

An alternative definition of the quasi-sliding mode was proposed in 1995 by Gao et al. [23]. The authors defined three key requirements for the sliding mode to emerge. The representative point of the system should monotonically approach the sliding plane and cross it in finite time. After crossing the sliding plane for the first time, the sign of the sliding variable must change in each subsequent time instant and its absolute value may not exceed an a priori known value representing the width of the quasi-sliding mode band. Moreover, the authors proposed a reaching law-based design method, similar to the one applied for continuous time systems [24]. The proposed reaching function consisted of a proportional term, mostly responsible for the convergence to the sliding plane, and a signum component, used to ensure recrossing of the sliding surface. Their clear definition of the sliding mode has found many followers ever since and numerous switching-type reaching laws have been proposed to this day [25,26,27,28,29,30,31,32]. However, the switching requirement enforced high-frequency changes of the sign of the control input, which not only brings back the chattering problem but also may be hard to realize in practice. Therefore, the method, although inspired, is nowadays rarely applied.

Due to growing importance of the discrete time-control systems, this study is concerned with the design of a new power-rate reaching law for the quasi-sliding mode control of sampled time systems. The motivation behind this study is to provide an energetically efficient control law, which provides reasonably fast convergence to the sliding surface and enables control limitations to be satisfied. Driven by the need to further reduce the control effort and save energy in the control system, the novel reaching law realizes the non-switching type of motion [17,20]. In other words, it is required that the representative point of the system starting from any initial position monotonically approaches the switching hyperplane. After the representative point enters a specified band around the sliding plane it will never leave this band again. However, crossing the sliding surface is not required, which on the contrary to the Gao’s method, eliminates the chattering problem. Moreover, the novelty of the proposed reaching law lies in the application of a hyperbolic tangent function, which allows for the rate of change of the sliding variable in the reaching phase to be restricted. When the representative point of the system is in significant distance from the sliding surface, the convergence pace is relatively small in order to reduce the control input, and as follows, the energy consumption. However, once the representative point is closer to the sliding surface, the convergence pace may be increased. Finally, for the purpose of reducing the disturbance impact, the control is implemented through a model-following scheme, recently proposed in [33,34]. This allows one to limit the disturbance influence on the system to one sampling step only. The results of this study, as well as the proof of stability of the system, are obtained by the means of mathematical analysis. The results are further confirmed through computer simulations.

Compared to the previous studies of the model-following-based quasi-sliding mode [33,34], the novelty of this paper lies in the application of a power-rate reaching law for the model. Utilizing a hyperbolic tangent function in the reaching law allows for the pace of convergence of the reference sliding variable profile to the sliding surface to be adapted in real time. Thanks to the hyperbolic tangent properties, the proposed reaching law simultaneously provides faster convergence and smaller control magnitudes compared to previously known quasi-sliding mode control strategies. These attributes are consequently transferred to the control of the real system through the common reference trajectory following control scheme.

The remainder of this manuscript is organized as follows. Section 2 introduces the novel reaching law for the model of the system. In this case no external disturbances or modeling uncertainties are considered. Next, Section 3 proposes a model following control scheme for a system subject to matched external disturbances and modeling uncertainties. In this case, the system is forced to follow the trajectory generated in Section 2 and its stability and robustness are proved. Finally, Section 4 provides a simulation example for both cases.

2. Ideal Quasi-Sliding Mode Control System

This study is concerned with the quasi-sliding mode control design for a linear, discrete time system. In the first part of the paper, we consider the ideal quasi-sliding mode, which may be ensured, when the modeling parameters of the system are exactly known and there are no external disturbances. Therefore, we introduce a mathematical model of the system and propose a new rsecteaching law approach, which limits the pace of convergence to the sliding surface and provides a zero-width quasi-sliding mode band. Next, in Section 3, the obtained model’s sliding variable profile will serve as a reference to control a discrete time system subject to external disturbances and modeling uncertainties.

2.1. Model of the Ideal System

To begin, we introduce the mathematical model of the plant

$$x_m\left(k+1\right)=A_m{x}_m\left(k\right)+b_m{u}_m\left(k\right),$$

(1)

where x_m(k) is the model’s state vector of dimensions n × 1, A_m denotes the n × n state matrix, b_m represents the input distribution vector of dimensions n × 1 and u_m(k) is a scalar control signal, designed further in this paper. Let x₀ denote the initial state vector x_m(0). As this section is concerned with the ideal sliding mode, there are no disturbances to be considered.

For system (1), we define the sliding variable s_m(k) = c^T[x_d − x_m(k)], where c is a designer defined n × 1 vector and x_d denotes the demand state vector. Physically, the sliding variable denotes the distance between the representative point of the system and the sliding hyperplane:

$$s_m\left(k\right)=c^T\left[x_d-x_m\left(k\right)\right]=0.$$

(2)

Let s₀ denote the initial sliding variable value s_m(0).

The concept of the sliding mode control is to drive the representative point of the system to the sliding surface (2) in finite time and ensure that the states remain bounded to this surface for the rest of the control process. Therefore, the control problem is traditionally divided to two stages: the reaching phase and the sliding phase. In the sliding phase the dimensions of the dynamical problem are reduced to the sliding surface only and its choice becomes crucial to ensure stability of the system. Once the representative point hits the sliding surface for the first time at some k = k₀, considering (1) and (2), it is satisfied that

$$s_m\left(k+1\right)=c^T\left[x_d-A_m{x}_m\left(k\right)-b_m{u}_m\left(k\right)\right]=0,$$

(3)

For any k ≥ k₀. The control signal, which ensures such motion, is expressed as

$$u_m\left(k\right)={\left(c^T{b}_m\right)}^{-1}{c}^T\left[x_d-A_m{x}_m\left(k\right)\right].$$

(4)

Substituting the control Equation (4) into the state Equation (1), we obtain

$$x_m\left(k+1\right)=\Phi x_m\left(k\right)+b_m{\left(c^T{b}_m\right)}^{-1}{c}^T{x}_d,$$

(5)

where

$$\Phi =\left[1_{n\times n}-b_m{\left(c^T{b}_m\right)}^{-1}{c}^T\right]A_m$$

(6)

denotes the new state matrix of the closed-loop system. Its eigenvalues z_i may be found from the solution of the characteristic polynomial

$$M\left(z\right)=\det \left(1_{n\times n}z-\Phi \right)=0.$$

(7)

As vector c is designer-defined, it may be used to place the eigenvalues of the closed-loop system in the desired position, so that the stability of the system will be ensured.

2.2. Novel Power-Rate Reaching Law

The control signal (4), developed in the previous section may be successfully applied in the sliding phase, to force the representative point of the system to slide along the sliding surface (2). However, the application of the same control signal in the reaching phase will force the system to reach the sliding surface in one control step—so at k₀ = 1, which usually requires a high control magnitude. Therefore, it is common to introduce a reaching function to define the evolution of the sliding-variable profile in the entire course of motion. The reaching law approach, originally proposed by Gao et al. [23], allows for the prolonging of the reaching phase, reducing the control magnitudes at each time step.

The general form of the novel power-rate reaching law proposed in this paper is

$$s_m\left(k+1\right)=\left\{\begin{array}{cc}{s}_m\left(k\right)-q^{\alpha \left(k\right)}{s}_m\left(k\right){\left|s_m\left(k\right)\right|}^{-\alpha (k)}& \mathrm{for}\hspace{0.33em}\hspace{0.33em}\left|s_m\left(k\right)\right|\hspace{0.33em}>\epsilon \\ 0& \mathrm{for}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\left|s_m\left(k\right)\right|\hspace{0.33em}\le \epsilon \end{array}\right.,$$

(8)

where α(k) ∈ (0, 1), ε is a constant selected so that ε ≥ 1 and parameter q = 1 is introduced to keep consistent dimensions. Therefore, the dimension of q is the same as the dimension of the sliding variable. The choice of α(k) determines the pace of convergence to the sliding surface. Let us consider two cases. In the first case, α(k) is selected as a positive constant; and in the second case, α(k) is a function of the sliding variable.

For any selection of α, application of the reaching law (8) to the system (1), for any |s_m(k)| > ε ≥ 1, generates the following control signal:

$$u_m\left(k\right)={\left(c^T{b}_m\right)}^{-1}\left[c^T{x}_d-c^T{A}_m{x}_m\left(k\right)-s_m\left(k\right)+q^{\alpha \left(k\right)}{s}_m\left(k\right){\left|s_m\left(k\right)\right|}^{-\alpha \left(k\right)}\right],$$

(9)

whereas for any |s_m(k)| ≤ ε, control (4) is applicable. Next, we will show that the proposed reaching law guarantees finite time convergence to the sliding surface and ideal quasi-sliding motion along this surface.

Theorem 1.

Application of the reaching law (8) to the discrete time system (1) guarantees that there exists a finite moment k₀, when the representative point of the system belongs to the sliding plane (2), hence s_m(k₀) = 0, and it will remain on this plane for any k ≥ k₀.

Proof of Theorem 1.

When the first term of the reaching law (8) acts, so for |s_m(k)| > ε ≥ 1, then

$$s_m\left(k+1\right)=s_m\left(k\right)-q^{\alpha \left(k\right)}\frac{s_m\left(k\right)}{{\left|s_m\left(k\right)\right|}^{\alpha \left(k\right)}}=\mathrm{sgn}\left[s_m\left(k\right)\right]\left|s_m\left(k\right)\right|\left[1-q^{\alpha \left(k\right)}\frac{1}{{\left|s_m\left(k\right)\right|}^{\alpha \left(k\right)}}\right].$$

(10)

As |s_m(k)| > ε ≥ 1, q = 1 and α(k) ∈ (0, 1) we conclude that |s_m(k)|^α(k) > 1 and

$$0<\frac{1}{{\left|s_m\left(k\right)\right|}^{\alpha \left(k\right)}}<1.$$

(11)

Consequently, the sign of the sliding variable does not change and the absolute value decreases in each step. Hence, it holds that

$$\mathrm{sgn}\left[s_m\left(k+1\right)\right]=\mathrm{sgn}\left[s_m\left(k\right)\right]\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{and}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\left|s_m\left(k+1\right)\right|<\left|s_m\left(k\right)\right|.$$

(12)

The rate of change of the sliding variable is expressed as

$$\Delta s_m\left(k\right)=\left|s_m\left(k\right)\right|-\left|s_m\left(k+1\right)\right|=q^{\alpha \left(k\right)} {\left|s_m\left(k\right)\right|}^{1-\alpha \left(k\right)}>0.$$

(13)

Considering q = 1, (13) leads to

$$\Delta s_m\left(k\right)>0.$$

(14)

As follows from (14), there exists a finite time instant k₀ – 1, such that s_m(k₀ − 1) ≤ ε. For that case, the second term of the reaching law (8) acts. Therefore, there always exists such k₀ that

$$s_m\left(k_0\right)=0$$

(15)

and the representative point of the system remains on the sliding surface for and k ≥ k₀, which ends the proof. □

As shown by (13), the choice of parameter α determines the pace of convergence of the system to the sliding surface. As mentioned before, two cases will be considered. First let α be a positive constant, such that α ∈ (0, 1). From (13), we may state that for large α, 1 − α becomes small, which leads to a smaller rate of change of the sliding variable Δs_m(k). Consequently, for large α we obtain slow convergence to the sliding surface. On the other hand, when α decreases, then 1 − α increases and the pace of convergence rises. However, we must always bear in mind that faster convergence requires larger control effort. One shall also notice that the largest values of the control signal (9) occur when the representative point of the system is far from the sliding surface, so at the beginning of the control process. Therefore, it would be beneficial to choose large α at this stage of motion. Once the representative point approaches the sliding surface, the sliding variable values are smaller, so the control signal takes smaller values as well and parameter α could be increased. Therefore, the choice of parameter α is a compromise between the convergence pace and the control magnitudes.

As an attempt to provide some solution to this dilemma, we propose to replace constant α with a function α(k) = tanh[|s_m(k)|/s₀], where s₀ denotes the initial sliding variable value s_m(0). As the hyperbolic tangent function takes values from 0 to 1, condition α(k) ∈ (0, 1) is satisfied with such a selection. Substituting the hyperbolic tangent function into (14) gives

$$\Delta s_m\left(k\right)={\left|s_m\left(k\right)\right|}^{1-\tanh \left[\frac{\left|s_m\left(k\right)\right|}{s_0}\right]}>0.$$

(16)

The evolution of hyperbolic tangent is highly beneficial in restricting the control magnitude. The function proposed for α(k) takes the maximum value at k = 0, when

$$\alpha \left(k\right)=\tanh \left[\frac{s_m\left(0\right)}{s_0}\right]\approx 0.76.$$

(17)

and α(k) decreases as the representative point approaches the sliding hyperplane. As follows, the evolution of 1 − α(k) is opposite, which is presented in Figure 1.

As may be seen, for large s_m(k), 1 − α(k) is relatively small, which guarantees a small rate of change of the sliding variable. Therefore, at the beginning of the control process the control effort is significantly reduced. However, when the sliding-variable values are smaller, the hyperbolic tangent value decreases, which provides faster convergence to the sliding surface without exceeding the control limits. In other words, the proposed reaching law ensures that the convergence pace is adjusted during the control process and allows for more restrictive control limitations to be satisfied than other known sliding-mode control strategies. This further results in improved energetic efficiency of the closed-loop system, which constitutes the main achievement of this study.

3. Disturbance Consideration

3.1. Non-Ideal Control System

This section is concerned with a modification of the proposed strategy for a non-ideal type of systems. In this case, we are considering a discrete time plant subject to external disturbances and modeling uncertainties, described as

$$x\left(k+1\right)=A_{m}x\left(k\right)+\Delta x\left(k\right)+b_{m}u\left(k\right)+d\left(k\right),$$

(18)

where x(k) denotes the n × 1 system’s state vector, Δx(k) is a column vector representing parameter uncertainty, and d(k) denotes external disturbances. We assume both modeling uncertainties and disturbances are upper- and lower-bounded. Let the initial state be the same as for the model in Section 2, so x₀ = x(0) = x_m(0) and the demand state vector is denoted with x_d.

As there are external disturbances and parameter uncertainties acting on the system (18), the ideal sliding mode, defined by (8), is not achievable in this case. Instead, we propose a model following control scheme, which allows for the quasi-sliding mode band to be reached in finite time. We intend to use the ideal evolution of s_m(k) obtained from the model as the reference and drive the system’s representative point to the reference position in each discrete time instant. With that approach, the characteristic dynamics provided by the reaching law (8) shall be preserved with the accuracy of single-step disturbance impact. Therefore, the quasi-sliding mode band shall be reached in finite time, at some k = k₀ and for any k ≥ k₀ the value of the sliding variable may not exceed an a priori known constant. Following the same reasoning as for the model, we begin by defining the sliding variable s(k) = c^T[x_d − x(k)] and the sliding surface as

$$s\left(k\right)=c^T\left[x_d-x\left(k\right)\right]=0.$$

(19)

With this definition, the impact of external disturbance d(k) and parameter uncertainty Δx(k) on the sliding variable may be denoted with

$$S\left(k\right)=c^T\Delta x\left(k\right),\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}D\left(k\right)=c^{T}d\left(k\right).$$

(20)

Moreover, let the disturbances be bounded with

$$\left|S\left(k\right)\right|\le S_{\max },\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\left|D\left(k\right)\right|=D_{\max },$$

(21)

where S_max and D_max are known positive constants. For simplicity, their joined effect on the sliding variable will be denoted with

$$F_{\max }=D_{\max }+S_{\max }.$$

(22)

3.2. Model-Following Control Strategy

We propose to control the system according to the reference trajectory generated with the model and denoted with s_m(k). For that purpose, we apply a model-following reaching law:

$$s\left(k+1\right)=s_m\left(k+1\right)-D\left(k\right)-S\left(k\right).$$

(23)

From the reaching law (23), (18), and (19) we derive the system’s control law:

$$u\left(k\right)={\left(c^{T}b\right)}^{-1}\left[c^T{x}_d-c^{T}Ax\left(k\right)-s_m\left(k+1\right)\right].$$

(24)

Directly from (23), one may notice that the representative point of the system (18) is driven to the demand position, denoted with s_m(k + 1) in each step, with accuracy of the disturbance and uncertainty impact from one last time step. However, in order to ensure that the quasi-sliding mode band is approached monotonically the reference sliding-variable profile must be slightly adjusted to alleviate the disturbance influence. Therefore, we modify parameter ε in (8) into

$$\epsilon \ge \max \left\{1,\hspace{0.33em}2F_{\max }\right\}$$

(25)

and the model’s reaching law becomes

$$s_m\left(k+1\right)=\left\{\begin{array}{cc}{s}_m\left(k\right)-q^{\alpha \left(k\right)}{s}_m\left(k\right){\left|s_m\left(k\right)\right|}^{-\alpha (k)}& \mathrm{for}\hspace{0.33em}\hspace{0.33em}{\left|s_m\left(k\right)\right|}^{1-\alpha \left(k\right)}>\epsilon \\ 0& \mathrm{for}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}{\left|s_m\left(k\right)\right|}^{1-\alpha \left(k\right)}\le \epsilon \end{array}\right..$$

(26)

Following the same reasoning as presented in Section 2, when the first term of (26) operates, then

$$\mathrm{sgn}\left[s_m\left(k+1\right)\right]\left|s_m\left(k+1\right)\right|=\mathrm{sgn}\left[s_m\left(k\right)\right]\left|s_m\left(k\right)\right|\left[1-\frac{q^{\alpha \left(k\right)}}{{\left|s_m\left(k\right)\right|}^{\alpha \left(k\right)}}\right].$$

(27)

Taking into account that

$${\left|s_m\left(k\right)\right|}^{1-\alpha \left(k\right)}>\epsilon \ge 1$$

(28)

and α(k) ∈ (0, 1), we conclude that |s_m(k)| > 1. As follows, with q = 1, at this stage of motion it holds that

$$0<1-\frac{1}{{\left|s_m\left(k\right)\right|}^{\alpha \left(k\right)}}<1.$$

(29)

Consequently, for any k < k₀, the sign of s_m(k) remains constant and the absolute value decreases. In other words,

$$\mathrm{sgn}\left[s_m\left(k+1\right)\right]=\mathrm{sgn}\left[s_m\left(k\right)\right]\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{and}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\left|s_m\left(k+1\right)\right|<\left|s_m\left(k\right)\right|.$$

(30)

For any k ≤ k₀ − 2, the absolute value of the model’s sliding variable satisfies

$$\left|s_m\left(k\right)\right|>\epsilon .$$

(31)

Moreover, as stated in Theorem 1, there exists such finite time instant k₀ − 1 that

$$\left|s_m\left(k_0-1\right)\right|\le \epsilon$$

(32)

and such finite k₀ that for any k ≥ k₀,

$$\left|s_m\left(k\right)\right|=0.$$

(33)

The condition imposed on the rate of change of the model’s sliding variable in (26) implies that for k ≤ k₀ − 2,

$$\Delta s_m\left(k\right)=\left|s_m\left(k\right)\right|-\left|s_m\left(k+1\right)\right|=q^{\alpha \left(k\right)}{\left|s_m\left(k\right)\right|}^{1-\alpha \left(k\right)}>\epsilon \ge \max \left\{1,\hspace{0.33em}2F_{\max }\right\},$$

(34)

which is crucial to ensure monotonic convergence of the system (18) to the sliding surface.

Let us now consider the evolution of the system’s sliding variable s(k). For any k ≤ k₀ − 2, so when (31) holds, from (23) and (26) we obtain

$$\mathrm{sgn}\left[s\left(k\right)\right]\left|s\left(k\right)\right|=\mathrm{sgn}\left[s_m\left(k\right)\right]\left|s_m\left(k\right)\right|-D\left(k\right)-S\left(k\right).$$

(35)

Considering the bounds of the external disturbances and modeling uncertainties (22), with ε ≥ 2F_max we obtain

$$\left|D\left(k\right)\right|+\left|S\left(k\right)\right|\le F_{\max }<\epsilon .$$

(36)

Consequently, for any k ≤ k₀ – 2,

$$\mathrm{sgn}\left[s\left(k\right)\right]=\mathrm{sgn}\left[s_m\left(k\right)\right]=\mathrm{sgn}\left[s\left(0\right)\right]$$

(37)

and the absolute value of s_m(k) satisfies

$$\left|s_m\left(k\right)\right|-F_{\max }\le \left|s\left(k\right)\right|\le \left|s_m\left(k\right)\right|+F_{\max }.$$

(38)

Physically, (38) means that the representative point of the system is within the band of F_max radius around the reference position. The rate of change of the sliding variable in the reaching phase, for any k ≤ k₀ − 2, may be expressed as

$$\Delta s\left(k\right)=\left|s\left(k\right)\right|-\left|s\left(k+1\right)\right|=\Delta s_m\left(k\right)+\left|D\left(k\right)+S\left(k\right)-D\left(k-1\right)-S\left(k-1\right)\right|.$$

(39)

As the signs of external disturbances are unknown, we conclude that

$$\left|D\left(k\right)+S\left(k\right)-D\left(k-1\right)-S\left(k-1\right)\right|\le 2F_{\max }.$$

(40)

Considering (34), we conclude that at this stage Δs_m(k) > ε > 2F_max, so

$$\Delta s\left(k\right)>0.$$

(41)

Therefore, the representative point of the system (18) monotonically approaches the sliding surface (19). Next, for k = k₀ − 1, (32) holds. As follows,

$$\left|s\left(k\right)\right|\le \epsilon +F_{\max }$$

(42)

and the sign of the sliding variable might change. Finally, for any k ≥ k₀, the model’s sliding variable is bounded to the sliding surface so that (33) holds. Consequently, the system’s sliding variable becomes bounded to the quasi-sliding mode band of F_max radius. In other words,

$$\left|s\left(k\right)\right|\le F_{\max }.$$

(43)

Hence, the non-switching type of quasi-sliding mode occurs.

As defined in Section 2, parameter α(k) may be chosen as a positive constant or as a hyperbolic tangent function. According to (39), the rate of change of the sliding variable s(k) directly depends on the rate of change of the model’s sliding variable s_m(k). Hence, the system’s control signal u(k), defined with (24), depends on it as well. For optimal energy consumption, it would be beneficial to keep the control signal limited at the beginning of the control process, so when the representative point of the system is in significant distance from the sliding surface. At that stage of the control process the model’s sliding variable has large values, which normally causes large control magnitudes. Whereas, when the values of the model’s sliding variable are reduced, they have smaller impact on the control magnitude. Therefore, it is beneficial to select large α(k) at the beginning of the control process and smaller α(k) at the final stages of the reaching phase. Such evolution is provided with

$$\alpha \left(k\right)=\tanh \left[\frac{\left|s_m\left(k\right)\right|}{s_0}\right],$$

(44)

which has already been discussed in the previous section.

4. Simulation Example

This section presents the findings of the study with a simulation example. The aim is to demonstrate and compare the properties of the proposed reaching law (8) with α(k) chosen as a constant value and as a hyperbolic tangent function. For that purpose, we consider a second-order discrete time system,

$$x\left(k+1\right)=\left[\begin{array}{cc}1.2& 0.1\\ 0& 0.6\end{array}\right]x\left(k\right)+\Delta x\left(k\right)+\left[\begin{array}{c}0\\ 1\end{array}\right]u\left(k\right)+d\left(k\right),$$

(45)

where Δx(k) and d(k) are column vectors representing modeling uncertainty and external disturbances, respectively. We consider

$$\Delta x\left(k\right)+d\left(k\right)=\left[\begin{array}{l}0\\ f\end{array}\right]$$

(46)

and f varies from −1 to 1. We begin by creating a model of the system (45) defined by

$$x_m\left(k+1\right)=\left[\begin{array}{cc}1.2& 0.1\\ 0& 0.6\end{array}\right]x_m\left(k\right)+\left[\begin{array}{c}0\\ 1\end{array}\right]u_m\left(k\right).$$

(47)

Let the initial conditions be x(0) = x_m(0) = x₀ = [20 − 6]^T. The aim is to drive the system to the demand state vector x_d = [0 0]^T.

The control design process begins with the definition of the sliding surface for the model:

$$s_m\left(k\right)=c^T{x}_d-c^T{x}_m\left(k\right)=0,$$

(48)

where c^T = [c₁ c₂]. Considering (47) and (6) the closed-loop system’s state matrix Φ becomes

$$\Phi =\left[\begin{array}{cc}1.2-0.1\frac{c_1}{c_2}& 0\\ 0& 0.6\end{array}\right].$$

(49)

Therefore, the choice of vector c determines stability of the closed-loop system. We choose c₁ = 8 and c₂ = 1, which gives a stable closed-loop system:

$$x_m\left(k+1\right)=\left[\begin{array}{cc}0.4& 0\\ 0& 0.6\end{array}\right]x_m\left(k\right).$$

(50)

However, any other stable pole placement is acceptable. For such definition of the sliding variable for the model, we apply the reaching law (8) with the control (9). We select ε = 2 and, as defined above, we consider α(k):

$$\alpha \left(k\right)=0.75\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{and}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\alpha \left(k\right)=\tanh \left[\frac{\left|s_m\left(k\right)\right|}{s_0}\right],$$

(51)

where s₀ = 154. Value for α = 0.75 was chosen so that for both cases the control is restricted by

$$\left|u_m\left(k\right)\right|\le u_{\max }=34.$$

(52)

As we are dealing with a discrete time system, the simulations were carried out in Matlab Simulink with a discrete solver and a fixed step of 1 unit. For the sake of comparison, the plots obtained with constant α will be shown in blue and with hyperbolic tangent function in red. It may be clearly seen from Figure 2 that the reaching law (8) provides monotonic convergence to the sliding surface and the non-switching type of quasi-sliding motion. However, the application of the hyperbolic tangent function provides much faster convergence. For constant α the sliding phase begins at k₀ = 57, whereas for the hyperbolic tangent function k₀ = 22. It is shown in Figure 3 that this significant improvement is achieved without exceeding the control limitation (52). Moreover, faster convergence guarantees a reduction in the control effort in the sliding phase, which results in saving energy in the whole regulation process.

In the next step, we utilize the obtained s_m(k) profile as a reference to control system (45). We define the sliding variable and the sliding plane using the same vector c as for the model:

$$s\left(k\right)=c^T\left[x_d-x\left(k\right)\right]=\left[\begin{array}{cc}8& 1\end{array}\right]\left[x_d-x\left(k\right)\right]=0.$$

(53)

Therefore, the impact of external disturbances and modeling uncertainties is bounded by F_max = 1. Parameter ε = 2 chosen for the model satisfies condition (25). With the application of control (24) we obtained the evolution of the sliding variable presented in Figure 4 and Figure 5. It is clearly visible that the proposed control strategy ensured monotonic convergence of the representative point of the system to the sliding surface and non-switching movement along this surface. The width of the quasi-sliding mode band is 2F_max = 2, as stated in (43). For the constant α case, the sliding phase begins at k₀ = 56 and for the hyperbolic tangent case at k₀ = 22, which may be verified in Figure 5. Figure 6 presents the control signal in both cases, which shows that the control limit (52) has not been exceeded. Finally, Figure 7 and Figure 8 present the evolution of the system’s state variables, which both converge to the demand state vector x_d in finite time.

From the presented figures, one may notice that the presented model-following control strategy combined with the novel power-rate reaching law provides a non-switching-type quasi-sliding mode in the presence of external disturbances and modeling uncertainties. Moreover, the choice of α(k) = tanh[|s_m(k)|/s₀] guarantees significantly faster convergence to the sliding surface while satisfying the same control limit.

5. Conclusions

This paper was aimed to provide an energetically efficient control strategy for discrete time systems subject to external disturbances and modeling uncertainties. The study proposed a novel model reference-based sliding-mode control scheme. Firstly, a model of the considered control system was created by neglecting any external disturbances. For the model, an ideal quasi-sliding mode was obtained with a new power-rate reaching law. The main achievement of this study is the construction of the reaching law, which utilized a hyperbolic tangent function in order to limit the control input in the reaching phase. In the initial stage of the control process, when the representative point of the system is in significant distance from the sliding surface, the convergence pace is limited thanks to the hyperbolic tangent function properties. However, when the representative point gets closer to the switching surface, the value of the sliding variable has a smaller impact on the control signal. Therefore, the convergence pace may be increased without unnecessarily large control magnitudes. As results, the novel reaching law allows for the adjustment of the convergence pace during the control process. Moreover, the proposed reaching law realizes the non-switching type of quasi-sliding mode, so the chattering problem is eliminated. In fact, as no external disturbances are considered at this point, in the sliding phase, for all k ≥ k₀, the representative point of the model belongs to the sliding surface. In the further part of the study, the ideal trajectory generated with the reference model serves as the demand profile for the sliding variable of the actual system, considering external disturbances and modeling uncertainties. It is proved, by the means of mathematical analysis, that the representative point of the system follows the desired path with accuracy of the disturbance and modeling uncertainty of a single step. Moreover, for any k ≥ k₀, the sliding variable becomes bounded by ±F_max, so a non-switching quasi-sliding mode occurs. Although the model-following quasi-sliding mode control strategies have been widely studied in the recent years, the novelty of this paper lies in the use of the hyperbolic tangent function in the model’s trajectory generator. On the contrary to other known reaching laws, the properties provided by the power-rate reaching law proposed for the model, such as adaptation of the convergence pace and resulting reduction of the control input, play the main role in performance improvement. The model’s convergence is slower at the beginning of the control process and its pace increases as the representative point approaches the sliding surface. This property is subsequently transferred to the real system, subject to external disturbances and modeling uncertainties. Consequently, the system’s control magnitudes are reduced, which directly results in energy saving.

The proposed control method may turn out applicable in the control of power electronic systems, power converters or electrical motor drives, where avoiding extensive power consumption and losses is important from the point of view of the remaining parts of the system. Moreover, elimination of chattering turns out useful in electromechanical applications, as continuous switching of the control signal may lead to mechanical damages.