Design and analysis of higher-order repetitive sliding mode controller for uncertain linear systems with time-varying periodic disturbances

Abstract

In this article, a discrete-time design of higher-order repetitive sliding mode controller (HO-RSMC) for uncertain linear systems with time-varying periodic disturbances is presented. In the design, a higher-order repetitive control (HORC) is employed to significantly suppress the periodic exogenous disturbance with period variation. The HORC-based control system offers a slow transient response due to the multiple control delays. Moreover, since the system stability is easily affected by model uncertainties, an output-feedback sliding mode controller (SMC) is added to overcome the shortcomings of HORC. Stability analysis of the resulting closed-loop system is rigorously provided. The simulation results and comparison studies confirm the effectiveness and good performance of the proposed design.

Keywords

Higher-order repetitive control (HORC)sliding mode control (SMC)uncertain linear systems time-varying periodic disturbances

Introduction

Periodic compensation is a common problem found in many control applications, for example, power electronics (Pandove and Singh, 2019; Zheng et al., 2018), energy (Liu et al., 2020; Ma et al., 2019b), mechatronics (Mondal et al., 2014; Nie et al., 2021), and biomedical (Laurent et al., 2009; Page and Freeman, 2020). Repetitive control (RC) as an internal model–based control strategy is well known for its capability of compensating periodic signals. The internal model–based control, or sometimes referred to internal model principle (IMP), is a control method originated by Francis and Wonham (1975) stating that a perfect disturbance cancelation is achieved when the feedback control loop contains the dynamics model of the disturbance signal.

The traditional RC is mostly designed by assuming a constant period (trial, batch, cycle, pass, repetition) of the disturbance which leads to the fixed integer number of samples per disturbance period, N. The integer number N is later defined as the delay length of the RC system. In practice, disturbances may be time-varying in period, where the time-varying periodic disturbances may appear in rotating machines (Yang et al., 2020), steel casting processes (Ma et al., 2019c), active suspension systems (Zhao et al., 2022), hypersonic flight vehicles (Dong et al., 2020), and many others. When the actual disturbance period is subject to variation, the RC system operates with a period mismatch. This condition makes the RC gains at a fundamental frequency and its harmonics drastically drop to a low-level magnitude. Consequently, the perfect disturbance cancelation is no more applicable, and the tracking performance is significantly degraded.

To deal with the limitation of traditional RC against uncertain periodic disturbances, an adaptive RC (ARC) and a higher-order RC (HORC) have been developed. In the presence of period variation, ARC works by tuning the sampling time $T_{s}$ to maintain the constant number $N$ per disturbance period (Olm et al., 2010; Kurniawan et al., 2014a, 2014b; Faro et al., 2022). By adjusting the sampling time $T_{s}$ , the discrete-time plant model also changes, which may result in an unstable RC closed-loop system. This poses another challenge in ARC-controlled system.

Moreover, the ARC gives rise to system’s complexity due to the use of multi-rate control, and the ARC is also not practical for the disturbance with rapid period variation. Another approach is HORC method, which attempts to modify the structure of the internal model and thus gives the robustness against period variation. Some examples of this approach were found in Steinbuch et al. (2007), Ramos et al. (2013), Flores and Flores (2018), Jamil et al. (2020) and Kurniawan et al. (2021), where an internal model with multiple periodic generators was used. The HORC keeps the sampling time $T_{s}$ remain fixed, but the use of multiple periodic generators increases the RC gains at the frequencies around the harmonics. Hence, the disturbance rejection performance at the neighboring frequencies should be improved by applying the HORC.

Unlike the traditional RC which involves a single delay length $N$ , the HORC comprises multiple delay lengths (i.e. $N$ , $2 N$ , …, $hN$ ), where $h$ is the order of HORC. As the consequence, the HORC introduces a longer control delay, resulting in slower convergence than the traditional RC. Besides the transient issues, the performance of HORC is also greatly affected by unmodeled dynamics, nonlinearities, and other uncertainties. To overcome the shortcomings of HORC, one may complement HORC with other control strategy such as sliding mode controller (SMC). SMC is a nonlinear control technique that is capable of providing fast transient response, strong robustness against plant uncertainties, and precise reference tracking (Yu et al., 2012). Similar to RC, SMC as well-known control strategy has been widely exploited and applied in many control applications (Bartoszewicz and Lesniewski, 2016; Brahmi et al., 2020; Chen et al., 2022; Ma et al., 2019a; Song et al., 2022a, 2022b).

In this paper, we address the problem of tracking control of uncertain linear systems affected by time-varying periodic disturbances. A novel strategy to incorporate HORC with SMC is presented to achieve fast transient response, good rejection of uncertain periodic disturbances, and robustness against plant uncertainties. Here, the HORC is applied to significantly suppress the uncertain periodic disturbances, while the SMC is utilized to provide fast transient response and robustness against plant uncertainties. In this study, the stability of the controlled system is analyzed, and the proposed controller is proved to be implementable, resulting in a stable closed-loop system. Simulation results and comparison studies validate the effectiveness of the proposed design. In order to emphasize the originality of our research work, our main contributions are listed as follows:

A systematic control strategy integrating the HORC and the SMC rendering high-order repetitive sliding mode control (HO-RSMC) is developed in this work. The HORC is chosen over the traditional RC to suppress the time-varying periodic disturbances significantly. However, HORC poses slower transient responses and instability due to model uncertainties. Hence, SMC is complemented to provide with faster transient responses and stronger robustness against model uncertainties.

The inclusion of the SMC makes the proposed controller becomes nonlinear. Then, the stability analysis based on SMC’s reaching condition is established. It is proved that the proposed controller is implementable and results in a bounded and stable closed-loop system.

The remainder of this paper is organized as follows: “Proposed method” section describes the proposed method covering problem formulation and HO-RSMC design method. Stability analysis of the HO-RSMC controlled system is presented in “Stability analysis” section. In “Results and discussions” section, simulation results and comparison study are provided. “Conclusion” section concludes the paper.

Proposed method

Problem formulation

In this work, we consider an uncertain linear time-invariant (LTI) system as follows

y_{k} = z^{- m} [G (z)] u_{k} + v_{k}

(1)

where $k, m \in Z_{> 0}$ ; $y_{k}$ , $u_{k}$ , and $v_{k} \in R$ denote the plant output, control input, and time-varying periodic exogenous disturbance, respectively; $z^{- m}$ is the input delay with length $m$ ; and $G (z)$ is the uncertain plant model. Here, the notation $z$ refers to $Z$ -transform variable and also the forward shifting operator, for example, $z u_{k} = u_{k + 1}$ . In addition, the notation $z^{- 1}$ represents a backward shifting operator, for example, $z^{- 1} u_{k} = u_{k - 1}$ .

The uncertain plant model $G (z)$ is expressed in the following form

G (z) = \frac{B (z) + Δ_{B} (z)}{A (z) + Δ_{A} (z)}, A (z) + Δ_{A} (z) \neq 0, A (z) \neq 0

(2)

where $B (z)$ and $A (z) \in C$ are nominal plant numerator and denumerator, respectively; $Δ_{B} (z) \in C$ is the parametric uncertainty of $B (z)$ ; and $Δ_{A} (z) \in C$ is the parametric uncertainty of $A (z)$ .

The plant numerator $B (z)$ and denumerator $A (z)$ are polynomials, which are given by

B (z) = b_{0} + b_{1} z^{- 1} + b_{2} z^{- 2} + \dots + b_{n_{b}} z^{- n_{b}}

(3)

A (z) = 1 + a_{1} z^{- 1} + a_{2} z^{- 2} + \dots + a_{n_{a}} z^{- n_{a}}

(4)

where $n_{b,} n_{a} \in N$ , and $n_{b} < n_{a}$ .

The tracking error is defined as the difference between the reference signal $r_{k}$ and the plant output $y_{k}$

e_{k} = r_{k} - y_{k}

(5)

The design objective is to synthesize the control input $u_{k}$ such that (1) the time-varying periodic disturbance $v_{k}$ is significantly suppressed, (2) the tracking error $e_{k}$ converges to a nearly zero steady-state value, (3) the fast transient response is obtained, and (4) the closed-loop system is both stable and robust against plant uncertainties. Several assumptions are made before proceeding the design of the proposed controller:

Assumption 1. The time-delay m in (1) is assumed to be known and constant. In addition, $B (z)$ and $A (z)$ are stable polynomials implying that all roots of $B (z)$ and $A (z)$ are inside the unit circle. Note that the design is intended for minimum-phase plant, which requires the plant $G (z)$ to be minimum-phase definitely. A different approach is required to formulize the control law design for either the non-minimum system or the system with unknown/uncertain time-delay m.

Assumption 2. The disturbance $v_{k}$ is periodic. The nominal period of the disturbance $v_{k}$ , that is, ${\bar{T}}_{v}$ , is known. However, the actual disturbance period $T_{v}$ may vary from its nominal value (i.e. $T_{v} = {\bar{T}}_{v} \pm Δ_{T}$ ).

Higher-order repetitive sliding mode controller

We first formulate an error dynamic $e_{k}$ derived from the tracking error (5) and system dynamics (1). Substituting equation (2) into equation (1), the system dynamics (1) can be rewritten in the following form

[A (z) + Δ A (z)] y_{k} = z^{- m} [B (z) + Δ_{B} (z)] u_{k} + [A (z) + Δ A (z)] v_{k}

(6)

Further rearranging equation (6), we obtain

y_{k} = z^{- m} [\frac{B (z)}{A (z)}] u_{k} + v_{k} - [\frac{Δ_{A} (z)}{A (z)}] y_{k} + z^{- m} [\frac{Δ_{B} (z)}{A (z)}] u_{k} + [\frac{Δ_{A} (z)}{A (z)}] v_{k}

(7)

The overall disturbances affecting the plant is defined as

d_{k} = v_{k} - [\frac{Δ_{A} (z)}{A (z)}] y_{k} + z^{- m} [\frac{Δ_{B} (z)}{A (z)}] u_{k} + [\frac{Δ_{A} (z)}{A (z)}] v_{k}

(8)

From equation (8), we notice that the disturbance $d_{k}$ comprises the exogenous disturbance $v_{k}$ and other disturbances generated due to the existence of plant uncertainties $Δ_{A} (z)$ and $Δ_{B} (z)$ .

Now, equation (7) can be expressed as

y_{k} = z^{- m} [\frac{B (z)}{A (z)}] u_{k} + d_{k}

(9)

Substituting equation (9) into equation (5) yields an error dynamics $e_{k}$ , that is

e_{k} = r_{k} - z^{- m} [\frac{B (z)}{A (z)}] u_{k} - d_{k}

(10)

In the design, a high-order RC is used to suppress the exogenous disturbance $v_{k}$ . For the simplicity, let the HORC with second-order form is considered. The second-order RC is expressed by the following internal model

I_{m} (z) = \frac{1}{1 - Q (z) D (z)}

(11)

where $D (z)$ is a sum of the weighted repetitive delays and $Q (z)$ is a low-pass filter. The term $D (z)$ in equation (11) has the following expression

D (z) = w_{1} z^{- N} + w_{2} z^{- 2 N}

(12)

where $w_{1}$ and $w_{2}$ are the delay weights and $N$ is an integer number defined as a fundamental delay length. The integer number $N$ is obtained from

N = \frac{{\bar{T}}_{v}}{T_{s}}

(13)

where ${\bar{T}}_{v}$ is the nominal period of disturbance $v_{k}$ , and $T_{s}$ is a sampling period. Note that for the traditional/first-order RC, equation (12) has a simple form as

D (z) = z^{- N}

(14)

The Q-filter is generally chosen as a zero-phase low-pass filter given by

Q (z) = q_{1} z^{- 1} + q_{0} + q_{1} z^{1}, with q_{0} + 2 q_{1} = 1

(15)

The filter (15) gives a unity gain at the frequencies below the filter’s bandwidth (i.e. $f < f_{Q}$ ). In addition, $Q (z)$ contributes zero-phase for all frequency components (i.e. $0 < f < f_{N}$ , where $f_{N}$ is Nyquist frequency). The weights $w_{1}$ and $w_{2}$ are properly chosen to ensure that the gains of HORC at fundamental frequency and its harmonics remain infinite. Hence, the weights $w_{1}$ and $w_{2}$ are picked to satisfy the following conditions (Steinbuch et al., 2007)

w_{1} + w_{2} = 1

(16)

w_{1} + 2 w_{2} = 0

(17)

The magnitude responses of the traditional RC and the HORC at the targeted fundamental frequency $1 Hz$ are shown in Figure 1. It can be seen that the high gain peaks at the harmonics are extended to larger regions. Thus, the disturbance rejection performance at the neighboring frequencies should be improved by applying the HORC.

Now, we define the repetitive model $R_{m} (z)$ which is a denumerator of (11), and later is used in the construction of control law $u_{k}$ . That is

R_{m} (z) \overset{Δ}{=} 1 - Q (z) D (z)

(18)

The HORC-controlled system offers a slow transient response due to the delay term $D (z)$ , and the system stability is easily affected by model uncertainties. Hence, the SMC technique is complemented to overcome the shortcomings of HORC. An output-feedback SMC with a linear sliding function is employed, that is

s_{k} = H (z) e_{k}

(19)

where $H (z)$ is a stable polynomial.

The following discrete-time reaching law (Gao et al., 1995; Sun et al., 2005) is also used in the design

s_{k + m} - s_{k} = - c_{1} T_{s} s_{k} - c_{2} T_{s} sat (\frac{s_{k}}{c_{3}})

(20)

We can represent the reaching law equation (20) as

s_{k + m} = (1 - c_{1} T_{s}) s_{k} - c_{2} T_{s} sat (\frac{s_{k}}{c_{3}})

(21)

From equation (21), the similar expression can be written as follows

s_{k} = (1 - c_{1} T_{s}) s_{k - m} - c_{2} T_{s} sat (\frac{s_{k - m}}{c_{3}})

(22)

where $c_{1}$ , $c_{2}$ , and $c_{3}$ are positive coefficients, $T_{s}$ is sampling time, and sat $(s_{k - m} / c_{3})$ refers to a saturation function of $(s_{k - m} / c_{3})$ . The saturation function is given by

sat (\frac{s_{k - m}}{c_{3}}) = {\begin{matrix} 1, s_{k - m} > c_{3} \\ \frac{s_{k - m}}{c_{3}}, - c_{3} < s_{k - m} < c_{3} \\ - 1, s_{k - m} < c_{3} \end{matrix}

(23)

Now, we can construct the HO-RSMC law based on the repetitive model (18), sliding function (19), and reaching law (20). Multiplying both sides of (19) by the repetitive model (18), we obtain

[1 - Q (z) D (z)] s_{k} = [1 - Q (z) D (z)] H (z) e_{k}

(24)

Substituting the error dynamics (10) into (24) yields

[1 - Q (z) D (z)] s_{k} = [1 - Q (z) D (z)] H (z) {r_{k} - z^{- m} [\frac{B (z)}{A (z)}] u_{k} - d_{k}}

(25)

Rearranging equation (25), we get

\begin{matrix} s_{k} - [Q (z) D (z)] s_{k} = [{1 - Q (z) D (z)} H (z)] r_{k} \\ - [{1 - Q (z) D (z)} H (z)] d_{k} \\ - z^{- m} [{1 - Q (z) D (z)} H (z)] [\frac{B (z)}{A (z)}] u_{k} \end{matrix}

(26)

Consider equation (26) for the nominal plant, that is, $d_{k} = v_{k}$ . Note that any periodic signal with period of ${\bar{T}}_{v}$ filtered by repetitive model (18) results in $[1 - Q (z) D (z)] v_{k} \approx 0$ . Hence, the term $[{1 - Q (z) D (z)} H (z)] d_{k}$ in equation (26) can be neglected, and then equation (26) can be simplified to

\begin{matrix} s_{k} - [Q (z) D (z)] s_{k} = [{1 - Q (z) D (z)} H (z)] r_{k} \\ - [\frac{H (z) B (z)}{A (z)}] [1 - Q (z) D (z)] u_{k - m} \end{matrix}

(27)

We define $R_{H} (z)$ and $G_{H} (z)$ , respectively, as

R_{H} (z) \overset{Δ}{=} {1 - Q (z) D (z) H (z)}

(28)

G_{H} (z) \overset{Δ}{=} [\frac{H (z) B (z)}{A (z)}]

(29)

Then, equation (27) can be rewritten as

\begin{matrix} s_{k} - [Q (z) D (z)] s_{k} = [R_{H} (z)] r_{k} - [G_{H} (z)] \\ {u_{k - m} - [Q (z) D (z)] u_{k - m}} \end{matrix}

(30)

Using equation (12), equation (30) can be expressed as

\begin{matrix} s_{k} - w_{1} [Q (z)] s_{k - N} - w_{2} [Q (z)] s_{k - 2 N} = [R_{H} (z)] r_{k} \\ - [G_{H} (z)] {u_{k - m} - w_{1} [Q (z)] u_{k - N - m} - w_{2} [Q (z)] u_{k - 2 N - m}} \end{matrix}

(31)

We can rearrange equation (31) as

\begin{matrix} [G_{H} (z)] u_{k - m} = [G_{H} (z)] {w_{1} [Q (z)] u_{k - N - m} + w_{2} [Q (z)] u_{k - 2 N - m}} \\ + [R_{H} (z)] r_{k} + w_{1} [Q (z)] s_{k - N} + w_{2} [Q (z)] s_{k - 2 N} - s_{k} \end{matrix}

(32)

Multiplying equation (32) by $G_{H}^{- 1} (z)$ , we have

\begin{matrix} u_{k - m} = w_{1} [Q (z)] u_{k - N - m} + w_{2} [Q (z)] u_{k - 2 N - m} + [\frac{R_{H} (z)}{G_{H} (z)}] r_{k} \\ + w_{1} [\frac{Q (z)}{G_{H} (z)}] s_{k - N} + w_{2} [\frac{Q (z)}{G_{H} (z)}] s_{k - 2 N} - [\frac{1}{G_{H} (z)}] s_{k} \end{matrix}

(33)

Forward shifting equation (33) by $m$ steps, we obtain

\begin{matrix} u_{k} = w_{1} [Q (z)] u_{k - N} + w_{2} [Q (z)] u_{k - 2 N} + [\frac{R_{H} (z)}{G_{H} (z)}] r_{k + m} \\ + w_{1} [\frac{Q (z)}{G_{H} (z)}] s_{k - N + m} \\ + w_{2} [\frac{Q (z)}{G_{H} (z)}] s_{k - 2 N + m} - [\frac{1}{G_{H} (z)}] s_{k + m} \end{matrix}

(34)

Replacing $s_{k + m}$ by equation (21), we obtain the proposed HO-RSMC as follows

\begin{matrix} u_{k} = w_{1} [Q (z)] u_{k - N} + w_{2} [Q (z)] u_{k - 2 N} + [\frac{R_{H} (z)}{G_{H} (z)}] r_{k + m} \\ + w_{1} [\frac{Q (z)}{G_{H} (z)}] s_{k - N + m} \\ + w_{2} [\frac{Q (z)}{G_{H} (z)}] s_{k - 2 N + m} - [G_{H}^{- 1} (z)] \\ {(1 - c_{1} T_{s}) s_{k} - c_{2} T_{s} sat (\frac{s_{k}}{c_{3}})} \end{matrix}

(35)

Similarly, we can represent the HO-RSMC law (35) as

\begin{matrix} u_{k} = {w_{1} [z^{- N} Q (z)] + w_{2} [z^{- 2 N} Q (z)]} u_{k} + [\frac{R_{H} (z)}{G_{H} (z)}] r_{k + m} \\ + {w_{1} [\frac{z^{- N + m} Q (z)}{G_{H} (z)}] + w_{2} [\frac{z^{- 2 N + m} Q (z)}{G_{H} (z)}]} s_{k} \\ - [G_{H}^{- 1} (z)] {(1 - c_{1} T_{s}) s_{k} - c_{2} T_{s} sat (\frac{s_{k}}{c_{3}})} \end{matrix}

(36)

Based on equation (36), the interconnection between the proposed controller and the open-loop plant can be depicted in Figure 2.

Remark 1. The low-pass filter $Q (z)$ (15) is originally represented in a non-causal form, but it is realizable due to the terms $z^{- N}$ , $z^{- 2 N}$ , $z^{- N + m}$ , and $z^{- 2 N + m}$ as shown in (36).

Remark 2. According to (19), $H (z)$ is defined as a stable polynomial. Hence, the term $R_{H} (z)$ as given in (28) is also a stable polynomial since the repetitive model $[1 - Q (z) D (z)]$ is a stable polynomial.

Remark 3. Following Assumption 1 that $B (z)$ and $A (z)$ are both stable polynomials, and $H (z)$ is chosen to be stable. Then, the term $[G_{H}^{- 1} (z)]$ which is an inverse form of (29) is a proper and stable transfer function.

Remark 4. Note that the proposed control law (35) is still limited for single-input single-output (SISO) system. Extending the work for a multivariable system requires further investigation. This is because the controller is an output feedback–based and involves the inverse dynamics in the synthesis. Some challenges arise when applying RC-based controller for the multivariable system. To obtain the inverse dynamics (i.e. invertible), the multivariable system must be in square form. This implies that the number of inputs is equal to the number of outputs (Kurniawan et al., 2016). In addition, the inverse dynamics should be causal and stable. Hence, a further analysis is required to extend the presented work to the multivariable system.

Based on Remarks (1)–(3), we conclude that the proposed controller is realizable. Moreover, the resulting controller guarantees the convergence of sliding function $s_{k}$ and the tracking error $e_{k}$ . We will prove this claim in the next section.

Stability analysis

The inclusion of reaching law (21) makes the controller (35) to be nonlinear. Thus, a linear stability analysis is no longer suitable in this case. To assess the stability of controlled system, the discrete-time reaching condition (Gao et al., 1995; Sun et al., 2005) is employed, which is given by

| s_{k} | < | s_{k - m} |

(37)

The reaching condition equation (37) is equivalent to

[s_{k} - s_{k - m}] sgn (s_{k - m}) < 0

(38)

[s_{k} + s_{k - m}] sgn (s_{k - m}) > 0

(39)

Next, we use the condition equations (38) and (39) to prove the stability of the controlled system with HO-RSMC. Since the stability analysis uses the sliding function $s_{k}$ , it is obvious to first obtain the sliding dynamics. To obtain the sliding dynamics, we begin by substituting $e_{k}$ in the sliding function (19) with the error dynamics (10). This gives

s_{k} = [H (z)] r_{k} - z^{- m} [\frac{H (z) B (z)}{A (z)}] u_{k} - [H (z)] d_{k}

(40)

Using equation (29), equation (40) can be further derived as

s_{k} = [H (z)] r_{k} - [G_{H} (z)] u_{k - m} - [H (z)] d_{k}

(41)

Now, we can substitute equation (32) into equation (41). Then, equation (41) becomes

\begin{matrix} s_{k} = [H (z)] r_{k} - [H (z)] d_{k} - [G_{H} (z)] \\ {w_{1} [Q (z)] u_{k - N - m} + w_{2} [Q (z)] u_{k - 2 N - m}} \\ - [R_{H} (z)] r_{k} - w_{1} [Q (z)] s_{k - N} - w_{2} [Q (z)] s_{k - 2 N} + s_{k} \end{matrix}

(42)

Using equation (12) and substituting equation (21) into equation (42), we can rewrite equation (42) as

\begin{matrix} s_{k} = [H (z)] r_{k} - [H (z)] d_{k} - [Q (z) D (z) G_{H} (z)] u_{k - m} - [R_{H} (z)] r_{k} \\ - [Q (z) D (z)] s_{k} + (1 - c_{1} T_{s}) s_{k - m} - c_{2} T_{s} sat (\frac{s_{k - m}}{c_{3}}) \end{matrix}

(43)

Multiplying equation (41) by $Q (z) D (z)$ gives

\begin{matrix} [Q (z) D (z)] s_{k} = [Q (z) D (z) H (z)] r_{k} - [Q (z) D (z) G_{H} (z)] u_{k - m} \\ - [Q (z) D (z) H (z)] d_{k} \end{matrix}

(44)

Subtracting equation (44) from equation (43), we obtain a final form of the sliding dynamics as follows

s_{k} = (1 - c_{1} T_{s}) s_{k - m} - c_{2} T_{s} sat (\frac{s_{k - m}}{c_{3}}) + [{Q (z) D (z) - 1} H (z)] d_{k}

(45)

From equation (45), we notice that the sliding dynamics consists of two error terms defined as SMC error and RC error as follows

e_{vk} = (1 - c_{1} T_{s}) s_{k - m} - c_{2} T_{s} sat (\frac{s_{k - m}}{c_{3}})

(46)

e_{rk} = [{Q (z) D (z) - 1} H (z)] d_{k}

(47)

Now, we can simplify equation (45) as

s_{k} = (1 - c_{1} T_{s}) s_{k - m} - c_{2} T_{s} sat (\frac{s_{k - m}}{c_{3}}) + e_{rk}

(48)

Theorem 1. Consider case 1 when the actual disturbance period $T_{v}$ is equal to the known nominal period ${\bar{T}}_{v}$ (i.e. $T_{v}$ = ${\bar{T}}_{v}$ ), and the plant uncertainties are not present $(Δ_{B} (z) = Δ_{A} (z) = 0)$ ; then, the RC error $e_{rk}$ in sliding dynamics (48) can be neglected due to that the total disturbance $d_{k}$ is perfectly canceled.

Proof. Based on equation (8), the total disturbance $d_{k}$ is equal to the periodic exogenous disturbance $v_{k}$ . This is due to the plant uncertainties are not present $(Δ_{B} (z) = Δ_{A} (z) = 0)$ . Thus, $d_{k}$ is a periodic signal with period of ${\bar{T}}_{v}$ . From equation (47), the filtered total disturbance $d_{k}$ is defined as $f_{k}$

f_{k} = [H (z)] d_{k}

(49)

It is known from equation (19) that $H (z)$ is stable polynomial. Hence, $f_{k}$ is also a periodic signal with period of ${\bar{T}}_{v}$ , but its magnitude and phase may differ from the ones in $d_{k}$ . Now, we can rewrite equation (47) as

e_{rk} = [Q (z) D (z) - 1] f_{k}

(50)

Applying equations (12) and (50) can be further expanded as

e_{rk} = w_{1} [z^{- N} Q (z)] f_{k} + w_{2} [z^{- 2 N} Q (z)] f_{k} - f_{k}

(51)

Based on the facts that $Q (z)$ is zero-phase filter with unity gain at frequencies below the filter’s bandwidth, and the actual disturbance period $T_{v}$ is equal to the nominal period ${\bar{T}}_{v}$ , the following properties hold

P 1 : [Q (z)] f_{k} = f_{k}, for 0 < ω_{v} < ω_{Q}

(52)

P 2 : f_{k} - f_{k - iN} = 0, for i \in {1, 2} and k > 2 N

(53)

Using the property equations (52) and (53) and the weights condition equation (16), equation (51) can be derived to

\begin{matrix} e_{rk} = w_{1} [z^{- N} Q (z)] f_{k} + w_{2} [z^{- 2 N} Q (z)] f_{k} - f_{k} \\ = w_{1} [z^{- N}] f_{k} + w_{2} [z^{- 2 N}] f_{k} - f_{k} \\ = w_{1} f_{k - N} + w_{2} f_{k - 2 N} - f_{k} \\ = w_{1} f_{k} + w_{2} f_{k} - f_{k} \\ = (w_{1} + w_{2} - 1) f_{k} \\ = 0 \end{matrix}

(54)

Then, it follows from equation (54) that the RC error $e_{rk}$ can be neglected as the filtered disturbance $f_{k}$ is canceled, which implies that the exogenous disturbance $v_{k}$ is also rejected.

Theorem 2. Consider case 2 when the actual disturbance period $T_{v}$ slightly varies from the known nominal period ${\bar{T}}_{v}$ (i.e. $T_{v} = {\bar{T}}_{v} \pm Δ_{T}$ ), and the plant uncertainties are not present (i.e. $Δ_{B} (z) = Δ_{A} (z) = 0$ ); then, the RC error $e_{rk}$ in (48) is no longer zero, but it is relatively small as the total disturbance $d_{k}$ is significantly suppressed by HORC model $[1 - Q (z) D (z)]$ .

Proof . When $T_{v} = {\bar{T}}_{v} \pm Δ_{T}$ , the property (53) does not hold, but changes to

P 3 : f_{k} - f_{k - iN} = α_{i}, for i \in {1, 2} and k > 2 N

(55)

where $α_{1}$ and $α_{2}$ are small, bounded mismatch values. Then, we can derive equation (51) to

\begin{matrix} e_{rk} = w_{1} f_{k - N} + w_{2} f_{k - 2 N} - f_{k} \\ = w_{1} f_{k} - w_{1} α_{1} + w_{2} f_{k} - w_{2} α_{2} - f_{k} \\ = (w_{1} + w_{2} - 1) f_{k} - (w_{1} α_{1} + w_{2} α_{2}) \\ = - (w_{1} α_{1} + w_{2} α_{2}) \\ = \sum_{i = 1}^{2} - w_{i} α_{i} \end{matrix}

(56)

From equation (56), it is straightforward that the RC error $e_{rk}$ is no longer zero, but it is relatively small by the factor of $\sum_{i = 1}^{2} - w_{i} α_{i}$ . This implies that the perfect cancelation does not apply, but the disturbance $v_{k}$ is still significantly suppressed.

Theorem 3. Consider case 3 when $T_{v} = {\bar{T}}_{v} \pm Δ_{T}, Δ_{B} (z) \neq 0, Δ_{A} (z) \neq 0$ and the RC error $e_{rk}$ is bounded by $| e_{rk} | < ξ$ , then the sliding dynamics (48) meets the reaching conditions (38), (39) for

ξ < c_{2} T_{s}, c_{2} T_{s} < c_{3} (1 - c_{1} T_{s}), 0 < (1 - c_{1} T_{s}) < 1

(57)

Proof. We first prove the reaching condition equation (38) for any sliding function $s_{k - m} > c_{3}$ . Given that $s_{k - m} > c_{3}$ , we express equation (38) as

s_{k} - s_{k - m} < 0

(58)

Rearranging equation (48), we get

s_{k} - s_{k - m} = - c_{1} T_{s} s_{k - m} - c_{2} T_{s} sat (\frac{s_{k - m}}{c_{3}}) + e_{rk}

(59)

Using equations (23), (57), equation (59) can be derived as

\begin{matrix} s_{k} - s_{k - m} = - c_{1} T_{s} s_{k - m} - c_{2} T_{s} + e_{rk} \\ < - c_{1} T_{s} s_{k - m} - c_{2} T_{s} + ξ \\ < - c_{2} T_{s} + ξ \\ < 0, for ξ < c_{2} T_{s} \end{matrix}

(60)

From equation (60), it is found that equation (58) is met, which implies that condition equation (38) is satisfied. Next, we prove that the reaching condition equation (39) holds for $s_{k - m} > c_{3}$ . Since $s_{k - m} > c_{3}$ , equation (39) can be expressed as

s_{k} + s_{k - m} > 0

(61)

Let us add both sides of equation (48) by $s_{k - m}$ , then we get

s_{k} + s_{k - m} = (2 - c_{1} T_{s}) s_{k - m} - c_{2} T_{s} sat (\frac{s_{k - m}}{c_{3}}) + e_{rk}

(62)

Similarly, we utilize equations (23) and (57) to obtain

\begin{matrix} s_{k} + s_{k - m} = (2 - c_{1} T_{s}) s_{k - m} - c_{2} T_{s} + e_{rk} \\ > (2 - c_{1} T_{s}) c_{3} - c_{2} T_{s} + e_{rk} \\ = c_{3} + (1 - c_{1} T_{s}) c_{3} - c_{2} T_{s} + e_{rk} \\ > c_{3} + (1 - c_{1} T_{s}) c_{3} - c_{3} (1 - c_{1} T_{s}) + e_{rk}, \\ for c_{2} T_{s} < c_{3} (1 - c_{1} T_{s}) \Rightarrow c_{2} T_{s} < c_{3} \\ = c_{3} + e_{rk} \\ > c_{2} T_{s} + e_{rk} \\ > ξ + e_{rk}, for ξ < c_{2} T_{s} \\ > 0 \end{matrix}

(63)

Equations (63) and (61) hold and it implies that equation (39) is satisfied for $s_{k - m} > c_{3}$ . Next, we prove the condition equations (38) and (39) for any $s_{k - m} < - c_{3}$ . Since $s_{k - m} < - c_{3}$ , we have

s_{k} - s_{k - m} > 0

(64)

s_{k} + s_{k - m} < 0

(65)

Now, we can derive equation (59) as

\begin{matrix} s_{k} - s_{k - m} = - c_{1} T_{s} s_{k - m} + c_{2} T_{s} + e_{rk} \\ > c_{1} T_{s} c_{3} + c_{2} T_{s} + e_{rk}, for s_{k - m} < - c_{3} \Rightarrow - s_{k - m} > c_{3} \\ > c_{1} T_{s} c_{3} + ξ + e_{rk}, for ξ < c_{2} T_{s} \\ > c_{1} T_{s} c_{3} + ξ - ξ, for | e_{rk} | < ξ \\ = c_{1} T_{s} c_{3} \\ > 0 \end{matrix}

(66)

Similarly, equation (62) can be expressed as

\begin{matrix} s_{k} + s_{k - m} = (2 - c_{1} T_{s}) s_{k - m} + c_{2} T_{s} + e_{rk} \\ < - (2 - c_{1} T_{s}) c_{3} + c_{2} T_{s} + e_{rk} \\ = - c_{3} - (1 - c_{1} T_{s}) c_{3} + c_{2} T_{s} + e_{rk} \\ < - c_{3} - (1 - c_{1} T_{s}) c_{3} + (1 - c_{1} T_{s}) c_{3} + e_{rk} \\ = - c_{3} + e_{rk} \\ < - c_{3} + ξ, for | e_{rk} | < ξ \\ < - c_{2} T_{s} + ξ, for c_{2} T_{s} < c_{3} (1 - c_{1} T_{s}) \Rightarrow - c_{2} T_{s} > - c_{3} \\ < 0 \end{matrix}

(67)

This completes the proof.

The proofs shown in equations (60), (63), (66), and (67) indicate that the proposed control law ensures the convergence of sliding function $s_{k}$ and the robustness of the controlled system against uncertain periodic disturbance $v_{k}$ and the plant uncertainties $Δ_{A} (z)$ and $Δ_{B} (z)$ .

Results and discussions

Controller design

Consider a discrete-time LTI system with the nominal plant model (Kurniawan et al., 2021) as follows

y_{k} = z^{- 1} \frac{0.0763 + 0.0717 z^{- 1}}{1 - 1.753 z^{- 1} + 0.9015 z^{- 2}} u_{k} + v_{k}

(68)

The reference $r_{k}$ is triangle signal with period of $1 s$ . The disturbance $v_{k}$ is a periodic signal with an amplitude of $0.1$ . The disturbance $v_{k}$ has a period of $1 s$ , and its period changes to $1.05 s$ when the time reaches $10 s$ . The disturbance $v_{k}$ is illustrated in Figure 3, which is modeled as

v_{k} : = {\begin{matrix} 0.1 \sin (2 π k T_{s}), for k T_{s} \leq 10 \\ 0.1 \sin (2 π k T_{s} / 1.05), for k T_{s} > 10 \end{matrix}

(69)

Note that the sampling time used in the example is $T_{s} = 0.005 s$ . We also consider that the nominal period of disturbance $v_{k}$ is ${\bar{T}}_{v}$ = 1 s. This gives the integer number $N$ as

N = \frac{{\bar{T}}_{v}}{T_{s}} = \frac{1}{0.005} = 200

(70)

The following information is obtained from equation (68)

\begin{matrix} A (z) = 1 - 1.753 z^{- 1} + 0.9015 z^{- 2}, B (z) = 0.0763 \\ + 0.0717 z^{- 1}, m = 1 \end{matrix}

(71)

The associated controller parameters $D (z)$ , $Q (z)$ , $H (z)$ , $c_{1}$ , $c_{2}$ , and $c_{3}$ are chosen and listed in Table 1. Note that the polynomial $D (z)$ behaves as the HORC delays and is designed based on equations (12), (13), (16), and (17). The $Q (z)$ is chosen as a zero-phase low-pass filter given in equation (15). The polynomial $H (z)$ is basically required to have stable zeros in order that the sliding function $s_{k}$ is stable and significantly attenuated as compared with the tracking error $e_{k}$ . In the simulation, we chose $H (z)$ to be a second-order polynomial containing two zeros at $z = 0.6$ . The constant $c_{3}$ is part of a saturation function equation (23) that is used to reduce the chattering phenomenon and is usually referred to as a boundary layer that smooths the SMC control signal. Reducing the value of the boundary layer ( $c_{3}$ ) to a very small number will speed up the transient response, but at the expense of rendering the chattering to occur. Therefore, we carefully tuned $c_{3}$ and set it to 0.1. Next, we chose $c_{1}$ , and $c_{2}$ such that conditions in equation (57) hold.

Table 1.

The associated controller parameters.

Parameters	Description	Refers toequations
$D (z)$	$2 z^{- N} - z^{- 2 N}$	(12), (13), (16), and (17)
$Q (z)$	$0.25 z^{- 1} + 0.5 + 0.25 z$	(15)
$H (z)$	$1 - 1.2 z^{- 1} + 0.36 z^{- 2}$	(19)
$c_{1}$ , $c_{2}$ , $c_{3}$	198, 10, 0.1	(57)

Utilizing Table 1, we can now construct $R_{H} (z)$ equation (28) and $G_{H} (z)$ equation (29) as follows

\begin{matrix} R_{H} (z) = 1 - 1.2 z^{- 1} + 0.36 z^{- 2} - 0.5 z^{- 199} - 0.4 z^{- 200} \\ + 0.52 z^{- 201} + 0.24 z^{- 202} \\ - 0.18 z^{- 203} + 0.25 z^{- 399} - 0.26 z^{- 401} - 0.12 z^{- 402} \\ + 0.09 z^{- 403} \end{matrix}

(72)

G_{H} (z) = 0.076 \frac{1 - 0.25 z^{- 1} + 0.776 z^{- 2} + 0.342 z^{- 3}}{1 - 1.753 z^{- 1} + 0.902 z^{- 2}}

(73)

Finally, we can synthesize the proposed HO-RSMC equation (35), which is given by

\begin{matrix} u_{k} = 2 [Q (z)] u_{k - 200} + [Q (z)] u_{k - 400} + [\frac{R_{H} (z)}{G_{H} (z)}] r_{k + 1} \\ + 2 [\frac{Q (z)}{G_{H} (z)}] s_{k - 199} \\ + [\frac{Q (z)}{G_{H} (z)}] s_{k - 399} - [G_{H}^{- 1} (z)] {0.01 s_{k} - 0.05 sat (\frac{s_{k}}{0.1})} \end{matrix}

(74)

Tracking performance of HO-RSMC

The tracking output $y_{k}$ of the LTI system (68) with the controller (74) is shown in Figure 4, while the sliding function $s_{k}$ and tracking error $e_{k}$ can be seen in Figure 5. Figure 4 shows that the plant output $y_{k}$ accurately track the reference $r_{k}$ with no sign of transient response. In addition, we hardly notice the discrepancy between $r_{k}$ and $y_{k}$ . It is noticeable from the tracking error $e_{k}$ plotted in Figure 5 that a very small transient response occurs during the first $2 s$ . During the transient period, the tracking error $e_{k}$ has a maximum amplitude of $0.06$ , which is relatively small compared with the magnitudes of the disturbance $v_{k}$ and reference $r_{k}$ . It is also clearly shown from Figure 5 that the tracking error $e_{k}$ converges to zero after about $2 s$ . This implies that the effect of disturbance $v_{k}$ is perfectly canceled after $2 s$ . Here, we can consider the first $2 s$ as the learning period of the proposed HO-RSMC. The perfect disturbance cancelation is achievable due to the actual disturbance period $T_{v}$ is equal to the known nominal period ${\bar{T}}_{v}$ . This performance confirms the Theorem 1 as stated in “Proposed method” section.

Varying the disturbance period to $1.05 s$ at $t = 10 s$ introduces the new transient response and results in a relatively small periodic steady-state error. This behavior corresponds to Theorem 2, which states that when the actual period disturbance varies from the known nominal period, then the RC error $e_{rk}$ converges to a small bounded value. From Figure 5, we also notice that the sliding function $s_{k}$ has a similar pattern to tracking error $e_{k}$ , except that its magnitude is smaller than $e_{k}$ . This is reasonable because the sliding function $s_{k}$ is basically the tracking error $e_{k}$ attenuated by the filter H(z) (see equation (19)).

Simulation is also carried out for tracking sinusoidal and square reference waves under the similar disturbance $v_{k}$ as in equation (69). The tracking performance of the LTI system (68) under a sinusoidal reference is depicted in Figures 6 and 7, while the tracking performance under a square reference is illustrated in Figures 8 and 9. We can observe from Figures 6 –9 that the similar behaviors to the one using triangular reference signal are obtained. These results imply that the proposed HO-RSMC is capable of accurately tracking any periodic reference signal, either smooth signal (sinusoidal) or sharp signal (triangular and square), and simultaneously suppresses the time-varying disturbance $v_{k}$ .

Comparison studies

To verify the superiority of our proposed method, we compare its performance with high-order RC (HORC) (Steinbuch et al., 2007). The HORC (Steinbuch et al., 2007) has the following input–output relation

\frac{U (z)}{E (z)} = [\frac{Q (z) D (z)}{1 - Q (z) D (z)}] L (z)

(75)

Here, $L (z)$ behaves as a compensator chosen as a zero-phase tracking error controller (ZPTEC) (Tomizuka et al., 1989) formulated by

L (z) = k_{r} z^{m} \frac{A (z)}{B (z)}

(76)

Note that $k_{r}$ is an HORC learning gain and set to $1$ , $A (z)$ , $B (z)$ , and $m$ are equivalent to equation (71), and $Q (z) and D (z)$ are shown in Table 1. The compensator $L (z)$ is now given by

L (z) = z \frac{1 - 1.753 z^{- 1} + {0.9015}^{- 2}}{0.0763 + 0.0717 z^{- 1}}

(77)

Next, we can express equation (75) as

\begin{matrix} u_{k} = 2 [Q (z)] u_{k - 200} + [Q (z)] u_{k - 400} + 2 [Q (z) L (z)] e_{k - 200} \\ + [Q (z) L (z)] e_{k - 400} \end{matrix}

(78)

Since $L (z)$ has an improper form by the factor of $z^{m} = z$ , we can multiply $L (z)$ by $z^{- 1}$ and represent equation (78) as

\begin{matrix} u_{k} = 2 [Q (z)] u_{k - 200} + [Q (z)] u_{k - 400} + 2 [Q (z) z^{- 1} L (z)] e_{k - 199} \\ + [Q (z) z^{- 1} L (z)] e_{k - 399} \end{matrix}

(79)

In this way, the HORC law now becomes realizable. Observing the HO-RSMC equation (74) and HORC equation (79), we notice some important facts here:

Fact 1 (F1): Both HO-RSMC and HORC have a similar form of the delayed control signal, that is $2 [Q (z)] u_{k - 200} + [Q (z)] u_{k - 400}$ .

Fact 2 (F2): The HO-RSMC involves the delayed sliding function, the reference signal, and the reaching law for correcting the control signal, while HORC only uses delayed tracking error as part of the correction.

Fact 3 (F3): The HO-RSMC uses $G_{H}^{- 1} (z)$ for compensating the system model, while HORC uses $L (z)$ .

Now, let us evaluate the tracking performance of the HORC with respect to that of the HO-RSMC when control signal (79) is applied to LTI system (68) and required to simultaneously track the reference $r_{k}$ and reject the disturbance $v_{k}$ . It can be observed from Figure 10 that, the plant output $y_{k}$ indicates small response during the first $1 s$ , and then it rises to larger peak value with the amplitude of twice than the amplitude of the reference $r_{k}$ . We also notice that the precise tracking is achieved after $t = 2 s$ . The transient response of HORC can be clearly seen from the tracking error plotted in Figure 11. Similar to the HO-RSMC, the HORC attains a zero-tracking error at $t = 2 s$ . Note that the duration of the transient responses of both HO-RSMC and HORC can be estimated from the largest delay step of $D (z)$ , which corresponds to Fact 1 (F1). Here, the largest delay step of $D (z)$ is $2 N = 400$ and thus the corresponding transient errors vanish after $400 \times 0.005 = 2 s$ .

In contrast to HO-RSMC, the HORC exhibits poor transient error, which is significantly larger than that of the HO-RSMC. These behaviors agree with Facts F1 and F2, where the HORC relies on the delayed control and the delayed error for producing the current control input. Hence, the HORC generates incorrect control signal during the first $2 s$ . In this period, we consider that HORC operates in the learning state. In vice versa, HO-RSMC takes advantage of the SMC’s reaching law for generating control signal while the learning process is in progress. Therefore, the HO-RSMC offers better transient response compared with the HORC. When the disturbance period changes to $T_{v} = 1.05 s$ at $t = 10 s$ , the HORC demonstrates similar performance to that of the HO-RSMC, where the tracking error $e_{k}$ converges to a small periodic value.

Next, we examine the robustness of HO-RSMC and HORC against plant parameter variations. For this purpose, the plant parameters in equation (68) are decreased by $2.5 %$ resulting in the following transfer function

G (z) = \frac{0.0744 + 0.0699 z^{- 1}}{1 - 1.7092 z^{- 1} + 0.8790 z^{- 2}}

(80)

The tracking errors of the controlled system with HO-RSMC and HORC are shown in Figures 12 and 13, respectively. As shown in Figure 12, the resulting closed-loop system by HO-RSMC method remains stable in the presence of plant uncertainties, but the corresponding tracking error $e_{k}$ becomes slower to converge compared with that in Figure 5. From Figure 13, we find out that the tracking error $e_{k}$ of HORC slowly diverges, which implies that the controlled system with the HORC is lack of stability robustness and is thus sensitive toward the plant parametric variations. When the actual plant model deviates from the known nominal model, dynamics mismatch between the actual plant and the designed controller will occur. Here, the HORC with a compensator $L (z)$ (76) fails to compensate the plant dynamics (80) leading to unstable closed-loop system (see F3).

A further comparison is made to proportional–integral–derivative (PID) controller with plug-in RC, which has the configuration shown in Figure 14.

The PID controller with plug-in RC is later referred as PID-RC. Here, the $L_{c} (z)$ in RC shown in Figure 14 is generally designed as

L_{c} (z) = k_{r} P_{c}^{- 1} (z) \Rightarrow P_{c} (z) = \frac{P (z) C (z)}{1 + P (z) C (z)}

(81)

where $k_{r} \in [0 < k_{r} < 2]$ is also a learning gain. The PID-RC is similarly simulated for tracking triangular reference wave with time-varying disturbance (69) under two cases, that is, nominal plant and 2.5% plant variation. The tracking output and error of the controlled system with PID-RC are given in Figures 15 and 16, respectively. The tracking error of the PID-RC at 2.5% plant variation is shown in Figure 17. It can be seen from Figure 17 that the controlled system with PID-RC remain stable under 2.5% plant variation. Here, we can also notice that PID-RC has a smaller transient error and better robustness compared with that in HORC. However, the transient and steady-state performances are still inferior to the proposed HO-RSMC. From all above results, we thus infer that a correct integration of the HORC and SMC methods greatly improves the transient response, disturbance rejection performance, and robustness of uncertain linear systems with time-varying disturbances.

Conclusion

A novel strategy to synthesize a higher-order repetitive sliding mode controller for uncertain linear systems with time-varying periodic disturbances has been presented in this article. The higher-order repetitive model, linear sliding function, and discrete-time reaching law are the main parts in the construction of HO-RSMC law. Stability analysis has been provided to prove the convergence of sliding function and the robustness of the controlled system against plant uncertainties and time-varying periodic disturbance. Numerical simulations have been conducted to show the superiority in performance compared with the HORC method, in terms of precise tracking, good disturbance rejection, and robustness capabilities. Finally, extending the proposed HO-RSMC for either non-minimum phase or multivariable systems can be some potential research work in the future.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Edi Kurniawan

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