Volume 17 Number 5
September 2020
Article Contents
Ze-Wen Wang, Jin-Hua She, Guang-Jun Wang. Adaptive Equivalent-input-disturbance Approach to Improving Disturbance-rejection Performance. International Journal of Automation and Computing, 2020, 17(5): 701-712. doi: 10.1007/s11633-020-1230-7
Cite as: Ze-Wen Wang, Jin-Hua She, Guang-Jun Wang. Adaptive Equivalent-input-disturbance Approach to Improving Disturbance-rejection Performance. International Journal of Automation and Computing, 2020, 17(5): 701-712. doi: 10.1007/s11633-020-1230-7

Adaptive Equivalent-input-disturbance Approach to Improving Disturbance-rejection Performance

Author Biography:
  • Ze-Wen Wang received the B. Sc. degree in engineering from Central China Normal University, China in 2017. He is currently a master student in control engineering from China University of Geosciences, China. His research interests include the application of control theory, and robust control. E-mail: 709922342@qq.com ORCID iD: 0000-0002-6633-2556

    Jin-Hua She received the B. Sc. degree in engineering from Central South University, China in 1983, and the M. Sc. and Ph. D. degrees in engineering from Tokyo Institute of Technology, Japan in 1990 and 1993, respectively. In 1993, he joined School of Engineering, Tokyo University of Technology, where he is currently a professor. He is a member of the Society of Instrument and Control Engineers, the Institute of Electrical Engineers of Japan, the Japan Society of Mechanical Engineers, and the Asian Control Association. He received the International Federation of Automatic Control Control Engineering Practice Prize Paper Award in 1999 (jointly with M. Wu and M. Nakano). His research interests include the application of control theory, repetitive control, process control, Internet-based engineering education, and assistive robotics.E-mail: she@stf.teu.ac.jp (Corresponding author) ORCID iD: 0000-0003-3165-5045

    Guang-Jun Wang received the B. Sc. and M. Sc. degrees from Central China Normal University, China in 1992, and the Ph. D. degree from the Huazhong University of Science and Technology, China in 2001. In 2002, he was with The Chinese University of Hong Kong as a visiting researcher. He was with the University of New Brunswick, Canada, as a Visiting Researcher from 2009 to 2010. He was a lecturer with School of Mechanical Engineering and Electronic Information, China University of Geosciences, China from 1994 to 1999, and an associate professor from 1999 to 2005, where he is currently a professor. His research interests include electronic information technology, pattern recognition, and intelligent systems. E-mail: gjwang@cug.edu.cn

  • Corresponding author: J. She, she@stf.teu.ac.jp
  • Received: 2020-01-08
  • Accepted: 2020-03-09
  • Published Online: 2020-05-22
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Adaptive Equivalent-input-disturbance Approach to Improving Disturbance-rejection Performance

Abstract: This paper presents an adaptive equivalent-input-disturbance (AEID) approach that contains a new adjustable gain to improve disturbance-rejection performance. A linear matrix inequality is derived to design the parameters of a control system. An adaptive law for the adjustable gain is presented based on the combination of the root locus method and Lyapunov stability theory to guarantee the stability of the AEID-based system. The adjustable gain is limited in an allowable range and the information for adjusting is obtained from the state of the system. Simulation results show that the method is effective and robust. A comparison with the conventional EID approach demonstrates the validity and superiority of the method.

Ze-Wen Wang, Jin-Hua She, Guang-Jun Wang. Adaptive Equivalent-input-disturbance Approach to Improving Disturbance-rejection Performance. International Journal of Automation and Computing, 2020, 17(5): 701-712. doi: 10.1007/s11633-020-1230-7
Citation: Ze-Wen Wang, Jin-Hua She, Guang-Jun Wang. Adaptive Equivalent-input-disturbance Approach to Improving Disturbance-rejection Performance. International Journal of Automation and Computing, 2020, 17(5): 701-712. doi: 10.1007/s11633-020-1230-7
    • A large number of control methods have been proposed to improve disturbance-rejection performance[13]. In particular, many methods have devised to actively compensate for disturbances, such as the active-disturbance-rejection control (ADRC)[46], the disturbance observer (DO)[7,8], and the equivalent-input-disturbance (EID) approach[9,10].

      The extended state observer (ESO) is widely used in the ADRC to deal with a disturbance[4]. Since the parameters in such a control system are hard to adjust, a linear method was presented to tune them in a simple way[11]. While the conventional ESO method is effective, it can handle only a chained system. This restricts its fields of application.

      The DO method is a frequency-domain method of estimating and rejecting to a disturbance[12]. The design of the DO is complicated because it requires a low-pass filter in the control system to ensure the causality and stability of the whole system, the disturbance-rejection performance of the system, and model matching between an actual and a nominal plant[7]. The DO method is simple and effective for a minimum-phase plant, but is hard to guarantee the stability of a non-minimum-phase plant.

      As one of the active disturbance-rejection methods, the EID approach makes use of the information on a state observer to create a control signal on the control input channel that produces the same effect on the output as a disturbance does[13]. It does not require the inverse dynamics of a plant, the availability of the system state, a model of exogenous disturbances, or the differentiation of measured outputs[14]. By taking nonlinearities as a state-dependent disturbance, it is able to simultaneously handle both disturbances and nonlinearities[15], and has been successfully applied to servo system[13,16], time-delay systems[1720], fractional-order systems[21], and so forth.

      On the other hand, the limitations, such as the maximum control input, of a real system may degrade control performance for a designed linear control law. Although the EID approach has advantages to disturbance rejection, a fixed linear controller may fail in exhibiting the ability of the EID approach. A way to solve this problem is to adaptively tune the gain of an EID estimator in a control system in accordance with the influence of a disturbance so it performs to the best of its potential[22,23].

      This paper presents an adaptive EID (AEID) approach that improves the disturbance-rejection performance. The gain of the EID estimator of the system is adaptively tuned based on the state of the low-pass filter and the error of the outputs between a plant and a state observer. Simulation results show that this improves the control performance of the system.

      The rest of the paper is organized as follows. The configuration of the AEID-based control system is presented and a stability condition is derived in Section 2. An algorithm of designing an AEID-based control system is presented in Section 3. Simulations show the validity of the method in Section 4. And some concluding remarks are given in Section 5.

      In this paper, $ {{\bf R}}^{n} $ is an $ n $ dimensional Euclidean space, $ {{\bf R}}^{n \times m} $ is the set of all $ n \times m $ real matrices, I and $ 0 $ stand for an identity matrix and a zero matrix with a suitable dimension, $ s $ is the Laplace operator, $ X(s) $ is the Laplace transform of $ x(t) $, the norm of a signal $ v(t) $ is $ \| v\| = \sqrt{\int_0^{\infty}v^{\rm{T}}(t)v(t){\rm{d}}t} $, and a symmetric matrix $ \left[ {\begin{array}{*{20}{c}}A&B\\B^{\rm{T}}&C\end{array}} \right] $ is denoted by $ \left[ {\begin{array}{*{20}{c}}A &B\\ {\text{*}} & C\end{array}} \right] $.

    • This section first shows the configuration of the AEID-based control system, then carries out the analysis of system stability.

    • For simplicity, this paper considers a single-input single-output (SISO) case. But the obtained results are easy to extend to the multi-input multi-output case.

      Consider a time-invariant plant in Fig. 1.

      Figure 1.  Configuration of AEID-based control system

      $ \left\{\begin{aligned} & \dot{x}_o(t) = Ax_o(t)+Bu(t)+B_d d(t) \\ & y_o(t) = Cx_o(t) \end{aligned}\right. $

      (1)

      where $ x_o(t)\; (\in {{\bf R}}^{n}) $ is the state of the plant, $ y_o(t)\; (\in {{\bf R}}) $ is the output of the plant, $ u(t)\; (\in {{\bf R}}) $ is the control input, $ d(t)\; (\in {{\bf R}}^{n_d}) $ is a disturbance, $ A \in {{\bf R}}^{n \times n} $, $ B \in {{\bf R}}^{n \times 1} $, $ B_d \in {{\bf R}}^{n \times n_d} $, and $ C \in {{\bf R}}^{1 \times n} $.

      The AEID approach creates a signal on the control input channel (an EID) that produces a output belonging to a set[9]:

      $ \Phi = \left\{\sum\limits_{i = 0}^{r}p_i(t){\rm sin}(\omega_it + \phi_i)\right\},\; i = 0,\; \cdots,\; r,\; r<\infty $

      (2)

      where $ \omega_i\; (\geq0) $ and $ \phi_i $ are constants, and $ p_i(t) $ is a polynomial in time $ t $.

      Assumptions 1−3 are made about the plant, P(s) = $ C(sI-A)^{-1}B $:

      Assumption 1. $ P(s) $ is controllable and observable.

      Assumption 2. $ P(s) $ has no zeros on the imaginary axis.

      Assumption 3. $ d(t) $ is unknown, but bounded.

      Assumptions 1 and 2 are made for convenience and are standard. And Assumption 3 is true for many real applications.

      Note that the disturbance, $ d(t) $, may be imposed on a channel different from the input channel. So, B and $ B_d $ may have different dimensions. However, there always exists an EID, $ d_e(t) $, on the input channel, which affects the output as the same as $ d(t) $ does. Using the concept of EID describes the plant as[9]

      $ \left\{\begin{aligned} &\dot{x}(t) = Ax(t)+B[u(t)+d_e(t)] \\ & y(t) = Cx(t) \end{aligned}\right. $

      (3)

      where $ d_e(t) $ $ (\in {{\bf R}}) $ is the EID, $ x(t)\; (\in {{\bf R}}^{n}) $ is the equivalent state of the plant, $ y(t)\; (\in {{\bf R}}) $ is the equivalent output of the plant.

      The AEID-based control system has three parts: the plant, a Luenberger state observer, and the AEID estimator that contains an EID estimator and a mechanism to adaptively tune an adjustable gain $ K $.

      The state observer is

      $ \dot{\hat{x}}(t) = A\hat{x}(t)+Bu_f(t)+L[y(t)-C\hat x(t)] $

      (4)

      to reproduce the state of the plant. Letting

      $ \Delta x(t) = x(t)-\hat{x}(t) $

      (5)

      and substituting it into (3) yield

      $ \dot{\hat{x}}(t)+ \Delta \dot{x}(t) = A[\hat x(t)+\Delta x(t)]+B[u(t)+d_{e}(t)]. $

      (6)

      Assume that there exists a control input $ \Delta d(t) $ that satisfies[24]

      $ B\Delta d(t) = - \Delta \dot x(t)+A \Delta x(t)+ (K-1)LC\Delta x(t) $

      (7)

      where $ K $ is a parameter that tunes control performance. Let an estimate of the EID be

      $ \hat d(t) = d_e(t)- \Delta d(t). $

      (8)

      Combining (7) and (8) gives an expression of the plant

      $ \dot{\hat{x}}(t) = A\hat{x}(t)+B[u(t)+\hat d(t)]-(K-1)LC\Delta x(t). $

      (9)

      And (4), (5), and (9) yield

      $ \hat{d}(t) = KB^{+}LC\Delta x(t)+u_f(t)-u(t) $

      (10)

      where

      $ B^{+} = (B^{\rm{T}}B)^{-1} B^{\rm{T}}. $

      (11)

      An angular-frequency band, $ \Omega_c $, for disturbance estimation is selected by a filter $ F(s) $:

      $ \left\{\begin{aligned} & \dot{x}_{f}(t) = A_{f}x_{f}(t)+B_{f}\hat d(t) \\ & \widetilde d(t) = C_{f}x_{f}(t). \end{aligned}\right. $

      (12)

      Let the highest angular frequency for disturbance rejection be $ \omega_d $. The cutoff angular frequency of $ F(s) $ is chosen to be $ 5 \,– 10 $ times larger than $ \omega_d $, which ensures

      $ F(j\omega) \approx 1, \;\;\; \forall \omega \in \Omega_c. $

      (13)

      The filtered estimate $ \widetilde d(t) $ is given by

      $ \widetilde D(s) = F(s)\hat D(s). $

      (14)

      An improved control law is

      $ u(t) = u_f(t)- \widetilde d(t). $

      (15)

      $ u_f(t) $ is chosen to be

      $ u_f(t) = K_P \hat{x}(t) $

      (16)

      where $ K_P $ is the state-feedback gain.

    • Redrawing Fig. 1 yields Fig. 2. We divide the whole system into two parts: Subsystems 1 and 2 (above and under the dotted line). Note that the subsystems are connected in series. The system is stable if Subsystems 1 and 2 are stable[25]. This allows us to deal with the stability issue of the whole system by considering the stability of Subsystems 1 and 2, separately.

      Figure 2.  Block diagram for analysis of system stability

      Assume that a suitable state-feedback gain $ K_P $ is designed to guarantee the stability of Subsystem 1 in Fig. 2. Rearranging Subsystem 2 gives Fig. 3. The state-space representation is

      Figure 3.  Block diagram of Subsystem 2

      $ \left\{ \begin{aligned} & \left[ {\begin{array}{*{20}{c}} {\Delta \dot x(t)}\\ {{{\dot x}_f}(t)} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {A - LC}&{ - B{C_f}}\\ {{B_f}K{B^ + }LC}&{{A_f} + {B_f}{C_f}} \end{array}} \right]\times \\ & \quad\quad\quad\quad\quad\quad\, \left[ {\begin{array}{*{20}{c}} {\Delta x(t)}\\ {{x_f}(t)} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} B\\ 0 \end{array}} \right]{d_e}(t)\\ & \widetilde{d}(t) = \left[ 0 \quad {{C_f}}\right]\left[ {\begin{array}{*{20}{c}} {\Delta x(t)}\\ {{x_f}(t)} \end{array}} \right]. \end{aligned} \right. $

      (17)

      The transfer function from $ d_e(t) $ to $ \widetilde d(t) $ is

      $\begin{array}{l} {G_{\widetilde d{d_e}}}(s) = \\ \left[0 \quad {{C_f}}\right]{\left\{ {sI - \left[ {\begin{array}{*{20}{c}} {A - LC}&{ - B{C_f}}\\ {{B_f}K{B^ + }LC}&{{A_f} + {B_f}{C_f}} \end{array}} \right]} \right\}^{ - 1}}\left[ {\begin{array}{*{20}{c}} B\\ 0 \end{array}} \right]. \end{array}$

      (18)

      Note that K in Fig. 3 is a gain that only affects Subsystem 2. The bigger K is, the smaller the error between $ d_e(t) $ and $ \widetilde d (t) $ is. So, we adjust it to ensure the precision of the EID estimate as high as possible.

      Since very large K may cause Subsystem 2 to become unstable, it is necessary to find the upper bound of the gain[26]. The root locus method[27] is used for this purpose.

      The open-loop transfer function of Subsystem 2 is

      $ G_{op}(s) = K\frac{C_f(sI-A_f)^{-1}B_fB^{+}LC\left[sI-(A-LC)\right]^{-1}B}{1-C_f[sI-A_f]^{-1}B_f}. $

      (19)

      $ K_{\max} $ is the gain when the root locus intersects the imaginary axis to ensure that the poles of the closed-loop system are all in the left half plane. To obtain the better performance of the subsystem, choose $ K = \lambda $ in $ (0, K_{\max}) $ to make $ G_{\widetilde dd_e}(j\omega)\approx1 $, $ \omega\leq\omega_d $. Moreover, considering that a high gain may result in high noise amplification or even resonance[28] and in overestimation, choose a suitable constant $ \lambda\; (<K_{\max}) $ and adjust K in $ (0, \lambda] $ to evade such situations. Summarizing the above explanations gives the following Theorem 1.

      Theorem 1. With a fixed K, the AEID-based system is stable if both of Subsystems 1 and 2 are stable. Moreover,

      1) Subsystem 1 is stable if $ A+BK_P $ is Hurwitz;

      2) Subsystem 2 is stable if the following conditions hold:

      2.1) $ A-LC $ is Hurwitz,

      2.2) $ A_f+B_fC_f $ is Hurwitz,

      2.3) $ 0<K \leq \lambda <K_{\max} $.

      Remark 1. Define $ G_{o}(s) $ in Subsystem 2 as

      $ G_{o}(s) = B^+LC[sI-(A-LC)]^{-1}B. $

      (20)

      When $ G_{o}(s) $ is a positive real function, $ K_{\rm max} $ cannot be bounded. However, if $ G_{o}(s) $ is not strictly positive real function, K is bounded. A reasonable $ \lambda $ guarantee the performance and the stability of the system, when $ K $ reaches the upper bound.

    • In this section, the gains of the state feedback, $ K_P $, and the state observer, $ L $, are first designed for a fixed K based on a linear matrix inequality (LMI). Then, the adaptive rate of K is designed to adjust K based on the combination of the root-locus method and Lyapunov stability theory. The design procedure is finally summarized in an algorithm.

    • Assume that the singular-value decomposition of a matrix Π is

      $ \Pi = \bar U\left[ {\begin{array}{*{20}{c}} {\bar S}&0 \end{array}} \right]{\bar T^{\rm{T}}} $

      (21)

      where $ \bar S $ is a diagonal matrix with positive diagonal elements in decreasing order, and $ \bar U $ and $ \bar T $ are unitary matrices.

      Lemma 1[29] is used to derive a stability condition for the design of $ K_P $ and $ L $.

      Lemma 1. For a given matrix $ \Pi \in {{\bf R}}^{p \times n} $ with $ {\rm rank}(\Pi) = p $, there exists a matrix $ \bar{X} \in {{\bf R}}^{p \times p} $ such that

      $ \Pi X = \bar X\Pi $

      (22)

      holds for any $ X\in {{\bf R}}^{n \times n} $ if and only if $ X $ can be decomposed as

      $ X = \bar T\left[ {\begin{array}{*{20}{c}} {{{\bar X}^{(11)}}}&0\\ 0&{{{\bar X}^{(22)}}} \end{array}} \right]{\bar T^{\rm{T}}}$

      (23)

      where $ \bar T\in {{\bf R}}^{n \times n} $ is a unitary matrix, $ \bar X_{11}\in {{\bf R}}^{p \times p} $, and $ \bar X_{22}\in {{\bf R}}^{(n-p) \times (n-p)} $.

      Combining (3), (4), (5), (12), (15) and (16) yields the following state-space representation of the system:

      $ \dot\varphi(t) = \bar A\varphi(t)+\bar Bd_e(t) $

      (24)

      where

      $ \begin{split} & \varphi (t) = {\left[ {\begin{array}{*{20}{c}} {{{\hat x}^{\rm{T}}}(t)}&{\Delta {x^{\rm{T}}}(t)}&{x_f^{\rm{T}}(t)} \end{array}} \right]^{\rm{T}}}\\ & \bar A = \left[ {\begin{array}{*{20}{c}} {A + B{K_P}}&{LC}&0\\ 0&{A - LC}&{ - B{C_f}}\\ 0&{{B_f}K{B^ + }LC}&{{A_f} + {B_f}{C_f}} \end{array}} \right]\\ & \bar B = \left[ {\begin{array}{*{20}{c}} 0\\ B\\ 0 \end{array}} \right]. \end{split}$

      (25)

      First, the highest angular frequency for disturbance rejection, $ \omega_d $, is selected based on design specifications. Then, the low-pass filter, $ F(s) $, is chosen to ensure Condition 2.2) in Theorem 1 and to satisfy (13).

      For a fixed K[9],

      $ K = K(0) = 1 .$

      (26)

      A stability condition with respect to $ K_P $ and L is given below.

      Theorem 2. For a bounded disturbance $ d(t) $, a fixed K in (26), a suitably selected filter $ F(s) $, positive numbers $ K(0) $, $ \alpha_1 $, $ \alpha_2 $, $ \gamma_1 $, $ \gamma_2 $ and $ \gamma_3 $, positive-definite matrices $ X_1 $, $ X_{11} $, $ X_{22} $ and $ X_{3} $, and appropriate matrices $ W_1 $ and $ W_2 $, the system is stable if the following inequality holds:

      $ \left[ {\begin{array}{*{20}{c}} {{\psi _{11}}}&{{\psi _{12}}}&0\\ * &{{\psi _{22}}}&{{\psi _{23}}}\\ * & * &{{\psi _{33}}} \end{array}} \right] < 0 $

      (27)

      where

      $ \begin{split} {\psi _{11}} =\;& {X_1}{(A + {\alpha _1}I + {\gamma _1}I)^{\rm{T}}} + (A + {\alpha _1}I + {\gamma _1}I){X_1} +\\ & B{W_1} + {(B{W_1})^{\rm{T}}}\\ {\psi _{12}} =\;& {W_2}C\\ {\psi _{22}} =\;& {X_2}{(A + {\alpha _2}I + {\gamma _2}I)^{\rm{T}}} + (A + {\alpha _2}I + {\gamma _2}I){X_2}-\\ & {W_2}C - {({W_2}C)^{\rm{T}}}\\ {\psi _{23}} =\;& - B{C_f}{X_3} + {({B_f}K(0){B^ + }{W_2}C)^{\rm{T}}}\\ {\psi _{33}} =\;& ({A_f} + {B_f}{C_f} + {\gamma _3}I){X_3} + {X_3}{({A_f} + {B_f}{C_f} + {\gamma _3}I)^{\rm{T}}}\\ {X_2} =\;& T\left[ {\begin{array}{*{20}{c}} {X_2^{(11)}}&0\\ 0&{X_2^{(22)}} \end{array}} \right]{T^{\rm{T}}}. \end{split} $

      Moreover, let the singular-value decomposition of the output matrix C be

      $ C = U\left[ {\begin{array}{*{20}{c}} S&0 \end{array}} \right]{T^{\rm{T}}}$

      (28)

      where S is a positive definite matrix, and U and T are unitary matrices. The gains of the state feedback and the state observer are

      $ \left\{\begin{array}{l} K_{P} = W_1X^{-1}_1 \\ L = W_2US{X_2^{(11)}}^{-1}S^{-1}U^{\rm{T}}.\\ \end{array}\right. $

      (29)

      Proof. For a fixed K, take $ \bar A $ to be $ \bar A_1 $:

      $ {\bar A_1} = \left[ {\begin{array}{*{20}{c}} {A + B{K_P}}&{LC}&0\\ 0&{A - LC}&{ - B{C_f}}\\ 0&{{B_f}K(0){B^ + }LC}&{{A_f} + {B_f}{C_f}} \end{array}} \right]. $

      (30)

      Define $ \bar A_\alpha = \bar A_1 +N $, where

      $ N = \left[ {\begin{array}{*{20}{c}} {{\alpha _1}I}&0&0\\ 0&{{\alpha _2}I}&0\\ 0&0&0 \end{array}} \right]. $

      (31)

      Choose a Lyapunov functional candidate to be

      $ V(\varphi(t)) = \varphi^{\rm{T}}(t)P\varphi(t) $

      (32)

      where $ P\; ( = X^{-1}) $ is a symmetric, positive-definite matrix, and X and P are

      $ X = \left[ {\begin{array}{*{20}{c}} {{X_1}}&0&0\\ 0&{{X_2}}&0\\ 0&0&{{X_3}} \end{array}} \right],P = \left[ {\begin{array}{*{20}{c}} {{P_1}}&0&0\\ 0&{{P_2}}&0\\ 0&0&{{P_3}} \end{array}} \right]. $

      (33)

      The derivative of $ V(\varphi(t)) $ is

      $ \begin{split} \dot V(\varphi(t)) =& \varphi^{\rm{T}}(t)(P\bar A_1+\bar A_1^{\rm{T}}P)\varphi(t)+\\ &\left[\bar Bd_e(t)\right]^{\rm{T}}P\varphi(t)+ \varphi^{\rm{T}}(t)P\bar Bd_e(t). \end{split} $

      (34)

      Since $ d(t) $ is bounded, $ d_e(t) $ is bounded as well. Thus, for a large enough $ \varphi(t) $, the following holds:

      $ \|\varphi^{\rm{T}}P\bar Bd_e\| \leq\|\varphi^{\rm{T}}\Gamma P\varphi\|,\; \forall t\geq 0 $

      (35)

      where $ \Gamma $ is a symmetric, non-negative definite matrix and $ \Gamma $ is defined as

      $ \Gamma = \left[ {\begin{array}{*{20}{c}} {{\gamma _1}I}&0&0\\ 0&{{\gamma _2}I}&0\\ 0&0&{{\gamma _3}I} \end{array}} \right]. $

      (36)

      Thus,

      $ \begin{split} \dot V(\varphi(t))\; & \leq\varphi^{\rm{T}}(t)(P\bar A_1+\bar A_1^{\rm{T}}P+2\Gamma P)\varphi(t)\\ & \leq\varphi^{\rm{T}}(t)(P\bar A_1+\bar A_1^{\rm{T}}P+M)\varphi(t) \end{split} $

      (37)

      where

      $ M = \left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {2{\gamma _1}{P_1}}&0&0\\ 0&{2{\gamma _2}{P_2}}&0\\ 0&0&{{P_3}({\gamma _3}I - {A_f} - {B_f}{C_f}) + {{({\gamma _3}I - {A_f} - {B_f}{C_f})}^{\rm{T}}}{P_3}} \end{array}} \end{array}} \right]. $

      (38)

      Since

      $ \begin{split} & \varphi^{\rm{T}}(t)(PN+N^{\rm{T}}P)\varphi(t)>\\ & -x_f^{\rm{T}}(t)\left[P_3(A_f+B_fC_f)+(A_f+B_fC_f)^{\rm{T}}P_3\right]x_f(t) \end{split} $

      (39)

      the following inequality holds[30]:

      $ P\bar A_1+\bar A_1^{\rm{T}}P+M<P\bar A_\alpha+\bar A_\alpha^{\rm{T}}P+2\Gamma P<0. $

      (40)

      Pre-multiplying and post-multiplying (40) by $ X\; ( = P^{-1}) $ yields

      $ \bar A_\alpha X+X\bar A_\alpha^{\rm{T}}+2\Gamma X<0. $

      (41)

      Applying Lemma 1 for $ X_2 $ in (41) gives

      $ CX_2 = \bar X_2C $

      (42)

      and

      $ \bar X_2 = USX_{11}S^{-1}U^{\rm{T}}. $

      (43)

      Thus, the gains of the state feedback and the state observer are

      $ K_P = W_1 X_1^{-1},\;L = W_2\bar X_2^{-1}. $

      (44)

      Remark 2. Since K is set to be one in a conventional EID estimator, we select the initial value of K to be $ K(0) = 1 $. A different method of choosing $ K(0) $ is given in [24] to achieve satisfied disturbance-rejection performance.

      Remark 3. The system is linear for a fixed K. Thus, the stability of the system is independent of an external disturbance.

    • According to Subsection 2.2, K is limited in $ (0, \lambda] $. To improve the disturbance-rejection performance of the conventional EID method, K is adaptively adjusted in $ [1, \lambda] $ based on Theorem 3.

      Theorem 3. For a suitably selected positive number $ \mu $, an adaptive law for K is given by

      $ \left\{\!{\begin{aligned} & {\dot K(t) \!=\! \frac{{x_f^{\rm{T}}(t){P_3}{B_f}{B^ + }L\Delta y(t) \!+\!\!{{\left[ {{B_f}{B^ + }L\Delta y(t)} \right]}^{\rm{T}}}\! {P_3}{x_f}(t)}}{{2\mu }}}\!+\\ & \quad\quad\quad\;\; \frac{{x_f^{\rm{T}}(t)\left[ {{{({A_f} \!+\! {B_f}{C_f})}^{\rm{T}}}{P_3}\! +\! {P_3}({A_f}\! + \!{B_f}{C_f})} \right]{x_f}(t)}}{{2\mu K(t)}}\\ & {K(0) = 1} \end{aligned}} \right. $

      (45)

      where

      $ \Delta y(t) = y(t)-\hat y(t). $

      (46)

      Moreover, for a positive number $ \lambda\; (<K_{\max}) $, $ K(t) $ has to satisfy

      $ 1 \leq K(t)\leq K_b(t) $

      (47)

      where

      $ K_b(t) = \min \left\{ \lambda, 1+\sqrt{\frac{x_f^{\rm{T}}(t)P_3x_f(t)}{\mu}} \right\}. $

      (48)

      Proof. Writing the adjustable gain, $ K(t) $, as

      $ K(t) = K(0) + \int^t_0 \dot K(t) {\rm{d}}t $

      (49)

      where $ K(0) = 1 $. And let $ \bar A = \bar A_1+\bar A_2 $, where

      $ {\bar A_2} = \left[ {\begin{array}{*{20}{c}} 0&0&0\\ 0&0&0\\ 0&{{B_f}\int\nolimits_0^t {\dot K} (t){\rm{d}}t{B^ + }LC}&0 \end{array}} \right]. $

      (50)

      A Lyapunov functional candidate is used to derive an adjusting law for $ K(t) $. It is chosen to be

      $ V(\varphi(t)) = \varphi^{\rm{T}}(t)P\varphi(t)-\mu [K(t)-K(0)]^2 $

      (51)

      where $ \mu $ is a positive number for which $ V(\varphi(t))>0 $. A condition to guarantee $ V(\varphi(t))>0 $ is

      $ x_f^{\rm{T}}(t)P_3x_f(t)-\mu [K(t)-K(0)]^2 \geq 0. $

      (52)

      It provides us

      $ K(t)\leq1+\sqrt{\frac{x_f^{\rm{T}}(t)P_3x_f(t)}{\mu}}. $

      (53)

      On the other hand, $ K(t) $ is restricted in $ [1, \lambda] $. Therefore, if $ K_b(t) $ is (48), $ V(\varphi(t))>0 $.

      The derivative of $ V(\varphi(t)) $ is

      $ \begin{split} \dot V(\varphi(t)) =\;& \varphi^{\rm{T}}(t)(P\bar A+\bar A^{\rm{T}}P)\varphi(t)+[\bar Bd_e(t)]^{\rm{T}}P\varphi(t) + \\ & \varphi^{\rm{T}}(t)P\bar Bd_e(t)-2\mu \int^t_0 \dot K(t){\rm{d}}t\dot K(t)=\\ \;& \varphi^{\rm{T}}(t)(P\bar A_1+\bar A_1^{\rm{T}}P+P\bar A_2+\bar A_2^{\rm{T}}P)\varphi(t)+ \\ & [\bar Bd_e(t)]^{\rm{T}}P\varphi(t)+\varphi^{\rm{T}}(t)P\bar Bd_e(t)-\\ & 2\mu \int^t_0 \dot K(t){\rm{d}}t\dot K(t).\\[-15pt] \end{split} $

      (54)

      Note that, from the proof of Theorem 2, we know that

      $ \begin{split} & \varphi^{\rm{T}}(t)(P\bar A_1+\bar A_1^{\rm{T}}P)\varphi(t)+[\bar Bd_e(t)]^{\rm{T}}P\varphi(t)+\\ & \quad\varphi^{\rm{T}}(t)P\bar Bd_e(t)- x_f^{\rm{T}}(t)[(A_f+B_fC_f)^{\rm{T}}P_3+\\ & \quad P_3(A_f+B_fC_f)] x_f(t)\leq\\ & \quad\varphi^{\rm{T}}(t)(P\bar A_\alpha+\bar A_\alpha^{\rm{T}}P+2\gamma P)\varphi(t)<0 \end{split} $

      (55)

      holds. Thus, if

      $ \begin{split} & \varphi^{\rm{T}}(t)(P\bar A_2+\bar A_2^{\rm{T}}P)\varphi(t)-2\mu \int^t_0 \dot K(t){\rm{d}}t\dot K(t)+\\ & \quad x_f^{\rm{T}}(t)[(A_f+B_fC_f)^{\rm{T}}P_3+P_3(A_f+B_fC_f)]x_f(t)\leq 0 \end{split} $

      (56)

      holds, $ \dot V(\varphi(t))<0 $, and the closed-loop system is stable. Since $ A_f+B_fC_f $ is Hurwitz and $ K(t)>\int^t_0 \dot K(t){\rm{d}}t>0 $, substituting (33) and (50) into (56) yields

      $ \begin{split} & \frac{x^{\rm{T}}_f(t)P_3B_fB^{+}LC\Delta x(t)+[B_fB^{+}LC\Delta x(t)]^{\rm{T}}P_3x_f(t)}{2\mu} +\\ & \quad\frac{x_f(t)^{\rm{T}}[(A_f+B_fC_f)^{\rm{T}}P_3+P_3(A_f+B_fC_f)]x_f(t)}{2\mu\int^t_0 \dot K(t){\rm{d}}t}< \\ & \quad\frac{x^{\rm{T}}_f(t)P_3B_fB^{+}LC\Delta x(t)+[B_fB^{+}LC\Delta x(t)]^{\rm{T}}P_3x_f(t)}{2\mu}+ \\ & \quad\frac{x_f(t)^{\rm{T}}[(A_f+B_fC_f)^{\rm{T}}P_3+P_3(A_f+B_fC_f)]x_f(t)}{2\mu K(t)} \leq\\ & \quad \dot K(t). \end{split} $

      (57)

      Therefore, we choose $ \dot K(t) $ to be (45). □

      Remark 4. Since the control gain, $ K(t) $, changes with the control error caused by $ d_e(t) $, the AEID-based system ensures good disturbance-rejection performance even for a big disturbance.

      Remark 5. A function $ V(x) $ is a Lyapunov function of a dynamic system with its state $ x(t) $ having an equilibrium at the origin if it fulfills the following properties[31]:

      1) $ V(x) = 0 $ for $ x = 0 $;

      2) $ V(x)>0 $ for $ x \neq 0 $;

      3) $ V(x) \to \infty $ as $ \|x\| \to \infty $;

      4) $ \dot{V} (x) < 0 $ for $ x \neq 0 $.

      Clearly, $ V(\varphi(t)) $ is a Lyapunov function. A negative term, $ -\mu [K(t)-K(0)]^2 $, is used in $ V(\varphi(t)) $. It provides us a new way to derive a simple adaptive adjusting law for $ K(t) $.

    • An algorithm of designing the AEID-based control system is given below.

      Design Algorithm:

      Step 1. Choose the maximum angular frequency for disturbance rejection, $ \omega_{d} $, the cutoff angular frequency of $ F(s) $, $ \omega_{c} $.

      Step 2. Choose a filter $ F(s) $ satisfying (13).

      Step 3. Choose positive numbers $ \alpha_1 $, $ \alpha_2 $, and $ K(0) $, and a suitable positive definite matrix $ \Gamma $. Find a feasible solution of the LMI (27) and calculate $ K_P $ and L from (29).

      Step 4. Calculate the open-loop transfer function of Subsystem 2, (43). Draw the root locus plot of the system and find $ K_{\max} $ (or judge whether $ K_{\max} $ exists).

      Step 5. Choose $ \lambda $ in range $ [K(0),K_{\max}) $. Let $ K = \lambda $, and draw the Bode plot of $ G_{\widetilde dd_e} $. Check if

      $ G_{\widetilde dd_e}(j\omega)\approx1,\; \forall \omega \in \Omega_c $

      (58)

      holds. Moreover, check if the phase lag is small. If it is not, adjust $ \lambda $ if necessary.

      Step 6. Choose $ \mu $ and calculate the law for adjusting $ K(t) $ in (45).

    • In this section, the effectiveness of the AEID approach is verified through simulations. A two-mass system in a mechanical system has been widely used to investigate vibration, nonlinearities, and many other control problems.

      In this study, the speed control of a two-mass system was concerned. A torque disturbance was imposed on the controlled motor from a channel other than control input channel. The parameters in plant (1) were [9].

      $ \left\{\begin{aligned} & A = \left[\!\!\!\begin{array}{*{20}{c}} -31.31& 0 & -2.833 \times 10^4 \\ 0 & -10.25 & 8001 \\ 1 & -1 & 0 \end{array}\!\!\!\right],\; B = \left[\!\!\!\begin{array}{*{20}{c}} 28.06 \\0\\0 \end{array}\!\!\!\right] \\ & B_d = \left[0 \quad 7.210 \quad 0\right]^{\rm{T}}, C = \left[1 \quad 0 \quad 0\right]. \end{aligned}\right.$

      (59)

      Disturbance

      $ d(t) = \left\{\!\!\!\begin{array}{ll} \sin \pi t + 0.5 \sin 2\pi t + 0.25 \sin 3\pi t, & {\rm if} \;0<t \leq 9\\ 0, & {\rm if} \;9<t \leq 10 \end{array}\right. $

      (60)

      or

      $ d(t) = \left\{\!\!\!\begin{array}{ll} 2(\sin \pi t \!+\! 0.5 \sin 2\pi t \!+\! 0.25 \sin 3\pi t), & {\rm if}\;0<t \leq 9 \\ 0, & {\rm if}\;9 < t \leq 10 \end{array}\right. $

      (61)

      in Fig. 4 was added to the system.

      Figure 4.  Disturbance $ d(t) $ in (60) and (61)

    • From (60), the maximum angular frequency for disturbance rejection, $ \omega_d $, was chosen to be $ 3\pi $. Then, $ \omega_c $ was chosen to be 101 rad/s. This gives the parameters of low-pass filter $ F(s) $ in (12) as

      $ A_f = -101,\; B_f = 100,\; C_f = 1. $

      (62)

      The choice of positive numbers $ \alpha_1 $ and $ \alpha_2 $ had the influence on the solution of the LMI. For simplicity, we chose

      $ \alpha_1 = \alpha_2 = 0.05 $

      (63)

      and

      $ \Gamma = \left[ {\begin{array}{*{20}{c}} {0.05{I_3}}&0&0\\ 0&{0.05{I_3}}&0\\ 0&0&{0.1} \end{array}} \right]. $

      (64)

      We obtained a feasible solution of the LMI (27) and the gains

      $\begin{split} & {K_P} = \left[ {\begin{array}{*{20}{c}} { - 0.97}&{1.38}&{734.00} \end{array}} \right],L = \left[ {\begin{array}{*{20}{c}} {27.01}\\ { - 24.91}\\ { - 6.23} \end{array}} \right],\\ & {P_3} = 0.38.\end{split} $

      (65)

      Simple verification showed that $ G_o(s) $ is a strictly positive real function. Thus, $ K_{\max} $ is unbounded (Fig. 5). $ \lambda $ was selected to satisfy the requirement (58). The Bode plots (Fig. 6) showed that (58) is satisfied when $ K \leq 5 $. When K continued to increase, the phase lag was relatively large. Therefore, to balance the disturbance-rejection performance, we finally chose

      Figure 5.  Root loci of $ G_{op}(s)$

      Figure 6.  Bode plots of $ G_{\widetilde dd_e}(s) $ for different $ \lambda\; (K = \lambda) $: (a) Magnitude and (b) Phase.

      $ \lambda = 5 $

      (66)

      in this study. For such a $ \lambda $ and the selection of the adaptive

      $ \mu = 0.1 $

      (67)

      law (45), was used to adjust $ K(t) $ in a real-time fashion.

    • A conventional-EID-based control system is designed for comparison. $ F(s) $, $ K_P $, and L are the same as those in the AEID-based control system ((12), (29), (62) and (65), respectively). The only difference is that K was fixed to 1 in the conventional-EID-based control system (Fig. 7) while it is adaptively adjusted in the AEID-based one.

      Figure 7.  Configuration of EID-based control system

      Simulation results for the disturbance (60) and (61) are shown in Figs. 8 and 9. For disturbance (60), the largest peak-to-peak value (PPV) of the control input is 3.87 for the AEID approach and 4.12 for the conventional EID approach. The largest PPV of the output is 0.49 for the AEID approach and 0.62 for the conventional EID approach. On the other hand, the disturbance in (61), the largest PPV of the control input is 7.50 for the AEID approach and 8.23 for the conventional EID approach, and the largest PPV of the output is 0.85 for the AEID approach and 1.24 for the conventional EID approach. Note that, since $ d(t) $ is much smaller in Fig. 8 than in Fig. 9, the range of changes in $ K(t) $ is smaller for (60) than for (61). In fact, $ K(t) $ was adjusted using the adaptive law (45), and $ K(t) $ changed in a range with its maximum less than $ \lambda $ in (66) for a small disturbance (60) (Fig. 8 (c)). The large disturbance (61) resulted in a very large gain and saturated at 5 around the peak values of the disturbance estimates (Fig. 9 (c)).

      Figure 8.  Simulation results for (60)

      Figure 9.  Simulation results for (61)

      The performance index

      $ J = \int_0^{10}y^{\rm{T}}(t)y(t){\rm{d}}t $

      (68)

      was used to compare the disturbance-rejection performance of the two methods. $ J_{EID} $ is the performance index for EID-based system and $ J_{AEID} $ is the performance index for AEID-based system. A simple calculation gives

      $ \left\{ \begin{array}{ll} \rm{For}\; (60): & J_{\rm{EID}} = 0.26,\; J_{\rm{AEID}} = 0.14 \\ \rm{For}\; (61): & J_{\rm{EID}} = 1.05,\; J_{\rm{AEID}} = 0.32. \end{array} \right. $

      (69)

      Clearly, the performance index is smaller for the AEID approach than for the conventional EID approach for both (60) and (61). Moreover, the improvement for (60) is less obvious than that of (61). One of the most important reasons is that the disturbance in (60) is smaller than that in (61). It accords with our design requirement when designing $ K $. Larger disturbance brings larger gain, and larger gain improves the disturbance-rejection performance of the system.

      To determine if the method is practical, simulations were carried out with measurement noise (peak value: 0.12). In order to show the difference between the AEID approach and the conventional EID approach clearly, a larger disturbance (61) was considered in the system. The simulation results for (61) (Fig. 10) show that, the estimated disturbance $ \widetilde d(t) $ is different from the conventional EID approach and the AEID approach. While the PPV of the output is $ 1.76 $ for the conventional EID approach, it is 1.41 for the AEID approach. And the performance index is $ J_{EID} = 1.14 $ and $ J_{AEID} = 0.43 $ for the EID and AEID approaches, respectively. This shows that the AEID approach is superior to the conventional EID approach.

      Figure 10.  Comparison of EID-based and AEID-based approaches with measurement noise

    • This paper described an AEID approach that provides us a way of improving disturbance-rejection performance. The gains of the state feedback and the state observer were designed based on an LMI. The adaptive law for an adjustable gain factor was presented based on the combination of the root locus method and Lyapunov stability theory. It is simple and easy to implement. Simulation results show that the AEID approach is effective and superior to the conventional EID approach.

    • This work was supported by National Natural Science Foundation of China (No. 61873348), National Key R&D Program of China (No. 2017YFB1300900), Hubei Provincial Natural Science Foundation of China (No. 2015CFA010), and the 111 Project, China (No. B17040).

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