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Performance analysis of active disturbance rejection tracking control for a class of uncertain LTI systems$ Wenchao Xue n, Yi Huang Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 18 October 2014 Received in revised form 17 April 2015 Accepted 3 May 2015 This paper was recommended for publication by Dr. Jeff Pieper

The paper considers the tracking problem for a class of uncertain linear time invariant (LTI) systems with both uncertain parameters and external disturbances. The active disturbance rejection tracking controller is designed and the resulting closed-loop system's characteristics are comprehensively studied. In the time-domain, it is proven that the output of closed-loop system can approach its ideal trajectory in the transient process against different kinds of uncertainties by tuning the bandwidth of extended state observer (ESO). In the frequency-domain, different kinds of parameters' influences on the phase margin and the crossover frequency of the resulting control system are illuminated. Finally, the effectiveness and robustness of the controller are verified through the actuator position control system with uncertain parameters and load disturbances in the simulations. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Extended state observer (ESO) Active disturbance rejection control (ADRC) Tracking Transient performance Uncertain systems

1. Introduction Dealing with multifarious uncertainties such that the closed-loop system can keep desired performance is always one of the longstanding fundamental control objects [1]. In the past decades, lots of effective approaches have been substantially developed to cope with the two basic kinds uncertainties: internal (parametric or unstructured) uncertainties and external disturbances. Robust control and adaptive control are well-known areas where parametric and/or bounded unstructured uncertainties of the plant are the main concern. To handle external disturbance, internal model principle [2] and disturbance observer based control [3,4] suggest to cancel the disturbances' effect with that the former embeds the disturbances' model in controller and the latter timely estimates the disturbances by constructing special observers. Since physical plants usually contain both internal uncertainties and external disturbances, dealing with mixed uncertainties is of great engineering necessity and is becoming an attractive field. Actually, many control methods have shown their power of estimating and attenuating the mixed uncertainties, such as the active disturbance rejection control (ADRC) [5,6], the nonlinear disturbance observer based control [7–9], the embedded model control [10], the composite hierarchical anti-disturbance control (CHADC) [11], and the internal model control (IMC) [12]. These approaches mainly share the structure of two degrees of freedom, i.e., one to achieve disturbance rejection and the other to regulate the closed-loop characteristics. Since such structure can ensure the closed-loop system to have desired performance in the entire control process despite uncertainties, it is appealing to practioners. The paper pays particular attention to ADRC, which is less dependent on the model due to its unique idea, i.e., using extended state observer (ESO) to estimate the “total disturbance” including both internal uncertainties and external disturbances of the plant [6]. Moreover, the strong robustness and the outstanding performance of ADRC have been shown in the application research for lots of industrial processes involving the servo systems [13–15], MEMS systems [16], robotic systems [17,18], power systems [19,20], and other plants(see [21,22]). In the past years, large progress has been made on the performance analysis of ADRC in time-domain [13,23–26] and the promising frequency characteristics of ADRC have been discovered in case studies [27]. The stability of linear ADRC and nonlinear ADRC based closedloop system is respectively presented in [23,24] by showing that arbitrarily small tracking error can be reached against unknown dynamics

☆ Supported by the National Basic Research Program of China under Grant no. 2014CB845303 and the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences. n Corresponding author. E-mail addresses: [email protected] (W. Xue), [email protected] (Y. Huang).

http://dx.doi.org/10.1016/j.isatra.2015.05.001 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Xue W, Huang Y. Performance analysis of active disturbance rejection tracking control for a class of uncertain LTI systems. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.001i

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after a finite time. Ref. [25] analyzes the performance of linear ADRC based closed-loop system for a class of lower-triangular systems. Ref. [13] proposes the generalized PI observer (GPIO), which is viewed as the generalized ESO, and shows the convergence of closed-loop system for the ADRC with GPIO in motor control system. Ref. [26] proves the capability of LADRC for LTI systems with unknown but bounded relative degrees. Although the performance of ADRC in time-domain has been discussed from several perspectives, the existing literatures usually put conservative restrictions on the uncertainties of plant, especially on the variation of the control input gain. Since many case studies shown that ADRC can tolerate large uncertainties on the control input gain (see [28,29]), it is important to provide stability analysis under much milder restrictions than those in existing work. On the other hand, few literatures focus on ADRC's frequency-domain analysis, which is of fundamental importance since the stability margin of the control system is mainly read from the frequency response of its loop transfer function. Ref. [27] found that the bandwidth and stability margin of ADRC based system are nearly unchanged against large parametric uncertainties in a simulation of 2nd order system. Such discover shows the promising frequency characteristics of ADRC based system which need to be adequately studied. This paper is devoted to the comprehensive and rigorous study on the performance of ADRC based control system from the views of both time-domain and frequency-domain. The main contributions include: (i) The ADRC design for the general tracking control problem is given. The paper shows the capability of ADRC to keep consistent performance against significant variation on the parameters of plant with the size of the uncertainty on the control input gain being much larger than that in the previous work. In addition, the physical meanings of ADRC's parameters are illustrated. (ii) The paper evaluates the frequency response of ADRC based control system, and illuminates that the phase margin as well as the crossover frequency is insensitive to the variations of parameters except the plant's control input gain which has major influence on the frequency characteristics of ADRC based control system. (iii) The robustness of ADRC based tracking controller is verified by the actuator position control system in the way of realistic computer simulations. The rest of this paper is arranged as follows. Section 2 presents the ADRC based tracking controller design. In Section 3, the performance of ADRC based control system from the aspects of both time-domain and frequency-domain is demonstrated. The simulations on a motion control system are carried out in Section 4 and some conclusions are given in Section 5.

2. Active disturbance rejection tracking controller Although ADRC is proposed to deal with general nonlinear uncertain systems [6], this paper limits the discussion to the following class of uncertain LTI systems to study the characteristics in both time-domain and frequency-domain: 8_ z 1 ¼ z2 ; > > > > z_ 2 ¼ z3 ; > > > > > > ⋮ > > > > z_ ¼  pT z  p > > m þ 1 x1 þ d1 ðtÞ; < m x_ 1 ¼ x2 ; t Z t0 ð1Þ > > > > x_ 2 ¼ x3 ; > > > > ⋮ > > > > > > x_ n ¼ aT x þ qT z þ d2 ðtÞ þ bu; > > : y ¼ x1 ; where x ¼ ½x1 ; x2 ; …; xn T A Rn is the state vector, y A R is the output, u A R is the control input. All the parameter vectors or parameters p ¼ ½pm ; pm  1 ; …; p1 T , a ¼ ½ an ;  an  1 ; …;  a1 T , q ¼ ½qm ; qm  1 ; …; q1 T , pm þ 1 and b have uncertainties with b is the known nominal value of b. z ¼ ½z1 ; z2 ; …; zm T A Rm can be viewed as another internal state vector causing uncertain dynamics which affects x. d1 ðtÞ and d2 ðtÞ are unknown external disturbances and the initial values ðzðt 0 Þ; xðt 0 ÞÞ also have uncertainties. This paper is interested in the tracking problem of (1) with the signal r(t) being timely tracked by the output y(t). In practice, the output yðtÞ ¼ x1 ðtÞ is usually required to have desired performance despite various uncertainties in the transient process. For instance, small overshoot and fast rise time of y(t) are needed when r(t) is constant. The performance requirements on y(t) indicate that the trajectory of y(t) should approach an ideal trajectory for t A ½t 0 ; 1Þ besides limt-1 j yðtÞ  rðtÞj -0. Note that xi is the ði 1Þ th-order derivative of y(t), then the paper is interested in the following control object: J xðtÞ  xn ðtÞ J r ε;

8 t A ½t 0 ; 1Þ;

ð2Þ

T

where x ¼ ½x1 ; x2 ; …; xn  , and ε 4 0 is the desired bound of the error between x and its ideal trajectory xn ¼ generated by 8 n x_ ¼ xn2 ; > > > 1 > > n > _n > < x 2 ¼ x3 ; ⋮ xn ðt 0 Þ ¼ xðt 0 Þ > n > X > n > > ki ðxnn þ 1  i  r ðn  iÞ Þ þ r ðnÞ ; x_ ¼  > > : n i¼1 where the polynomial sn þ engineering requirements.1

Pn

i¼1

½xn1 ; xn2 ; …; xnn T ,

which is

ð3Þ

ki sn  i is stable. Thus, the dynamic performance of ðxn1  rÞ can be tuned by ki ; i A n to meet specific

Remark 1. The object (2) means that the trajectory of x(t) should be close to the ideal trajectory xn ðtÞ in the entire transient process against both uncertain parameters ða; q; p; pm þ 1 ; bÞ and external disturbances ðd1 ðtÞ; d2 ðtÞÞ. Assume that the system (1) is the typical motion 1

In this paper, n denotes the set of f1; 2; …ng and r ðiÞ denotes the ith-order derivative of r(t).

Please cite this article as: Xue W, Huang Y. Performance analysis of active disturbance rejection tracking control for a class of uncertain LTI systems. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.001i

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control system with x1 ðtÞ being the position, x2 ðtÞ being the velocity, u(t) being the input acceleration and r(t) being the constant, then xn ðtÞ is generated by ( n x_ 1 ¼ xn2 ; ð4Þ xn ðt 0 Þ ¼ xðt 0 Þ x_ n2 ¼  k2 ðxn1  rÞ  k1 xn2 ; pffiffiffi R1 where xn1 ðtÞ is the ideal trajectory of position. Let ðk1 ¼ 2c; k2 ¼ c2 Þ; c 4 0 or ðk1 ¼ 2c; k2 ¼ c2 Þ; c 40 such that the index t0 tj xn1 ðtÞ  rðtÞj dt R1 n or t0 ðx1 ðtÞ  rðtÞÞ2 dt is minimized, respectively. Moreover, larger c means faster rise time of xn1 ðtÞ and more control energy to generate the acceleration signal k2 ðxn1  rÞ þk1 xn2 . Therefore, the dynamics of xn ðtÞ is suggested to be designed according to the performance requirements on the states of interest and the maximal acceleration provided by the actuator. Next, the ADRC based tracking controller will be designed to achieve the control object (2). Firstly, the following equivalent dynamic systems of xn ðtÞ and x(t) are introduced: 8 n 8 > e_ 1 ¼ en2 ; > > e_ 1 ¼ e2 ; > > n > > n > > _ ¼ e ; e > > 3 < 2 < e_ 2 ¼ e3 ; ⋮ ; ð5Þ … > > n > > X > > > _n >_ n > : ki en þ 1  i ; e ¼ > e n ¼ en þ 1 þ bu; > : n i¼1 where eni ðtÞ ¼ xni ðtÞ  r ði  1Þ ðtÞ; i A n is the ideal trajectory of the tracking error ei ðtÞ ¼ xi ðtÞ  r ði  1Þ ðtÞ, and en þ 1 ðx; z; t; uÞ 9 aT e þ qT z 

n X

ai r ðn  iÞ ðtÞ  r ðnÞ ðtÞ þ d2 ðtÞ þ ðb bÞu

ð6Þ

i¼1

can be viewed as the “total disturbance” affecting the dynamics of the tracking errors ei ðtÞ; i A n. Furthermore, by using (5), the control object (2) equals to J eðtÞ  en ðtÞ J r ε;

8 t A ½t 0 ; 1Þ; T

n

ð7Þ n

n

n T

n

where e ¼ ½e1 ; e2 ; …; en  and e ¼ ½e1 ; e2 ; …; en  . Eq. (5) shows that to force the tracking error e(t) to approach its ideal trajectory e ðtÞ for 8 t A ½t 0 ; 1Þ, the information of both the states e(t) and the “total disturbance” en þ 1 ðÞ in real time is needed. The key of ADRC is designing an extended state observer (ESO) to timely estimate e(t) and en þ 1 ðÞ from the input data u(t) and the output data y(t) (or e1 ðtÞ) of the system (1). The linear form of ESO is [30] n ð8Þ e^_ 1 ¼ e^ 2 þ β1 ðe1  e^ 1 Þ⋮e^_ i ¼ e^ i þ 1 þβ2 ðe1  e^ 1 Þ; ⋮e^_ n ¼ e^ n þ 1 þ bu þ βn ðe1  e^ 1 Þe^_ n þ 1 ¼ βn þ 1 ðe1  e^ 1 Þ: where βi ðÞ; i A n þ1, satisfy sn þ 1 þ

n þ1 X

βi sn þ 1  i ¼ ðs þ ωe Þn þ 1 ;

ð9Þ

i¼1

and ωe 4 0 is viewed as the bandwidth of ESO. The outputs e^ i ðtÞ; i A n þ 1, are desired to estimate ei ðtÞ; i A n þ 1, respectively. In addition, the initial conditions of (8) are set by e^ 1 ðt 0 Þ ¼ yðt 0 Þ  rðt 0 Þ;

e^ i ðt 0 Þ ¼ x i ðt 0 Þ  r ði  1Þ ðt 0 Þ;

2 r i r n;

e^ n þ 1 ðt 0 Þ ¼ 0

ð10Þ

Therefore, the following control law u¼ 

e^ n þ 1 b



k1 e^ n þ ⋯ þ kn  1 e^ 2 þkn e^ 1 b

ð11Þ

will push the dynamics of e(t) to approach en ðtÞ; 8 t A ½t 0 ; 1Þ. Remark 2. If r ðiÞ ðtÞ; i ¼ 1; ‥; n, are all timely available , then the following ESO n x^_ 1 ¼ x^ 2 þ β1 ðx1  x^ 1 Þ⋮x^_ i ¼ x^ i þ 1 þβ2 ðx1  x^ 1 Þ; ⋮x^_ n ¼ x^ n þ 1 þ bu þ βn ðx1  x^ 1 Þx^_ n þ 1 ¼ βn þ 1 ðx1  x^ 1 Þ:

ð12Þ

is suggested to estimate the states xi ði ¼ 1; ‥; nÞ and the “total disturbance” xn þ 1 9 aT x þ qT z þ d2 ðtÞ þ ðb  bÞu. Furthermore, the control law becomes P x^ n þ 1 k1 x^ n þ ⋯ þ kn  1 x^ 2 þ kn x^ 1  ni¼ 1 ki r ðn  iÞ  r ðnÞ u¼   : ð13Þ b b Note that the high order derivatives of r(t) may not be available in physical system, then ADRC (8)–(11) can deal with more general tracking problems. 3. Main results Before the performance of the ADRC (8)–(11) based control system is discussed, we firstly give the following assumptions on the uncertainties of the system (1) and the reference signal r(t): P i1 (A1) the polynomial sm þ m pm þ 1  i is stable, i¼1s (A2) j d1 ðtÞj r α1 , j d2 ðtÞj rα1 , j d_ 2 ðtÞj r α1 ; 8 t Z t 0 , (A3) j ai j r α2 ; i A n, j pj j r α2 ; j A m þ1 and j qk j r α2 ; k A m, Please cite this article as: Xue W, Huang Y. Performance analysis of active disturbance rejection tracking control for a class of uncertain LTI systems. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.001i

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(A4) b=b A ½α3 ; α4   ð0; 2 þ 2=nÞ, (A5) j zi ðt 0 Þj rρ1 ; i A m, j x1 ðt 0 Þj r ρ2 and j rðt 0 Þj r ρ3 , (A6) j xi ðt 0 Þ  x i ðt 0 Þj r ρ4 ; 2 r i r n; j r ðiÞ ðtÞj r α5 ; 8 t Z t 0 ; 0 r ir n þ 1, where α1 ; α2 ; α3 ; α4 ; α5 ; ρ1 ; ρ2 ; ρ3 ; ρ4 are known positives, x i ðt 0 Þ is the known nominal value of xi ðt 0 Þ. Remark 3. From (A1)–(A2), z(t) is bounded if x(t) is bounded. Hence, the control object (2) or (7) implies that all the states of system (1) are bounded. Also, the system (1) with (A1)–(A6) can represent many practical plants in different industrial sectors, such as the following typical ones.

 Actuator in [31]:

8 _ > < x 1 ¼ x2 ; x_ 2 ¼  a2 x1  a1 x2 þ bu; > :y¼x ;

t Z t0

1



where y ¼ x1 is the actuator position, x2 is the velocity, u is the voltage input of the amplifier corresponding to the thrust force generated by the voice-coil motor and the parameters ða1 ; a2 Þ have uncertainties. Pitch channel of flight control in [32]: 8 _ > < z 1 ¼  p1 z 1  p2 x 1 ; x_ 1 ¼  a1 x1  q1 z1 þ dðtÞ þbu; t Z t 0 > :y¼x ; 1 where y ¼ x1 is the pitch rate, u is the roll rudder deflection, z1 is the internal state generated by the stable zero of transfer function from u to y, d(t) is the external disturbance, the parameters ðp1 ; p2 ; a1 ; q1 Þ have uncertainties and p1 40.

Although the applications of ADRC for several practical plants [13]–[22] show the satisfactory performance of the closed-loop systems against internal and external uncertainties, these previous literatures are in short of the comprehensive analysis from the views of either frequency-domain or time-domain. This paper aims to analyze the properties of ADRC in both time-domain and frequency-domain as well as give several crucial guidelines on tuning of ADRC.

3.1. Performance of ADRC in the time-domain Usually, ωe, i.e., the bandwidth of ESO (8), is tuned large such that the “total disturbance” en þ 1 ðÞ can be estimated and compensated for as soon as possible. Thus the estimation error of ESO may be large in a very short initial phase if ei ðt 0 Þ  e^ i ðt 0 Þ ¼ xi ðt 0 Þ  x i ðt 0 Þ a0ð2 r i r nÞ or ρ4 a 0 [33]. To avoid this peaking phenomenon of ESO, ADRC is suggested to be designed as 8 t o tu; > < 0; e^ n þ 1 þ k1 e^ n þ ⋯ þ kn  1 e^ 2 þ kn e^ 1 u¼ ð14Þ ; t Z tu; > : b where tu is the end time of peaking for ESO (8). The proof of Theorem 1 will show   lnðρ4 ωe Þ ;0 ; t u ¼ t 0 þ 2ðn  1Þ J P 2 J max ωe

ð15Þ

where P 2 is the unique positive solution of the equation 2 3 ϕ1 1 … 0 6 7 0 … 6 ϕ2 7 T 7: A 2 P 2 þ P 2 A 2 ¼  I; A 2 ¼ 6 6 … … … 17 4 5 ϕn þ 1 0 … 0 Remark 4. Eq. (15) means that if ρ4, i.e., the size of the initial estimation error j xi ðt 0 Þ  x i ðt 0 Þj ; 2 r ir n, satisfies ρ4 r 1=ωe , then there is no peaking and t u ¼ t 0 . The following Theorem 1 presents the properties of ADRC in the time-domain. Theorem 1. Consider the ADRC based closed-loop system (1), (8) and (14) under (A1)–(A6). There exist ωn 40 and γ ni 4 0; i ¼ 1; 2; 3; 4, such that 8 ωe Zωn , ðiÞ

J xðtÞ  xn ðtÞ J r γ n1

maxflnðρ4 ωe Þ; ln ωe ; 1g ; ωe

ðiiÞ

J e^ i ðtÞ ei ðtÞ J r γ n3

ðiiiÞ

J uðtÞ J r γ n4 ;

1 ; ωe

8 t Zt 0 þ γ n2

8 t Zt 0 ;

maxflnðρ4 ωe Þ; ln ωe ; 0g ; iA n þ 1; ωe

8 t Z t0 :

Please cite this article as: Xue W, Huang Y. Performance analysis of active disturbance rejection tracking control for a class of uncertain LTI systems. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.001i

ð16Þ

ð17Þ ð18Þ

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The proof of Theorem 1 is in Appendix A. In light of Theorem 1, the following significant properties of ADRC based closed-loop system are revealed: (a) Eqs. (16) show that the states x(t) of the closed-loop system are close to their reference trajectory xn ðtÞ for t A ½t 0 ; 1Þ despite uncertain parameters and external disturbances. The error J xðtÞ  xn ðtÞ J ; 8 t A ½t 0 ; 1Þ can be tuned by ωe, which means the object (2) or (7) can be achieved. Moreover, the parameters of ADRC are transparent for the engineers: ki ð1 ri r nÞ are designed to shape the desired trajectory xn ðtÞ by (3) and ωe plays the role of decreasing the disparity between the actual output and its desired trajectory. In addition, (17)–(18) mean that small estimation error of ESO and uniform boundness of the control input are ensured against all uncertainties in (A1)–(A6). (b) Theorem 1 implies a tuning method of ωe, that is increasing ωe from a small value until the error between the actual output and its desired trajectory meets practical requirements. (c) Obviously, the right hand of (16) is decreasing with respect to ρ4 which represents the size of initial estimation error for xi ðt 0 Þ; 2 r i rn. Thus, diminishing the initial estimation error of xi ðt 0 Þ is also helpful to regulate the performance of the trajectory xðtÞ; 8 t A ½t 0 ; 1Þ. (d) The uncertain parameter b belonging to b=b A ð0; 2 þ 2=nÞ can be permitted by ADRC. Additionally, the function 2 þ 2=n is decreasing with respect to n. Therefore, Theorem 1 proves that ADRC can deal with larger uncertainties of b for lower order plant. For example, (A4) becomes b=b A ð0; 4Þ in the case of n ¼1 and (A4) becomes b=b A ð0; 3Þ in the case of n ¼2. To our knowledge, the weakest condition on b to ensure the stability of ADRC based closed-loop system in the previous work is (see [34]) ðA4Þ0

b=b A ½α03 ; α04   ð0; 2Þ;

where α03 ; α04 are constants. Since ð2n þ 2Þ=n 4 2; 8 n Z 1, ðA4Þ0 is stronger than (A4). According to the above discussion in the time-domain, the origin of the ADRC based closed-loop system (1), (8) and (14) is stabilized despite various uncertainties. In controller design, engineers probably further concern with the stability margin of the ADRC based control system, which may be demonstrated from the point view of the frequency-domain in the next subsection.

3.2. Performance of ADRC in the frequency-domain The frequency characteristics such as the crossover frequency and stability margin are usually used to quantify the robustness of control system. Since the phase margin and crossover frequency are independent of the initial condition of the states, let the initial values of states in the system (1) to be zero. Consequently, (1) can be rewritten in the following form of transfer function: YðsÞ ¼

bMðsÞUðsÞ þMðsÞD2 ðsÞ þ HðsÞD1 ðsÞ ; MðsÞNðsÞ þ HðsÞpm þ 1

ð19Þ

where MðsÞ ¼ sm þ p1 sm  1 þ ⋯ þ pm ; NðsÞ ¼ sn þ a1 sn  1 þ ⋯ þ an ; HðsÞ ¼ q1 sm  1 þ q2 sm  2 þ ⋯ þ qm ; YðsÞ; UðsÞ; D1 ðsÞ and D2 ðsÞ are the Laplace Transformation of yðtÞ; uðtÞ; d1 ðtÞ and d2 ðtÞ respectively. Next, the ADRC (8) and (14) in the form of transfer function will be studied. There is no control action on t A ½t 0 ; t u Þ implies that the Laplace Transformation of u(t) only for t A ½t u ; 1Þ needs to be considered. Therefore, C 2 ðsÞ 1 C 2 ðsÞ 1 E1 ðsÞ ¼  ðYðsÞ RðsÞÞ; ð20Þ C 1 ðsÞ b C 1 ðsÞ b P P P P where C 1 ðsÞ ¼ ni¼ 0 ð iq ¼ 0 kq βi  q Þsn þ 1  i ; C 2 ðsÞ ¼ ni¼ 0 ð nq ¼i0 ki þ q βn þ 1  q Þsn  i . E1 ðsÞ and R(s) are the Laplace Transformation of e1 ðtÞ and r (t), respectively. Fig. 1 shows the block diagram of the ADRC based control system. Combination of (19) and (20) leads to the transfer function of the ADRC based closed-loop system:

UðsÞ ¼ 

b MðsÞC 2 ðsÞ HðsÞD1 ðsÞ þ MðsÞD2 ðsÞ RðsÞ þ ðMðsÞNðsÞ þ HðsÞpm þ 1 ÞC 1 ðsÞ MðsÞNðsÞ þ HðsÞpm þ 1 b : YðsÞ ¼ b MðsÞC 2 ðsÞ 1þ b ðMðsÞNðsÞ þ HðsÞpm þ 1 ÞC 1 ðsÞ

ð21Þ

Obviously, the stability of the closed-loop system (21) depends on the zeros of 1 þ LðsÞ, where LðsÞ 9

b

MðsÞC 2 ðsÞ

ð22Þ

b ðMðsÞNðsÞ þ HðsÞpm þ 1 ÞC 1 ðsÞ

is referred to the loop transfer function (LTF). The frequency response of LTF also decides the phase margin and crossover frequency which are important indexes to evaluate the performance of control systems. Usually, phase margin implicates that how much phase delay the system can tolerate to ensure the stability of the closed-loop system, and crossover frequency reflects the bandwidth of control system. Let

Fig. 1. The ADRC based control system in the transfer function form.

Please cite this article as: Xue W, Huang Y. Performance analysis of active disturbance rejection tracking control for a class of uncertain LTI systems. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.001i

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θm and ωθ denote the phase margin and crossover frequency of L(s), respectively. Then, θm ¼ π þ ∠Lðjωθ Þ;

j Lðjωθ Þj ¼ 1:

ð23Þ

Since L(s) has uncertain parameters ða; p; q; bÞ, it is necessary to study how the variation of ða; p; q; bÞ influences ðθm ; ωθ Þ. The influence on ðθm ; ωθ Þ caused by the uncertain parameters ða; p; q; bÞ will be discussed via the relationship between the responses of L(s) and L0 ðsÞ, which is defined as L0 ðsÞ ¼

b b sn þ 1 þ

β Pnn þ 1

nþ1i i ¼ 1 βi s

¼

b

ωne þ 1

b ðs þ ωe Þn þ 1  ωne þ 1

:

ð24Þ

Let θ m and ω θ denote the phase margin and crossover frequency of L0 ðsÞ, respectively. According to the definitions of phase margin and crossover frequency, ω θ and θ m satisfy j L0 ðjω θ Þj ¼ 1;

θ m ¼ π þ ∠L0 ðjω θ Þ:

ð25Þ

Since L0 ðjω θ Þ ¼

b

1

b ðjω θ =ωe þ 1Þn þ 1 1

;

(25)means that ω θ can be determined by ω θ ¼ σ ωe

where σ 4 0

b

and

1

b j ðjσ þ1Þn þ 1  1j

¼ 1:

ð26Þ

Similarly, θ m is determined by θm ¼ π þ∠

b

1 nþ1

b ðjω θ =ωe þ 1Þ

1

  ¼ π  ∠ ðjσ þ 1Þn þ 1  1 :

ð27Þ

The following theorem shows the relationship between ðωθ ; θm Þ and ðω θ ; θ m Þ. Theorem 2. Assume that (A1)–(A4) are satisfied. Then for 8 ωe Zωn , there is j θm =θ m 1j r γ n1 ðωe Þ;

j ωθ =ω θ  1j r γ n2 ðωe Þ;

ð28Þ

where ωn is the same as that of Theorem 1 and ðγ n1 ðωe Þ; γ n2 ðωe ÞÞ satisfies lim γ ni ðωe Þ ¼ 0;

ωe -1

i ¼ 1; 2:

ð29Þ

The proof of Theorem 2 is in Appendix B. According to Theorem 2, the following points on the robustness of the ADRC based control system from the view of frequency-domain are suggested: (a) Theorem 2 means that ðθm ; ωθ Þ can be estimated by ðθ m ; ω θ Þ which depends on b=b rather than ða; p; qÞ. Moreover, the estimation errors can be arbitrarily small by increasing the ESO's bandwidth, which is usually tuned larger than that of the dynamics of plant such that the uncertain dynamics can be estimated. Thus, the phase margin and crossover frequency of ADRC based system can be almost invariant despite ða; p; qÞ. This also reveals that the performance of the closed-loop system is robust to the uncertainties ða; p; qÞ, which is an advantage of ADRC. (b) The relationship revealed by (26)–(27) can help practioners to quantitatively know the time delay tolerated in the control signal of ADRC based control system. The critical value of the tolerated time delay, according to the knowledge of relative stability, is τd ¼ θm =ωθ : Since ðθm ; ωθ Þ can be estimated by ðθ m ; ω θ Þ, (26)–(27) mean that   nþ1 1 θm 1 π  ∠ ðjσ þ 1Þ ; τd  ¼ ω θ ωe σ

ð30Þ

ð31Þ

where σ 4 0 is computed by solving j ðjσ þ 1Þn þ 1  1j ¼ b=b. Also, note that ω θ ¼ σ ωe , then (31) reflects that larger ωe generates larger crossover frequency but smaller time-delay tolerated by ADRC based control system. On the other hand, (31) implies the upper bound of ωe to ensure that the tolerated time delay τd is larger than a given value τnd . For instance, consider the case of n ¼ 1. The explicit expression for the solution of (26)–(27) shows that vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi uv π 1 uu  tan Θ u b u 1 1 b2 1 θm 1  : τd  ¼ 2 ; Θb ¼ t t þ ð32Þ 4 16 b 2 2 ω θ ωe 2Θb Therefore, τd Z τnd equals to π 1 Θb 1 2  tan : ωe r n τd 2Θb Please cite this article as: Xue W, Huang Y. Performance analysis of active disturbance rejection tracking control for a class of uncertain LTI systems. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.001i

ð33Þ

W. Xue, Y. Huang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Since b=b satisfies (A4) and the right-hand of (33) is decreasing with respect to b=b, ωe has the following upper bound: π  tan  1 Θb : ωe r 2 2τnd Θb

7

ð34Þ

Remark 5. The frequency-domain discussions of this subsection explain why the limitation on b is stronger than the limitation on ða; p; qÞ in (A1)–(A6) and show that the information of b is more critical than ða; p; qÞ for ADRC. Also, the upper bound of ωe is quantitatively presented to ensure that the given time delay can be tolerated. The fact that these crucial conclusions are all achieved by frequencydomain analysis demonstrates the necessity of both frequency-domain and time-domain studies in this paper. 3.3. Comparison with IMC/DOBC In the last decades, many other control methods with the idea of estimating and canceling the uncertainties have also been substantially developed, such as IMC and DOBC. These approaches are shown to be effective in controlling practical systems with both external disturbance and part of uncertain model, although the stability of the corresponding closed-loop systems is only rigorously analyzed for some special cases [4,35]. The following comparison will demonstrate that the analysis of ADRC in this paper provides the theoretical foundation for a constructive design of IMC/DOBC to deal with more general model uncertainty and external disturbances. Firstly, define the Laplace transformations of ei ðtÞ and e^ i ðtÞ in ESO (11) as Ei ðsÞ and E^ i ðsÞ; iA n þ 1, respectively. Therefore, (5) and (8) mean that 8 sðE1  E^ 1 Þ ¼ E2  E^ 2 þ β1 ðE^ 1  E1 Þ > > > > sðEn  E^ n Þ ¼ En þ 1  E^ n þ 1 þ βn ðE^ 1  E1 Þ > > > : sE^ ¼ β ðE^  E Þ: nþ1

nþ1

1

1

Due to En þ 1 ðsÞ ¼ sn E1 ðsÞ  bUðsÞ, (35) leads to 0 1 !1 nþ1 bω b e @ E1 ðsÞ  UðsÞA: E^ n þ 1 ðsÞ ¼ sn ðs þ ωe Þn þ 1

ð36Þ

On the other hand, according to [12,4], the estimations of the uncertainty En þ 1 ðsÞ provided by IMC and DOBC can be equivalently written as Δ^ I ðsÞ ¼ Q I ðsÞðE1 ðsÞ  P I ðsÞUðsÞÞ

ð37Þ

and

  Δ^ D ðsÞ ¼ Q D ðsÞ P D 1 ðsÞE1 ðsÞ  UðsÞ

ð38Þ

respectively, where PI(s) or PD(s) stands for the nominal model of the plant, and ðQ I ðsÞ; Q D ðsÞÞ are stable proper transfer functions to be selected. By comparison between (36)–(38), (36) suggests that the IMC/DOBC with Q I ðsÞ ¼

sn ωne þ 1 ðs þωe Þ

nþ1

;

Q D ðsÞ ¼

bωne þ 1

ðs þ ωe Þn þ 1

;

P I ðsÞ ¼ P D ðsÞ ¼

b sn

ð39Þ

can be used to estimate the “total disturbance” En þ 1 ðsÞ. Note that design of ðP I ðsÞ; P D ðsÞ; Q I ðsÞ; Q D ðsÞÞ is usually key for the stability of IMC and DOBC [4], then the theoretical results of this paper provide a practical and safe choice (39) for IMC/DOBC to deal with general internal uncertainties and external disturbance for the system (1). Remark 6. By the last equation of (11), En þ 1 ðsÞ ¼ sn E1 ðsÞ  bUðsÞ and (36), the transfer function from En þ 1 ðsÞ to U(s) is UðsÞ 1 ωne þ 1 ¼ : En þ 1 ðsÞ b ðs þ ωe Þn þ 1

ð40Þ

Hence, the ESO's bandwidth ωe not only means faster estimation but also implies faster compensation for the “total disturbance”. More importantly, the transfer function  ð1=bÞωne þ 1 =ðs þ ωe Þn þ 1 also serves as a low pass filter to protect the control command from sharp varying of the “total disturbance”. 4. Simulations results Consider the following typical dynamics of an actuator: y€ ¼  a2 y  a1 y_ þ dðtÞ þbu;

t Z0

ð41Þ

where y¼p is the actuator measured position, y_ is the velocity, u is the voltage input of the amplifier corresponding to the thrust force generated by motor. a1 ; a2 and b are uncertain parameters related to the process of actuator, d(t) is the disturbance caused by unknown load, friction force and other unmodeled dynamics. The system (41) can be used to describe the fast tool servo system [31] and other practical motor control systems. Assume that the uncertainties of system (41) satisfy n _ _ ð42Þ yð0Þ ¼ 0; yð0Þ ¼ y_ ð0Þ ¼ 0j dðtÞj r 1; j dðtÞj r 0:5; j ai j r2; i ¼ 1; 2; b=b A ½0:5; 3; b ¼ 1: Please cite this article as: Xue W, Huang Y. Performance analysis of active disturbance rejection tracking control for a class of uncertain LTI systems. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.001i

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The output y(t) is desired to track the constant signal rðtÞ  1 and the ideal trajectory of y(t) is yn ðtÞ which is generated by 8 n > x_ ¼ xn2 > < 1 x_ n2 ¼  k2 ðxn1  rðtÞÞ  k1 xn2 > > : yn ¼ xn ; xn ð0Þ ¼ yð0Þ; xn ð0Þ ¼ yð0Þ; _ k1 ¼ 4; k2 ¼ 4: 1 1 2 The equation of the tracking error for the system (41) is ( e_ 1 ¼ e2 ; t Z0 e_ 2 ¼ e3 þ bu;

ð43Þ

ð44Þ

where e1 ¼ x1  r, e2 ¼ x2  r_ ¼ x2 and e3 9  a1 e2 a2 ðe1 þ 0:1Þ þdðtÞ þðb  bÞu is viewed as the “total disturbance” of the system (44). Furthermore, the ideal trajectory of e1 ðtÞ is en1 ðtÞ which satisfies 8 n > e_ ¼ en2 > < 1 e_ n2 ¼  k1 en2  k2 en1 > > : en ð0Þ ¼ e1 ð0Þ; en ð0Þ ¼ e2 ð0Þ: 1

n

ð45Þ

2

The ADRC design (11)–(14) means that the following ESO e^_ 1 ¼ e^ 2  β1 ðe^ 1  e1 Þ; e^_ 2 ¼ e^ 3  β2 ðe^ 1  e1 Þ þ buðtÞ; e^_ 3 ¼ β3 ðe^ 1  e1 Þ; :

e^ 1 ð0Þ ¼ yð0Þ  rð0Þ;

ð46Þ

e^ 2 ð0Þ ¼ 0; e^ 3 ð0Þ ¼ 0;

can timely estimate e1 ; e2 and e3, where β1 ¼ 3ωe ; β2 ¼ 3ω2e ; β3 ¼ ω3e . Since x2 ð0Þ ¼ 0, the definition of tu implies tu ¼0. Hence, u ¼  e^ 3  k1 e^ 2 k2 e^ 1 ;

t Z 0;

ð47Þ n

n

is designed to force y(t) approach y ðtÞ or e1 ðtÞ approach e1 ðtÞ. The parameter ωe is tuned by the method suggested by Theorem 1, i.e., increasing ωe from a small value until the satisfactory performance of the system's output is achieved. Then ωe is tuned to be ωe ¼ 30. Fig. 2 shows the block diagram of the control system (41)–(47). The responses of the closed-loop system (41)–(47) in both time-domain and frequency-domain will be investigated. 4.1. Response in the time-domain According to the scope of uncertainties in the system (41)–(42), several cases of uncertain parameters ða1 ; a2 ; bÞ and load disturbance d (t) in Table 1 will be tested. In addition, the simulation takes no disturbance for t o5 s and takes both step and sinusoidal disturbances for t Z 5 s such that the tracking performance of ADRC with and without disturbance is presented. From Fig. 3, it is evident that the actual outputs almost recover the target transient process of xni despite various uncertainties existing in parametric uncertainties and external disturbances. Such robustness can be explained in Figs. 4 and 5. Fig. 4 illustrates that ESO can timely and accurately estimate the “total disturbance” even it has sharp change at the time 5 s after which the external disturbances are added in the process. Consequently, the “total disturbance” can be actively compensated for by the control input, as shown in Fig. 5.

Fig. 2. The ADRC design based actuator position control system.

Table 1 The sets of ða1 ; a2 ; b; dðtÞÞ.

C1–1 C1–2 C1–3 C1–4 C2–1 C2–2 C2–3 C2–4 C3–1 C3–2 C3–3 C3–4

a1

a2

b

dðtÞ; t o 5 s

dðtÞ; t Z 5 s

 0.1 0.6 1.4  1.6  0.1 0.6 1.4  1.6  0.1 0.6 1.4  1.6

2.0  0.8 1.1  0.5 2.0  0.8 1.1  0.5 2.0  0.8 1.1  0.5

0.5 0.5 0.5 0.5 1.3 1.3 1.3 1.3 2.1 2.1 2.1 2.1

0 0 0 0 0 0 0 0 0 0 0 0

 sin ð0:08πtÞ sin ð0:1πtÞ 1 1  sin ð0:08πtÞ sin ð0:1πtÞ 1 1  sin ð0:08πtÞ sin ð0:1πtÞ 1 1

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The curves of the position and the reference trajectory 1.2

x1(C1−.1) x1(C1−.2)

1

x1(C1−.3) x (C1−.4)

0.8

1

0.6

x1(C2−.2) x1(C2−.3)

1

x (m)

x1(C2−.1)

0.4

x (C2−.4) 1

x1(C3−.1)

0.2

x1(C3−.2) x (C3−.3)

0

1

x1(C3−.4) −0.2

0

5

10

15

x*1

20

time(s) Fig. 3. The curves of the position and the reference trajectory.

Fig. 4. The total disturbance and its estimation under C1–1,C1–2,‥,C3–4.

The control input u(C1)

5 0 −5 0

5

10

20

u(Ci−1) u(Ci−2) u(Ci−3) u(Ci−4,i=1,2,3)

2 u(C2)

15

0 −2 0

5

0

5

10

15

20

10

15

20

u(C3)

2 0 −2 time(s) Fig. 5. The curves of the control input.

Furthermore, the relationship between ωe and tracking error x1 ðtÞ  xn1 ðtÞ, and the relationship between ωe and estimation error ^e 3 ðtÞ  e3 ðtÞ will be demonstrated. The following sets of bandwidth for ESO (46) for C1–1 are considered ωe ¼ 30;

ωe ¼ 35;

ωe ¼ 40:

Please cite this article as: Xue W, Huang Y. Performance analysis of active disturbance rejection tracking control for a class of uncertain LTI systems. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.001i

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Fig. 6. The curves of x1 ðtÞ  xn ðtÞ with different ωe for C1–1.

The curves of the position for r(t)=0 x (C1−.1) 1

0.25

x (C1−.2) 1

0.2

x (C1−.3) 1

0.15

x (C1−.4) 1

x (m) 1

0.1

x1(C2−.1)

0.05

x (C2−.2) 1

0

x1(C2−.3)

−0.05

x (C2−.4) 1

−0.1 −0.15

x1(C3−.1)

−0.2

x1(C3−.2) x (C3−.3)

−0.25

1

10

20

30

40

x1(C3−.4) 50 60 r

time(s) Fig. 7. The curves of the position for rðtÞ ¼ 0.

Fig. 8. The total disturbance and its estimation under C1–1,C1–2,‥,C3–4 for rðtÞ ¼ 0.

The corresponding simulation results are given in Fig. 6 which demonstrates that the larger ωe, the smaller error j x1 ðtÞ  xn1 ðtÞj in the whole transient process. That is because larger ωe leads to faster convergence and smaller steady estimation error of e^ 3 to its target e3 ðtÞ. Hence, the simulation results are quite accordance with the analysis result. Next, the performance of ADRC on pure disturbance rejection will be investigated by setting rðtÞ ¼ 0; t Z0 in the following simulation. Figs. 7–9 are the resulting simulation results for t A ½5; 60 s. From Fig. 7, the varying of the position x1 is always kept in ½  %5; %5 under various uncertainties of both parameters and external disturbances. Also, this strong capability on disturbance rejection of the proposed Please cite this article as: Xue W, Huang Y. Performance analysis of active disturbance rejection tracking control for a class of uncertain LTI systems. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.001i

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11

u(C1)

The control input for r(t)=0 4 2 0 −2 −4

10

20

30

40

u(C2)

2

50

60

u(Ci−1) u(Ci−2) u(Ci−3) u(Ci−4,i=1,2,3)

0 −2

10

20

30

40

50

60

10

20

30

40

50

60

u(C3)

2 0 −2

time(s) Fig. 9. The curves of the control input for rðtÞ ¼ 0.

The curves of the position and its reference trajectory under noise x (C1−.1) 1

x (C1−.2)

1

1

x (C1−.3) 1

x1(C1−.4)

0.8

x1 (m)

x (C2−.1) 1

0.6

x1(C2−.2) x (C2−.3)

0.4

1

x (C2−.4) 1

x1(C3−.1)

0.2

x1(C3−.2)

0 −0.2

x1(C3−.3) x1(C3−.4) 0

5

10 time(s)

15

x*1

20

Fig. 10. The curves of the position and the reference trajectory under noise.

Fig. 11. The total disturbance and its estimation under C1–1,C1–2,‥,C3–4 and noise.

controller is ensured by that the “total disturbance” is timely and accurately estimated by ESO and canceled by control feedback, as shown in Figs. 8 and 9. Finally, the effectiveness of ADRC in filtering with stochastic noise in measurement will be studied by assuming that the system (41) contains the random noise as follows: (

y€ ¼  a2 y a1 y_ þ dðtÞ þ bu; ym ¼ yþ n1

t Z0

ð48Þ

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The control input under noise u(C1)

5 0 −5

u(Ci−1) 15 u(Ci−2) u(Ci−3) u(Ci−4,i=1,2,3)

0

5

10

0

5

10

15

20

0

5

10

15

20

u(C2)

2

20

0 −2

u(C3)

2 0 −2 time(s) Fig. 12. The curves of the control input under noise.

Bode Diagrams of L(s) and L (s) (C1−1,2,3,4) Bode Diagrams of L(s) and L (s) (C2−1,2,3,4) 0

50

Phase (deg) Magnitude (dB)

Phase (deg)

Magnitude (dB)

0

0 −50 0 −90 −180 −270 −360

0

10

10

2

50 0 −50 0 −90 −180 −270 −360

0

10

Frequency (rad/s)

10

2

Frequency (rad/s)

Bode Diagrams of L(s) and L (s) (C3−1,2,3,4) Phase (deg) Magnitude (dB)

0

100

L(s), Ci−1 L(s), Ci−2

0

L(s), Ci−3 −100 0 −90 −180 −270 −360

L(s), Ci−4 L0(s), Ci−1,2,3,4, i=1,2,3. 0

10

10

2

Frequency (rad/s) Fig. 13. The Bode Plot for L(s) and L0 ðsÞ Under C1–1,…,C3–4.

where n1 are zero-mean normal white noise with the standard variance being 0.001. Consequently, the ADRC design for (48) becomes 8 > e^_ 1 ¼ e^ 2  β1 ðe^ 1 ðym  rÞÞ; > > > > < e^_ ¼ e^  β ðe^ ðy  rÞÞ þbuðtÞ; 2 3 2 1 m _ ¼ β ðe^  ðy  rÞÞ; > ^ > e 3 1 3 m > > > : u ¼  e^ 3 k1 e^ 2  k2 e^ 1 ;

e^ 1 ð0Þ ¼ yð0Þ  rð0Þ;

e^ 2 ð0Þ ¼ 0;

e^ 3 ð0Þ ¼ 0;

and e^ i is used to estimate ei ; i ¼ 1; 2; 3. All cases in Table 1 are tested and the corresponding results are shown in Figs. 10–12, which demonstrate that the ADRC based closed-loop system also ensures satisfied performance against certain measurement noise n1. It is important to point out that the theoretical study on the performance of ADRC based closed-loop system with stochastic noise should be an important further work.

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Table 2 ωθ =ω θ ; θm =θ m under the cases of Table 1.

C1–1 C1–2 C1–3 C1–4 C2–1 C2–2 C2–3 C2–4 C3–1 C3–2 C3–3 C3–4

j ωθ =ω θ  1j

ωθ

j θm =θ m  1j

θ m ðdegÞ

0.23 0.07 0.16 0.17 0.20 0.08 0.12 0.16 0.18 0.08 0.11 0.15

5.0 5.0 5.0 5.0 12.6 12.6 12.6 12.6 19.5 19.5 19.5 19.5

0.02 0.33 0.09 0.29 0.05 0.16 0.07 0.19 0.10 0.14 0.08 0.20

80.5 80.5 80.5 80.5 65.9 65.9 65.9 65.9 52.9 52.9 52.9 52.9

Table 3 ðωθ =ω θ ; θm =θ m Þ of C1–1 under different ωe.

j ωθ =ω θ  1j j θm =θ m  1j

ωe ¼ 30

ωe ¼ 35

ωe ¼ 40

0.23 0.17

0.20 0.14

0.18 0.12

The curves of the position and the reference trajectory under IMC (52)−(54) x1(C1−.1) x1(C1−.2)

1

x1(C1−.3) x1(C1−.4)

x1 (m)

0.8

x1(C2−.1)

0.6

x1(C2−.2) x1(C2−.3)

0.4

x1(C2−.4) x (C3−.1)

0.2

1

x (C3−.2) 1

0

x (C3−.3) 1

−0.2

x (C3−.4) 1

0

5

10 time(s)

15

x*

20

1

Fig. 14. The curves of the position and the reference trajectory under IMC (51)–(53).

Fig. 15. The total disturbance and its estimation under C1–1,C1–2,‥,C3–4 and IMC (51)–(53).

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The curves of the position and the reference trajectory under IMC (55) x1(C1−.1) x (C1−.2) 1

1

x (C1−.3) 1

x1(C1−.4)

0.8

x (C2−.1) x1(C2−.2) x (C2−.3)

1

x (m)

1

0.6

1

0.4

x (C2−.4) 1

x1(C3−.1)

0.2

x (C3−.2) 1

0

x (C3−.3)

−0.2

x1(C3−.4)

1

0

5

10 time(s)

15

x*1

20

Fig. 16. The curves of the position and the reference trajectory under IMC (51), (53)–(54).

Fig. 17. The total disturbance and its estimation under C1–1,C1–2,‥,C3–4 and IMC (51), (53)–(54).

4.2. Response in the frequency-domain The transfer function of the motion control system (41) can also be YðsÞ b ¼ PðsÞ ¼ 2 : UðsÞ s þ a1 s þ a2

ð49Þ

According to (20), ADRC (46)–(47) gives UðsÞ ¼ 

1 ðβ3 þ k1 β2 þ k2 β1 Þs2 þ ðk1 β3 þ k2 β2 Þs þ k2 β3 YðsÞ: s3 þ ðk1 þ β1 Þs2 þ sðβ2 þk1 β1 þ k2 Þ b

Thus the loop transfer function of the ADRC based closed-loop system (41), (46)–(47) is LðsÞ ¼

b ðβ3 þ k1 β2 þ k2 β1 Þs2 þ ðk1 β3 þ k2 β2 Þs þ k2 β3 : s3 þ ðk1 þ β1 Þs2 þ sðβ2 þ k1 β1 þ k2 Þ b

Also, construct L0 ðsÞ ¼ ðb=bÞω2e =s2 þ 2ωe s to compute ω θ and θ m by (26)–(27). To verify the relationship between ðωθ ; θm Þ and ðω θ ; θ m Þ shown in Theorem 2, the 12 cases of ða1 ; a2 ; bÞ in Table 1 will again be tested and Fig. 13 is their Bode plots. C1–1,2,3,4 share b ¼0.5, thus they have different L(s) but the same L0 ðsÞ. Similarly, C2–1,2,3,4 share b¼1.3 and C3–1,2,3,4 share b¼ 2.1. Obviously, the plants with Table 1 can be classified as three groups by different b, that is C1–1,2,3,4, C2–1,2,3,4 and C3–1,2,3,4. Table 2 lists the ωθ =ω θ ; ω θ ; θm =θ m and θ m under the cases in Table 1. From Fig. 13 and Table 2, there is 1. ðωθ ; θm Þ belonging to each group are almost the same although ða1 ; a2 Þ are different. This result indicates that the performance of the closed-loop system is robust to the uncertain ða1 ; a2 Þ, which is an outstanding advantage of ADRC. Additionally, the larger the b=b is, the larger the ωθ and the smaller the θm. Thus, the information on b will be helpful for increasing robustness of ADRC. 2. ðωθ ; θm Þ is evidently different for different b even ða1 ; a2 Þ are the same. Thus, although the time-domain responses in Fig. 3 show that the variations of both ða1 ; a2 Þ and b hardly change the dynamic performance of the output, the frequency-domain responses demonstrate that the stability margin of the system is more sensitive to b than ða1 ; a2 Þ. This also indicates that the investigation of ADRC from the views of both time-domain and frequency-domain is necessary. The simulation on the relationship between ωe and j ωθ =ω θ  1j , and the relationship between ωe and j θm =θ m  1j is carried out as follows. Using the parameters of C1–1, the following sets of bandwidth for ESO Please cite this article as: Xue W, Huang Y. Performance analysis of active disturbance rejection tracking control for a class of uncertain LTI systems. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.001i

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15

(46) are considered: ωe ¼ 30;

ωe ¼ 35;

ωe ¼ 40:

The results are given in Table 3, which illustrates that the larger the ωe, the smaller the error j ωθ =ω θ  1j and j θm =θ m  1j . 4.3. Comparison with IMC/DOBC The discussion of Section 3.3 demonstrates that ADRC and IMC/DOBC have the similar structure of estimating and canceling uncertainties. Moreover, our theoretical results provide a practical and safe choice (39) for IMC/DOBC to deal with the general uncertain system (1). Next, the simulation of the comparison between ADRC and IMC/DOBC will be carried out for the example (41). Note that IMC and DOBC can be equivalently transformed by (37)–(38), we only need to consider IMC. Firstly, the transfer function form of (44) equals to E1 ðsÞ ¼ P I ðsÞðUðsÞ þ E3 ðsÞ=bÞ;

b ¼1

ð50Þ

where P I ðsÞ ¼ b=s is the nominal model of the system (41). According to [35], IMC control usually equals to 2

^ I ðsÞ ¼ Q ðsÞðE1 ðsÞ  P I ðsÞU I ðsÞÞ D I

ð51Þ

as the estimation of E3 ðsÞ, where QI(s) is a proper and stable transfer function to be designed. Since the existing tuning methods for QI(s) are mainly proposed under the condition that the nominal model is stable [12,35], they are probably not applicable to the system (41) with its nominal model being P I ðsÞ ¼ b=s2 . Next, the method (39) suggested in Section 3.3 will be used to design QI(s), that is Q I ðsÞ ¼ s2 ω3e =ðs þ ωe Þ3 :

ð52Þ

Consequently, (51)–(52) based IMC is ^ I ðsÞ  k1 E2 ðsÞ  k2 E1 ðsÞÞ=b; UðsÞ ¼ ð  D

k1 ¼ 4; k2 ¼ 4:

ð53Þ

where E1 ðsÞ and E2 ðsÞ, or e1 ðtÞ and e2 ðtÞ, are assumed to be known in IMC design. The corresponding responses of the position are given in Fig. 14 which shows that the IMC (51)–(53) is also capable of recovering the target transient process of xni despite both parametric uncertainties and external disturbances. In addition, Fig. 15 demonstrates that the estimation (51) in IMC can track the “total disturbance” well if IMC is designed according to the method (39), which is suggested from the analysis results of ADRC. It is important to point out that the choice of QI(s) in IMC is crucial for the performance of the resulting closed-loop system. As a comparison, Figs. 16 and 17 are the simulation results in the case that QI(s) being designed as Q I ðsÞ ¼

sn ωne þ 1

5

ðs þωe Þn þ 1 s þ 5

:

ð54Þ

These figures evidently demonstrate that the tracking performance as well as the estimation accuracy of “total disturbance” is largely deteriorated although the closed-loop system still has stability. Therefore, according to the theoretical analysis and the simulation results of this paper, the design method (39) proposed by our paper can be used as a practical and valid scheme for IMC to deal with general uncertain systems.

5. Conclusions This paper discusses the ADRC based tracking controller for a class of LTI systems with both uncertain parameters and external disturbances. The rigorous analysis comprehensively evaluates the resulting control system's performance in both time-domain and frequency-domain. It is shown that the parameters ki ð1 r i rnÞ in ADRC controller can be used to shape the desired trajectory of the closed-loop output, and the ESO's bandwidth can be used to reduce the error between the output and its desired trajectory in the entire transient process despite uncertain dynamics and external disturbances. Furthermore, the size of uncertainty on the control input gain b, which can be permitted by the ADRC, is much larger than that in the previous work. The phase margin as well as the crossover frequency of the ADRC based control system is shown to be insensitive to the variations of parameters except the control input gain b, which has major influence on the frequency characteristics of ADRC based control system. In addition, the relationship between ωe and the time delay tolerated by the control system is quantitatively discussed. The consistent performance and strong robustness of ADRC based closedloop system are verified through a motion control system which has uncertain parameters and sinusoidal load disturbances.

Acknowledgment The authors would like to thank Professor Zhiqiang Gao for his helpful suggestions.

Appendix A Firstly, introduce the following transformation for the ESO (8): 2 3 2 3 21 3 1 … 0 ξ1 e1  e^ 1 ωne 6 7 6 7 6 7 1 6 ξ2 7 6 e2  e^ 2 7 6 7 7 ¼ T  16 7; T ¼ 6 0 ωne  1 … 0 7: ξ¼6 6 ⋮ 7 6 7 6 7 ⋮ 4 5 4 5 4⋮ ⋮ … ⋮5 ξn þ 1 en þ 1  e^ n þ 1 0 0 … 1 Please cite this article as: Xue W, Huang Y. Performance analysis of active disturbance rejection tracking control for a class of uncertain LTI systems. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.001i

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16

Hence, the ADRC based closed-loop system composed of (1), (8) and (14) satisfies that 8 t A ½t 0 ; t u Þ; #     " B11 d3 ðtÞ z z_ ¼ A1 ; t o tu þ B12 Δ1 ðz; e; d3 ðtÞÞ e e_ ξ_ ¼ ωe A 2 ξ þ B2 Δ2 ðÞ;

t o tu

ð56Þ

#   " B11 d3 ðtÞ z ¼ A1 ; þ B12 K Te Tξ e e_

  z_

ξ_ ¼ ωe A2 ξ þ B2 Δ3 ðÞ;

ð57Þ

t Z tu

t Z tu

ð58Þ

where

2 "

e ¼ ½e1 ⋯ en  1 en T ; 2 3 0 6⋮7 6 7 B11 ¼ 6 7 405 1 " A1 ¼

A1 ¼ 2

;

Az

B11 Cpm þ 1

0

Ae

…

0 6 6 Ae ¼ 6 40

1 0

…

0

0

…

m 1

Az

 B11 Cpm þ 1

0

Ae

# ;

0

7 7 7 15

…

 ϕ1

6 ϕ 2 6 6 … A2 ¼ 6 6 6 ϕ 4 n  ϕn þ 1

1

0

…

0

1

…

0

0

…

0

0

…

;

0 n n 2 3 0 6⋮7 6 7 B12 ¼ 6 7 ; 405 0

3

07 7 7 7; 7 17 5 0

;

6 6 Az ¼ 6 6 4

0 n 1 2 3 0 6⋮7 6 7 B2 ¼ 6 7 405

n 1

2

3 kn 6 7 6 kn  1 7 7 Ke ¼ 6 6 ⋮ 7; 4 5 k0

ðn þ 1Þ! b b ϕi ¼ ; b~ ¼ ; i!ðn þ 1  iÞ! b m n X X z i qm þ 1  i  ei an þ 1  i þ d4 ðtÞ; Δ1 ðÞ ¼ i¼1

#

2 3T 1 6⋮7 6 7 C¼6 7 405

3

1 2

ð55Þ

;

0

…

1

… 0

0

…

 qm

 qm  1

…

0

3

7 7 7; 1 7 5  q1

d3 ðtÞ ¼ d1 ðtÞ  pm þ 1 rðtÞ; 2

0

6 6 Ae ¼ 6 0 4 0  kn  kn  1 1 ðn þ 1Þ 1 2  ϕ1 1 6 0  ϕ2 6 6 … k0 ¼ 1; A2 ¼ 6 6 6 0  ϕn 4 ϕ ðb~ þ 1Þ 0 ;

nþ1

d4 ðtÞ ¼  r ðnÞ ðtÞ 

i¼1

n X

… …

1

0

3

7 7 7; 1 5 …  k1 3 0 … 0 1 … 07 7 7 7; 7 0 … 17 5 0 … 0 …

ai r ðn  iÞ ðtÞ þ d2 ðtÞ;

i¼1

Δ2 ðÞ ¼ aT ðA e e þ B12 Δ1 Þ þ qT ðAz z þ B11 ðd3  pm þ 1 e1 ÞÞ þ d_ 4 ; Δ3 ðÞ ¼ aT Ae e  b~

n X

ki en þ 2  i k1

i¼2

n X

! ki en þ 1  i  b~ k1 K Te Tξ þ

i¼1

n X ϕ ξ1 ξi þ 1 i

i¼1

ωne  i

! kn þ 1  i

þaT B12 K Te Tξ þ qT ðAz z þB11 ðd3  pm þ 1 e1 ÞÞ þ d_ 4 : It is easily verified that Δ1 ðÞ, Δ2 ðÞ and Δ3 ðÞ satisfy that j Δ1 ðÞj r γ 1 þ ð J e J þ J z J Þγ 2 ; j Δ2 ðÞj r γ 3 þ ð J e J þ J z J Þγ 4 ;



j Δ3 ðÞj r γ 5 þ ð J e J þ J z J þ J ξ J Þγ 6 1=ωe

ð59Þ

where γ i 4 0; i ¼ 1; …; 5, are positives and γ 6 ðÞ is a polynomial of 1=ωe . Next, the properties of A1 , A 2 and A2 will be shown. Note that Az and Ae are Hurwitz, then A1 is Hurwitz, thus there exists positive T matrix P1 such that AT1 P 1 þ P 1 A1 ¼  I. Since A 2 is Hurwitz, there exists positive matrix P 2 such that A 2 P 2 þ P 2 A 2 ¼  I. Lemma 1. There exist a constant matrix P T2 ¼ P 2 40 and a constant c0 4 0 such that ~ T P þ P A ðbÞ ~ r c P ; A2 ðbÞ 2 2 2 0 2

~ A ½α ; α : 8 bðÞ 3 4

ð60Þ

Proof of Lemma 1. Define Q 1 ðsÞ ¼ C T ðsI A 2 Þ  1 B, then Q 1 ðjωÞ ¼ cos n þ 1 θð cos ððn þ1ÞθÞ j sin ððn þ1ÞθÞÞ;

θ ¼ a tan ðωÞ;



where ωA ð  1; 1Þ. Next, proof of that the Nyquist plot of Q 1 ðjωÞ in the complex plane lies in the closed disk D  n=ðn þ 2Þ; 1 will be given, whose diameter is the line segment connecting the points  n=ðn þ 2Þ þ j0 and 1 þ j0. Due to that the plots of Q 1 ðjωÞ and

D n=ðn þ 2Þ; 1 are both symmetric with respect to the x-axis, it is sufficient to consider ωA ½0; 1Þ, i.e., θ A ½0; π=2Þ. Please cite this article as: Xue W, Huang Y. Performance analysis of active disturbance rejection tracking control for a class of uncertain LTI systems. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.001i

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17

Using the fact that 8 θ A ½0; π=2Þ, 8 > 1  cos 2ðn þ 1Þ θ 1 > > ¼ ; > > 1 nþ1 nþ3 n 1  cos n þ 1 θ cos ððn þ 1ÞθÞ > cos θ  cos θX > i sin ððn iÞθÞ > < 1þ cos θ sin θ 1  cos 2ðn þ 1Þ θ i¼0   >   >   n  1 > n þ 1 n þ 3 X > θ  cos θ  1 sin ððn  iÞθÞ  nðn þ 1Þ  >  cos > ; cos i θ r >  r n þ1 ;  >  : sin θ 2 1  cos 2ðn þ 1Þ θ i¼0 it can be safely concluded that 1 1 1  cos 2ðn þ 1Þ θ r n þ 2 2 1  cos n þ 1 θ cos ððn þ 1ÞθÞ

ð61Þ

which is equivalent to

2

1 nþ1 2 þ cos 2ðn þ 1Þ θ sin 2 ððn þ 1ÞθÞ r cos n þ 1 θ cos ððn þ 1ÞθÞ  nþ2 nþ2

ð62Þ

Thus the circle criterion means that Q 2 ðsÞ 9 ð1 þ α4 Q 1 ðsÞÞ=ð1 þ α3 Q 1 ðsÞÞ is strictly positive real for ½α3 ; α4   ð  1; ðn þ 2Þ=nÞ. Since Q 2 ðsÞ has the following state-space realization: ( ζ_ ¼ ðA 2  α4 BC T Þζ þ Bu n y n ¼ ðα3  α4 ÞC T ζ þ u n ; according to the Kalman–Yakubovich–Popov theorem, there exist P T2 ¼ P 2 40; c0 4 0 and L such that ( ðA 2 α4 BC T ÞT P 2 þ P 2 ðA 2  α4 BC T Þ ¼  LLT  c0 P 2 pffiffiffi P 2 B ¼ ðα3  α4 ÞC  2L

ð63Þ

ð64Þ

~ with b~ A ½α ; α  satisfies Consequently, it is safe to conclude that A2 ðbÞ 3 4 ~ T ÞT P þ P ðA  bBC ~ TÞ AT2 P 2 þ P 2 A2 ¼ ðA 2  bBC 2 2 2 pffiffiffi T T T T ~ ~ ¼  LL  c0 P 2  2ðα4  bÞðα 4  α3 ÞCC þ 2ðα4  bÞðLC þ CL Þ pffiffiffi p ffiffiffi ~  LÞT ð 2ðα  bÞC ~  LÞ r  c P :□ r  c0 P 2  ð 2ðα4  bÞC 4 0 2

ð65Þ

Proof of Theorem 1. Step 1: t A ½t 0 ; t u Þ. The system (55) is a linear time invariant system with bounded time-varying input d1 ðtÞ, thus for 8 ωn 40 there exists ρn1 4 0 such that 8 ωe A ½ωn ; 1Þ J ½zðtÞT ; eðtÞT  J r ρn1 ;

t A ½t 0 ; t u Þ ¼ ½t 0 ; t 0 þ μðωe ÞÞ

ð66Þ n

where μðωe Þ ¼ 2ðn 1Þ J P 2 J maxflnðρ4 ωe Þ=ωe ; 0g. Moveover, μðωe Þ is finite means that there exists ρ~ 1 such that (55) satisfies J eðtÞ  eðt 0 Þ J r ρ~ n1 μðωe Þ;

8 t A ½t 0 ; t u Þ

ð67Þ

n

Since e_ ¼ Ae en and Ae is Hurwitz, there is 8 t A ½t 0 ; t u Þ, J en ðtÞ J r ρn2 ;

ð68Þ

and J en ðtÞ  en ðt 0 Þ J r ρn2 J Ae J μðωe Þ;

ð69Þ

n

n

where ρ2 is a positive. Combination of (67)–(69) and eðt 0 Þ ¼ e ðt 0 Þ yields 8 t A ½t 0 ; t u Þ J eðtÞ  en ðtÞ J r ρn3 μðωe Þ; n

n

ð70Þ

n

where ρ3 ¼ ρ~ 1 þ ρ2 J Ae J . Next, the properties of ξðtÞ; t A ½t 0 ; t u Þ, will be discussed. By using en þ 1 ðt 0 Þ ¼ aT eðt 0 Þ þqT zðt 0 Þ þ ðb  bÞuðt 0 Þ þ d4 ðtÞ;

uðt 0 Þ ¼ 0;

ð71Þ

and (A1)–(A3), there is j en þ 1 ðt 0 Þj rα2 ρ0 þ ðnα2 þ 1Þα5 þ α1 P where ρ0 ¼ mρ1 þ ρ2 þ ρ3 þ ni¼ 2 j x i ðt 0 Þ  r ði  1Þ ðt 0 Þj þ ðn  1Þρ4 . Thus,  T J ξðt 0 Þ J r J T  1 e1 ðt 0 Þ  e^ 1 ðt 0 Þ; e2 ðt 0 Þ  e^ 2 ðt 0 Þ; …en þ 1 ðt 0 Þ  e^ n þ 1 ðt 0 Þ J rmaxfωe ; ωne  1 gðn 1Þρ4 þ α2 ρ0 þ ðnα2 þ 1Þα5 þ α1 : Define V 2 ¼ ξT P 2 ξ and let c 21 and c 22 are the maximal and minimal eigenvalues of P 2 respectively, then there is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi V 2 ðξðt 0 ÞÞ r c 22 maxfωe ; ωne  1 gðn  1Þρ4 þρn4 pffiffiffiffiffiffiffi where ρn4 ¼ c 22 ðα2 ρ0 þ ðnα2 þ 1Þα5 þα1 Þ.

ð72Þ

ð73Þ

ð74Þ

Please cite this article as: Xue W, Huang Y. Performance analysis of active disturbance rejection tracking control for a class of uncertain LTI systems. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.001i

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18

Using Lemma 1 and (56), there is V_ 2 ðξÞ r  ωe J ξ J 2 þ 2ðγ 3 þ γ 4 ρn1 Þ J P 2 B2 J J ξ J :

ð75Þ

It follows that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ωe J ξðtÞ J 2 2ðγ 3 þ γ 4 ρn1 Þ J P 2 B2 J qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 2 ðξðtÞÞ r  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ dt 2 V 2 ðξðtÞÞ 2 V 2 ðξðtÞÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðγ þ γ 4 ρn1 Þ J P 2 B2 J ωe pffiffiffiffiffiffiffi V 2 ðξðtÞÞ þ 3 r : 2c 22 c 21

ð76Þ

Applying the Gronwall–Bellman Inequality means that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c 22 ðγ 3 þ γ 4 ρn1 Þ J P 2 B2 J pffiffiffiffiffiffiffi þ V 2 ðξðt 0 ÞÞe  ðωe =2c 22 Þðt  t0 Þ : V 2 ðξðtÞÞ r ωe c 21

ð77Þ

Combining t u  t 0 ¼ 2ðn  1Þc 22 μ1 ðωe Þ

ð78Þ

and (77), it can be concluded that qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lim V 2 ðξðtÞÞ r ρn5

ð79Þ

t-t u

where 2c 22 ðγ 3 þ γ 4 ρ1 Þ J P 2 B2 J pffiffiffiffiffiffiffi ρ5 ¼ þ ωe c 21 n

pffiffiffiffiffiffiffi c 22 maxfωe ; ωne  1 gðn  1Þρ4 þ ρn4

maxfωe ρ4 ; 1gn  1 ( ) 2c 22 ðγ 3 þ γ 4 ρ1 Þ J P 2 B2 J pffiffiffiffiffiffiffi 1 1 pffiffiffiffiffiffiffi r ; ; 1 þ ρn4 : þ c 22 ðn  1Þ max ðωe ρ4 Þn  2 ρn4  2 ωe c 21

ð80Þ

Step 2: t A ½t u ; 1Þ. Firstly, the initial condition of the closed-loop system (57)–(58) on t A ½t u ; t 1 Þ, i.e., ðzðt u Þ; eðt u Þ; ξðt u ÞÞ is required to be analyzed. Using the continuous of zðtÞ; eðtÞ at tu, there is J ½eT ðt u Þ; zT ðt u Þ J r ρn1 :

ð81Þ

Note ξi ðt u Þ; i o n þ 1 are continuous at tu. Therefore, vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n uX ρn5 V 2 ðξðtÞÞ t ξi ðt u Þ2 r lim pffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffi : t-t u c 21 c 21 i¼1

ð82Þ

Rewrite j ξn þ 1 ðt u Þj as j ξn þ 1 ðt u Þj ¼ j en þ 1 ðt u Þ  e^ n þ 1 ðt u Þj ¼ j aT eðt u Þ þ qT zðt u Þ þ d4 ðt u Þj þ b~ j

n X

ki en þ 1  i ðt u Þ þ

i¼1

n X

ki

i¼1

ξn þ 1  i ðt u Þ ωie

þ e^ n þ 1 ðt u Þj þ j e^ n þ 1 ðt u Þj :

ð83Þ

Since

    j e^ n þ 1 ðt u Þj ¼  lim e^ n þ 1 ðtÞ ¼ j aT eðt u Þ þqT zðt u Þ þ d4 ðt u Þj þ j ξn þ 1 ðt u Þj t-t u ρn5 rα2 ρn1 þ ðnα2 þ 1Þα5 þ α1 þ pffiffiffiffiffiffiffi ; c 21

ð84Þ

there is

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi un þ 1 uX ξi ðt u Þ2 r ρn6 J ξðt u Þ J r t i¼1

where ρn5 ρn6 ¼ ð2 þ α4 Þðα2 ρn1 þ ðnα2 þ1Þα5 þ α1 Þ þ ð1 þ α4 Þpffiffiffiffiffiffiffi c 21  !   n ρn5 1 1 þ α4 max ki ρ1 þ pffiffiffiffiffiffiffi max ; : ωe ωne c 21

Please cite this article as: Xue W, Huang Y. Performance analysis of active disturbance rejection tracking control for a class of uncertain LTI systems. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.001i

ð85Þ

W. Xue, Y. Huang / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

19

Next, the properties of ðzðtÞ; eðtÞ; ξðtÞÞ; t A ½t u ; 1Þ will be studied. Define  T   z z V ¼ V 1 ðz; eÞ þV 2 ðξÞ; V 1 ðz; eÞ 9 P1 ; V 2 ðξÞ 9 ξT P 2 ξ: e e Let c11 and c12 be the maximal and minimal eigenvalues of P 1 respectively. Let c21 and c22 be the maximal and minimal eigenvalues of P 2 respectively. Since c11 and c12 are continuous with respect to pi, thus cn11 9 min j pi j r α2 c11 and cn12 9 max j pi j r α2 c12 exist. Consequently,   1 V_ r  n V 1 ðz; eÞ  c0 ωe V 2 ðξÞ þ 2 J zT ; eT J J P 1 B1 K Te T J J ξ J c12 þ2 J ξ J J P 2 B2 J ðγ 5 ð J ½zT ; eT  J þ J ξ J Þ þ γ 6 Þ:

ð86Þ

n

n

Thus there exist positives ω ; γ 7 ; γ 8 such that 8 ωe Z ω , pffiffiffiffiffiffiffiffiffi V_ ðtÞ r  γ 7 VðtÞ þ γ 8 V ðtÞ: Consequently, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2γ pffiffiffiffiffiffiffiffiffi VðtÞ r e  ðγ 7 =2Þðt  t u Þ V 1 ðxðt u Þ; zðt u ÞÞ þ V 2 ðξðt u ÞÞ þ 8 : γ7 Combination of (81) and (85) shows that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffi n pffiffiffiffiffiffiffi n V 1 ðeðt u Þ; zðt u ÞÞ þV 2 ðξðt u ÞÞ r cn12 ρ1 þ c22 ρ6 :

ð87Þ

ð88Þ

ð89Þ

Therefore, qffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2γ VðtÞ r ρn7 9 cn12 ρn1 þ c22 ρn6 þ 8 : γ7

ð90Þ

Next, the dynamics of ξðtÞ will be considered. Using (58), there is V_ 2 ðξðtÞÞ r  c0 ωe V 2 ðξðtÞÞ þ2 J P 2 B2 J J ξ J ðγ 5 ð J x J þ J z J þ J ξ J Þ þ γ 6 Þ ! ! pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρn7 ρn7 V 2 ðξðtÞ ffi þ γ 5 pffiffiffiffiffiffi þ γ r  c0 ωe V 2 ðξðtÞÞ þ 2 J P 2 B2 J pffiffiffiffiffiffiffi p ffiffiffiffiffiffi ffi 6 c21 c21 cn11

ð91Þ

which leads to pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρn6 ρn ffiþ 8; V 2 ðξðtÞÞ r e  c0 ωe ðt  tu Þ=2 pffiffiffiffiffiffi c21 ωe pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi n where ρ8 ¼ ð2=c0 c21 Þ J P 2 B2 J ðγ 5 ðρn7 = cn11 þ ρn7 = c21 Þ þγ 6 Þ. n Next, the dynamics of vðtÞ 9 eðtÞ  e ðtÞ will be deduced. By using (5) and (57), there is _ ¼ Ae vðtÞ þ B1 K Te TξðtÞ; vðtÞ

ð92Þ

J vðt u Þ J r ρ2 μðωe Þ:

ð93Þ

eAe ðt  τÞ B12 K Te TξðτÞ dτ:

ð94Þ

Hence, vðtÞ ¼ eAe ðt  tu Þ vðt u Þ þ

Z

t

tu

Define the following Lyapunov function for (93): V v ¼ vT P 3 v;

ð95Þ ATe P 3 þ P 3 Ae

¼  I. Consequently, pffiffiffiffiffiffi 1 Vv V_ v ¼  J v J 2 þ 2vT P 1 B12 K Te TξðtÞ r  V v þ 2pffiffiffiffiffiffiffi J P 1 B12 K Te J J TξðtÞ J c32 c31 where P3 is the unique positive solution of

where c32 and c31 are the maximal and minimal eigenvalues of P 3 respectively. Before the bound of J vðtÞ J is given, the convergence of J ξðtÞ J will be discussed. Define   ln ωe ;0 : t 1 ¼ t u þ 2c0 max ωe According to (92), there is 8 t Z t 1 n

ρ6 ρn8 1 J ξðtÞ J r þ pffiffiffiffiffiffi ffi : c21 c21 ωe

ð96Þ

ð97Þ

ð98Þ

Thus, pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi Z V v ðtÞ r eð1=2c32 Þðt  tu Þ V v ðt u Þ þ Z þ

t1 1

eð1=2c32 Þðt  τÞ

t1 tu

eð1=2c32 Þðt  τÞ

J P 1 B12 K Te J J TξðτÞ J dτ pffiffiffiffiffiffiffi c31

J P 1 B12 K Te J J TξðτÞ J dτ: pffiffiffiffiffiffiffi c31

ð99Þ

Please cite this article as: Xue W, Huang Y. Performance analysis of active disturbance rejection tracking control for a class of uncertain LTI systems. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.001i

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Since J vðt u Þ J r ρn3 μðωe Þ, there is pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi eð1=2c32 Þðt  t u Þ V v ðt u Þ r c32 ρn3 μðωe Þ:

ð100Þ

By using (90), it can be concluded that Z

1 t1

n

ρ6 eð1=2c32 Þðt  τÞ 1 2c32 1 pffiffiffiffiffiffiffi J P 1 B12 K Te J J TξðτÞ J dτ r pffiffiffiffiffiffiffi J P 1 B12 K Te J J T J þ pffiffiffiffiffiffiffiρn8 : ωe c31 c21 c31 c21

ð101Þ

According to (97) and (90), there is   Z t1 ð1=2c32 Þðt  τÞ e J P 1 B12 K Te J J T J ρn7 ln ωe pffiffiffiffiffiffiffi J P 1 B12 K Te J J TξðτÞ J dτ r pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2c0 max ;0 : ωe c31 c21 c31 tu

ð102Þ

Combination of (99)–(104) shows that there exists γ n1 such that (16) holds, i.e., J vðtÞ J rγ n1

maxflnðρ4 ωe Þ; ln ωe ; 1g : ωe

ð103Þ

Thus, (17) is satisfied by setting   n

ρ6 ρn8 1 þ pffiffiffiffiffiffi γ n2 ¼ 2ðn 1Þ J P 2 J þ 2c0 ; γ n3 ¼ ffi max n; 1 : n c21 c21 ðω Þ

ð104Þ

Since for t Z t u ; P P e^ n þ 1 þ ni¼ 1 ki e^ n þ 1  i en þ 1 þ ðe^ n þ 1  en þ 1 Þ þ ni¼ 1 ki ðen þ 1  i þ ðe^ n þ 1  i  en þ 1  i ÞÞ ¼ u¼  b b and en þ 1 ¼ aT e þ qT z 

n X

ai r ðn  iÞ ðtÞ  r ðnÞ ðtÞ þ d2 ðtÞ þ ðb bÞu;

i¼1

u equals to u¼ 

aT e þ qT z 

Pn

i¼1

ai r ðn  iÞ ðtÞ  r ðnÞ ðtÞ þ d2 ðtÞ þ ðe^ n þ 1  en þ 1 Þ þ b

Pn

i¼1

kn þ 1  i ðei þ ðe^ i  ei ÞÞ

Note that (A2)–(A4) and (16)–(17), then there exists γ n3 such that (18) is meet.□ Appendix B Proof of Theorem 2. According to (22), there is ðNðjω θ Þ þ Hðjω θ Þpm þ 1 =Mðjω θ ÞÞC 1 ðjω θ Þ L0 ðjω θ Þ ωne þ 1 ¼ C 2 ðjω θ Þ Lðjω θ Þ ðjω θ þ ωe Þn þ 1 ωne þ 1 ! Pi  ðq  1Þ 0 Pn q ¼ 1 kq ϕi  q ωe pm þ 1 Pm qi ðjσ Þ  i ðjσ Þn þ 1  i i ¼ 0 ϕi þ B  i i¼1 ω X e 1 n ai ðjσ Þ ðjσ ωe Þn ωie B þ ¼ Pn ! B1 þ Pn  i  ðq  1Þ nþ1i i i¼1 @ ωe P pi ðjσ Þ  i Pn i ¼ 0 ϕi ðjσ Þ q ¼ 1 ki þ q ϕn þ 1  q ωe 1þ m ðjσ ωe Þ  i i¼1 i ¼ 0 ki þ ωie ωe

ðψ 1 þ ψ 3 =ωe Þ ψ6 1 1þ ψ4 þ ¼ ¼ 1 þ ψ 7 =ωe ; ψ 1 ð1 þ ψ 2 =ωe Þ 1 þ ψ 5 =ωe ωe

1 C C C A

where 8 n X > > > ϕi ðjσ Þn þ 1  i ; > ψ1 ¼ > > > i ¼ 0 > > ! > > n n ni X X > 1 X > > > ψ2 ¼ kq ϕn  q ωe ðq  1Þ þ ki þ ki þ q ϕn  q ωe ðq  1Þ ðjσ Þ  i ωe ði  1Þ ; > > ωe q ¼ 1 < q¼1 i¼1 ! n i n m X X X X > pi ðjσ Þ  i > > > kq ϕi  q ωe ðq  1Þ ðjσ Þn þ 1  i ; ψ 4 ¼ ai ðjσ Þ  i ωe ði  1Þ ; ψ 5 ¼ ; ψ3 ¼ > > ωie 1 > > i¼0 q¼1 i¼1 i¼1 > > >

m > > pm þ 1 X qi ðjσ Þ  i ðψ 1 þ ψ 3 =ωe Þ ψ6 ψ 3 þ ψ 1ψ 2 > > ψ : ψ ¼ ; ψ ¼ þ þ > 6 7 4 n > : ψ ð1 þ ψ =ω Þ 1 þ ψ =ω ψ ð1 þ ψ =ω Þ ðjσ ω Þ ωi  1 e

i¼1

e

1

2

e

5

e

1

2

e

(26)leads to j ψ 1 j ¼ b=b. Since (A1), there is MðjωÞ a0 which means ð1 þ ψ 5 =ωe Þ ¼ MðjωÞωe m ðjσ ωe Þ  m a 0; Moreover, lim

n ωe -1 C 2 ðjωÞ=ωe

8 ωe 4 0: Pn ¼ ðjωÞ þ i ¼ 1 ki ðjωÞn  i means that there exists ωn1 such that 8 ωe Z ωn1 , n

ð1 þ ψ 2 =ωe Þ ¼ C 2 ðjωÞωe n ðjσ ωe Þ  n a 0: Please cite this article as: Xue W, Huang Y. Performance analysis of active disturbance rejection tracking control for a class of uncertain LTI systems. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.001i

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Since ψ 2 ; ψ 4 ; ψ 3 and ψ6 are the polynomials of 1=ωe with all coefficients are bounded under (A1)–(A5), there exists constant γ 1 such that for 8 ωe Z ωn1 , j ψ 7 j oγ 1   which leads to L0 ðjω θ Þ=Lðjω θ Þ  1 o γ 1 n =ωe . Next, the property of ωθ =ωe  ω θ =ωe will be studied. Let σ ¼ ωθ =ωe , then       ! Pn  i  ðq  1Þ   k ϕ ω  Pn  q ¼ 1 iþq nþ1q e  i  ki þ ðjσωe Þ  ωe    b i ¼ 0 1  1 ¼ Lðjσωe Þ ¼  !  Pi  ðq  1Þ b P  pm þ 1 Pm qi ðjσÞ  i k ϕ ω  q e iq q¼1  n ðjσÞn þ 1  i  i  i¼1 i ¼ 0 ϕi þ Pn ai ðjσÞ ðjσωe Þn ωie ωe   þ 1 þ i ¼ 1 ωi P pi ðjσÞ  i  e 1þ m  i¼1 ωie

                 

ð105Þ

The following text will prove that there exist ωn2 Z ωn1 and γ 2 4 0 such that 8 ωe Zωn2 , σ r γ 2:

ð106Þ

The methods of Reduction ad absurdum is used. Assume there exists fωe;l g1 l ¼ 1 satisfying liml-1 ωe;l ¼ 1 such that σ l 9 ωθ =ωe;l . There is       ! P  ðq  1Þ ni    Pn  q ¼ 1 ki þ q ϕn þ 1  q ωe;l  ðjσ l ωe;l Þ  i  i ¼ 0 ki þ  ωe   1 b  lim Lðjσ l ωe;l Þ ¼ lim   ! Pi  ðq  1Þ  l-1 l-1b pm þ 1 Pm qi ðjσ l Þ  i k ϕ ω  Pn q ¼ 1 q i  q e;l   nþ1i ϕ þ Þ ðjσ n   i¼1 l i i¼0 ωie;l   Pn ai ðjσ l Þ  i ðjσ l ωe;l Þ ωe;l þ 1 þ i ¼ 1  ωie;l P pi ðjσ l Þ  i  1þ m i¼1  ωi e;l

liml-1 σ l ¼ 1 where           ¼0         

ð107Þ

which leads to a contradiction to (105). Thus the Eq. (106) holds. Next, the following equation will be proven: lim σωe ¼ 1:

ð108Þ

ωe -1

The methods of Reduction ad absurdum is again used. Assume there exist γ 2 and fωe;l g1 satisfying liml-1 ωe;l ¼ 1 such Pn Pn l ¼ 1 Pn  i q i i that liml-1 σ l ωe;l r γ 2 where σ l 9 ωθ =ωe;l . Note that a 0, liml-1 i ¼ 0 ki ðjσ l ωe;l Þ i ¼ 0 ðki þ q ¼ 1 ki þ q ϕn þ 1  q ωe;l Þðjσ l ωe;l Þ = Pn i ¼ 1 and liml-1 σ l r liml-1 γ 2 =ωe;l ¼ 0, then there is i ¼ 0 ki ðjσ l ωe;l Þ                P  Pn  i  q n  i       b i ¼ 0 ki þ q ¼ 1 ki þ q ϕn þ 1  q ωe;l ðjσ l ωe;l Þ  1   lim Lðjσ l ωe;l Þ ¼ lim  ð109Þ  ¼1 Pi Pn  P  q p q n þ 1  i   m mþ1 i l-1 l-1b ðϕ þ k ϕ ω Þðjσ Þ  q l i i  q e;l i¼0 q¼1   n i¼1 i  ðjσ l ωe;l Þ ðjσ l ωe;l Þ  ai 1 þ Pn þ i¼1   Pm pi ðjσ l ωe;l Þi   1þ i ¼ 1   i ðjσ ω Þ l

which leads to a contradiction to (105). Thus the Eq. (108) holds. Taking the limit on the right-hand of (105) and using (108), there is         P  Pn  i q n  i   ðjσωe Þ     b i ¼ 0 ki þ q ¼ 1 ki þ q ϕn þ 1  q ωe 1    lim Lðjσωe Þ ¼ lim  Pi  pm þ 1 Pm qi q nþ1i ωe -1 ωe -1b Pn   ðjσÞ ϕ þ k ϕ ω q e i iq i¼0 q¼1 i¼1  ðjσωe Þn ai ðjσωe Þi 1 þ Pn þ  i¼1 P pi  ðjσωe Þi 1þ m  i¼1 ðjσωe Þi     b     b : ¼ P

 n  i  ni ¼ 01 ϕi jlimωe -1 σ   

e;l

               

ð110Þ

Please cite this article as: Xue W, Huang Y. Performance analysis of active disturbance rejection tracking control for a class of uncertain LTI systems. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.001i

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Consequently,     b     b P  ¼ 1:  n n þ 1  i  i ¼ 0 ϕi ðjlimωe -1 σÞ    Note that     b     b P ¼1  n n þ 1  i  i ¼ 0 ϕi ðjσ Þ    and σ can be viewed as a function being continuous with respect to ωe, then for 8 ωe A ½ωn ; ωn2  or 8 ωe A ½ωn2 ; ωn , σ is bounded and limωe -1 σ ¼ σ . Using the definition of σ , σ is the positive independent of ωe. Therefore, ( γ n1 ðωe Þ such that for 8 ωe Z ωn j ωθ =ω θ  1j ¼ j σ=σ 1j r γ n1 ðωe Þ;

ð111Þ

n

where limωe -1 γ 1 ðωe Þ ¼ 0. Next, the error j θm =θ m 1j will be discussed. Simple computation shows     

        lim θm  θ m  ¼  lim ∠L0 ðjσωe Þ ¼  lim ∠ 1 þ 1 ψ 7  ¼ 0:  ωe -1  ωe -1 Lðjω θ Þ  ωe -1 ωe

ð112Þ

Similar to (111), ( γ n2 ðωe Þ such that j θm =θ m 1j r γ n2 ðωe Þ;

8 ω e Z ωn :

ð113Þ

n

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Please cite this article as: Xue W, Huang Y. Performance analysis of active disturbance rejection tracking control for a class of uncertain LTI systems. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.05.001i