On the Conditions of Exponential Stability in Active ...

May 3, 2017

International Journal of Control

Shao˙Gao20170503

To appear in the International Journal of Control Vol. 00, No. 00, Month 20XX, 1–21

On the Conditions of Exponential Stability in Active Disturbance Rejection Control Based on Singular Perturbation Analysis S. Shaoa∗ and Z. Gaob a

b

Department of Mathematics, Cleveland State University, Cleveland, OH 44115, USA; Center for Advanced Control Technologies, Cleveland State University, Cleveland, Ohio 44115, USA (February 2016, revised June 2016)

Stability of the active disturbance rejection control (ADRC) is analyzed in the presence of unknown, nonlinear, and time-varying dynamics. In the framework of singular perturbations, the closed-loop error dynamics are semi-decoupled into a relatively slow subsystem (the feedback loop) and a relatively fast subsystem (the extended state observer), respectively. It is shown, analytically and geometrically, that there exists a unique exponential stable solution if the size of the initial observer error is sufficiently small, i.e. in the same order of the inverse of the observer bandwidth. The process of developing the uniformly asymptotic solution of the system reveals the condition on the stability of the ADRC and the relationship between the rate of change in the total disturbance and the size of the estimation error. The differentiability of the total disturbance is the only assumption made. Keywords: active disturbance rejection control, extended state observer, singular perturbation theory

1.

Introduction

Imagine an ideal process to be controlled, in the form of the chained integrators for example, and treat anything other than the ideal dynamics as disturbance in totality, to be estimated and canceled in real time. This is the idea known as active disturbance rejection control (ADRC), conceived and further developed by Han(Han, 1989, 1995, 1998, 1999, 2008, 2009). It has since become a phenomenon when the ADRC algorithm is hardwired into the general purpose control chips made by industry giants such as Texas Instruments (Texas Instruments, 2013) and Freescale Semiconductor (Freescale, 2015), when it is adopted by global manufacturers such as Parker (Zheng & Gao, 2012), and when it is implemented successfully by the utility industry in power plants (Sun et al., 2016). Moreover, various research groups in the world put this technology to the test in various domains of engineering (Castaneda et al., 2015; Lei et al., 2016; Madoski et al., 2014; Ramirez-Neria et al., 2015, 2014, 2016; Sun et al., 2016; Sun et al. , 2015; Xue et al., 2015), to name just a few recent studies. As ADRC emerges as a viable general solution for industrial control, research interests have been intensified, which can be seen in the recent survey and position papers (Gao, 2014, 2015, 2016; Guo & Zhao, 2015; Huang & Xue, 2014; Xue & Huang, 2015), and (Chen et al., 2016). Implementation issues in the industrial setting were also addressed (Herbst, 2016). In particular, the transition of ADRC from an academic idea to an emerging technology was recently documented (Gao, 2015). Crucial in the process is the simplification and parameterization (Gao, 2003) that allows seamless integration of the time domain formulation and the frequency domain insights (Zheng & Gao, 2016). This led to the initial attempts at a rigorous stability analysis (Zheng et al., 2009; Zheng, Gao & ∗ Corresponding

author. Email: [email protected]

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Gao, 2007, 2012), where not only the stability characteristics but also the bandwidth-error bound relationship are established. The initial analysis of the ADRC was followed by various groups using more advanced mathematical techniques, the scope of which has been well documented recently in (Gao, 2015; Guo & Zhao, 2015; Huang & Xue, 2014; Xue & Huang, 2015), and (Chen et al., 2016). Furthermore, the problem of convergence in nonlinear ADRC for a class of MIMO nonlinear and uncertain systems, under several assumptions on the total disturbance, was examined in (Guo & Zhao, 2011) and (Guo & Zhao, 2013). More recently, a rigorous study was performed on a new kind of ADRC where the extended state observer (ESO) is used in conjunction with the projective gradient method to estimate both the total disturbance and of the input gain (Jiang, Huang & Guo, 2015). Closely related to the current work, the performance of ADRC based control system was studied by Xue and Huang (Xue & Huang, 2015), where the tracking problem is considered for a class of uncertain linear time invariant (LTI) systems with both uncertain parameters and external disturbances. The output of closed-loop system is shown to approach its ideal trajectory in the transient process against different kinds of uncertainties and the peaking phenomenon can be avoided if the upper bound for the initial estimation error is less and equal than the inverse of the bandwidth of the ESO. While the previous work represents significant progress in solidifying the theoretical foundation for ADRC, particularly the conditions of stability, the mathematical sophistication may prevent engineers from having a clear and intuitive understanding of the assumptions made and the practical implications. In this paper we revisit the problem of analysis using the singular perturbation theory and the multi-time-scale techniques, which were first introduce to control engineering in the late 1960s by Kokotovic, Sannuti, Khalil and Reilly (Kokotovic, Khalil & Reilly, 1986; Kokotovi’c & Sannuti, 1968), as can be seen from the perspective of systems and control (Zhang et al., 2014). The ensuing separation of the control action into a slow and a fast component (Siciliano & Book, 1988) seems to be particularly promising in providing a powerful conceptual framework for the two-step design strategy commonly used in ADRC. The techniques have reached a high level of maturity as demonstrated in the publications of many excellent research and survey papers (see for example,(Zhang et al., 2014)). Zhou et al. in (Zhou, Shao & Gao, 2009) took advantage of the existing body of work on nonlinear control, particularly the work of Khalil (Khalil, 2002), and proposed a simpler framework based on the singular perturbation methods from (Kokotovic, Khalil & Reilly, 1986) and (Khalil, 2002). However, the results were obtained under several rather restrictive assumptions and there were some defects in the proof that need to be corrected. Our goals in this paper are two-fold: (1) reinterpret and reformulate the problem of stability in ADRC, and (2) find the stability conditions with fewer assumptions. In doing so we show that the closed-loop error dynamics in the ADRC can be reduced to a set of time varying singularly perturbed linear system of differential equations with unknown dynamics. To this end, the classic theory of singular perturbation by R. O’Malley (OMalley, 1991) and F. Verhulst (Verhulst, 1990) can be used to simplify the task of the analysis. It alao brings down the level of mathematical sophistication in studying the condition on the exponential stability and obtaining the upper bounds for both the tracking and the estimation errors. These results are also strongly corroborated by the geometrical interpretations, which show that the region of slow manifold of the system exists and is exponentially stable when the size of the initial estimated error is sufficiently small. Finally, most assumptions made in the previous studies have been removed except the one that is implicit in the ADRC formulation: the total disturbance is differentiable. The rest of the paper is organized as follows: problem formulation and the generalized separation principle are presented in Section 2; the asymptotic solution of the error system of the ADRC is provided along with the stability condition in Section 3; the region of the slow manifold of the 2

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system and their stability properties as well as the relationship between the rate of change in total disturbance and the size of the estimation error are presented in Section 4; and, finally, the concluding remarks are given in Section 5.

2.

Problem formulation and the generalized separation principle

In this section, the problem is first formulated as the problem of control design for a class of single input and single output (SISO) systems where the plant dynamics could be nonlinear, mostly unknown, and time-varying. The problem is made more challenging by the presence of the external disturbances which is also unknown. The design objective is to meet all design specifications amid such uncertainties in ADRC as perhaps the only general solution is introduced in the standard form as described in (Gao, 2003, 2006). By reformulating the error dynamics of the closed-loop system it becomes clear, for the first time, why the controller design and the disturbance rejection can be carried out separately, which makes the ADRC a very convenient solution for practitioners. In fact, the reformulation of the error dynamics in the ADRC demonstrates that the well-known separation principle in the linear system theory can be readily generalized to the nonlinear and uncertain systems discussed here, as suggested recently in (Gao, 2006). More importantly, this allows the stability condition and asymptotic solution to be derived via singular perturbation analysis, to be carried out in the next section. Consider the SISO plant in the form

y (n) (t) = f (y(t), y(t), ˙ · · · , y (n−1) (t), w(t)) + bu(t),

(1)

where u(t), y(t) and w(t) are the input, the output and the external disturbance of the system, respectively, and b is the given input gain. Here f (y(t), y(t), ˙ ..., y (n−1) (t), w(t)) represents the collective effect of both the nonlinear, time-varying and unknown dynamics of the plant and the external disturbance, and it is denoted as the total disturbance of the plant. For this plant (1), the only information that is assumed to be given, is the order of the plant and the parameter b. For the sake of simplicity, we denote f (·) = f (y(t), y(t), ˙ ..., y (n−1) (t), w(t)). Unlike the conventional model-based design principles, where the detailed mathematical description of f (y(t), y(t), ˙ ..., y (n−1) (t), w(t)) is assumed to be given, ADRC as a design principle only assumes that the first derivative of f (·) exists with respect to t. Numerous simulation studies, laboratory tests, and industrial case studies point to the simple fact that f (·) can be treated as a state variable and estimated in real time using a state observer, as fˆ(·). Then it can be canceled in real time in the control law of the form

u = (−fˆ(·) + u0 )/b

(2)

which reduces plant (1) to the cascade integral form:

y (n) (t) = f (·) − fˆ(·) + u0 ≈ u0 .

(3)

That is, at the core of the ADRC is an elegant and yet effective way of canceling the complications from the nasty nonlinearity, uncertain dynamics and external disturbances, forcing the plant to 3

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behave like the ideal, chained pure integrators. Our objective in this section is to answer the common question: How can this be possible? Or specifically, on what ground can f (·) be effectively estimated as a state? We start by extending the state definition to the total disturbance f (·) in the augmented state vector x1 = y, x2 = y, ˙ . . . , xn = y (n−1) , and, most importantly, the extended state xn+1 = f . System (1) can now be written in the augmented stat space form as x˙ i = xi+1 , 1 ≤ i ≤ n − 1, x˙ n = xn+1 + bu, x˙ n+1 = η, y = x1

(4)

with x = [x1 , x2 , ..., xn+1 ]T ∈ Rn+1 , u ∈ R and y ∈ R are the state, the input and the output of the system, respectively; η = η(·) = f˙(·) = f˙. In turn, system (4) can be rewritten in matrix form as x˙ = Ax + Bu + Eη, y = Cx, where     A=  

0 1 0 0 0 1 .. .. .. . . . 0 0 0 0 0 0

··· ··· .. . ··· ···

0 0 .. .





0 0 .. .

      , B=     b 1  0 (n+1)×(n+1) 0

      

(5)



0 0 .. .

   , E=   0 1 (n+1)×1

      



1 0 .. .

   , C=   0 0 (n+1)×1

      

,

(n+1)×1

(6)

which is observable, meaning that the state can be estimated by using a state observer, to which we now turn. 2.1

The ESO

With the total disturbance f (·) defined as the extended state, xn+1 , in system (4), it can now be estimated in real time using a state observer. Such state observer is denoted as the ESO and it can be designed in various forms. Han’s original ESO uses nonlinear gains to maximize the performance. Gao (Gao, 2006) simplified it with a linear ESO (LESO), where all observer gains are parametrized as functions of the observer bandwidth for the ease of tuning in practice. The LESO also makes the analysis much easier as is shown here. Let z1 , ..., zn , zn+1 be the estimates of x1 , ..., xn , xn+1 , respectively. The LESO of system (4) is given as z˙i = zi+1 + li (y − z1 ), 1 ≤ i ≤ n − 1, z˙n = zn+1 + bu + ln (y − z1 ), z˙n+1 = ln+1 (y − z1 ), 4

(7)

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where l1 , l2 , ..., ln+1 are the observer gains and they are chosen as [l1 , l2 , ..., ln+1 ] = [β1 ωo , β2 ωo 2 , ..., βn+1 ωo n+1 ].

(8)

Here βi , i = 1, 2, ..., n + 1 are selected such that the characteristic polynomial sn+1 + β1 sn + · · · + βn s+βn+1 is Hurwitz. That is, all the roots of the characteristic polynomial are in the open left-half complex plane. For the sake of simplicity, let sn+1 + β1 sn + · · · + βn s + βn+1 = (s + 1)n+1 where the binomial coefficients βi , i = 1, 2, ..., n + 1 are defined as

βi =

(n + 1)! , i!(n + 1 − i)!

1 ≤ i ≤ n + 1.

(9)

Since the characteristic polynomial of system (7) has the following form

|sI − (A − lC)| = sn+1 + l1 sn + · · · + ln s + ln+1 = (s + ωo )n+1 , the observer bandwidth ωo becomes the only tuning parameter of the observer, which makes tuning of the LESO a much easier task (see (Gao, 2003)). This is because the users, by adjusting just one parameter ωo , can quickly find the right trade-off between how fast the observer tracks the states and how sensitive it is to noises. Remark 2.1: The concept of the extended state is perhaps the most crucial one in ADRC. It allows the total disturbance to be estimated by the ESO and canceled by the control signal, making an otherwise complicated problem of controller design almost trivial. The question many people had was, of course, does this outside-the-box idea work? Can theoretical analysis explain the engineering success? By reformulating the error dynamics in the LESO, it is shown below that the problem of analysis can be solved in a more straightforward manner using the well-established singular perturbation theory.

2.2

The error dynamics of the LESO

Consider plant (5), the corresponding LESO is given by z˙ = Az + Bu + l(y − yˆ), yˆ = Cz,

(10)

where z = [z1 , z2 , ..., zn+1 ]T ∈ Rn+1 , and l = [l1 , l2 , ..., ln+1 ]T ∈ Rn+1 is the observer gain vector defined in (8). Let the estimation errors be defined as e˜i = xi − zi , 1 ≤ i ≤ n + 1, we have e˜ = [˜ e1 , e˜2 , ..., e˜n+1 ]T = x − z.

(11)

Then the error dynamics of the LESO has the following form e˜˙ = (A − lC)˜ e + Eη, 5

(12)

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where A, l, C, E and η are defined as in (6) and (8), respectively. For convenience, we define the state transformation as  e˜1 = ωo −n ξ1      .. .   e˜n = ωo −1 ξn    e˜n+1 = ξn+1 ,

(13)

which can also be rewritten in matrix form as e˜i = ωo i−(n+1) ξi ,

or e˜ = Λξ,

(14) −(n−1)

where e˜ = [˜ e1 , e˜2 , . . . , e˜n+1 ]T , ξ = [ξ1 , ξ2 , . . . , ξn+1 ]T , and Λ = diag[ωo−n ,ωo (n + 1) × (n + 1) diagonal matrix.

, · · · , ωo −1 , 1] is an

Substitute (14) into (12), we have Λξ˙ = (A − lC)Λξ + E.

(15)

 T With the parametrized observer gain vector l = β1 ωo , β2 ωo2 , · · · , βn ωon , βn+1 ωon+1 as defined in (8), the error dynamics of LESO can be written as: 1 1 ˙ ξ = Az ξ + Eη, ωo ωo

(16)

where 

−β1 −β2 .. .

   Az =    −βn −βn+1

1 0 ··· 0 1 ··· .. .. . . . . . 0 0 ··· 0 0 ···

0 0 .. .



     1  0 (n+1)×(n+1).

(17)

By the definitions of βi , i = 1, 2, ..., n + 1 in (9) and Az in (17), the eigevalues λi {Az } of Az are obtained as follows, λi {Az } = −1 < 0,

i = 1, 2, ..., n, n + 1,

(18)

which implies that Az is Hurwitz and invertible. With the observer bandwidth ωo > 0 as the tuning parameter and let ε=

1 , ωo

6

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the error dynamics of the LESO has the following form εξ˙ = Az ξ + εEη.

(19)

Equation (19) shows that the effect of the uncertainty represented by η is counterbalanced by ε. In other words, the increase in ωo will diminish the effect of η in the observer error dynamics. 2.3

The error dynamics of the feedback loop

With the total disturbance estimated by LESO, the control law 1 u = [−zn+1 + u0 ] b

(20)

is to applied to system (4) to reduce it approximately to the nth order pure integral plant, y (n) = (f (·) − zn+1 ) + u0 ≈ u0 ,

(21)

where u0 is the control signal. At this point, all model-based design methods can be used to meet the design specifications but perhaps the one most oftenly used is the parametrized ProportionalDerivatives (PD) controller with a feedforward term u0 =

n X

ki (yr(i−1) − zi ) + yr(n) ,

(22)

i=1

where yr is the reference signal, ki , i = 1, 2, ..., n are controller gains parametrized as sn + k1 sn−1 + · · · + kn = (s + ωc )n . Here ωc is defined as the controller bandwidth. The corresponding binomial coefficients can be obtained as

ki =

n! ωi , i!(n − i)! c

1 ≤ i ≤ n.

(23)

Note that the controller bandwidth ωc is the only tuning parameter to be adjusted by the users for the proper trade-off between the tracking performance and its costs. To obtain the error dynamics in such control loop, we let the reference vector be defined as xr = [yr , y˙ r , · · · , yr(n−1) ]T

(24)

and the tracking error vector is given as e = [e1 , e2 , · · · , en ]T = x − xr = [y − yr , y˙ − y˙ r , ..., y (n−1) − yr(n−1) ]T .

(25)

Then the error dynamics of the feedback loop can be described as e˙ = Ae e + Be u0 + Bf (f − yr(n) ),

7

(26)

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International Journal of Control

    where Ae =   

0

1

0

0 0 1 .. . . . . . . . 0 0 ··· 0 0 ···

Shao˙Gao20170503

··· .. . .. . 0 0

  0 0 ..   0 .     , Be =  ... 0     0 1  b 0 n×n





     

0 0 .. .

   , Bf =    0 1 n×1

      

.

n×1

Now the complete control law can be written in details as u0 = k1 [yr − (x1 − e˜1 )] + k2 [y˙ r − (x2 − e˜2 )] + · · · + kn [yr(n−1) − (xn − e˜n )] + yr(n) = k1 (yr − x1 ) + k1 e˜1 + k2 (y˙ r − x2 ) + k2 e˜2 + · · · + kn (yr(n−1) − xn ) + kn e˜n + yr(n) = −k1 e1 + k1 e˜1 − k2 e2 + · · · − kn en + kn e˜n +

(27)

yr(n) .

From equation (11) we have zn+1 = f (·) − e˜n+1 ,

(28)

then the total control law can be written as 1 u = [Kf e˜ − Ke − f (·) + yr(n) ], b

(29)

where K = [k1 , k2 , ..., kn ], and Kf = [k1 , k2 , ..., kn , 1]. Substituting (29) into (26), the error dynamics of the feedback loop can shown as

e˙ = Af e + Bf Kf Λξ,

(30)

where Af = Ae − Bf K implies that 

0

1

0

   Af =   

0 .. .

0 .. .

1 .. .

0 0 ··· −k1 −k2 · · ·

··· .. . .. .

0 .. .

0 0 1 −kn−1 −kn

       n×n

whose eigenvalues, by (23), can be determined as λi {Af } = −ωc < 0,

i = 1, 2, ..., n.

This implies matrix Af is both Hurwitz and invertible. Using the definitions of Bf , Kf and Λ in (26), (29), and (14), respectively, we obtain

8

(31)

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    Bf Kf Λ =        =  

=

0 0 .. .

0 0 .. .

0 0 .. .

··· ··· .. .

0 0 .. .



     0  1 n×(n+1)

0 0 0 ··· −(n−1) −(n−2) −n k1 ωo k2 ωo k3 ωo ···  0 0 0 ··· 0 0 0 0 ··· 0   .. .. .. ..  .. . .  . . .  0 0 0 ··· 0  k1 εn k2 εn−1 k3 εn−2 · · · 1 n×(n+1)

n X

(32)

εm Dm kn+1−m ,

m=0

where Dm = (dij )m is a constant n by (n + 1) matrix for each m = 0, 1, 2, ..., n, and dij is defined by ( 1, dij = 0,

if i = n and j = n + 1 − m, otherwise.

(33)

Note that matrices Dm are independent of ε for m = 0, 1, 2, ..., n, and set kn+1 = 1. 2.4

Dynamics of the combined closed-loop system

Combine the feedback loop error dynamics (30) and the observer error dynamics (19), we have the error dynamics of the entire system

e˙ = Af e + [

n X

εm Dm kn+1−m ]ξ,

m=0

(34)

εξ˙ = Az ξ + εEη, which is a standard singularly perturbed linear system of differential equations and will be thereby served as the starting point for the stability analysis in the next section. Remark 2.2: System (34) of the closed-loop error dynamics gives us insight on why the ADRC works. It is naturally separated into two kinds of dynamics: the fast dynamics of the LESO which is mostly governed by the observer bandwidth ωo and the slow dynamics of the feedback loop which is mostly governed by the controller bandwidth ωc . Both dynamics can be seen as the ideal dynamics by design plus a disturbance term, the effect of which is the subject of the study as illustrated in the following section.

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3.

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The asymptotic solution and the conditions of the stability

In this section, we divide our discussion into two cases: (i) f (·) is given, and (ii) f (·) is unknown. The stability properties of system (34) for case (i) is obvious and can be obtained easily by applying the well known theory in singular perturbation methods in control (Kokotovic, Khalil & Reilly, 1986) after a simple transformation. For case (ii), we first apply the singular perturbation theory to the error dynamics of system (34) to obtain a uniformly asymptotic solution and the stability properties. Then we show that the closed-loop system is exponentially stable and uniformly asymptotically stable as ε → 0 if the size of the initial observer error ξ(0) = ξ0 is sufficiently small, i.e., kξ0 k = O(ε) = O( ω1o ), where k · k denotes the Euclidean norm. This result will be corroborated by the exponential closeness of the slow manifold in geometric terms shown in Section 4, which also suggests the exponential stability of system (34). Consider the initial value problem of the singularly perturbed system of linear differential equations as in (34), e˙ = Af e + [

n X

εm Dm kn+1−m ]ξ,

e(0) = e0 ,

m=0

εξ˙ = Az ξ + εEη,

(35)

ξ(0) = ξ0 ,

where ε = 1/ωo is a small positive parameter, Af , Az , Dm and E are the constant matrices defined in (30), (16) and (33), respectively; and kξ0 k = O(ε). Case (i) f (·) is given. ¯ is a solution of the system (35) for all t ≥ 0. Making change of variables by defining Suppose (¯ e, ξ) ¯ we have eˆ = e − e¯, ξˆ = ξ − ξ, ˆ = ξ(0) − ξ(0) ¯ = ξ0 − ξ0 = 0, ξ(0)

eˆ(0) = e(0) − e¯(0) = e0 − e0 = 0,

system (35) is transformed to the following singularly perturbed linear homogeneous system with the origin (0, 0) being an equilibrium point of the transformed system at t = 0, n

X dˆ e ˆ = Af eˆ + [ εm Dm kn+1−m ]ξ, dt

eˆ(0) = 0,

m=0

dξˆ ˆ ε = Az ξ, dt

(36) ˆ = 0. ξ(0)

Since matrices Af and Az are Hurwitz and invertible with λi {Af } = −ωc < 0,

λi {Az } = −1 < 0,

i = 1, 2, ..., n,

i = 1, 2, ..., n, n + 1,

(see (31) and (18)), we have kAf k ≤ m1 ,

kAz k ≤ m2 ,

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for some positive numbers m1 and m2 on [0, ∞). The stability property of system (36) is well known, that is, there exists an ε1 > 0 such that for all ε ∈ (0, ε1 ) system (36) is uniformly asymptotically stable (see (Kokotovic, Khalil & Reilly, 1986)). Case (ii) f (·) is unknown. In this case, we are not able to make a simple transformation to obtain the stability properties of system (35) as in case (i). To show the conditions on the stability properties of system (35), we shall now perform the procedure of the O’Malley/Hoppensteadt construction in the sense of the Tikhonove’s Theorem (Kokotovic, Khalil & Reilly, 1986; OMalley, 1991; Verhulst, 1990) to obtain the asymptotic solution (e(t, ε), ξ(t, ε)) of system (35), in the following form e(t, ε) = es + εφ(τ, ε), e(0, ε) = e0 , ξ(t, ε) = ξs + ϕ(τ, ε), ξ(0, ε) = ξ0 ,

(37)

where (es , ξs ) is the outer solution of system (35), (ε, φ, ϕ) → 0 as τ = t/ε → ∞. Remark 3.1: (1) It is important to note that the initial values of system (37) are crucial for the solution of system (37). We will explain it in the later stage following system (38) and further emphasize it in Remark 3.3. (2) In general, it is required to have the assumption of differentiability of the functions on the right-hand side of differential equations of system (35) in order to obtain the higher order terms of ε in the asymptotic expansion of system (37). In our case, it means that we may need to assume that η is infinitely differentiable with respect to t. This is, however, not necessary for the error dynamics in system (35) since Eη is multiplied by ε, which means that η is involved only in the ‘higher order’of ε in the asymptotic solution, which is usually ignored since ε is small. This is very important in the analysis of the ADRC because the unknown function η = f˙ drops out from the outer solution of system (35). Hence, it could be ignored in the asymptotic solution of system (35). Therefore, there is no need to make any further assumptions, such as the differentiability or boundedness of η as in the previous work. The initial conditions of es , ξs , φ and ϕ from (37) must satisfy the following equations e0 = e(0, ε) = es (0, ε) + εφ(0, ε), ξ0 = ξ(0, ε) = ξs (0, ε) + ϕ(0, ε),

(38)

thus, solution (e(t, ε), ξ(t, ε)) of system (37) is completely specified by the initial condition e0 and ξ0 (OMalley, 1991) with kξ0 k = O(ε) lies in the stable initial manifold for ξ0 (see Remark 3.2). Furthermore, the outer solution (es , ξs ) of system (35) satisfies the following equations e˙ s = Af es + [

n X

εm Dm kn+1−m ]ξs ,

m=0

(39)

εξ˙s = Az ξs + εEη. Remark 3.2: If ξ0 lies off its stable initial manifold, the blowup of ξ as ε → 0 for t > 0 is expected (OMalley, 1991). But when the ADRC system is commissioned in industrial applications, the common practice is to first make sure that the LESO tracks the target sufficiently close before its estimates can be used in the control law, and this can be determined from the observer error. 11

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It is therefore reasonable to assume that kξ0 k = O(ε), which means that the initial observer error ξ(0) = ξ0 lies in the stable initial manifold for ξ0 and that there will be no danger of large magnitude transients (‘peaking’) the system. This is an example of mathematical analysis that starts with assumptions that reflect the common engineering practice, and then lead to results that are quite meaningful to engineers, as below. Let es = a0 (t) + εa1 (t) + O(ε2 ),

ξs = b0 (t) + εb1 (t) + O(ε2 ).

(40)

The leading term (a0 (t), b0 (t)) of (es , ξs ) is the slow solution of system (35), which satisfies a˙ 0 = Af a0 + D0 kn+1 b0 , 0 = Az b0 .

a0 (0) = e0 ,

(41)

The stability properties (18) and (31) imply that system (41) has a unique solution a0 (t) = exp(Af t)e0 , b0 (t) = 0,

(42)

which is unique, exponentially decay and uniformly asymptotic stable for finite t > 0. To obtain the initial layer correction (εφ, ϕ) of system (35), let t = τ ε, then (εφ, ϕ) of system (37) satisfies the homogeneous system, n

X dφ = εAf φ + [ εm Dm kn+1−m ]ϕ, dτ m=0

(43)

dϕ = Az ϕ. dτ Let φ = φ0 (τ ) + εφ1 (τ ) + O(ε2 ),

ϕ = ϕ0 (τ ) + εϕ1 (τ ) + O(ε2 )

(44)

whose terms must decay to zero as τ → ∞. Thus, the leading term (φ0 (τ ), ϕ0 (τ )), which is the fast solution of system (35), must satisfy the following equations with constant coefficients, dφ0 = D0 kn+1 ϕ0 , dτ dϕ0 = Az ϕ0 , ϕ0 (0) = ξ0 , dτ

(45)

where the initial condition ϕ0 (0) = ξ0 is obtained from ξ0 = ξ(0, 0) = b0 (0) + ϕ0 (0) and b0 (t) = 0. Since matrix Az is Hurwitz and invertible, as shown above, the decay can be described as τ → ∞ as ϕ0 (τ ) = exp(Az τ )ξ0 , φ0 (τ ) = D0 kn+1 Az −1 exp(Az τ )ξ0 ,

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where kϕ0 (0)k = kξ0 k = O(ε) lies in the domain of attraction of the equilibrium solution of the ϕ0 -equation. Therefore, we conclude that the fast solution (φ0 , ϕ0 ) of system (35) is uniformly asymptotic stable in ξ0 and it is unique and exponential decay as τ → ∞. From (38), the matching initial conditions for φ1 , ϕ1 , a1 and b1 satisfy the following equations, a1 (0) = −φ0 (0), ϕ1 (0) = −b1 (0),

(47)

which implies that φ0 (0) = D0 kn+1 Az −1 ξ0 and a1 (0) = −D0 kn+1 Az −1 ξ0 . For the coefficient of ε, we have the following initial value problems (48) and (49), a˙ 1 = Af a1 + D0 kn+1 b1 + D1 kn b0 , b˙ 0 = Af b1 + Eη

a1 (0) = −D0 kn+1 Az −1 ξ0 ,

(48)

and dφ1 = Af φ0 + D0 kn+1 ϕ1 + D1 kn ϕ0 , dτ dϕ1 = Az ϕ1 , ϕ1 (0) = −b1 (0), dτ

(49)

their solutions can be found easily as b1 (t) = −Az −1 Eη, a1 (t) = − exp(Af t)D0 kn+1 Az

−1

Z ξ0 − D0 kn+1

t

exp(Af (t − s))Az −1 Eη(s)ds,

(50)

0

and ϕ1 (τ ) = − exp(Az τ )Az −1 Eη(0), φ1 (τ ) = −[Af D0 kn+1 Az −2 exp(Az τ )ξ0 + D0 kn+1 Az −1 exp(Az τ )Az −1 Eη(0) + D 1 k n Az

−1

(51)

exp(Az τ )ξ0 ].

Continuing the similar processes, solution (37) could be completely determined. From the above analysis, we have the following theorem. Theorem 3.1: If kξ0 k = O(ε), the formal approximation (e(ε, t), ξ(ε, t)) of the initial value problem of system (35) has the following form t e(ε, t) = exp(Af t)e0 + ε[a1 (t) + φ0 ( )] + O(ε2 ), ε t t ξ(ε, t) = exp(Az )ξ0 + ε[b1 (t) + ϕ1 ( )] + O(ε2 ), ε ε

13

(52)

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where a1 (t) = − exp(Af t)D0 kn+1 Az

−1

Z ξ0 − D0 kn+1

t

exp(Af (t − s))Az −1 Eη(s)ds,

0

b1 (t) = −Az −1 Eη, t t φ0 ( ) = D0 kn+1 Az −1 exp(Az )ξ0 , ε ε t t −1 ϕ1 ( ) = − exp(Az )Az Eη(0), ε ε and the following statements hold: (1) the asymptotic solutions (52) is uniformly valid for all finite time, say 0 ≤ t ≤ L < ∞ with L being finite; (2) the critical point of the initial layer equation e˙ = Af e + D0 kn+1 ξ, 0 = Az ξ,

(53)

of system (35) is asymptotically stable in the linear approximation, uniformly in the parameters e and t; (3) the fast and the slow solutions of system (35) are exponentially decay as ε → 0; (4) there exists an ε∗ > 0 such that for all ε ∈ [0, ε∗ ] the system (35) is exponentially stable as ε → 0.

Remark 3.3: We show that the closed-loop system is exponentially stable and uniformly asymptotically stable if the norm of the initial observer error kξ0 k = O(ε) = O(1/ωo ) (the result holds, for stability within the stable initial manifold for ξ0 , it is not the stability in the Lyapunov sense), which is the order of the inverse proportion of the observer bandwidth. In particular, Theorem 3.1 shows that if the initial observer error ξ0 lies in the stable initial manifold (see the geometry interpretations in Section 4), there will be no danger of large magnitude transients (‘peaking’) in the system as ε tends to zero as we discussed in Remark 3.1. When the size of the initial observer error equals to 1/ωo , it becomes the same condition that Xue and Huang obtained in (Xue & Huang, 2015). Our condition on the initial observer error is less restrictive and hence it enlarges the error bounds as we show later in Theorem 4.1. As long as f˙(·) exists, all the results of Theorem 3.1 hold if the norm of the initial observer error kξ0 k is equal to O(ε) = O(1/ωo ).

4.

Geometrical interpretations of the condition on the stability and the size of the estimation error

Our results in Section 3 could be further verified using the geometric properties obtained from the singular perturbation theory as it is applied to system (35). To show the existence of the slow manifold Mε , we notice that the slow motion of system (35) is to be primarily determined by system (39) and since matrix Az is Hurwitz, it lies on the n-dimensional stable manifold M0 : ξ = 0, where M0 is the first-order approximation of the n-dimensional slow manifold Mε . By Fenichel’ results (Verhulst, 1990), there exist stable manifolds of Mε , smooth continuations of the corresponding

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manifolds of M0 , on which the flow is fast. The slow manifold corresponds with the critical point of the ϕ-equation in the fast system (43) and it is asymptotically stable. For the behavior near the slow manifold Mε , we notice that M0 is a compact and stable manifold. According to Fenichel, Hirsch, Pugh, and Shub (Verhulst, 1990), an invariant manifold Mε exists in a O(ε)-neighborhood of M0 for ε sufficiently small (Verhulst, 1990). The dynamics of Mε is slow (in terms of ε) and the transversal dynamics is fast. Theorem 1 shows that the solution of the initial value problem of system (35) remains in an O(ε) neighborhood of the approximate slow manifold M0 . We shall show in the following theorem that the nearby slow manifold Mε is approached closely and exponentially. The upper bounds of the solution of (35) and the relationship between the rate of change in the total disturbance and the size of the estimation error are also provided. Let us define H1 = [−ρ1 , ρ1 ]n ⊂ Rn and H2 = [−ρ2 , ρ2 ]n+1 ⊂ Rn+1 , where ρi = γi max{ωc , ω1c , kAf k, kAz k} for some positive constants γi , i = 1, 2. Recall that ωc is the controller bandwidth. Then H1 and H2 are compact in Rn and Rn+1 , respectively. Theorem 4.1: Consider the initial value problem of system (35), n X

e˙ = Af e + [

εm Dm kn+1−m ]ξ,

e(0) = e0 ,

m=0

εξ˙ = Az ξ + εEη,

ξ(0) = ξ0 ,

for all e ∈ H1 ⊂ Rn and ξ ∈ H2 ⊂ Rn+1 , there exist positive constants K2 and L, both independent of ε, such that kξ0 k = O(ε) ⇒ kξ(t)k = O(ε),

0 ≤ t ≤ L,

(54)

and kξ0 k = O(ε) ⇒ kηk = kf˙k ≤ K2 kξk,

(55)

here η = f˙. Moreover, solution (e(t), ξ(t)) of system (35) satisfies kξ(t)k ≤ εC2 exp[−(1/2 − εC2 K2 )t/ε], ke(t)k ≤ C1 exp(−ωc t)e0 + εC1 C2 K1 exp(−(1/2 − εC2 K2 )t/ε)[exp(−ωc t) − 1]/ωc ,

(56)

where ωc is the controller bandwidth, C1 , C2 and K1 are constants and satisfy the following equations C1 =

√

n+

n−1 X

Lj 2 k(Af + ωc In )j k, j!

(57)

n 1 X Lj 2 k(Az + I(n+1) )j k, ωc j!

(58)

j=1

C2 =

√

n+1+

j=1

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v u n X u K1 = t1 + km 2 ,

(59)

m=1

here In is the n by n identity matrix, and k · k denotes the Euclidean norm.

Proof. By the assumption of kξ0 k = O(ε), according to Theorem 1, and the O’Malley-Vasil’eva expansion theorem (Verhulst, 1990), we have kξ(t)k = O(ε),

0≤t≤L

(60)

for some positive constant L which is independent of ε. This implies kεEηk ≤ εK2 kξk

(61)

for some positive constant K2 . Since kεEηk = εkηk = εkf˙k, we obtain kf˙k ≤ K2 kξk.

(62)

Now consider the ξ-equation of system (35) ˙ = Az ξ(t) + εEη, εξ(t)

(63)

ξ(0) = ξ0 .

By the variation of constants formula, we have t

Z

(64)

exp(Az (t − s)/ε)Eηds.

ξ(t) = exp(Az t/ε)ξ0 + 0

For kξ0 k = O(ε) and all the eigenvalues of Az , λi {Az } = −1 < 0, Theorem A1 in the Appendix to obtain exp(Az t/ε) = exp(−t/ε)[In+1 +

n X (t/ε)j j=1

which implies k exp(Az t/ε)k ≤ C2 exp(−t/ε) with C2 = Together with (55), we obtain Z

j!

i = 1, 2, ..., n, n + 1, we apply

2

(Az + In+1 )j ],

√ n+1 +

1 ωc

Lj j=1 j! k(Az

Pn

(65) 2

+ In+1 )j k.

t

kξ(t)k ≤ εC2 exp(−t/ε) + C2

exp(−(t − s)/ε)(K2 kξ(s)k)ds. 0

In the spirit of (Verhulst, 1990), let z(t) = kξ(t)k exp(−t/2ε),

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(66)

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we obtain Z

t

exp(−(t − s)/ε)K2 z(s)ds z(t) ≤ εC2 exp(−t/2ε) + C2 0 Z t K2 z(s)ds, ≤ εC2 + C2

(67)

0

≤ εC2 exp(C2 K2 ). By the Gronwell’s inequality (Verhulst, 1990), we obtain the first inequality in (56) kξ(t)k ≤ εC2 exp(−(1/2 − εC2 K2 )t/ε).

(68)

Similarly, we consider the e-equation of system (35), e(t) ˙ = Af e(t) + h1 (ξ(t), ε),

e(0) = e0 ,

(69)

P where h1 (ξ(t), ε) = [ nm=0 εm Dm kn+1−m ]ξ, which implies kh1 (ξ(t), ε)k ≤ K1 kξk.

(70)

q

P Let K1 = 1 + nm=1 km 2 , since all the eigenvalues of Af , λi {Af } = −ωc < 0, Theorem A1 in the Appendix A implies exp(Af t) = exp(−tωc )[In +

n−1 X tj j=1

j!

i = 1, 2, ..., n,

2

(Af + ωc In )j ].

(71)

which implies k exp(Af t)k ≤ C1 exp(−ωc t). where C1 =

P √ n + n−1 j=1

Lj j! k(Af

(72)

2

+ ωc In )j k.

Combine the result in (68) and (70), we obtain the second inequality in (56), Z

t

ke(t)k ≤ C1 exp(−ωc t)e0 + C1

exp(−ωc (t − s)/)(K1 kξ(s)k)ds, Z t = C1 exp(−ωc t)e0 + C1 K1 εC2 exp(−(1/2 − εC2 K2 )t/ε) exp(−ωc (t − s))ds, 0

(73)

0

≤ C1 exp(−ωc t)e0 + εC1 C2 K1 exp(−(1/2 − εC2 K2 )t/ε)[exp(−ωc t) − 1]/ωc . (66) and (73) give (56). We finish the proof. Remark 4.1: (1) Since kξ0 k = O(ε) and exp(Az εt ) is exponentially decay, Theorem 4.1 shows that kξ(t)k = O(ε). That is, the magnitude of the observer error is monotonic decreasing with respect to the observer bandwidth ωo .

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(2) From equation (56), we show that the upper bounds of the solution (e(t), ξ(t)) of system (35), which is the size of the estimation error of the ADRC. The upper bound of ε is provided, i.e. ε < 2C21K2 . To guarantee the exponential stability property, we need to choose the observer √ P j 2 bandwidth ωo > 2C2 K2 with C2 = n + 1 + ω1c nj=1 Lj! k(Az + I(n+1) )j k. (3) There is no prior restriction on the interval bound L. Since system (35) is linear, the restriction of the solution (e, ξ) of system (35) to be defined on some compact sets could be removed, and the existence and uniqueness of the slow manifold can be shown (Verhulst, 1990). For our case, L can be any finite number. (4) The exponential closeness of the nearby slow manifold Mε shows the exponential stability of system (35), which implies that the dynamics of the feedback loop error and the dynamics of the observer error of the ADRC are exponentially stable if kξ0 k = O(ε). The proofs of Theorem 3.1 and Theorem 4.1 are based on the premise that f˙(·) exists, i.e. the total disturbance is differentiable. The relationship between kf˙(·)k and the size of the estimation error kξk is obtained in (55) in Theorem 4.1.

5.

Concluding Remarks

The mathematical analysis based on singular perturbation theory helps to establish formally the generalized separation principle recently postulated (Gao, 2006) as the underlining principle of ADRC. The engineering proficiency was confirmed by the theoretical analysis in that the controller design, for the purpose of meeting the design specifications, can be indeed carried out independently from the observer design. Through the singular perturbation analysis for the entire closed-loop system, a uniform asymptotic solution is obtained and the exponential stability in the error dynamics is established under the condition that the initial observer error is reasonably small. Our results hold under a very mild condition, namely the total disturbance is differentiable with respect to time t. Also shown are the upper bonds of the errors in both the control loop and the observer, and the fact that they decrease with their respective bandwidths. Furthermore, the process of developing the uniformly asymptotic solution of the system reveals the condition on the stability of the ADRC and the relationship between the rate of change in the total disturbance and the size of the estimation error.

Acknowledgment The authors would like to sincerely thank the senior editor and the reviewers for their valuable and insightful comments, the revised manuscript benefited a great deal in overall presentation and clarity.

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Appendix A. Theorem A1 Theorem A.1: (Putzer, 1966) Let A be an n × n constant matrix, let λ1 , λ2 , ..., λn be the eigenvalues of A, then exp (At) =

n−1 X

rj+1 (t)Pj ,

(A1)

j=0

where P0 = In ;

Pj =

j Y

(A − λk In )j ,

k=1

20

j = 1, .., n,

(A2)

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where In denotes the n by n identity matrix, and r1 (t), ..., rn (t) is the solution of the triangular system r˙1 = λ1 r1 , r˙j = rj−1 + λj rj , j = 2, ..., n, r1 (0) = 1, ; rj (0) = 0, j = 2, ..., n.

(A3)

Applying Theorem A1 to exp(Af t), we have P0 = In ;

Pj =

j Y

(Af + ωc In )j ,

j = 1, .., n,

k=1

r1 (t) = exp(−ωc t), r2 (t) = t exp(−ωc t), rn (t) =

(A4)

tn−1 exp(−ωc t), (n − 1)!

note that the eigenvalues of Af is −ωc . It follows that exp (Af t) = exp(−ωc t)(In +

n−1 X

tj 2 [ (Af + ωc In )j ]), j!

(A5)

n X τj 2 [ (Az + In+1 )j ]). j!

(A6)

j=1

Similarly, we have exp (Az τ ) = exp(−τ )(In+1 +

j=1

21