Frequency Performance of the Extended State Observer for General Systems

: The observing capability of the Active Disturbance Rejection Control (ADRC) for the general systems is discussed in this study. The frequency performance of the Linear Extended State Observer (LESO) is analyzed. And the theoretical result is verified by simulations. It is shown that ESO can estimate the required states at the designed speed, in spite of the different total uncertainties.


INTRODUCTION
Control design for the systems with uncertainties is a longstanding fundamental issue in automatic control. The uncertainties usually stem from 2 sources: the internal (parameter or structure) uncertainty and the external (disturbance) uncertainty. Hence, many disturbance estimating techniques appeared, such as Unknown Input Observer (UIO) (Basile and Marro, 1969), Perturbation Observer (POB) (Kwon and Chung, 2003), the Disturbance Observer (DOB) (Schrijver and Dijk, 2002). And the Active Disturbance Rejection Control (ADRC), which is based on the use of Extended State Observer (ESO), attracted many researchers' attention due to less dependence on the model information, strong capabilities for disturbance rejection and its simple control structure. The key of ADRC is to use ESO to estimate the total uncertainty, which lumps the internal nonlinear and uncertain dynamics and the external disturbance. Then the uncertainties of the system can be compensated actively.
The idea of ESO can be demonstrated in the following Single-Input and Single-Output (SISO) model: where, n : The order of the plant model y : The output, u is the input b : A constant w(t) : The external disturbance f(.) : The total uncertainty (or total disturbance) of the system, containing both internal and external uncertainties Introduce e(t) = df/dt . Then the total uncertainty f(.) can be taken as an extended state of the system (1) and the Eq. (1) can be written in the state form as: is the state vector of the system. The general form of ESO estimating both the states and the extended state for the uncertain system (1) is given in Gao et al. (2001) and Han (1995). As a special form of ESO, the following Linear ESO (LESO) is discussed in many studies, such as Zheng et al. (2007a, b): is the state vector of ESO , 1,2, … 1 are the observer gains And ESO (3) is designed to have the property that: In Yang and Huang (2009), the capability of LESO (3) is analyzed for a wider scope of uncertain function f (t). It is shown that LESO can estimate the total uncertainty with bounded error if either the derivative of the total uncertainty is bounded or the total uncertainty itself is bounded. If the plant model is unknown, it will be difficult to discuss the error bound of ESO generally, since the total uncertainty may involve the high order derivatives of the system output. Sometimes, the orders of involved derivatives are even higher than the plant order. In this study, the capability of LESO is analyzed in frequency domain for the plants with unknown model. The main result is that the LESO can track the derivatives of the system output and the total uncertainty at the designed speed, while the stability of ESO is guaranteed.
In this study, the capabilities of ESO for estimating uncertainties are analyzed. The main result implies that ESO (6) can be used to observe the required states and the total uncertainty while the stability of ESO (6) is guaranteed. Moreover, the theoretical results are verified in detail by the simulations.

PROBLEM FORMULATION
In the following discussion, a symbol may represent both a time-domain variable and its Laplace transform when there is no confusion. Consider the following general system: where, , and u represent the state vector, the output and the control input, respectively. The design objective is to build an LESO (6) which can estimate the 1 st to (n-1) th order derivatives of y and the total uncertainty, guarantee y(t) to be controlled to follow the command signal v(t). And assume that the 1st to (n-1) th order derivatives of v(t) can be obtained in real time.
Due to the lack of the plant knowledge, the plant (4) may be modeled by the following dynamic system while designing the ADRC scheme: where, 1 n ≥ : An estimate of the order of the plant (4) b e : An estimate of the gain for the control input f(.) : The total uncertainty, which contains all the unknown dynamics: If n is greater than the plant order, f(.) will involve the derivative(s) of order(s) higher than the plant order Remark 1: Since the gain for the control input of the system (4) may be unknown or even varying, b e is different from b in (1) and (3). If the plant (4) is a linear system, b e is an estimate of the plant's NZC, which is short for the numerator's zero-order coefficient (Zhao and Huang, 2011;Zhao, 2011).
The key in ADRC is to design the Extended State Observer (ESO) to estimate the total uncertainty f(.) Based on (5), the LESO (3) is modified as follows: where, 1 1 are positive parameters. And 1 1 should be designed such that 1 tracks 1 and z n+1 tracks f(.). Then the following control law is proposed for the tracking problem: where, the parameters 0 for 1 , v (0) = v is the reference signal and 1 represents the i th order derivative of v. Based on (6), there is: 1 is stable, then its zero-frequency gain is F yi (0) = 1.

MAIN RESULTS
for any fixed 0 and 0. Before the proof of Theorem1, the following lemma shown in Zhao (2010) will be introduced: is Hurwitz immediately. Then, D i (s) = s n+1-i + is Hurwitz from Lemma 3, provided that D i+1 (s) is Hurwitz and is big enough. Thus, the induction is completed and the stability is proved.
Furthermore, (10) will be achieved by modifying the former induction. Assume 1 and 1 without loss of generality. Let 0,1 be small enough such that: Hence, taking U as disturbance, ESO (6) can be used for the uncertain minimum-phase system ( ) H s to observe the required states. Moreover, ESO (6) plus the controller (7) can guarantee the close-loop stability. Then, the practical performances will be shown by simulation in next section.

SIMULATIONS
Consider the following group of plants: