Chapter 1
Introduction

In this chapter, we introduce some necessary background about the active disturbance rejection control (ADRC). Some notation and preliminary results are also presented.

1.1 Problem Statement

In most control industries, it is hard to establish accurate mathematical models to describe the systems precisely. In addition, there are some terms that are not explicitly known in mathematical equations and, on the other hand, some unknown external disturbances exist around the system environment. The uncertainty, which includes internal uncertainty and external disturbance, is ubiquitous in practical control systems. This is perhaps the main reason why the proportional-integral-derivative (PID) control approach has dominated the control industry for almost a century because PID control does not utilize any mathematical model for system control. The birth and large-scale deployment of the PID control technology can be traced back to the period of the 1920s-1940s in response to the demands of industrial automation before World War II. Its dominance is evident even today across various sectors of the entire industry. It has been reported that 98% of the control loops in the pulp and paper industries are controlled by single-input single-output PI controllers [18]. In process control applications, more than 95% of the controllers are of the PID type [9].

Let us look at the structure of PID control first. For a control system, let the control input be and let the output be . The control objective is to make the output track a reference signal . Let be the tracking error. Then PID control law is represented as follows:

where , and are tuning parameters. The PID control is a typical error-based control method, rather than a model-based method, which is seen from Figure 1.1.1 for its advantage of easy design. The nature of independent mathematical model and easy design perhaps have explained the partiality of control engineers to PID.

Figure 1.1.1 PID control topology.

However, it is undeniable that PID is increasingly overwhelmed by the new demands in this era of modern industries where an unending efficiency is pursued for systems working in more complicated environments. In these circumstances, a new control technology named active disturbance rejection control (ADRC) was proposed by Jingqing Han in the 1980s and 1990s to deal with the control systems with vast uncertainty [58 59, 60 62, 63]. As indicated in Han's seminal work [58], the initial motivation for the ADRC is to improve the control capability and performance limited by PID control in two ways. One is by changing the linear PID (1.1.1) to nonlinear PID and the other is to make use of "derivative" in PID more efficiently because it is commonly recognized that, in PID, the "D" part can significantly improve the capability and transient performance of the control systems. However, the derivative of error is not easily measured and the classical differentiation most often magnifies the noise, which makes the PID control actually PI control in applications, that is, in (1.1.1), .

In automatic principle of compensation, the differential signal for a given reference signal is approximated by in the following process:

where represents the Laplace transform of , is a constant, and represents the inertial element with respect to (see Figure 1.1.2).

Figure 1.1.2 Classical differentiation topology.

The time domain realization of (1.1.2) is

If is contaminated by a high-frequency noise with zero expectation, the inertial element can filter the noise ([62], pp. 50-51):

That is, the output signal contains the magnified noise . If is small, the differential signal may be overwhelmed by the magnified noise.

To overcome this difficulty, Han proposed a noise tolerant tracking differentiator:

whose state-space realization is

The smaller is, the quicker tracks . The abstract form of (1.1.6) is formulated by Han as follows:

where is the tuning parameter and is an appropriate nonlinear function. Although a convergence of (1.1.7) is first reported in [59], it is lately shown to be true only for the constant signal . Nevertheless, the effectiveness of a tracking differentiator (1.1.7) has been witnessed by many numerical experiments and control practices [64 147, 152 153]. The convergence proof for (1.1.7) is finally established in [55 and 52]. In Chapter 2, we analyze this differentiator, and some illustrative numerical simulations and applications are also presented.

The second key part of the ADRC is the extended state observer (ESO). The ESO is an extension of the state observer in control theory. In control theory, a state observer is a system that provides an estimate of the internal state of a given real system from its input and output. For the linear system of the following:

where is the state, is the control (input), and is the output (measurement). When , the whole state is measured and the state observer is unwanted. If , the Luenberger observer can be designed in the following way to recover the whole state by input and output:

where the matrix is chosen so that is Hurwitz. It is readily shown that the observer error as . The existence of the gain matrix is guaranteed by the detectability of system (1.1.8). If it is further assumed that system (1.1.8) is stabilizable, then there exists a matrix such that the closed-loop system under the state feedback is asymptotically stable: as . In other words, is Hurwitz. When the observer (1.1.9) exists, then under the observer-based feedback control , the closed-loop system becomes

It can be shown that as and, moreover, the eigenvalues of (1.1.10) are composed of , which is called the separation principle for the linear system (1.1.8). In other words, the matrices and can be chosen separately.

The observer design is a relatively independent topic in control theory. There are huge works attributed to observer design for nonlinear systems; see, for instance, the nonlinear observer with linearizable error dynamics in [87 and 88], the high-gain observer in [84], the sliding mode observer in [24 26, and 130], the state observer for a system with uncertainty [22], and the high-gain finite-time observer in [103 109, and 116]. For more details of the state observer we refer to recent monograph [14].

A breakthrough in observer design is the extended state observer, which was proposed by Han in the 1990s to be used not only to estimate the state but also the "total disturbance" that comes from unmodeled system dynamics, unknown coefficient of control and external disturbance. Actually, uncertainty is ubiquitous in a control system itself and the external environment, such as unmodeled system dynamics, external disturbance, and inaccuracy in control coefficient. The ubiquitous uncertainty in systems explains why the PID control technology is so popular in industry control because PID control is based mainly on the output error not on the systems' mathematical models. Since the ESO, the "total disturbance" and the state of the system are estimated simultaneously, we can design an output feedback control that is not critically reliant on the mathematical models. Let us start from an th order SISO nonlinear control systems given by

which can be rewritten as

where is the control (input), is the output (measurement), is the system function, which is possibly unknown, and is unknown external disturbance; is called the "total disturbance" or "extended state" and are the tuning parameters. The ESO designed in [60] is as follows:

By appropriately choosing the nonlinear functions and tuning the parameters , we expect that the states of the ESO (1.1.12) can approximately recover the states and the extended state , that is,

In Chapter 3, we have a principle of choosing the nonlinear functions and tuning the gain parameters . The convergence of the ESO is established. We also present some numerical results to show visually the estimations of state and extended state. In particular, if the functions in (1.1.12) are linear, the ESO is referred to as the linear extended state observer (LESO). The LESO is also called the extended high-gain observer in [35].

The final key part of the ADRC is the TD and the ESO-based feedback control. In the feedback loop, a key component is to compensate (cancel) the "total disturbance" by making use of its estimate obtained from the ESO. The topology of the active disturbance rejection control is blocked in Figure 1.1.3.

Figure 1.1.3 Topology of active disturbance rejection control.

Now we can describe the whole picture of the ADRC for a control system with vast uncertainty that includes the external disturbance and unmodeled dynamics. The control purpose is to design an output feedback control law that drives the output of the system to track a given reference signal . Generally speaking,...