Transfer function analysis and implementation of active disturbance rejection control

Gernot Herbst

Abstract

Keywords Active disturbance rejection control (ADRC) · Frequency-domain analysis · Digital implementation

1 Introduction

1.1 Motivation and related work

Since its inception, active disturbance rejection control(ADRC) attracted widespread attention among both scholars and practitioners. Starting with Han’s nonlinear controller[1] and streamlined in Gao’s linear variant [2], it has found the way into numerous application domains. An overview is given in [3] and, more recently, in [4].

What are ADRC’s main selling points? It is a seriousand probably the most popular-contender to overcome the“theory versus practice hassle” [5] that has been tackled also by other model-free controller approaches [6]. The “paradigm shift” [7] is enabled by a combination of modern control elements with pragmatic, minimal plant modeling and control loop tuning efforts. At the same time, it can easily be equipped with controller features desirable in industrial practice [8].

As a general purpose controller, ADRC competes with the ubiquitous PI and PID controllers. In many practical settings, these controllers are-and, which is a good thing,can be-tuned and implemented by people with expertise in the application domain, rather than dedicated control systems specialists. Therefore, to reach a wider audience with ADRC, we should adopt a suitable perspective and language to lower existing barriers for application experts. We believe this can be achieved with a frequency-domain view and implementation of linear ADRC.

In recent years, different studies have been tackling the connection of ADRC and transfer functions from various perspectives: [9] on the compensator properties of ADRC,[10] with a PID interpretation, [11] on the internal model control (IMC) representation of ADRC, [12] and [13] on a transfer function representation and a generalization incorporating additional plant model information, and [14] even on the opposite direction, implementing linear controllers with ADRC.

To achieve compatibility with existing traditional solutions in practical applications, simplifying ADRC to a onedegree of freedom (1DOF) error-based transfer function was proposed in [15, 16], with an approximate discrete-time implementation being given as well [17]. In contrast, retaining the original 2DOF characteristics of ADRC and providing exact continuous- and discrete-time transfer function representations are the guiding ideas pursued in this article.

Existing frequency-domain studies have put their focus on loop gain analysis [18-21], disturbance behavior [22, 23],or the modified plant [24], but all without realizable transfer functions [9, 10, 25-29]. Departing from that, and to the best of the author’s knowledge, this work presents the first study putting emphasis on realizable transfer functions, enabling both a comprehensive frequency-domain analysis of ADRC and its implementation at the same time.

1.2 Structure and contributions of this article

To establish the notation and some necessary design equations, a review of linear ADRC is presented in Sect. 2. Afterwards, the main contributions of this article are presented in Sects. 3, 4, and 5:

· Implementing ADRC using transfer functions. Section 3 derives a transfer function representation of continuoustime ADRC, that-in contrast to existing works on this subject-has realizability in mind and seeks for similarity to traditional control loop setups. This will facilitate understanding and implementing ADRC using classical feedback controller and filter structures, with a special focus on first- and second-order ADRC as the contenders to PI and PID controllers.

· Understanding ADRC from a frequency-domain perspective. In Sect. 4, a detailed frequency-domain analysis is performed for continuous-time ADRC; including pole/zero analyses of its feedback controller transfer function and the influence of ADRC’s tuning parameters on the closed-loop behavior. Again, a decidedly classical control systems perspective on ADRC is chosen to break down barriers hindering the adoption of state-space methods in real-world control systems.

· Efficiently implementing discrete-time ADRC using transfer functions. For discrete-time ADRC, an exact transfer function representation is derived in Sect. 5 for the first time, enabling the implementation of discretetime ADRC using digital filter structures. One of the main benefits is a reduction of the computational burden of the control law in comparison to the state-space form,which will increase the attractiveness of ADRC especially in resource-constrained embedded systems.

2 Summary of continuous?time linear ADRC

2.1 Plant model

As an introductory example we will, following related work[2, 30], consider a simple plant withnth order low-pass behavior, outputy, inputu, and disturbance inputd. First,the input gainb=b0+Δbis being split into a knownb0and unknown Δbpart. Then, a generalized disturbance termfis being introduced such that only a disturbednth order integrator chain remains of the original plant model:

Considering any plant to be controlled as a disturbed integrator chain-regardless of the actual plant structure and parameters-is the gist of ADRC, and a distinct departure from model-based approaches [31]. Therefore, apart from the order of the systemn,b0remains as the only parameter that needs to be known for plant modeling in the context of ADRC. Due to its importance,b0is also known as thecritical gain parameter[32].

2.2 Control law

The control law of linear ADRC is based on three ingredients:

A visualization of the observer-based control law of linear ADRC is given in Fig. 1. With a reference signalr(t) and controller outputu(t), the control law can be given as

A Luenberger observer of order (n+1) is being set up in order to estimate the (n) states of the plant model (integrator chain), extended by the estimate+1of the generalized disturbance. In the context of ADRC, it is usually denoted as theextended state observer[31, 33] (ESO):

Fig. 1 Continuous-time statespace implementation of linear ADRC. The parameters for controller and extended state observer (ESO) are given in (5)and (4)

2.3 Controller and observer design

Throughout this article, the “bandwidth parameterization” approach [2] will be used, which is the most common method for designing the controller/observer in the context of linear ADRC. For the controller design we assume full rejection of the total (generalized) disturbance in (1). Therefore, the characteristic polynomial of the closed control loop is being reduced to the following form (state feedback control of annth order integrator plant), and the gainskican be obtained from parameterization with a desired closed-loop bandwidthωCL:

To provide sufficiently fast state estimation and disturbance rejection, the closed-loop poles of the observer must be placed far enough left in thes-plane. We will follow the notation of [8, 30] in bandwidth parameterization of the observer with common poles atkESO·ωCL, withkESObeing the relative factor (typically within the range 3-10) of the observer poles compared to the desired poles of the closed control loop:

Solving (6) and (7) using binominal expansion of leads to the parameterization equations forkTandl. Forn=1 andn=2 , which are the most relevant cases in practice, the controller and observer gains can be found in Table 1. The general terms are given in (8) and (9).

Table 1 Continuous-time controller and observer gains using bandwidth parameterization. The tuning parameters are ωCL (desired closed-loop bandwidth) and kESO (observer bandwidth factor, i.e., relative position of the observer poles compared to the poles of the closed control loop)

3 Transfer function representation

3.1 Approach

Aim of this section is to derive a set of realizable transfer functions that implement ADRC such that a comparison with traditional controller and filter transfer functions becomes easily possible. This will also allow to analyze ADRC with traditional (frequency domain) means. We start with the closed-loop observer dynamics by putting the Laplace transform of (2) in (3):

In this article, we propose a generic approach to represent continuous-time ADRC in frequency domain using three realizable transfer functions, namely a feedback controllerCFB(s) , a reference signal prefilterCPF(s) , and a feedforward of the reference signal to the control signal viaCFF(s) , cf.Fig. 2:

The main benefits of the proposed transfer function approach are

· There is only one transfer function with an integrator component: the feedback controllerCFB(s) , and its position in the control loop is exactly the same as in conventional control loops; and

· by introducing the feedforward termCFF(s) , the reference signal prefilter transfer functionCPF(s) can be made realizable in the form of a conventional lead-lag filter.Otherwise, its numerator polynomial order would exceed the denominator polynomial.

3.2 Derivation of the transfer functions

Fig. 2 Control loop with transfer function based ADRC implementation consisting of feedback controller CFB(s) , reference signal prefilter CPF(s) , and reference signal feedforward CFF(s) . Disturbances (at plant input, d, and plant output, n) are present in order to derive the gang-of-six transfer functions in Sect. 4.3

Table 2 Continuous-time transfer function implementation of first- and second-order ADRC. The α , β , γ coefficients as well as the gains KI and KI∕ln+1 for the first- ( n=1 ) and second-order case ( n=2 ) can be obtained from Table 3 and Table 4, respectively

Table 3 Continuous-time transfer function parameters for first-order ADRC. Bandwidth parameterization is based on b0 (gain parameter of the plant model), ωCL (desired closed-loop bandwidth), and kESO(observer bandwidth factor)

Since (15) is not strictly proper,CFB(s) could not be made realizable withoutCFF(s) . Therefore,CFF(s) will be chosen to carry thesn+1term from the numerator polynomial of (15):

In summary, the three transfer functions required to implement a realizable form of annth order ADRC can be formulated as follows:

Table 4 Continuous-time transfer function parameters for secondorder ADRC. Bandwidth parameterization is based on b0 (gain parameter of the plant model), ωCL (desired closed-loop bandwidth),and kESO (observer bandwidth factor)

Forn=1 andn=2 (i.e., first- and second-order ADRC),the transfer functions as well as their parameters are given in Table 2, Table 3, and Table 4, respectively. Note that the coefficients in Table 3 and Table 4 are given both depending on the design parameterskESOandωCLwhen using bandwidth parameterization, and in general terms using the controller/observer parameterskandl. The latter allows to use alternative tuning approaches such as the recently proposed half-gain tuning for ADRC [34].

4 Frequency?domain analysis

4.1 Plant model: The critical gain parameter in frequency domain

For a frequency-domain interpretation of the important plant model parameterb0(also known as thecritical gain parameter[32]), we will start by rewriting (1) as a frequencydomain transfer function:

Fornth order plants that can be represented by (21), this means that the correspondingb0parameter can be found from the plant’s Bode (magnitude) plot by extending the straight-line approximation of the -n·20 dB/decade segment in order to find its crossover frequencyωX. A visualization of this relation for first- and second-order cases is given in Fig. 3.

4.2 Analysis and comparison of the feedback controller transfer function

4.2.1 Feedback controller of first?order ADRC

Let us consider the feedback controller transfer function from Table 2 using bandwidth parameterization as given in Table 3:

Fig. 3 Frequency-domain illustration of the critical gain parameter b0 for first- (left-hand side) and second-order (right-hand side diagram)plant variations with the same value of b0 (in these examples: b0 =1 ).The critical gain parameter can be obtained from magnitude plots of the plant transfer function by extending the straight-line approximation (red dashed line) of the -20 dB/decade (first-order case) or-40 dB/decade segment (second-order case) in order to find the 0 dB crossover frequency (encircled in the diagrams). The crossover angular frequency amounts to b0 in the first-order case andn the second-order case

This controller structure (PI controller with first-order noise filter, or interpretation as integrator with lead/lag filter) is a common choice for example in power electronics applications, known as “Type 2” controller [35]. It is also in line with the recommendation of H?gglund to add a firstorder noise filter to PI controllers [36].

A frequency-domain visualization of the impact of the tuning parameterkESOon the feedback controller transferfunction, similar to the analysis performed in [27], is given in Fig. 4, while the effect ofb0andωCLonKIand the pole/zero frequencies can be easily grasped from (24). For high observer gains (increasing values ofkESO), the low-pass filter cut-off frequency moves to infinity; therefore, the feedback controller transfer function (24) converges to a standard PI controller with a zero at the desired closed-loop bandwidth:

Fig. 4 Bode plots of the feedback controller CFB of first- and second-order ADRC with variation of kESO using bandwidth parameterization.The exponentially increasing values of kESO are ranging from kESO =1 (purple) through kESO =5 (black) to kESO =25(orange). The remaining tuning parameters are fixed at b0 =1 and ωCL =2π (i.e., 1 Hz)

4.2.2 Feedback controller of second?order ADRC

The feedback controller transfer functionCFB,2from Table 2 can be interpreted as an integrator with second-order lead/lag filter, a structure known as “Type 3” controller in power electronics applications [35]. Alternatively, it can be viewed as a PID controller with a second-order low-pass noise filter,as also pointed out in [10]. As a side note, this structure is recommended for PID by H?gglund [36].

The poles ofCFB,2(s) consist of an integrator (pole at origin) and the poles defined by the roots of (1+α1s+α2s2) .Rewriting the latter part of the denominator polynomial using bandwidth parameterization as given in Table 4 leads to (26).part of second-order ADRC converges to a PID controller. Note that this relation was independently reported very recently in [10], as well.

Table 5 Gang-of-six transfer functions for a closed control loop with plant P(s) and continuous-time ADRC in a transfer function implementation with feedback controller CFB(s) , reference signal prefilter CPF(s) , and feedforward term CFF(s)

The impact ofkESOonCFB,2(jω) is illustrated in Fig. 4. A report on the resulting phase margins can be found in [9].

4.3 Influence of tuning parameters on closed-loop behavior

ADRC is a two-degree-of-freedoms controller, and as such,a control loop with ADRC is only being fully characterized by six transfer functions known as the “gang of six” [38].They relate the inputsr,d,nof a control loop as depicted in Fig. 2 to the signalsyandu, cf. (30). For the realizable transfer function representation of continuous-time ADRC derived in Sect. 3, the gang-of-six transfer functions are given in Table 5.

With the full picture the gang-of-six transfer functions provide, it is possible to gain deeper insights into the tradeoffs that must be resolved when tuning the parameters of ADRC. When the desired closed-loop bandwidthωCLis chosen (depending on the plant dynamics as well as the requirements of the application), there are two questions remaining:How to choose the relative bandwidthkESOof the observer?What happens if the plant model parameterb0is above or below the actual value of the plant, i.e., is it better to over- or to underestimateb0?

4.3.1 Influence of kESO variations

Clearly, higher values ofkESO(i.e., faster observers) help rejecting low-frequency disturbances at plant input (d) and output (n), and maintain the desired dynamics between reference signal (r) and controlled variable (y). On the other hand, a price is to pay in increasing high-frequency gains of transfer functions from input and output disturbances to the control action (u). Especially fromGunit can be seen that the impact of high-frequency measurement noise on the control action will be a limiting factor (upper bound) for selectingkESO.

When tuningkESOit is advisable to start with a singledigit value. From Fig. 5, the following practical guidelines can be inferred:

Table 6 Discrete-time controller and observer gains using bandwidth parameterization. Note that the controller gains do not change compared to the continuous-time version, cf. Table 1; only the observer dynamics have to be discretized [30]. The tuning parameters are T(sample time of the discretized implementation) and zESO =e-kESOωCLT(observer pole locations in z-domain, based on the desired closedloop bandwidth ωCL and the observer bandwidth factor kESO)

4.3.2 Influence of b0 variations

Fig. 5 Sensitivity of the gang-of-six transfer functions to variations of kESO for second-order ADRC. Normalized plant in this example:P(s)= Fixed tuning parameters are b0 =1 and ωCL =0.4π (i.e., 0.2 Hz). The exponentially increasing values of kESO are ranging from kESO =1 (purple) through kESO =5 (black) to kESO =25 (orange)

Fig. 6 Sensitivity of the gang-of-six transfer functions to variations of b0 for second-order ADRC. Normalized plant in this example:P(s)= . Fixed tuning parameters are ωCL =0.4π (i.e.,0.2 Hz) and kESO =5 . The exponentially increasing values of b0 are ranging from b0 =0.2 (purple) through b0 =1 (black, nominal value)to b0 =5 (orange)

Fig. 7 Sensitivity of the gang-of-six transfer functions to an additional plant zero for second-order ADRC, covering both right half plane (RHP) and left half plane plane zeros. Normalized plant in this example (without the additional zero): P(s)= . Fixed tuning parameters are b0 =1 , ωCL =0.4π (i.e., 0.2 Hz), kESO =5 . The time constant of the additional zero ranges from -0.2 s (RHP zeros,purple) to 0.2 s (orange)

Figure 6 shows the effect of under- and overestimatingb0on the closed-loop dynamics using the same exemplary plant,both by a factor of up to five. Overestimatingb0will lead to a less aggressively tuned controller with lower noise sensitivity, of course being also less effective in compensating low-frequency disturbances at the plant input or output. A compromise has to be found forb0since, on the other hand,underestimatingb0-while providing better disturbance rejection at the cost of increased high-frequency noise sensitivity-can induce pronounced medium-frequency oscillations in both control signal (u) and plant output (y), which will eventually lead to instability. Note that the frequency of the oscillations that will occur when underestimatingb0can be read from any of the gang-of-six plots in Fig. 6.

As a guideline when choosingb0in cases its value cannot exactly be inferred from a plant model, the following procedure is recommended in light of the insights provided by the gang-of-six analysis:

· Start with overestimatingb0to obtain a stable loop. A too large value ofb0can be identified by a reduced bandwidth of the closed loop compared to the design valueωCL(cf. orange plots ofGyrin Fig. 6).

· Decreaseb0to bring the closed-loop behavior closer to the intended design value, but only as long as the resulting noise sensitivity of the the control signal is tolerable and oscillations do not appear (cf. purple plots in Fig. 6).Further properties ofb0and its role were extensively studied in [13], also for the case of a generalized ADRC incorporating the full plant model instead of the integrator chain approach of conventional linear ADRC.

4.4 Influence of an additional plant zero

The gang-of-six approach can also be used to analyze the behavior in cases the actual plant exhibits dynamics beyondthe simplified modeling approach of ADRC. As an example with practical relevance, the case of an additional plant zero(right half plane (RHP) or left half plane) will be examined here. This was not covered yet by previous studies such as[30] (which, however, already reported on ADRC’s performance when faced with varying plant parameters such as damping and gain). RHP zeros occur, for instance, in boost or buck/boost-derived DC-DC converters in the power electronics application domain [39].

Fig. 8 Discrete-time state-space implementation of linear ADRC using (31) and (32) for the extended state observer (ESO)

As visible in Fig. 7, especially RHP zeros will induce medium-frequency oscillations, which will be visible in the control signal earlier and more pronounced than in the controlled variable. When becoming too dominant, RHP zeros will finally render the closed loop unstable. On the other hand, unmodeled LHP zeros are quite well tolerated in the range examined here.

5 Discrete?time transfer function implementation

5.1 Discretization of the extended state observer

To establish the notation and introduce the design equations,a brief summary of discrete-time ADRC will be given here.To obtain discrete-time ADRC, only the observer equations must be discretized from their continuous-time counterparts [30]. The state-of-the-art approach for discretizing the extended state observer (ESO) is to employ a current observer approach [33]:

Fig. 9 Control loop with discrete-time transfer function based ADRC implementation consisting of feedback controller CFB(z) and reference signal prefilter CPF(z)

For the cases most relevant to industrial practice (n=1 andn=2 ), the resulting observer gains are given in Table 6.A visualization of the discretized control law is presented in Fig. 8, which is the discrete-time counterpart of the continuous-time case in Fig. 1.

5.2 Derivation of discrete-time transfer functions

We follow a similar approach as in Sect. 3.1 to obtain the transfer function representation of discrete-time ADRC based on the current observer approach from Sect. 5.1,starting with thez-transform of control law and observer dynamics:

In constrast to continuous-time case, which required three transfer functions for a realizable representation of the control law (as introduced in Sect. 3.1), discrete-time discrete-time ADRC can be implemented using using only two transfer functions: a feedback controllerCFB(z) , and a reference signal prefilterCPF(z) (cf. Fig. 9). The approach for the control law is

For a discrete-time implementation of thenth order ADRC, these two transfer functions can be represented using the following coefficients:

Note that thez-1-based numerator polynomial ofCPF,n(z)is of order (n+1) compared to ordernin the denominator.What was a realizability hurdle in the continuous-time case poses no problem in a digital filter implementation, hence only two instead of three transfer functions are necessary here.

Moreover, note that the denominator coefficients of the feedback controllerCFB,n(z) are given as. This shall indicate an intermediate result, since a modification of the feedback controller with additional benefits will be introduced in the following section.

Table 7 Discrete-time transfer function implementation of first- and second-order ADRC.The α , β , γ coefficients for the first- and second-order case can be obtained from Table 8 and Table 9, respectively

Table 8 Discrete-time transfer function parameters for first-order ADRC, given in terms of the controller and discrete-time observer gains (see Table 1 and Table 6, respectively), and in terms of bandwidth tuning parameters. The latter are ωCL (desired closed-loop bandwidth) and zESO =e-kESOωCLT (observer pole locations in z-domain, based on the observer bandwidth factor kESO ). Common parameters are b0 (gain parameter of the plant model) and T (sample time of the discretized implementation)

Table 9 Discrete-time transfer function parameters for second-order ADRC, given in terms of the controller and discrete-time observer gains (see Table 1 and Table 6, respectively), and in terms of bandwidth tuning parameters. The latter are ωCL (desired closed-loop bandwidth) and zESO =e-kESOωCLT (observer pole locations in z-domain, based on the observer bandwidth factor kESO ). Common parameters are b0 (gain parameter of the plant model) and T (sample time of the discretized implementation)

5.3 Feedback controller with factored-out accumulator

In a transfer function implementation of annth order ADRC,the feedback controller transfer function includes an integrator. Therefore,CFB,n(z) will have a pole atz=1 , and we can factor this discrete-time accumulator out:

The benefits of implementing the feedback controller transfer functionCFB,n(z) with a factored-out accumulator are

· Compared to a state-space implementation of ADRC,fewer multiplications in the control law are required,decreasing the computational burden of the control algorithm to be implemented (first-order: 7 instead of 11, second-order: 11 instead of 19; and this is already using the matrices given in (32) for the observer (31)).Additionally, factoring out the accumulator saves oneαcoefficient to be stored for the transfer function implementation.

· With the factored-out accumulator, a straightforward solution is possible for clamping the controller output:the accumulator has to be simply replaced by a clamped accumulator, thereby avoiding windup issues [40].

· The use of one common accumulator allows to switch between different controllers easily [40], for example when using an “override control” scheme [41] with two controllers acting on the same plant input.

· A better equivalence to the continuous-time case is achieved, where the feedback controller transfer function includes a factored-out integrator as well, cf. (20).

5.4 Summary

Discrete-time linear ADRC can be implemented using transfer functions with the following control law:

where the feedback controller transfer functionCFB(z) and the reference signal prefilterCPF(z) for thenth order ADRC are given as

For the first- and second-order ADRC, the discrete-time transfer functions are given again in detail in Table 7, while the accordingα,β, andγcoefficients can be found in Table 8 and Table 9.

6 Conclusions

To foster a more widespread adoption of linear ADRC in industrial practice, this article has made three contributions.First, a representation of ADRC using realizable transfer functions was introduced. This allows to compare ADRC to an existing “classical” solution more easily, for example by analyzing the pole/zero placement of the feedback controller part.

Second, based on the transfer function implementation,a detailed frequency-domain analysis of linear ADRC was performed, including plant modeling and the feedback controller part. A frequency-domain inspection of the tuning parameter impact on the “gang-of-six” transfer functions of a closed loop with ADRC supports the tuning process,e.g., when looking for a compromise between low-frequency disturbance rejection and high-frequency noise sensitivity.

Finally, an exact and low-footprint transfer function realization of discrete-time linear ADRC was derived, with ready-to-use coefficient tables given for first- and secondorder ADRC - the most relevant contenders to existing PI and PID controller solutions. With the results presented in this article, an efficient implementation of linear ADRC even on low-cost target hardware should have become a low-effort and straightforward task.