Erschienen in:

Open Access 2025 | OriginalPaper | Buchkapitel

5. Visual Tour

verfasst von : Gernot Herbst, Rafal Madonski

Erschienen in: Active Disturbance Rejection Control

Verlag: Springer Nature Switzerland

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Abstract

A picture is worth a thousand words. This chapter aims at analyzing and showcasing ADRC in a predominantly visual manner, in both time and frequency domains, going from modeling to tuning and its closed-loop behavior. What can one expect from ADRC regarding tracking performance and disturbance rejection? What is the influence of its modeling and tuning parameters? And what if we set them too low or too high?

5.1 Plant Modeling

5.1.1 Obtaining b₀ in Time Domain

We started introducing the core concepts of ADRC in Sect. 3.1 with its simple plant model (3.2) that—apart from the plant order N—only has one parameter b₀, the critical gain parameter. In this section, we want to demonstrate how b₀ can experimentally be obtained given a step response of a plant following a stepwise change in its input signal u(t) = u_∞⋅ σ(t), where σ(t) is the Heaviside step function and u_∞ the steady-state value of the input—if applicable, relative to its initial value. This will be shown using three simple examples:

Plant with pure integrator behavior. If the plant can be described as an integrator with a gain value K_I,
$$\displaystyle \begin{aligned} P(s) = \frac{K_{\mathrm{I}}}{s} = \frac{b_0}{s} , \end{aligned}$$
its reaction to an input signal u(t) = u_∞⋅ σ(t) will be
$$\displaystyle \begin{aligned} y(t) = K_{\mathrm{I}} \cdot t \cdot u_\infty \cdot \sigma(t) = b_0 \cdot t \cdot u_\infty \cdot \sigma(t) . \end{aligned}$$

As shown in case (a) of Fig. 5.1, b₀ can be computed from the normalized slope:
$$\displaystyle \begin{aligned} b_0 = \frac{\varDelta y}{\varDelta t \cdot u_\infty} . \end{aligned} $$

(5.1)

Fig. 5.1
Step responses of three simple, commonly seen plant types: (a) integrator, (b) integrator plus first-order lag, and (c) first-order low-pass. Using the specified characteristics, the critical gain parameter b₀ can be computed from such a visual plant description that might be obtained from experiments or simulations

×
Plant with integrator plus first-order lag behavior. If a plant has integrator behavior with a first-order lag (time constant T)
$$\displaystyle \begin{aligned} P(s) = \frac{K_{\mathrm{I}}}{s} \cdot \frac{1}{T s + 1} = \frac{\frac{K_{\mathrm{I}}}{T}}{s^2 + \frac{1}{T} s} = \frac{b_0}{s^2 + a_1 s} , \end{aligned}$$
the reaction to an input signal u(t) = u_∞⋅ σ(t) will be
$$\displaystyle \begin{aligned} y(t) = K_{\mathrm{I}} \cdot \left( t - T \cdot \left( 1 - \mathrm{e}^{-\frac{t}{T}} \right) \right) \cdot u_\infty \cdot \sigma(t) , \end{aligned}$$
which, for t ≫ T, approaches a ramp:
$$\displaystyle \begin{aligned} y(t \gg T) \approx K_{\mathrm{I}} \cdot (t - T) \cdot u_\infty . \end{aligned}$$

As shown in case (b) of Fig. 5.1, b₀ can then be computed from the normalized slope and the value of T read off the intersection of the ramp with the time axis:
$$\displaystyle \begin{aligned} b_0 = \frac{K_{\mathrm{I}}}{T} = \frac{\varDelta y}{\varDelta t \cdot u_\infty \cdot T} . \end{aligned} $$

(5.2)
Plant with first-order low-pass behavior. For a self-regulating plant with first-order lag behavior (gain K and time constant T)
$$\displaystyle \begin{aligned} P(s) = \frac{K}{T s + 1} = \frac{\frac{K}{T}}{s + \frac{1}{T}} = \frac{b_0}{s + a_0} , \end{aligned}$$
the reaction to an input signal u(t) = u_∞⋅ σ(t) will be
$$\displaystyle \begin{aligned} y(t) = K \cdot \left( 1 - \mathrm{e}^{-\frac{t}{T}} \right) \cdot u_\infty \cdot \sigma(t) . \end{aligned}$$

There are two options to compute b₀. The first one uses K obtained from the steady-state value, $K = \frac {y_\infty }{u_\infty }$, and T from a tangent to y(t) as depicted in Fig. 5.1, case (c). The value of b₀ then simply results from
$$\displaystyle \begin{aligned} b_0 = \frac{K}{T} . \end{aligned} $$

(5.3)

The second option is to evaluate the step response at its onset, i.e., for t ≪ T. At its beginning, y(t) can be approximated by a linear response,
$$\displaystyle \begin{aligned} y(t) \approx \frac{K}{T} \cdot t \cdot u_\infty = b_0 \cdot t \cdot u_\infty ,\quad 0 \le t \ll T ; \end{aligned}$$
hence b₀ can be computed from the normalized slope:
$$\displaystyle \begin{aligned} b_0 = \frac{\varDelta y}{\varDelta t \cdot u_\infty} . \end{aligned} $$

(5.4)

This can be extended to more complex plant dynamics, notably with second-order low-pass behavior, for which numerous identification methods exist to experimentally obtain their parameters. For ADRC, these can then be evaluated to compute b₀.

5.1.2 Understanding b₀ in Frequency Domain

Let us now turn to the frequency domain for a different interpretation of b₀ and another option to extract it from experimentally obtained data. The plant model (3.1) can, without the disturbance input, be given as a frequency-domain transfer function:

$$\displaystyle \begin{aligned} P(\mathrm{j}\omega) = \frac{y(\mathrm{j}\omega)}{u(\mathrm{j}\omega)} = \frac{ b_0 }{ (\mathrm{j}\omega)^N + a_{N - 1} \cdot (\mathrm{j}\omega)^{N - 1} + \ldots + a_1 \cdot \mathrm{j}\omega + a_0 } . {} \end{aligned} $$

(5.5)

For higher frequencies, the denominator polynomial becomes dominated by its highest order term, and one can approximate

$$\displaystyle \begin{aligned} \lim_{\omega \to \infty} \left| P(\mathrm{j}\omega) \right| \approx \left| \frac{ b_0 }{ (\mathrm{j}\omega)^N } \right| = \frac{ b_0 }{ \omega^N } . \end{aligned}$$

Solving this high-frequency approximation for its crossover frequency ω_X leads us to a relation between ω_X and b₀:

$$\displaystyle \begin{aligned} \left| P(\mathrm{j}\omega_{\mathrm{X}}) \right| = 0\,\mathrm{dB} = 1 \quad \text{yields}\quad b_0 = \omega_{\mathrm{X}}^N \quad \text{or}\quad \omega_{\mathrm{X}} = \sqrt[N]{b_0} . {} \end{aligned} $$

(5.6)

If an Nth-order plant can be modeled by (5.5), b₀ can therefore be found from a Bode magnitude plot of $\left |P(\mathrm {j}\omega )\right |$ by approximating the upper-frequency range with an Nth-order integrator, i.e., a straight line. The 0 dB crossover frequency ω_X of this approximation then gives b₀ via (5.6). Various examples of first- and second-order plants with the same approximation to obtain b₀ are shown in Fig. 5.2.

5.2 Closing the Loop: A Nominal Example

In a simple simulation example, which will serve as the nominal case of further examples studied in this chapter, a normalized, critically damped second-order plant with a transfer function $P(s) = \frac {1}{s^2 + 2s + 1}$ shall be controlled. To do so, let us make use of the knowledge gained so far in the book and follow the “cooking recipe” for ADRC given on page 42:

Plant modeling: From P(s) we obtain the plant order N = 2, which means we need a second-order ADRC. Recalling the examples in Table 3.1, we also determine the critical gain parameter as b₀ = 1.

Controller tuning: Since we are not given certain design goals or restrictions, we arbitrarily select a bandwidth of 0.2 Hz, i.e., ω_CL = 2π ⋅ 0.2 rad∕s for this example. According to (3.21), this should result in a closed-loop settling time T_settle,98% ≈ 5 s. The controller gains k_1∕2 are then computed using the general design equation for bandwidth parameterization given by (3.19) or through consulting Table A.1.1.

Observer tuning: As a reasonable starting point, we select a relative observer bandwidth factor k_ESO = 5. To compute the observer gains l_1∕2∕3, we put ω_CL and k_ESO into the general design equation (3.24) or once more refer to Table A.1.1.

Implementation: In our model-based implementation, we use the state-space form of linear ADRC depicted in Fig. 3.5. We refer to this form as “standard” ADRC, whenever contrasting it with other or modified forms. It is the origin of all variants and implementations covered in this book.

Let us conduct a first test run. Note that the control loop for our simulation example, shown in Fig. 5.3, now includes additional inputs to model the effect of disturbances at the plant input (usually “load disturbance,” d) and plant output (measurement noise, n). At time t = 0, a stepwise change of the reference signal r(t) from r = 0 to r = 1 occurs, followed by a stepwise change of the input disturbance d(t) from d = 0 to d = 0.5 at t = 10 s. ¹

From the results shown in Fig. 5.4, one can nicely observe the closed-loop tracking behavior of y(t) exhibiting the desired settling time. After t = 10 s, the controller output includes the efforts required to compensate for the persisting input disturbance. Remember that the control law (3.15) is $u(t) = \frac {1}{b_0} \cdot \left ( k_1 \cdot r(t) - \begin {pmatrix} \boldsymbol {k}^{\mathrm {T}} & 1 \end {pmatrix} \cdot \hat {\boldsymbol {x}}(t) \right )$. It is therefore very interesting to monitor the observer state variables $\hat {\boldsymbol {x}}^{\mathrm {T}}(t) = \begin {pmatrix} \hat {x}_1(t) & \hat {x}_2(t) & \hat {x}_3(t) \end {pmatrix}$, with $\hat {x}_3(t)$ being the estimated total disturbance $\hat {f}(t)$:

First phase, $0\,\mathrm {s} \le t \lessapprox 5\,\mathrm {s}$: During the transient following the reference step, all estimated states contribute to the controller output u(t).

Fig. 5.4
Simulation example of the control loop from Fig. 5.3, showing the reaction of controlled variable y(t), controller output u(t), and observer state variables $\hat {x}_{1/2/3}(t)$ to a reference step and input disturbance step

×
Second phase, $5\,\mathrm {s} \lessapprox t < 10\,\mathrm {s}$: In the settled state, only the estimated total disturbance contributes to u(t), as $r = y = \hat {x}_1$ and $\hat {x}_2 = 0$. The state variable $\hat {x}_3$ acts as the integrator state of this controller.
Third phase, 10 s ≤ t: When the input disturbance d = 0.5 becomes active at t = 10 s, $\hat {x}_3(t)$ changes to include the estimated input disturbance, such that u(t) can compensate d(t).

We have now successfully applied a “cooking recipe” to implement and tune ADRC and gained insights into the role of the observer state variables during reference tracking and disturbance rejection. In the remainder of this chapter, we will build on this example several times to demonstrate the influence of ADRC’s tuning parameters and uncertainties to be expected in a not-so-well-known plant.

Please note that, in this nominal example, the closed loop is (approximately) only as slow or fast as the open-loop plant. This is a conscious decision on our side, as we will need some headroom to compare this design to faster ones and with other varying parameters.

5.3 The Role of ADRC’s Tuning Parameters

5.3.1 Preliminaries: The Gang of Six

Our focus in this chapter is to provide visual answers to “What happens (to the controller/control loop) if …” questions, in order to create some intuition for the tuning parameters, the abilities, and the limitations of ADRC.

A block diagram of the control loop from Fig. 5.3, now in its realizable transfer function representation introduced in Sect. 4.2.2, is provided in Fig. 5.5. When answering the “What happens if …” questions, the influence of the three possible input signals (reference r, load disturbance d, and measurement noise n) in Fig. 5.5 on both the controlled variable y and the controller output u will be of interest. We can give a compact representation of all possible transfer functions as follows, which are known as the gang of six:

$$\displaystyle \begin{aligned} \begin{pmatrix} y(s) \\ u(s) \end{pmatrix} = \begin{pmatrix} G_{\mathrm{yr}}(s) & G_{\mathrm{yd}}(s) & G_{\mathrm{yn}}(s) \\ G_{\mathrm{ur}}(s) & G_{\mathrm{ud}}(s) & G_{\mathrm{un}}(s) \end{pmatrix} \cdot \begin{pmatrix} r(s) \\ d(s) \\ n(s) \end{pmatrix}. {} \end{aligned} $$

(5.7)

The gang of six transfer functions can be computed from any transfer function representation of ADRC. Table 5.1 gives the gang of six in general form if the realizable representation from Sect. 4.2.2 is to be used. The particular transfer functions C_FB(s), C_PF(s), and C_FF(s) have then to be inserted. In the remainder of this chapter, we will make use of the gang of six, showing some or all of these transfer functions in frequency domain (mostly as Bode magnitude plots) or time domain in the form of step responses.

Table 5.1

General form of the gang of six transfer functions for a control loop with a plant P(s) and continuous-time ADRC using the realizable transfer function implementation consisting of feedback controller C_FB(s), reference signal prefilter C_PF(s), and feedforward C_FF(s). Note that we omitted “(s)” from these transfer functions in the table only for lack of space

	From: r (reference signal)	From: d (disturbance)	From: n (noise)
To: y	$\displaystyle G_{\mathrm {yr}}(s) = \frac {P \cdot \left ( C_{\mathrm {FF}} + C_{\mathrm {FB}} C_{\mathrm {PF}} \right )}{1 + P C_{\mathrm {FB}}}$	$\displaystyle G_{\mathrm {yd}}(s) = \frac {P}{1 + P C_{\mathrm {FB}}}$	$\displaystyle G_{\mathrm {yn}}(s) = \frac {1}{1 + P C_{\mathrm {FB}}}$
To: u	$\displaystyle G_{\mathrm {ur}}(s) = \frac {C_{\mathrm {FF}} + C_{\mathrm {FB}} C_{\mathrm {PF}}}{1 + P C_{\mathrm {FB}}}$	$\displaystyle G_{\mathrm {ud}}(s) = \frac {-P C_{\mathrm {FB}}}{1 + P C_{\mathrm {FB}}}$	$\displaystyle G_{\mathrm {un}}(s) = \frac {-C_{\mathrm {FB}}}{1 + P C_{\mathrm {FB}}}$

5.3.2 On Bandwidth: Reference Tracking Versus Disturbance Rejection

As described in Sects. 3.2.1 and 3.2.2, the process of tuning ADRC involves parameterization of an outer state-feedback controller and of an observer at its core, the latter providing estimated states for both the state-feedback controller and disturbance rejection. Using the bandwidth parameterization approach, this task is reduced to choosing a closed-loop bandwidth ω_CL and a relative bandwidth factor k_ESO for the observer.

In this section, we want to shed some light on the interplay of these two parameters and their impact on the tracking performance and disturbance rejection. We will do so with parameter sweeps of ω_CL and k_ESO in second-order simulation examples for the normalized plant $P(s) = \frac {1}{s^2 + 2s + 1}$ already introduced in Sect. 5.2. The default tuning and modeling parameters for ADRC with N = 2 in these examples remain the same with ω_CL = 2π ⋅ 0.2 rad∕s and k_ESO = 5, as well as b₀ = 1.

A first set of simulation results is shown in Fig. 5.6: reactions to reference signal and input disturbance steps, i.e., step responses of G_yr(s) and G_yd(s) from the gang of six. Clearly, increasing the bandwidth ω_CL results in both improved tracking performance and better disturbance rejection at the same time. ²

A second set of examples presented in Fig. 5.7 fixes ω_CL and varies k_ESO, i.e., only the observer speed. As visible from the step responses of G_yr(s), the tracking performance suffers and does not meet the desired design goals (settling time) if k_ESO is chosen significantly below k_ESO = 5. On the other hand, increasing k_ESO beyond the single-digit range does not endlessly improve the tracking performance, as the controller can obviously not do more than perfectly achieving the design goal. However, the impact of the input disturbance d(t) on the controlled variable y(t) is obviously reduced increasingly better with larger values of k_ESO.

What can we learn from these examples?

Tracking and rejection are both influenced by ω_CL and k_ESO. This means that ADRC, as introduced in Chap. 3, does not allow for a true two-degrees-of-freedom (2DOF) design with separate goals for tracking and disturbance rejection. This impression could, however, have arisen earlier when we introduced the transfer function representation in Sect. 4.2. Structurally, ADRC is a 2DOF controller, but the two design goals are intertwined in the tuning parameters. Note that this does not depend on a particular tuning approach, such as bandwidth parameterization. As can be seen in the coefficients of the transfer function representation in Appendix A.2, all transfer functions depend on both controller and observer gains. Modifying these gains therefore affects the performance of reference tracking and disturbance rejection at the same time.
Is it possible yet to conclude on a best choice for ω_CL and k_ESO? Harder, better, faster, stronger? The obvious benefits of increased bandwidth must be paid for with increased feedback gains, making the controller and observer more susceptible to measurement noise and stability problems. In Sect. 5.3.3 we will make this compromise much more apparent when analyzing the influence of the observer bandwidth in greater detail.

5.3.3 Influence of the Observer Bandwidth

Gang-of-Six Analysis

In Sect. 3.1, we explicated the central role of the observer within ADRC. To somewhat decouple its tuning procedure from that of the controller, the tuning factor k_ESO was introduced within the bandwidth parameterization approach outlined in Sect. 3.2.2, determining the relative bandwidth of the observer compared to the closed control loop. Its influence on ADRC’s performance will be examined in the following. This will also help answering the question raised in Sect. 5.3.2: What are the downsides of using a large relative observer bandwidth k_ESO, when the benefits of doing so appear so obvious?

To provide a full picture of k_ESO’s influence on the closed loop dynamics, a frequency-domain analysis of the gang of six transfer functions will be performed, with a parameter sweep of k_ESO for each of the transfer functions. This will be once more exemplified based on the nominal example described in Sect. 5.2, i.e., for a normalized plant $P(s) = \frac {1}{s^2 + 2s + 1}$ and second-order ADRC tuned with ω_CL = 2π ⋅ 0.2 rad∕s, k_ESO = 5, and b₀ = 1.

The results (Bode magnitude plots) are presented in Fig. 5.8. Note that the step responses previously shown in Fig. 5.7 are the time-domain counterparts of $\left | G_{\mathrm {yr}}(\mathrm {j}\omega ) \right |$ and $\left | G_{\mathrm {yd}}(\mathrm {j}\omega ) \right |$ in Fig. 5.8, which confirm the positive impact of increasing k_ESO values on achieving the desired closed-loop bandwidth and on rejecting low-frequency disturbances. The biggest price to be paid for these benefits can be seen from $\left | G_{\mathrm {un}}(\mathrm {j}\omega ) \right |$: With increasing k_ESO, the controller output u will be increasingly susceptible to high-frequency disturbances.

We have not yet elaborated on the influence and treatment of measurement noise—this topic will be covered in Sect. 9.3—but nonetheless hope all readers agree on our view by now that measurement noise will be inevitable in any practical setting. In the particular example of Fig. 5.8, increasing k_ESO from 5 to 25 (i.e., by a factor of 5) results in almost a 40 dB gain increase in $\left | G_{\mathrm {un}}(\mathrm {j}\omega ) \right |$ for high frequencies. With k_ESO = 25, the noise level in the controller output would be larger by almost two orders of magnitude. In practical settings, this can be a hard restriction on the range of possible values for k_ESO, and this problem becomes worse with increasing ADRC order N.

The compromise one has to find between good tracking performance and low-frequency disturbance rejection on the one hand, and susceptibility to high-frequency disturbances on the other, is, of course, nothing new: “Sensitivity reduction at low frequency unavoidably leads to sensitivity increase at higher frequencies.” ³

Frequency-Domain Analysis of the Feedback Controller

As visible from Fig. 5.5, the feedback controller C_FB(s) in the transfer function representation of ADRC plays a central role in stability, disturbance rejection, and susceptibility to measurement noise. The “what if” question in this section therefore is: What happens to the feedback controller C_FB(s) if one increases or decreases the tuning parameter k_ESO? The visual answer is given in the form of Bode plots in Fig. 5.9 for the first- and second-order cases.

To interpret these Bode plots, recall the transfer functions for the first- and second-order ADRCs from (4.22) in Sect. 4.2:

$$\displaystyle \begin{aligned} C_{{\mathrm{FB,1}}}(s) = \frac{K_{\mathrm{I}}}{s} \cdot \frac{ 1 + \beta_1 s }{ 1 + \alpha_1 s } \quad \text{and}\quad C_{{\mathrm{FB,2}}}(s) = \frac{K_{\mathrm{I}}}{s} \cdot \frac{ 1 + \beta_1 s + \beta_2 s^2 }{ 1 + \alpha_1 s + \alpha_2 s^2 } . \end{aligned}$$

Obviously the C_FB(s) transfer functions consists of an integrator $\frac {K_{\mathrm {I}}}{s}$ in series with Nth-order lead-lag filters. From Fig. 5.9, one can recognize that the latter only act as lead filters, increasingly providing phase boost (not exactly) around ω_CL for larger values of k_ESO. These feedback controllers are the “Type 2” and “Type 3” controllers, already mentioned in Sect. 4.2.1, which are, in turn, only special cases of PI and PID controllers with an additional first- or second-order low-pass filter, respectively. Note that the existence of this low-pass component can be attributed to the choice of using a full-order observer in the variants of ADRC considered in this book: The outer controller feeds back the estimated value $\hat {y}$ of the plant output y, even though the latter is measured and could be used directly, leading to a reduced-order observer. Estimating $\hat {y}$, on the other hand, comes with low-pass filter characteristics, which are beneficial for reducing the impact of measurement noise.

Let us now go into some more detail for first-order ADRC by putting the coefficients for bandwidth parameterization from Table A.2.2 in C_FB,1(s):

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-031-72687-3_5/MediaObjects/617308_1_En_5_Equ8_HTML.png

upper C Subscript upper F upper B comma 1 Baseline left parenthesis s right parenthesis equals ModifyingBelow StartFraction 1 Over b 0 EndFraction dot StartFraction k Subscript upper E upper S upper O Superscript 2 Baseline omega Subscript upper C upper L Superscript 2 Baseline Over 1 plus 2 k Subscript upper E upper S upper O Baseline EndFraction With bottom brace Underscript upper K Subscript normal upper I Baseline Endscripts dot StartFraction 1 Over s EndFraction dot StartFraction 1 plus ModifyingAbove StartFraction 2 plus k Subscript upper E upper S upper O Baseline Over k Subscript upper E upper S upper O Baseline omega Subscript upper C upper L Baseline EndFraction With top brace Overscript beta 1 equals 1 divided by omega Subscript normal upper Z Baseline Endscripts dot s Over 1 plus ModifyingBelow StartFraction 1 Over omega Subscript upper C upper L Baseline dot left parenthesis 1 plus 2 k Subscript upper E upper S upper O Baseline right parenthesis EndFraction With bottom brace Underscript alpha 1 equals 1 divided by omega Subscript normal upper P Baseline Endscripts dot s EndFraction period

(5.8)

It may become increasingly clear why we focus on analyzing the influence of k_ESO on C_FB(s). Changing b₀, as part of K_I, merely moves the magnitude plots up or down, while changing ω_CL moves the pole and zero frequencies ω_P and ω_Z to the left or the right. Both visually from Fig. 5.9 and analytically from (5.8), one can make the following observations regarding k_ESO:

As part of K_I, increasing/decreasing k_ESO moves the magnitude plot up/down. For larger values of k_ESO, this approximately becomes a proportional influence:
$$\displaystyle \begin{aligned} \lim_{k_{\mathrm{ESO}} \to \infty} K_{\mathrm{I}} = \lim_{k_{\mathrm{ESO}} \to \infty} \frac{1}{b_0} \cdot \frac{ k_{\mathrm{ESO}}^2 \omega_{\mathrm{CL}}^2 }{ 1 + 2 k_{\mathrm{ESO}} } = \frac{1}{b_0} \cdot \frac{ k_{\mathrm{ESO}} \omega_{\mathrm{CL}}^2 }{ 2 } . \end{aligned}$$
With increasing values of k_ESO, the angular frequency ω_Z of the zero in C_FB,1(s) converges to the closed-loop bandwidth ω_CL:
$$\displaystyle \begin{aligned} \lim_{k_{\mathrm{ESO}} \to \infty} \omega_{\mathrm{Z}} = \lim_{k_{\mathrm{ESO}} \to \infty} \frac{ k_{\mathrm{ESO}} \omega_{\mathrm{CL}} }{ 2 + k_{\mathrm{ESO}} } = \omega_{\mathrm{CL}} . \end{aligned}$$
With increasing values of k_ESO, the angular frequency ω_P of the pole in C_FB,1(s), i.e., the cutoff frequency of its low-pass filter component, goes to infinity:
$$\displaystyle \begin{aligned} \lim_{k_{\mathrm{ESO}} \to \infty} \omega_{\mathrm{P}} = \lim_{k_{\mathrm{ESO}} \to \infty} \omega_{\mathrm{CL}} \cdot (1 + 2 k_{\mathrm{ESO}}) \to \infty . \end{aligned}$$

Since the passband of the low-pass filter grows with k_ESO, the influence of high-frequency measurement noise on the controller output will increase—a behavior we could already spot in $\left | G_{\mathrm {un}} \right |$ of Fig. 5.8.

We can summarize these observations by stating that, for k_ESO →∞, the feedback controller C_FB,1(s) of LADRC would converge to a standard PI controller only consisting of an integrator and a zero at − ω_CL. A similar analysis can be performed for the feedback controller component of second-order ADRC:

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-031-72687-3_5/MediaObjects/617308_1_En_5_Equn_HTML.png

StartLayout 1st Row 1st Column upper C Subscript upper F upper B comma 2 Baseline left parenthesis s right parenthesis equals 2nd Column ModifyingBelow StartFraction 1 Over b 0 EndFraction dot StartFraction k Subscript upper E upper S upper O Superscript 3 Baseline omega Subscript upper C upper L Superscript 3 Baseline Over 1 plus 6 k Subscript upper E upper S upper O Baseline plus 3 k Subscript upper E upper S upper O Superscript 2 Baseline EndFraction With bottom brace Underscript upper K Subscript normal upper I Endscripts dot StartFraction 1 Over s EndFraction 2nd Row 1st Column Blank 2nd Column dot StartFraction 1 plus ModifyingAbove StartFraction 1 Over omega Subscript upper C upper L Baseline EndFraction dot left parenthesis StartFraction 3 Over k Subscript upper E upper S upper O Baseline EndFraction plus 2 right parenthesis With top brace Overscript beta 1 Endscripts dot s plus ModifyingAbove StartFraction 1 Over omega Subscript upper C upper L Superscript 2 Baseline EndFraction dot left parenthesis StartFraction 3 Over k Subscript upper E upper S upper O Superscript 2 Baseline EndFraction plus StartFraction 6 Over k Subscript upper E upper S upper O Baseline EndFraction plus 1 right parenthesis With top brace Overscript beta 2 Endscripts dot s squared Over 1 plus ModifyingBelow StartFraction 2 plus 3 k Subscript upper E upper S upper O Baseline Over omega Subscript upper C upper L Baseline dot left parenthesis 1 plus 6 k Subscript upper E upper S upper O Baseline plus 3 k Subscript upper E upper S upper O Superscript 2 Baseline right parenthesis EndFraction With bottom brace Underscript alpha 1 Endscripts dot s plus ModifyingBelow StartFraction 1 Over omega Subscript upper C upper L Superscript 2 Baseline dot left parenthesis 1 plus 6 k Subscript upper E upper S upper O Baseline plus 3 k Subscript upper E upper S upper O Superscript 2 Baseline right parenthesis EndFraction With bottom brace Underscript alpha 2 Endscripts dot s squared EndFraction period EndLayout

Similar to C_FB,1(s), C_FB,2(s) consists of an integrator and a lead-lag filter, this time of order 2. Solving the roots of the numerator and denominator polynomials of the latter gives a pair of conjugate complex poles s_Z1∕2 (for the zeros) and s_P1∕2 (for the poles):

$$\displaystyle \begin{aligned} s_{\mathrm{Z1/2}} = \frac{ k_{\mathrm{ESO}} \omega_{\mathrm{CL}} }{ 3 + 6 k_{\mathrm{ESO}} + k_{\mathrm{ESO}}^2 } \cdot \left[ -\left( \frac{3}{2} + k_{\mathrm{ESO}} \right) \pm \mathrm{j} \sqrt{ \frac{3}{4} + 3 k_{\mathrm{ESO}} } \right] ,\quad \text{and} \end{aligned}$$

$$\displaystyle \begin{aligned} s_{\mathrm{P1/2}} = \omega_{\mathrm{CL}} \cdot \left[ -\left( 1 + \frac{3}{2} k_{\mathrm{ESO}} \right) \pm \mathrm{j} \sqrt{ 3 k_{\mathrm{ESO}} + \frac{3}{4} k_{\mathrm{ESO}}^2 } \right] . \end{aligned}$$

As for the first-order case, the influence of k_ESO on the integrator gain and the position of zeros and poles of C_FB,2(s) can now be analyzed:

Increasing or decreasing k_ESO moves the magnitude plot up or down. This influence converges to a linear one for large values of k_ESO:
$$\displaystyle \begin{aligned} \lim_{k_{\mathrm{ESO}} \to \infty} K_{\mathrm{I}} = \lim_{k_{\mathrm{ESO}} \to \infty} \frac{1}{b_0} \cdot \frac{ k_{\mathrm{ESO}}^3 \omega_{\mathrm{CL}}^3 }{ 1 + 6 k_{\mathrm{ESO}} + 3 k_{\mathrm{ESO}}^2 } = \frac{1}{b_0} \cdot \frac{ k_{\mathrm{ESO}} \omega_{\mathrm{CL}}^3 }{ 2 } . \end{aligned}$$
With increasing values of k_ESO, the two zeros converge to a pair of real-valued zeros with a common angular frequency at the closed-loop bandwidth ω_CL:
$$\displaystyle \begin{aligned} \lim_{k_{\mathrm{ESO}} \to \infty} s_{\mathrm{Z1/2}} = -\omega_{\mathrm{CL}} . \end{aligned}$$
The poles s_P1∕2 remain conjugate complex for increasing values of k_ESO, with an approximately constant damping ratio that starts at D ≈ 0.791 for k_ESO = 1 and relatively quickly converges to $D = \sqrt {3/4} \approx 0.866$. ⁴ However, at the same time, the real part of s_P1∕2 moves toward minus infinity. This means that high-frequency measurement noise will increasingly affect the controller output.

Summarizing these findings we can therefore conclude that the feedback controller C_FB,2(s) of LADRC converges to a standard PID controller with k_ESO →∞.

5.3.4 Influence of the Critical Gain Parameter

In Sect. 3.1, we have introduced b₀ as the critical gain parameter—which is, besides the plant order N, the only modeling information required for ADRC. We have presented several ways to obtain b₀ apart from a mathematical model in Sect. 5.1. But even if modeling efforts are largely reduced that way, what happens if one over- or underestimates b₀?

As in Sect. 5.3.3, we want to give a visual answer to this question by applying a parameter sweep of b₀ to the gang of six transfer functions. This will, again, be based on the nominal example from Sect. 5.2, i.e., for a normalized plant $P(s) = \frac {1}{s^2 + 2s + 1}$ and second-order ADRC tuned with ω_CL = 2π ⋅ 0.2 rad∕s and k_ESO = 5. The exact value of b₀ is therefore b₀ = 1.

Figure 5.10 contains the Bode magnitude plots of the gang of six and Fig. 5.11 selected time-domain step responses to complement their interpretation. We can summarize these results in two major takeaways:

Overestimating b₀ will result in undermatching the desired bandwidth. This means that the tracking performance will suffer, and disturbances will be compensated more slowly.

Fig. 5.10
Influence of under- or overestimating b₀ on the gang of six transfer functions, in this example for second-order ADRC. Normalized plant: $P(s) = \frac {1}{s^2 + 2s + 1}$. Values of b₀ are logarithmically spaced, ranging from b₀ = 0.2 (blue, underestimation) through b₀ = 1 (thick black line, nominal value) to b₀ = 5 (red, overestimation). Other tuning parameters remain fixed at ω_CL = 2π ⋅ 0.2 rad∕s (i.e., 0.2 Hz) and k_ESO = 5

Fig. 5.11
Closed-loop responses to stepwise changes of the reference signal r and the input disturbance d, i.e., step responses of four of the six transfer functions from Fig. 5.10. Values of b₀ are logarithmically spaced, ranging from b₀ = 0.2 (blue, underestimation) through b₀ = 1 (thick black line, nominal value) to b₀ = 5 (red, overestimation)

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Underestimating b₀, on the other hand, will result in a more aggressively tuned controller. This could already be expected from the $\frac {1}{b_0}$ gain factor in the feedback controller transfer function C_FB(s). In the gang of six magnitude plots, one can recognize the effects of this: increased sensitivity of the controller output u to measurement noise n, and an increasingly pronounced resonance peak emerging in all closed-loop transfer functions. One can particularly well observe this in the time-domain step responses of the controller output u(t), where oscillations (in these examples roughly around 2 Hz) are increasingly dominating the response for significantly underestimated values of b₀. Ultimately, severe underestimation of b₀ will therefore result in instability.

5.4 Coping with Parametric and Structural Plant Uncertainties

In Sect. 5.3.4 we have already seen what happens if the estimated value of b₀ does not match the actual value of the plant. However, plant dynamics are not only dependent on this single, lumped parameter—others were just discarded in the process of ADRC’s simplified plant modeling approach. Besides unexpected parameter changes, there may also be unmodeled dynamics in a plant—such as additional poles or zeros. To develop an intuition of how ADRC will handle such cases, we will perform a deeper analysis based on time-domain examples in two parts in the following.

5.4.1 Variation of Plant Parameters

One can view b₀ as a lumped estimation of some of a plant’s parameters, cf. Table 3.1. What happens if individual plant parameters diverge from this estimation, regardless of if they are part of b₀?

Let us once more consider our normalized example from Sect. 5.2, a second-order plant $P(s) = \frac {1}{s^2 + 2s + 1}$. As per Table 3.1, this plant has a gain K = 1, damping D = 1, and time constant T = 1 s. We will now perform parameter sweeps, changing each of the plant’s parameters, and provide closed-loop step responses of G_yr(s) and G_ur(s)—i.e., the controlled variable and the controller output. All simulations are performed with second-order ADRC tuned with fixed settings ω_CL = 2π ⋅ 0.2 rad∕s, k_ESO = 5, and b₀ = 1.

Varying plant gain K: As visible from Fig. 5.12, an increasing value of K has the same effect on the closed-loop step response of y(t), as the underestimation of b₀ in Fig. 5.11. The effect on the controller output u(t) is different, of course, as its steady-state value now must cater to the varying plant gain. When the plant gain K exceeds the value assumed in b₀, the control loop may ultimately become unstable. A reduced plant gain, on the other hand, will result in a closed-loop bandwidth that does not match the intended design.

Fig. 5.12
Effect of varying gain K of a plant $P(s) = \frac {K}{s^2 + 2 s + 1}$ on the closed-loop step responses of controlled variable y and controller output u. Controller parameters remain fixed, designed for the nominal case K = 1. Values of K are logarithmically spaced, ranging from K = 0.2 (blue) through K = 1 (thick black line, nominal value) to K = 5 (red)

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Varying plant damping D: It is quite interesting to see in Fig. 5.13 that the control loop is relatively insensitive to changes of the plant damping D, even though this value was varied by a factor of five in both directions in these examples. The slowed-down, increasingly overshooting response for large values of D results from the large time constant emerging in the denominator of the plant transfer function: 1 + 2DTs + T²s² = (1 + T₁s) ⋅ (1 + T₂s) for D > 1, with $T_{\mathrm {1/2}} = \frac {T}{D \pm \sqrt {D^2 - 1}}$. For such heavily overdamped cases, it might be favorable to resort to first-order ADRC, focusing only on the larger time constant.

Fig. 5.13
Effect of varying gain D of a plant $P(s) = \frac {1}{s^2 + 2 D s + 1}$ on the closed-loop step responses of controlled variable y and controller output u. Controller parameters remain fixed. Values of D are logarithmically spaced, ranging from D = 0.2 (blue) through D = 1 (thick black line, nominal value) to D = 5 (red)

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Varying plant time constant T: As visible from Fig. 5.14, the effect of a varying plant time constant ranges from oscillations for reduced values of T (ultimately, instability) to slowed-down, overshooting responses for larger values of T. Note that the range of examples chosen for T variations was reduced to a factor of 3 in both directions, since $b_0 = \frac {K}{T^2}$, i.e., the time constant has a quadratic influence on the plant’s critical gain parameter.

Fig. 5.14
Effect of varying gain T of a plant $P(s) = \frac {1}{T^2 s^2 + 2 T s + 1}$ on the closed-loop step responses of controlled variable y and controller output u. Controller parameters remain fixed, designed for the nominal case T = 1. Values of T are logarithmically spaced, ranging from $T = \frac {1}{3}\,\mathrm {s}$ (blue) through T = 1 s (thick black line, nominal value) to T = 3 s (red)

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In Sect. 3.1 we started motivating the ingredients of ADRC by stating that unmodeled plant dynamics or parameter variations will be compensated by the estimated total disturbance—if the observer is fast enough. In the examples above we have clearly seen there are (even though reasonable) limits to this ability, at least when using a moderate observer bandwidth factor k_ESO = 5. Let us therefore briefly consider an obvious fix to this problem: speeding up the observer.

In Fig. 5.15 the largest deviations of D and T from Figs. 5.13 and 5.14 are picked up again, this time with increasing values of k_ESO. Going beyond the range considered in Sect. 5.3.3, k_ESO is increased up to a value of k_ESO = 50 in these examples. As can be seen, the resulting closed-loop step responses are brought back to the desired nominal response. Is increasing k_ESO the silver bullet? While looking like a great solution in theory, we already know the answer from Sect. 5.3.3: Increasing k_ESO beyond reasonable values will make the controller very susceptible to measurement noise and even more so for higher orders N. The maximum allowable observer bandwidth must therefore be individually found for each application.

5.4.2 Additional Plant Pole or Zero

Beyond unknown or varying parameters of a plant to be controlled, there will often also be unmodeled dynamics—i.e., there are more poles or zeros in the actual plant than considered when setting up the (coarse) model for ADRC. As an example, a process may exhibit a predominant second-order behavior, yet there are higher-order dynamics, such as a minor time constant introduced by a limited-bandwidth sensor.

What can one expect from the closed-loop behavior of ADRC when encountering such situations? We want to consider two scenarios, using—for the last time in this chapter— the nominal example introduced in Sect. 5.2.

Additional pole: In the first scenario, an additional (unknown) pole with a time constant T_P is being added to the normalized plant from Sect. 5.2, i.e., plant P(s) now reads
$$\displaystyle \begin{aligned} P(s) = \frac{1}{s^2 + 2 s + 1} \cdot \frac{1}{T_{\mathrm{P}} s + 1} . \end{aligned}$$

As T_P is assumed to be unknown when designing the controller, the plant is only modeled with N = 2 and b₀ = 1. Figure 5.16 shows several examples with values of T_P ranging up to T_P = 1 s—quite a large range of values considered, as the plant will have three identical poles at in the latter case, yet is treated as a second-order plant. One can attest that this control loop is pretty insensitive to additional minor dynamics, supporting the initial claim of ADRC that a coarse model for the dominant dynamics indeed is a reasonable starting point in the process of designing the controller.

Fig. 5.16
Effect of adding an additional pole with a time constant T_P on the closed-loop step responses of controller output u and output y of an otherwise normalized plant $P(s) = \frac {1}{s^2 + 2 s + 1}$ (nominal case without additional pole: thick black line). Controller parameters remain fixed. Values of T_P are logarithmically spaced, ranging from T_P = 0.01 s to T_P = 1 s (red colors, increasingly dark)

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Additional zero: Secondly, the presence of an additional zero in the plant dynamics shall be examined. P(s) now reads
$$\displaystyle \begin{aligned} P(s) = \frac{1}{s^2 + 2 s + 1} \cdot \frac{T_{\mathrm{Z}} s + 1}{1} = \frac{T_{\mathrm{Z}} s + 1}{s^2 + 2 s + 1} . \end{aligned}$$

Again, the zero assumed to be unknown, the plant continues to be modeled with N = 2 and b₀ = 1. In Fig. 5.17, the value of T_Z is being varied between − 0.2 s and 0.2 s, i.e., in a range of ± 20 % of the plant time constant T. This covers zeros in both the left and the right half of the s-plane (LHP and RHP). The latter case, found in practice in non-minimum phase systems such as certain DC-DC converter topologies, can be recognized from the characteristic undershoot in the step response of the controlled variable. RHP zeros severely restrict the achievable bandwidth of a control system, and the effects seen in Fig. 5.17 are no exception to that. The control loop with a fixed controller design is much more tolerant to LHP than RHP zeros.

Fig. 5.17
Effect of adding an additional zero with a time constant T_Z on the closed-loop step responses of controller output u and the output y of an otherwise normalized plant $P(s) = \frac {1}{s^2 + 2 s + 1}$ (nominal case without additional zero: thick black line). Controller parameters remain fixed. Values of T_Z are logarithmically spaced, ranging from $ \left | T_{\mathrm {Z}} \right | = 0.02\,\mathrm {s}$ (brighter colors) to $ \left | T_{\mathrm {Z}} \right | = 0.2\,\mathrm {s}$ (darker colors) for both right-half plane (RHP, T_Z < 0, blue colors) and left-half plane zeros (LHP, T_Z > 0, red colors)

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	From: r (reference signal)	From: d (disturbance)	From: n (noise)
To: y	\(\displaystyle G_{\mathrm {yr}}(s) = \frac {P \cdot \left ( C_{\mathrm {FF}} + C_{\mathrm {FB}} C_{\mathrm {PF}} \right )}{1 + P C_{\mathrm {FB}}}\)	\(\displaystyle G_{\mathrm {yd}}(s) = \frac {P}{1 + P C_{\mathrm {FB}}}\)	\(\displaystyle G_{\mathrm {yn}}(s) = \frac {1}{1 + P C_{\mathrm {FB}}}\)
To: u	\(\displaystyle G_{\mathrm {ur}}(s) = \frac {C_{\mathrm {FF}} + C_{\mathrm {FB}} C_{\mathrm {PF}}}{1 + P C_{\mathrm {FB}}}\)	\(\displaystyle G_{\mathrm {ud}}(s) = \frac {-P C_{\mathrm {FB}}}{1 + P C_{\mathrm {FB}}}\)	\(\displaystyle G_{\mathrm {un}}(s) = \frac {-C_{\mathrm {FB}}}{1 + P C_{\mathrm {FB}}}\)