Continuous reset element: Transient and steady-state analysis for precision motion systems

This paper addresses the main goal of using reset control in precision motion control systems, breaking of the well-known"Waterbed effect". A new architecture for reset elements will be introduced which has a continuous output signal as opposed to conventional reset elements. A steady-state precision study is presented, showing the steady-state precision is preserved while the peak of sensitivity is reduced. The architecture is then used for a"Constant in Gain Lead in Phase"(CgLp) element and a numerical analysis on transient response shows a significant improvement in transient response. It is shown that by following the presented guideline for tuning, settling time can be reduced and at the same time a non-overshoot step response can be achieved. A practical example is presented to verify the results and also to show that the proposed element can achieve a complex-order behaviour.

especially for high-tech industrial applications such as precision motion control.
One can interpret this effect by putting transient and steady-state response of the system on two sides of this infamous waterbed, which implicates that by improving one, you are sacrificing the other.
Reset control systems, first proposed by Clegg in [2], are proving themselves as 10 alternatives for linear control systems as they showed potential to outperform linear control systems by breaking waterbed effect limitation. Clegg proposed an integrator whose output will reset to zero whenever its input crosses zero. It was later established that based on Describing Function (DF) analysis, such an action will reduce the phase lag of the integrator by 52 • . Although this already 15 breaks the Bode's gain-phase relation for linear control systems, there are concerns while using Clegg's Integrator (CI) in practice, namely, the accuracy of DF approximation, limit-cycle, etc.
In order to address the drawbacks and exploiting the benefits, the idea was later extended to more sophisticated elements such as "First-Order Reset Ele-20 ment" [3,4] and "Second-Order Reset Element" [5] or using Clegg's integrator in form of PI+CI [6] or resetting the state to a fraction of its current value, known as partial resetting [7]. Reset control has also recently been used to approximate the complex-order filters [8,9]. Advantage of using reset control over linear control has been shown in many studies especially in precision motion 25 control [10,11,12,7,13,14,15,16,17,18]. However, these studies are mostly focused on solving one problem. For example they either improve transient [19] or steady-state response of the system while paying little or no attention to the other.
One of the recent studies introduces a new reset element called "Constant-in- 30 Gain, Lead-in-Phase" (CgLp) element which is proposed based on the loopshaping concept [9]. DF analysis of this element shows that it can provide broadband phase lead while maintaining a constant gain. Such an element is used in the literature to replace some part of the differentiation action in PID controllers as it will help improve the precision of the system according to loop- 35 shaping concept [18,12,13,9].
In [12,18], it is suggested that DF analysis for reset control systems can be inaccurate as it neglects the higher-order harmonics created in response of reset control systems. These studies also suggest that suppressing higher-order harmonics can improve the steady-state precision of the system. 40 One of the benefits of providing phase lead through CgLp is improving the transient response properties of the system, as it is shown that it reduces the overshoot and settling time of the system. However, the way to achieve this goal is not only through phase compensation around cross-over frequency. It is shown in [20] that since reset control systems are nonlinear systems, the sequence of el-45 ements in control loop affects the output of the system. It was shown that when the lead elements are placed before reset element, it can improve the overshoot of the system. However, no systematic approach is proposed there for further improving the transient response. In [21], it is shown that by changing the resetting condition of reset element to reset based on its input and its derivative, 50 overshoot limitation in linear control, systems can be overcome. This limitation has also been broken using the same technique in another hybrid control system called "Hybrid Integrator Gain System" (HIGS) [22]. However, in these studies the effect of such an action on steady-state performance of the system is not addressed. 55 Another important common properties of all reset elements in the literature is the discontinuity of the output signal. This properties is a cause for presence of high-frequency content in the signals and subsequent practical issues [18].
Continuous time implementation as opposed to discrete time implementation of reset control and also soft resetting were introduced in the literature to mitigate 60 this problem to some extent [23,24]. However, this paper proposes an approach which can also used in discrete time.
The main contribution of this paper is to propose a new architecture for CgLp element which has a continuous output as opposed to conventional reset elements. This element will drastically improves the transient response of the systems without jeopardizing the steady-state performance of the system by increasing higher-order harmonics. This paper shows that this architecture even reduces the higher-order harmonics by smoothing the reset jumps. Reset control systems are also known for having big jumps and peaks in their control input which can be a limiting factor in practical applications due to saturation. The 70 proposed architecture will also improve this drawback. A guideline for tuning the propose architecture will also be provided.
The remainder of this paper is organized as follows: The first section will present the preliminaries of the study. The following section will present the continuous reset architecture. Section 4 will study the open-loop steady-state properties of 75 the proposed architecture. The following two sections will numerically study the closed-loop transient and steady-state characteristics of the proposed controller.
Section 7, will verify the results by presenting the results of an experiment on a precision positioning system system and at last the paper concludes along with some tips for ongoing works.

Preliminaries
This section will discuss the preliminaries of this study.

Dynamics of Precision Motion Systems
The first stage in precise control of a mechatronic system is to determine the dynamics of motion. A friction-less moving mass is the most basic mechatronic 85 system. Its motion dynamics are represented by a double integrator. A DC motor or a voice-coil actuator are examples of such systems. In practice, the masses are usually constrained by springs and there is always some amounts of damping present, which creates a mass-spring-damper dynamics. Such dynamics in frequency domain has a constant spring line and a resonance peak in 90 addition to the negative-sloped mass line.
Most of the precision motion setups are well-designed systems which can be modeled as mass-spring-damper systems or a cascade of them [25,26,27,28].
Whether they are collocated or non-collocated systems, in practice, the crossover frequency to control them is usually placed along the -2 slope mass line. ical steady-state analysis will be carried out for general motion plants, for the transient numerical analysis for the sake of generality and simplicity, a mass plant will be assumed. However, it will be shown in experimental results that the study hold for a mass-spring-damper system with higher frequency modes.

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The general form of reset controllers used in this study is as following: where A r , B r , C r , D r denote the state space matrices of the Base Linear System (BLS) and reset matrix is denoted by A ρ = diag(γ 1 , ..., γ n ) which contains the reset coefficients for each state. e(t) and u(t) represent the input and output for the reset controller, respectively.
A special type of reset elements which is of concern in this paper is First Order

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Reset Element (FORE). In the literature, this element is typically shown as¨B γ 1 s/ωr+1 , where ω r is the corner frequency and the arrow indicates the resetting action and since element has only one resetting state, A ρ = γ.

H β condition
Among different criteria for stability of reset control systems [11,29,30,31,32,33], 115 despite of its conservativity, H β condition has gained attention because of simplicity and frequency domain applicability [7]. In [34], the H β condition has been reformulated such that the frequency response functions of the controllers and the plant can be used directly. This method especially includes the case where the reset element is not the first element in the loop.
where κ(jω) = 1 + O * (jω), O * (jω) is the conjugate of O(jω) and R(.) stands for the real part of a complex number. Let Then the h β condition for a reset control system is satisfied and its response is

Describing Functions
Describing function analysis is the known approach in literature for approximation of frequency response of nonlinear systems like reset controllers [35].
However, the DF method only takes the first harmonic of Fourier series decomposition of the output into account and neglects the effects of the higher order harmonics. This simplification can be significantly inaccurate under certain circumstances [12]. The "Higher Order Sinusoidal Input Describing Function" (HOSIDF) method has been introduced in [36] to provide more accurate information about the frequency response of nonlinear systems by investigation of higher-order harmonics of the Fourier series decomposition. In other words, in this method, the nonlinear element will be replaced by a virtual harmonic generator. This method was developed in [37] for reset elements defined by Eq.  as follows: where H n (ω) is the n th harmonic describing function for sinusoidal input with x 2 (t)

CgLp
CgLp is a broadband phase compensation reset element which has a first harmonic constant gain behaviour while providing a phase lead [9]. This element consists in a reset lag element in series with a linear lead filter, namely R and D, respectively. For FORE CgLp: where ω rα = αω r , α is a tuning parameter accounting for a shift in corner frequency of the filter due to resetting action, and [ω r , ω f ] is the frequency range where the CgLp will provide the required phase lead. The arrow indicates the resetting action as described in Eq. (1).

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CgLp provides the phase lead by using the reduced phase lag of reset lag element in combination with a corresponding lead element to create broadband phase lead. Ideally, the gain of the reset lag element should be canceled out by the gain of the corresponding linear lead element, which creates a constant gain behavior. The concept is depicted in Fig. 1.

Proposed Architecture for Continuous Reset (CR) Elements
The new architecture which this paper proposes consists of adding a firstorder lag element, R(s), after the reset element and adding the inverse of it, which is basically a lead element, after the reset element. Fig. 2 depicts the new architecture in which In the ideal case, L(s) = R −1 (s), however, in order to make L(s) proper and realizable, the presence of the denominator in L(s) is necessary. Nevertheless, In the context of linear control systems, adding these two elements would almost have no effect on the output of the system in lower frequencies and improve the noise attenuation behaviour at higher frequencies, provided the internal states stability.
However, in the context of nonlinear control systems, the output of the system will be changed significantly.
In this new architecture the resetting condition is changed from e(t) = 0 to x 1 (t) = 0. Again considering that ω h is large enough, the new resetting condition can be approximated as The new reset element resets based on a linear combination of e(t) andė(t), where ω l determines the weight of each. In closed loop, e(t) andė(t) are the error and its differentiation.  Proof. If the reset instants are {t k | k = 1, 2, 3, · · · }, from Eq. (1) and Fig. 2, it can be seen that It is readily obvious that and thus it is discontinuous. Nevertheless, for u(t) one can write It can be readily seen that In addition to making the reset element output continuous, other motivations to use this architecture can be described in terms of steady-state and transient response of system, which will be discussed in details in following sections.

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Frequency domain analysis is the popular approach for study of the steadystate response of a system. However, as mentioned earlier, because of the nonlinearity of reset elements, that is not directly possible. The DF and HOSIDF methods are two approaches to approximate a frequency response for a reset control systems, where DF can be regarded as a special case of HOSIDF in . . .

H3
Hn which, only the first-order harmonic is studied. In order to illustrate how the HOSIDF approach can be used for the CR architecture proposed, one can refer to Fig. 3.

Proposition 1.
For ω h = ∞, the CR architecture has the same DF as the R 165 alone.
Proof. Following the same reasoning as Proposition 1, one has where u n (t) is the n th harmonic of u(t), h n (ω) = |H n (jω)| and ϕ n = ∠H n (jω)+ where A n (ω) stands for |u n (t)|. In other terms, for large enough ω h , For ω ω l , and for ω ω l , reduces the higher-order harmonics which makes the DF approximation more accurate. It is shown in [12,18] that it improves the performance of the systems in terms of steady-state precision.
Moreover, the discontinuity of output signal in reset controllers creates practical 180 problems such as amplifier or actuator saturation and excitation of higher frequency modes for complex plants. The CR architecture will solve these problems by reducing the known peaks in the control input of the reset control systems.
In order to illustrate the effect of the CR architecture on HOSIDF of reset el- The superiority of CgLp control structures over other reset control strategies in precision motion control has been shown in many researches [9,13,18]. In the remainder of this paper, for the sake of conciseness, only CR CgLp architecture will be studied. However, the same approach can be used for other reset control 205 structures.
For the case of CR CgLp, the magnitude of higher-order harmonics for fre-quencies lower than ω c (where it matters the most for tracking and disturbance rejection [12,18]) is also affected by parameters other than ω l . These parameters are ω r and γ. However, unlike ω l , these two parameters also affect the 210 DF phase and consequently the amount phase lead created by CR CgLp. This creates a trade-off between reduction of higher-order harmonics magnitude and maximum achievable Phase Advantage (PA) of CR CgLp. Fig. 6 illustrates the trade-off. CgLp will be logically designed to provide phase lead at cross-over frequency, i.e., ω c . As ω r approaches ω c the integral of 3 rd harmonic magni-215 tude over frequencies below ω c decreases significantly. The reduction of integral value is an indication of the reduction of magnitude of higher-order harmonics in general. Furthermore, the peak of higher-order harmonics will also shift to higher frequencies when ω r approaches higher frequencies. Thus it seems logical to have this peak in frequencies where tracking and disturbance rejection per-220 formance is not a matter of concern, i.e., the frequencies after the bandwidth.
When ω r is in [ω c , 1.5ω c ], higher-order harmonics are very low and still a PA up to 35 • is achievable. This can be a general guideline for tuning ω r in CR CgLp. derivative. This will change transient response of the system as well [38,22,20].

Closed-Loop
In order to study the effect of parameters of CR architecture on transient response of a closed-loop precision motion control system, a data-based approach  has been used in this paper. Following the discussion in Section 2.1, the plant which is used for this databased study is a mass system, i.e., P (s) = 1/s 2 . In experimental validation, it will be shown that the analysis will also hold mass-spring-damper systems.
The H β condition for stability of the reset control systems necessarily requires the BLS to be stable. Thus, a PID controller is present in the loop. However, according to loop-shaping technique, to ensure the maximum steady-state precision performance for the system, the differentiation part of the PID should be as weak as possible to only guarantee the stability of the BLS. Normally, such a tuning for PID control system will perform poorly in terms transient response in absence of CR CgLp. Nevertheless, it will be shown that the presence CR Figure 7: The control loop used for precision motion control using CR CgLp. P (s) is the ä .
CgLp will significantly improve transient response without affecting the maximally precise steady-state performance of the system. In this study, using a rule of thumb, the PID is tuned such that the BLS has 5 • phase margin, which is enough to stabilize the BLS and since it has a weak differentiator, does not jeopardize the steady-state precision. The following equation shows the parameters chosen in this regard.
And consequently, k p can be determined according to ω c . According to the discussions in Section 4, without loss of generality, for this data-based study, This leaves the effect of γ and ω l to be studied. Since ω r and the parameters of PID are fixed, the only parameter which affects the phase margin of the designed system is γ. It has to be noted, that according to Proposition 1, CR architecture does not change the DF, thus ω l does not have an effect on phase 230 margin. Fig. 8 shows the open-loop DF of the system under study and also the effect of γ on phase margin. γ = 1 indicates the base linear system and as the value γ decreases the phase margin will increase. At ω = ω c , it can be seen that CR CgLp not only does not change the gain behavoiur, but also creates a positive slope in phase, which resembles the complex-order controllers. In the the closed-loop system will be shown.

Overshoot
As mentioned before, it is expected that the variation of phase margin caused by variation of γ and the variation on ω l create different transient responses for the closed-loop system. In order to do a data-based study, a unit step reference was given to the closed-loop system and the the response was simulated using Simulink environment of Matlab. The overshoot versus the variation of ω l and phase margin is depicted in Fig. 9.
From Fig. 9, it can be concluded that similar to linear controllers, with increase of the phase margin the overshoot decreases almost linearly. Furthermore, for a constant value of phase margin as ω l decrease the overshoot decreases and for some configurations a non-overshoot performance is realizable. It should be also noted that as ω l increases, it weakens the lead element L(s) and thus system gradually tends to the performance of the conventional CgLp. Overshoot of the system in the absence of CR CgLp, i.e., BLS, is 96%.
In the range of Phase Margin (PM) ∈ [10,30] and ω l /ω c ∈ [0.1, 1], the decrease of overshoot (OS) is almost linear with respect decrease of log(ω l ). A fitting operation reveals the following relation between the OS and PM and ω l . (25) where PM is in degrees.
In order to better illustrate the effect of these two parameters on overshoot and 240 in general transient response of the closed-loop system, one can refer to Fig. 10.
For this simulation ω c = 100 rad/s. Fig. 10a shows the reduction of overshoot by reduction of ω l , the non-overshoot response is shown to be realizable. However, too much reduction of ω l can result in long settling times as is the case for ω l = 10 rad/s. Obviously, since CgLp does not contain ω l , it has only one 245 response. The study shows the significant improve in transient response by CR CgLp. It worth mentioning that it will be showed later that this improvement in transient will not sacrifice the steady-state response.

Settling time 255
According to Fig. 9 and 10a, reduction of ω l generally decreases overshoot, it may have an adverse effect on settling time. In order to find a sweet spot where overshoot and settling time are improved simultaneously the same sweep as Fig. 9 has been done for settling time and depicted in Fig. 11. According to this figure, for a constant ω l the settling time decreases with increase of PM as 260 as like the case for linear controllers. However, there is no linear relation for Figure 9: The overshoot of the system to a unit step for phase margin in range of [5,22] and ω l /ωc ∈ [0.1, 1]. 5 • of the phase margin is provided through base linear system. The overshoot in the absence of the CR CgLp, i.e., BLS, is 0.962.
ω l /ω c and settling time.
As a rule of thumb, ω l /ω c ∈ [0.3, 0.6] and PM larger than 20 • shows a favorable settling time. In this range the settling time of the CR CgLp is shorter than CgLp and referring to Fig. 9, non-overshoot performance can also be achieved.

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Thus one can use this general rule of thumb as the tuning guideline of CR CgLp.   behaviour for the designed reset controllers, the presence of higher-order har-275 monics makes achieving it impossible. Thus, as discussed in Section 4, reducing higher-order harmonics brings the reset controller closer to the ideal behaviour.
It is shown that CR architecture and its tuning guidelines can reduce the magnitude of higher-order harmonics. Thus, it is expected that actual closed-loop steady-sate performance is very close to approximation created by DF. In order 280 to verify the latter, a comparison has been made. A series of simulations has been run to determine the actual sensitivity functions values for different frequencies. However, because of nonlinearity of the system, the output will not be sinusoidal. To approximate, the second norm of the signals has been used.
According to Fig. 12, the presence of either CgLp or CR CgLp reduces the 285 peak of sensitivity significantly, which is logical because both of them increase the phase margin of the system. At the same time because ω r is tuned to reduce the higher-order harmonics, it was expected that sensitivity of CgLp and CR CgLp, namely, S CgLp and S CR CgLp , closely match the sensitivity of the BLS and the sensitivity approximated by DF, i.e., S BLS and S DF . However, the 290 Parameter CgLp. This analysis indicates that the significant improvement in transient behaviour of the CR CgLp architecture not only has almost no negative effect on steady-state behaviour but also positively affects it by reducing the peak of sensitivity.

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To summarize the rule of thumb tuning guideline to CR CgLp elements the suggested values for different parameters are presented in Table 1.
The data-based analysis done in previous sections was for mass plants. However, the concepts and the procedure can be done for generalized for massspring-damper plants and the suggested rule of thumb tuning values roughly 300 stands for every mass-spring-damper plant. To verify, in the next section, a practical example CR CgLp will be designed and tested for a precision motion setup which has a mass-spring-damper plant with high-frequency modes.

Plant
The precision positioning stage "Spyder" is depicted in Fig. 13 is a 3 degrees of freedom planar positioning stage which is used for validation. Since reset

Controller Design Approach
In order to compare the performance of PID and CR CgLp and show the 315 superiority of the CR CgLp over PID in both steady-state and transient, four controllers were designed. PID controllers are tuned following the tuning rules presented in [26] and reset controllers are designed following the guidelines presented in the paper. The controller loop is already depicted in Fig 7. However, due to presence of noise in practice, a first order low-pass filter, 1 s/ωz+1 , has 320 been added to the loop. The parameters for designed controllers is presented in Table 2.
Since the input signal to L(s) is e(t), this element will amplify the noise present in e(t) and thus creates excessive zero crossings and thus excessive reset actions [20]. In order to avoid this phenomenon, ω h has chosen to be smaller 325 than the rule-of-thumb guidelines provided in previous sections to better attenuate the high-frequency content of the signal. This change in ω h increases the overshoot in step response, to compensate, ω l has chosen to be smaller than Parameter PID #1 can also be considered the BLS for the CR CgLp controller, since the 330 latter is simply PID #1 with CR CgLp element preceding it, as can be seen in For this purpose, sinusoidal signals between 1 and 500 Hz has been input as r(t) and e(t) 2 r(t) 2 has been calculated and plotted for each sinusoidal.

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In the range of [1,10] Hz, the sensitivity of all controllers seemed to be lower bounded by -60 dB, this effect is caused by the quantization and the precision of the sensor. However, comparing PID #1 and CR CgLp in range of [10,500] Hz reveals that performance of the CR CgLp closely matches PID #1 in lower frequencies and its peak of sensitivity is 1.5 dB lower. Thus, one can conclude

Comparison of the transient response
For comparison of the step responses of the controllers, a step input of 0.15µm height has been used. The response of the controllers are depicted in Fig. 17. As it can be seen the CR CgLp shows a no-overshoot performance where PID #1 shows an overshoot of 38%. It is noteworthy that according 365 to Fig. 16, these two controllers have matching sensitivity at lower frequencies.
The settling time has also improved by 25%. This example clearly demonstrates that by adding CR CgLp element to an existing PID linear loop, one can achieve a no-overshoot performance and generally significantly improved transient response while maintaining the steady-state precision.  CgLp is 10% lower than that of PID. This results validates that the transient performance of the reset controllers, especially the overshoot, is affected but not solely by PM and peak of sensitivity. The architecture and ω l also play role.
The reduction of overshoot for PID #2 compared to PID #1 was obvious due to wider band of differentiation and thus reduced peak of sensitivity. However, despite the fact that its peak of sensitivity is lower than CR CgLp, the overshoot is still larger than that of CR CgLp. Meanwhile steady-state precision 380 was already shown to be lower that CR CgLp. It has to be noted, because of the relatively high bandwidth which is chosen for the controllers, i.e., 400 Hz, and limitations of the actuator, control signal for the controllers come close to saturation in only one sample of time. Nevertheless, it will be shown later that this is not the case for lower bandwidths, even for larger references. In Fig. 17, ω l = 50 Hz. In order to validate the effect of ω l on transient response, the step response for different values of ω l while maintaining the other parameters is depicted in Fig. 18. It can be clearly seen that overshoot keeps decreasing with reduction of ω l /ω c . Furthermore, it can be also validated that 390 settling time will increase when ω l /ω c drops below a certain threshold. This phenomenon can be due to the fact that too much reset and resetting too soon can jeopardize the effect of integrator. It is noteworthy that according to Proposition 1, the value of ω l does not have an effect on DF and thus steady-state tracking performance of the system. 395

Complex-order behaviour
Another interesting behaviour of the CR CgLp controller is the ability to create a complex-order behaviour as depicted in Fig. 19. Two controllers have been designed for ω c = 100 Hz. In the case of gain variation of 5 dB, ω c will change to 150 Hz, in such a situation, PID loses 3 • of PM while CR CgLp 400 will show a complex-order behaviour, meaning the phase increases while gain decreases [8], and gain 5 • more PM. Thus the modulus margin for PID is expected to be decreased and for CR CgLp to be increased. Since ω c is increased and ω l has been kept constant, the ratio of ω l /ω c is subsequently reduced, which also helps the reduction of overshoot. Complex-order behaviour of CR CGLp

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A new architecture for reset elements, named "Continuous Reset Element" was presented in this paper. Such an architecture consists of having a linear lead and lag element, before and after of a reset element. It was shown that such an architecture not only does not influence the DF gain and phase of reset elements, but also reduces the magnitude of higher-order harmonics, which 415 will positively effect the steady-state tracking precision of the reset controllers.
Furthermore, it was shown that having a strictly proper lag element after the reset element will make the output of the reset element continuous as opposed to conventional reset elements.
Moreover, it was shown that such CR architecture also can significantly improve  Fig. 19  able to achieve a no-overshoot performance and a reduced settling time while matching the steady-state performance of the PID BLS at lower frequencies and a showing a reduced peak of sensitivity.
However, the presence of a lead element before a reset element can introduce excessive reset actions to the control because of noise. To avoid such a phe-435 nomenon a low-pass filter or in general term a shaping filter can be used to remove the high-frequency content of the signal. For which a more extensive research is required. The latter can be ongoing work of the propose design.