Speed- and Range-Based Filter Design for Dual-Stage Actuator Control

Illustration of the horizontal axis warping into the spatial domain, showing how time-domain signals at different frequencies but constant amplitude share the same spatial domain profile

Fig. 2

Illustration of the horizontal axis warping into the spatial domain, showing how time-domain signals at constant frequency but increasing amplitudes map to distinct spatial waveforms and increasing spatial frequency

Complementary spatial domain filters can therefore be used to filter high- and low-amplitude components and allocate them appropriately to LRA and SRA, respectively. Such filters, G₁(s′), G₂(s′), may have the same form as Eqs. (1) and (2), only with the Laplace variable s replaced by s′ to indicate the Laplace transform of a spatial domain, rather than time-domain, signal.

Although Eqs. – are defined in continuous time (or continuous displacement), it is more convenient to implement the filtering strategy in discrete time (or discrete displacement). The conventional time-domain filters, G₁(s), G₂(s) for example, may be discretized using the Tustin approximation

s \leftarrow \frac{2}{T} \frac{z - 1}{z + 1}

(4)

where T denotes the fixed/constant sample period and z⁻¹ a unit time delay. Substituting Eqs. in , and rewriting as a pair of difference equations then gives

\begin{aligned} y_{1} (n) & = - \frac{(T ω_{b} - 2)}{(T ω_{b} + 2)} y_{1} (n - 1) \\ + \frac{T ω_{b}}{(T ω_{b} + 2)} (x (n) + x (n - 1)) \end{aligned}

(5)

\begin{aligned} y_{2} (n) & = - \frac{(T ω_{b} - 2)}{(T ω_{b} + 2)} y_{2} (n - 1) \\ + \frac{2}{(T ω_{b} + 2)} (x (n) - x (n - 1)) \end{aligned}

(6)

where n denotes the discrete time-step. It is not immediately apparent, however, how to apply the Tustin approximation in the case of the spatial filters, G₁(s′), G₂(s′). For discrete signals, the warping in Eq. may be approximated as

\begin{aligned} σ (n) = \sum_{k = 1}^{n} D (k) \\ where D (n) = | y (n) - y (n - 1) | \end{aligned}

(7)

Note that this represents a warping of the time axis, so that samples obtained at a constant sample interval T in the time domain, now have a variable spatial sample interval defined by D(n), the distance traveled between samples. Most discretization methods assume a constant sampling interval and are therefore inapplicable in this instance. One advantage of the Tustin approximation, however, is that it is based on trapezoidal integration over a sample period and does not require all sample periods to be identical. In this case, therefore, the substitution (Eq. ) is replaced by

s^{'} \leftarrow \frac{2}{D (n)} \frac{z - 1}{z + 1}

(8)

Furthermore, note that the actual displacements, denoted as y(n) when expressed in the time-step domain, or y(σ(n)) when expressed in the spatial domain, are not changed by the warping of the time axis, i.e., y(n) = y(σ(n)). Substituting Eq. into expressions equivalent to Eqs. and for G₁(s′), G₂(s′), and rewriting the result as a difference equation in y(n), rather than y(σ(n)), then gives

\begin{aligned} y_{1} (n) & = - \frac{(D (n) ω_{b} - 2)}{(D (n) ω_{b} + 2)} y_{1} (n - 1) \\ + \frac{D (n) ω_{b}}{(D (n) ω_{b} + 2)} (x (n) + x (n - 1)) \end{aligned}

(9)

\begin{aligned} y_{2} (n) & = - \frac{(D (n) ω_{b} - 2)}{(D (n) ω_{b} + 2)} y_{2} (n - 1) \\ + \frac{2}{(D (n) ω_{b} + 2)} (x (n) - x (n - 1)) \end{aligned}

(10)

Equations (9) and (10) therefore represent a convenient discretization of the spatial filtering concept which can be applied directly to the time-domain samples of desired and actual trajectories. Although the derivation of the filter was not formally presented previously, the filter itself was first introduced in Ref. [18] and has been shown to be effective in allocating signals based on the range [18–21]. In this work, the spatial cutoff frequency ω_b was chosen such that ω_b = π/R_c where R_c is the desired range cutoff for the filter. The results of applying this filter with R_c = 100 to different amplitudes/ranges of the multi-tone sinusoid considered in Figs. 1 and 2 are shown in Fig. 3(a)). When the sinusoid amplitude A is significantly below the range cutoff, then all the signal is passed to the SRA and the LRA is essentially unused, (Fig. 3(a), left). The opposite behavior is observed if the sinusoid amplitude significantly exceeds the range cutoff (Fig. 3(a), right). The transition between the two occurs when the amplitude is equal to the range cutoff. In this case, the signal is split between the SRA and LRA (Fig. 3(a), center), with each actuator contributing to the final response. Note that these behaviors are independent of the frequency of the signal and (unlike conventional systems) the fact that low-frequency, low-amplitude signals are passed to the SRA provides finer control actuation for such signals. If the same signals are applied to the conventional system of Eqs. (3) and (4), as shown in Fig. 3(b)), then low-frequency signals are always passed to the LRA regardless of amplitude.

Fig. 3

Illustration showing how three different ranges of an input signal are allocated to the LRA and SRA using (row a) the proposed range-based strategy, and (row b) a conventional frequency-based strategy. The range-based strategy allocates small amplitude signals to the SRA but switches to the LRA as the range is increased. The frequency-based strategy always allocates low frequencies to the LRA and high frequencies to SRA regardless of range.

3 Interpretation as an Adaptive Speed-Based Filter

Although conceived as a range-based filter, the implementation defined by Eqs. (9) and (10) is arguably more easily seen as an adaptive speed-based filter. Comparing Eqs. (5) and (6) and Eqs. (9) and (10), for example, shows that both have the same functional form, only the bandwidth $ω_{b}^{'}$ is now adjusted dynamically according to the magnitude of changes in the commanded input

\begin{aligned} ω_{b}^{'} (n) & = \frac{D (n)}{T} ω_{b} = \frac{σ (n) - σ (n - 1)}{T} ω_{b} \\ = \frac{| x (n) - x (n - 1) |}{T} ω_{b} \end{aligned}

(11)

The dynamically changing bandwidth scaling factor can therefore be recognized as the difference in the spatial distance over a sample interval or the instantaneous speed of the input signal (m/s). Signals with low instantaneous speed (less than 1 m/s) and which therefore change only by a small amount over the sampling period cause the effective bandwidth ω′_b to be lowered, thereby increasing the percentage of the signal that is allocated to the high-frequency SRA. Conversely, signals with instantaneous speeds greater than 1 m/s increase the effective bandwidth and are therefore more likely to be allocated to the LRA. Unlike the conventional system, a low-frequency but low-amplitude signal will be allocated to the SRA as desired because the speed is low. As the amplitude of the signal increases, so does the speed or derivative of the signal, increasing the bandwidth and sending more of the signal to the LRA, exactly as observed in Fig. 3(a)).

This reinterpretation of the range-based filter also provides insight into some extreme cases of filter behavior when the input waveform contains constant (i.e., zero-speed) segments. If the speed at which the signal is changing decreases to the point at which it tends to a zero-speed constant value, then the effective filter bandwidth $ω_{b}^{'}$ decreases to zero and all of the signal is allocated to the SRA. The output of the LRA, therefore, does not change (it receives no allocation) and neither does that of the SRA (because the input difference is zero in this case).

When this behavior is combined with step transitions, some interesting results can be observed. Consider, for example, the case of a symmetric square wave of amplitude A, where the instantaneous velocity is either zero (most of the time) or a very high value (during the step transitions). Hence

ω_{b}^{'} (n) = {\begin{matrix} 0 & most of the time \\ \frac{2 A}{T} & \begin{matrix} during transitions : \\ i . e ., big for small T \end{matrix} \end{matrix}

(12)

One might therefore expect the transition to be fed to the LRA. However, closer inspection of Eq. shows that during the transitions, the LRA actuator filter reduces to

\begin{aligned} y_{1} (n) & = - \frac{(A ω_{b} - 1)}{(A ω_{b} + 1)} y_{1} (n - 1) + \frac{A ω_{b}}{(A ω_{b} + 2)} \times 0 \\ \approx - y_{1} (n - 1) for A ω_{b} >> 1 \end{aligned}

(13)

since x(n) + x(n − 1) = 0. Likewise, for the SRA filter, Eq. becomes:

\begin{aligned} y_{2} (n) & = - \frac{(A ω_{b} - 1)}{(A ω_{b} + 1)} y_{2} (n - 1) + \frac{1}{(A ω_{b} + 1)} (- 1)^{N} \times 2 \\ \approx - y_{2} (n - 1) f o r A ω_{b} >> 1 \end{aligned}

(14)

The bandwidth therefore does indeed adjust to feed almost all of the input signal to the LRA, and very little to the SRA, but because the square wave is symmetric, the input to the LRA filter is zero. Neither filter therefore receives any contribution from the input during the transition, and instead, the output of both filters simply toggle their current value. The result therefore depends entirely on the initial conditions of the filters. If the square wave was not symmetric so that x(n) + x(n−1) ≠ 0, then this sensitivity to initial conditions would decay with time.

4 Range-Based Filter Design

Although the analysis in Sec. 2 regards the filters (Eqs. and ) as a speed-based filtering technique, the properties of the trajectories to which they are applied also enable the filter to act as a range-based filter since speed is the distance traveled divided by time. For periodic waveforms, the instantaneous velocity is proportional to the frequency ω and range R (= 2A) of the signal. In the simplest case of a symmetric triangular trajectory, for example, the speed D(n)/T is constant and equal to the average speed over the cycle. The latter is readily found as the distance 2R traveled over one cycle divided by the cycle period 2π/ω, giving

\frac{D (n)}{T} = 2 R . \frac{ω}{2 π} = \frac{R ω}{π}

(15)

Hence for a periodic triangular wave of frequency ω, the effective filter bandwidth defined by Eq. becomes

ω_{b}^{'} (n) = \frac{R ω}{π} . ω_{b} = \frac{R}{R_{c}} ω

(16)

where the second equality follows by defining the spatial cutoff range, R_c, as

R_{c} = \frac{π}{ω_{b}}

(17)

The effective filter cutoff frequency is therefore equal to the frequency ω of the periodic input trajectory, scaled by the ratio of its range R to the spatial filter range cutoff R_c. If the range of the triangular waveform exceeds the cutoff range, then the frequency ω of the signal falls within the effective bandwidth ω′_b of the filter and the signal is passed to the low-frequency LRA. Conversely, if the range of the signal is smaller than the cutoff range, then the effective bandwidth of the filter is lowered. The frequency ω of the signal then no longer falls within this bandwidth and is therefore passed to the high-frequency SRA. In the special case where the range of the input waveform matches the cutoff range, then the effective filter cutoff frequency becomes identical to that of the input trajectory waveform and the signal is split between the actuators.

The behavior of the complementary range-based filters can be seen more explicitly by computing their frequency response characteristics in contrast to those of the conventional filters. The response of the linear filters (Eqs. and ) is readily computed according to

| G_{1} (e^{j ω T}) | = \frac{ω_{b} T \sqrt{(1 + \cos ω T)}}{\sqrt{{(ω_{b} T)}^{2} + ({(ω_{b} T)}^{2} - 4) \cos ω T + 4}}

(18)

| G_{2} (e^{j ω T}) | = \frac{2 \sqrt{(1 - \cos ω T)}}{\sqrt{{(ω_{b} T)}^{2} + ({(ω_{b} T)}^{2} - 4) \cos ω T + 4}}

(19)

and a plot of these characteristics is shown in Fig. 4, for the case ω_b = 0.01 T. Each filter has unity gain in its high-pass or low-pass band, respectively, and a gain of 1/√2 at the cutoff frequency. As expected, the response is independent of signal amplitude or range.

Fig. 4

Frequency response plots for conventional complementary low and high-pass filters. Low-frequency signals are allocated to the LRA, and high-frequency signals to the SRA, regardless of range amplitude.

Very different behavior is seen for the range-based filter design. Substituting from Eq. , in Eqs. and gives

G_{1} (e^{j ω T}) = \frac{\frac{R}{R_{c}} ω T \sqrt{(1 + \cos ω T)}}{\sqrt{{(\frac{R}{R_{c}} ω T)}^{2} + ({(\frac{R}{R_{c}} ω T)}^{2} - 4) \cos ω T + 4}}

(20)

G_{2} (e^{j ω T}) = \frac{2 \sqrt{(1 - \cos ω T)}}{\sqrt{{(\frac{R}{R_{c}} ω T)}^{2} + ({(\frac{R}{R_{c}} ω T)}^{2} - 4) \cos ω T + 4}}

(21)

Plots of these functions are presented in Fig. 5, showing their dependency (or lack thereof) on both frequency and the range ratio R/R_c. In this case, the gains of the two filters appear almost independent of frequency, except at the limits ωT = π or ωT = 0 when by inspection of Eqs. (20) and (21), G₁(jω), G₂(jω) tend to zero or unity, respectively. Instead, the filter characteristics are dominated by the range ratio. When this is low, the gain G₂(jω) is high and G₁(jω) is low, so most of the signal is sent to the SRA as desired. However, when the range of the signal exceeds the range cutoff, i.e., the range ratio is high, then the converse is true: the gain G₂(jω) is low and G₁(jω) is high, so most of the signal is sent to the LRA. If the range of the signal equals the range cutoff value, then both filters are at their crossover range “bandwidth” with a gain equal to 1/√2.

Fig. 5

Range-frequency response plots for the complementary range-based filter showing (a) G1(jω) and (b) G2(jω). The filter characteristics are now largely independent of frequency and instead vary primarily as a function of range.

Range-frequency response plots for the complementary range-based filter showing (a) G₁(jω) and (b) G₂(jω). The filter characteristics are now largely independent of frequency and instead vary primarily as a function of range.

Strictly speaking, the relationships (15) and (16) or (20) and (21) hold only for triangular trajectories i.e., signals with constant speeds, but the results should hold approximately for other waveforms (e.g., single sinusoid or raised sinusoid) that have two turning points per cycle and which spend equal amounts of time increasing and decreasing, and whose maximum speeds are not significantly in excess of the average speed over a cycle. However, other waveforms with more turning points, for example, may have a different relationship between amplitude and speed. For example, consider and input signal comprising a 3-Hz sinusoidal waveform, amplitude A = 2, multiplied by an amplitude (A/2) square wave signal, as depicted in Fig. 6. The composite signal has a total range R = 6, but because of multiple turnarounds, travels a much larger distance in a single cycle, so the speeds are nearly 5 times that of a pure sinusoid or triangular waveform with the same overall range. If Eq. (16) is used without adjustment, then a significant part of this signal is (incorrectly) directed to the LRA (see Fig. 6, top). If, however, the range cutoff (Eq. (17)) is redefined as R_c = 2π/ω_b, then signals for this form with a range less than R_c are now mostly passed to the SRA as indeed shown in Fig. 6 (bottom).

Fig. 6

Filter examples showing an input waveform whose distance travelled per cycle is more than that of a regular sinusoid or triangular waveform. If the cutoff range (17) is used a significant part of the signal is incorrectly allocated to the LRA (top plot). If the cutoff range is adjusted to reflect the increased speed, the signal is correctly allocated primarily to the SRA (bottom plot).

One should also proceed with caution in cases where there is asymmetry in the distance traveled (speed times time) in the positive and negative directions. Consider, for example, a sawtooth waveform that ramped upward with the same speed defined in Eq. (15) for the entire period, only stepping back to zero at the end of the cycle. The speed of the signal is sufficiently low that it will be passed to the SRA, causing the SRA to saturate even though the total distance traveled over the cycle is the same as that before, namely 2R. In this context, it is important to remember that the filter is ultimately a speed-based filter. No hard constraints are placed on the range which is only limited by the knowledge of the trajectory shape/properties and smart choices of the nominal filter bandwidth according to Eqs. (16) and (17).

In general, variations in the instantaneous velocity also affect the time constant of the filters and their response to transients since τ = 1/ω_b. Hence, from Eq. , the effective time constant is

τ^{'} (n) = \frac{T}{D (n)} τ = \frac{T}{| x (n) - x (n - 1) |} τ

(22)

The filters therefore respond rapidly (short time constant) when the instantaneous velocity is high and more slowly when the instantaneous velocity is low. Recall for the range cutoff choice and triangular waveforms of the type considered in Eqs. and , the effective time constant becomes

τ^{'} (n) = \frac{1}{ω_{b}^{'} (n)} = \frac{R_{c}}{R ω}

(23)

Given triangular waveforms, the settling time can be found using T_s = 4τ′. Consider the transient example illustrated in Fig. 7, which shows input triangular signals (x) with ranges R = 2 (top), R = 20 (middle), and R = 200 (bottom) as a solid line and the sum of the two filter outputs, i.e., y₁ + y₂, as a dashed line. Using a range cutoff of R_c = 10 and signal frequency ω = 2π rad/s, the estimated time constants and settling times and simulated settling times are shown in Table 1. As can be seen, the estimated and simulated settling times are in good agreement.

Fig. 7

Simulated input and output signals showing settling times for triangular signals of varying range. The time constant becomes shorter with increased signal speed or (for a given waveform shape, as in this case) with amplitude.

Table 1

Comparison between estimated and simulated settling times

Range	Time constant	Settling time—T_s(s)
R	τ(s)	Estimated	Simulated
2	0.796	3.18	3.10
20	0.080	0.32	0.31
200	0.008	0.03	0.03

5 Conclusions

Conventional dual-actuator filter designs typically allocate the command signal to the individual actuators using a fixed-frequency filter. Recent “range-based” filter designs have been shown to provide better signal allocation, particularly in the case of low-frequency, low-amplitude signals. In this work, a rigorous theoretical derivation of the filter has been obtained for the first time. It has also been shown that the filter is perhaps more accurately understood as a speed-based adaptive filter, retaining the concept of frequency-based allocation, but adapting the filter cutoff frequency according to the rate of change of the input trajectory. Unlike conventional fixed-frequency dual-actuator filters, the speed-based adaptive filter allocates a low-frequency but low-amplitude signal to the SRA as desired because the speed is low. As the amplitude of the signal increases, so does the speed or derivative of the signal, increasing the bandwidth and sending more of the signal to the LRA.

This interpretation not only explains the behavior of the filter clearly, but it provides a more robust strategy for incorporating range constraints in filter design for different desired trajectories. In particular, the frequency and range response characteristics for triangular waveforms have been derived, and the same results hold approximately for other signals whose instantaneous speeds do not deviate too much from the average speed of a triangular waveform. The analysis also highlights some weaknesses of the “range-based” filter design and of conventional filtering strategies since neither case guarantees that range constraints will be satisfied. Future work will aim to embed hard range constraints more explicitly in the filter design process.

Acknowledgment

The authors gratefully acknowledge financial support from the Briar Hill Foundation and from National Science Foundation (Grant Nos. CMMI 1537983 and 1537722).

Conflict of Interest

There are no conflicts of interest.

Nomenclature

t =: time
s =: continuous time Laplace transform variable
z =: discrete time z-transform variable
N =: discrete time-step
R =: spatial range of the signal
T =: time-domain sample period/sample interval
R_c =: filter range cutoff
x(t) =: desired trajectory
y₁(t), y₂(t) =: actuation signals sent to LRA and SRA, respectively
D(n) =: spatial domain sample interval (distance traveled between samples)
G₁(s), G₂(s) =: complementary transfer functions to allocate signals to LRA and SRA, respectively
σ =: cumulative distance traveled
τ =: filter time constant
ω_b =: filter frequency cutoff/bandwidth

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