H∞-Based Selective Inversion of Nonminimumphase Systems for Feedback Controls

Dan Wang and Xu Chen,

Abstract—Stably inverting a dynamic system model is fundamental to subsequent servo designs. Current inversion techniques have provided effective model matching for feedforward controls.However, when the inverse models are to be implemented in feedback systems, additional considerations are demanded for assuring causality, robustness, and stability under closed-loop constraints. To bridge the gap between accurate model approximations and robust feedback performances, this paper provides a new treatment of unstable zeros in inverse design. We provide first an intuitive pole-zero-map-based inverse tuning to verify the basic principle of the unstable-zero treatment. From there, for general nonminimum-phase and unstable systems, we propose an optimal inversion algorithm that can attain model accuracy at the frequency regions of interest while constraining noise amplification elsewhere to guarantee system robustness. Along the way, we also provide a modern review of model inversion techniques. The proposed algorithm is validated on motion control systems and complex high-order systems.

I. Introduction

GIVEN a linear time-invariant system modelG, the inversion ofGhas numerous practical implementations including iterative learning control (ILC) [1]–[3], repetitive control [4], [5], two-degree-of-freedom servo in feedforward control [6], [7], as well as Youla parameterization and disturbance observer in feedback control [8]–[12]. Here,Gcan be an open-loop plant model or a closed-loop control system. For a minimum-phase system,G?1is stable and ready to be implemented. However, for a system with nonminimum-phase(NMP or unstable) zeros,G?1is unstable and cannot be implemented directly. To find a stable, rational, and causal replacementsuch thatapproximates 1 is thus a fundamental challenge in inversion-based control designs. Such a challenge is more pronounced in discrete-time systems since 1) integrator-type plant dynamics1When actuators take forces or torques as the input and linear/angular position as the output, integrator-type plant dynamics with a relative degree not less than two show up., common in motion control,generate NMP zeros in their zero-order-hold (ZOH) equivalents when the sampling time is sufficiently small; 2) fractional-order delays induce unstable zeros after discretization [13].

Considering the importance and the challenge of model inversion, numerous strategies have been established in modern literature. Based on system representations and scopes of application, we can classify these strategies into two categories: frequency- and time-domain model inversions.The frequency-domain strategies focus on expressing the transfer functions of the stable inverses and hence can be used in both feedback and feedforward controls. Examples in this category include the approximate (e.g., NPZ-ignore, zerophase-error-tracking control (ZPETC), and zero-magnitudeerror-tracking control (ZMETC)) [14]–[17], the ILC-based[18]–[20], and theH∞-based [21]–[23] model inversions. On the other hand, the time-domain strategies [24]–[27] aim at identifying the optimal control signal that minimizes the error between a given reference and the output. These time-domain algorithms are mainly used as feedforward techniques since a preview of the reference is generally not available in feedback design.

This paper studies the analysis and design of model inversion strategies in the frequency domain. Current strategies in this category aim at achieving effective model matching betweenandG. Compared with the approximate and the ILC-based model inversions, theH∞-based model inversion can automatically identify the inverse model without knowing the exact NMP zeros, which particularly benefits systems with complicated pole-zero distributions. However,when the inverses are to be implemented in feedback systems,additional considerations are needed for assuring closed-loop stability and robustness. In pursuit of bridging the gap between accurate model approximations and robust feedback performances, this study builds a newH∞-based optimal inversion algorithm that advances the field by 1) mitigating control efforts at customized frequencies and thereby enhancing system robustness; 2) reaching high efficiency for complex high-order systems and unstable systems.

Before presenting the main algorithm, we first provide a pole-zero-map-based NMP-zero modulation by replacing high-frequency NMP zeros with stable ones in motion control applications. We verify the feasibility and limit of this intuitive modulation in achieving a stable inverse model and meanwhile capturing the low-frequency system dynamics for high-performance motion control. Then we extend this intuitive modulation to an optimal design of model inversion.There, replacing the manual adjustment with an automatic and optimal search, we develop a newH∞-based algorithm that can attain model accuracy at the frequency regions of interest while constraining noise amplification elsewhere to guarantee system robustness. The design goals are achieved by a multiobjective formulation and an all-pass factorization that consider model matching, gain constraints, causality of transfer functions, and factorization of unstable system modes in a unified scheme. The proposed algorithm is validated on motion control systems and complex high-order systems.Moreover, along the path, we unveil previously ignored features of existing inversion strategies by developing a general frequency-domain analysis method, which also gives new insights into comparing the performances of different strategies.

The main contributions of this paper are:

1) conducting an up-to-date review of model inversion strategies and proposing a new frequency-domain analysis method;

2) analyzing the effect of an intuitive NMP-zero modulation and developing a newH∞-based inversion algorithm;

3) validating the proposed algorithm by presenting detailed case studies with high-fidelity experimental data.

The remainder of this paper is structured as follows. Section II conducts an in-depth review of literature and proposes the new frequency-domain analysis method. Section III elucidates the effect of modulating NMP zeros. The proposed optimal inversion is presented and verified in Section IV. Section V concludes this paper.

II. Review and Comparison of Frequency-Domain Inversion Algorithms—an End-Point Perspective

The frequency-domain inversion algorithms aim at expressing the stable inverse modelsin thes- orz-domain (sandzare complex numbers in the Laplace transform andz-transform, respectively).is the minimumphase system model that approximatesGand has a stable inverse. An optimal inverse model is desired forto approximate 1. In this section, we review and compare three typical types of frequency-domain inversion algorithms. In addition, we unveil new features of existing algorithms by developing a general frequency-domain analysis method.

A. H∞-Based Model Inversion

1) Algorithm

The model inversion problem for NMP systems has been solved using theH∞formulation [21]–[23]. For a continuoustime NMP systemG(s)=(b?s)/(b+s) withb>0, under a cost functionwhere the weightingW(s)=(k+ξs)/(k+s) is a low-pass filter withk>0 and 0 ≤ξ<1, the optimal inverse ofG(s) that minimizesJis a lead filter [22]

that has high gains at high frequencies. The frequency response of the optimalis

where ? is in rad/s.

2) Frequency-Domain Analysis

To quickly capture the essence of, we examine the frequency response ofat the two frequency endpoints (0 and ∞ for a continuous-time system or 0 and π in rad for a discrete-time system) and evaluate the characteristics of model matching.

Consideringb=2 , ξ=0.3, and differentk’s, we depict in Fig. 1 the frequency responses of (2). As ? increases from 0 to ∞, the phase ofalways goes from 0 to?180?(the bottom plot of Fig. 1), and its magnitude goes fromtomonotonically. Therefore,depending on the values ofkand ξ,is a high-pass filter wheni.e.,k>ξb(0.6 in this example), a low-pass filter whenk<ξb, and has a constant magnitude whenk=ξb(the top plot of Fig. 1).

Fig. 1. Frequency responses of with b =2 , ξ =0.3, and different values of k.

B. Approximate Model Inversions

1) Algorithms

For discrete-time NMP systems, to obtain the basic structure of the inverse model, approximate model inversions [14]–[17]first factor out the unstable zeros of the system as

whereN(z) andD(z) are coprime polynomials ofz, andNs(z)andNu(z) contain, respectively, the stable and the unstable zeros. Here, we defineNu(z) as

wherez1,z2,...,znare outside the unit circle. Note that

has stable zeros. In addition,is zero-phase.

In the general case, the approximate inverse model ofG(z)in (3) has a structure of

TABLE I Y(z) R(z) in Approximate Model Inversions. and are Transfer Functions of the Output and Reference Signals Shown in Fig. 2.

TABLE I Y(z) R(z) in Approximate Model Inversions. and are Transfer Functions of the Output and Reference Signals Shown in Fig. 2.

whereis a design parameter.

Table I summarizes three approximate model inversions method (NPZ-ignore) [16], [17] replacesNu(z) with(1)at the cost of magnitude and phase mismatch inThe ZPETC [14] assigns insteadand achieves zero-phase error dynamics sinceis zero-phase. The ZMETC[16], on the other hand, eliminates all magnitude errors by converting the unstable zeros to their stable reciprocals,with different designs ofThe NMP zeros ignore namely,Note thatNu(1) in NPZ-ignore and[Nu(1)]2in ZPETC are added to create a unity DC gain of

Furthermore, to make the approximate inverse modelin (5) realizable and ready to be implemented as a block during feedback/feedforward implementation, a causal inverse model is obtained by multiplyingwithz?m

where

is the relative degree ofand theOrderfunction calculates the highest exponent in a transfer function. Next we will prove thatmis always larger than 0 in the NPZ-ignore,ZPETC, and ZMETC.

Proof:We can tell from Table I that the relative degree ofis 0 in each of the three designs. Thus, from (5), the expression ofmin (7) can be reduced to

Also, we haveOrder[D(z)]≥Order[Ns(z)]+Order[Nu(z)]from (3) andOrder[Nu(z)]>0 for NMP systems, yieldingOrder[D(z)]>Order[Ns(z)], that is,m>0 in (8).m>0z?m

Here, the result means the delay should always be accounted for to make the inverse model realizable in the feedback/feedforward applications of approximate model inversions. In feedforward applications where a preview of the desire outputyd(k) is available, the delayz?mcan be canceled out by lettingr(k)=yd(k+m).

2) Frequency-Domain Analysisthe reference signalr(k) to the output signaly(k) iswhich reflects the accuracy of the causal inverseF(z). Table I lists the transfer functions ofY(z)/R(z) in the three approximate model

Fig. 2 shows a block diagram to illustrate the goal of the model inversion design, wherer,u, andyrepresent the reference, the input, and the output signals, respectively. Note that subsequentlyFcan be implemented as a block in the feedback/feedforward controller designs, such as the examples in Section IV-C. In Fig. 2, the overall transfer function from inversions. We take the hard disk drive (HDD) system in Section III as an illustrative example. The transfer function of the system with a sampling frequency (1/Ts) of 26.4 kHz is

Fig. 2. Block diagram to illustrate the goal of the model inversion design.Note that F can be implemented as a feedback/feedforward controller.

Here,G(z) has one NMP zero at around ?2.5,Nu(z)=z+2.494311, andmin (8) is 4.Y(z)/R(z) arez?4(z+2.494311)/3.494311 for NPZ-ignore,z?4(z+2.494311)(z?1+2.494311)/3.4943112for ZPETC, andz?4(z+2.494311)/(z?1+2.494311)for ZMETC. Fig. 3 plots the frequency responses ofY(z)/R(z)of the three approximate designs. At low frequencies close to 0, i.e.,we get the desired resultY(z)/R(z)→1 for all three methods, and thereby the magnitude and phase responses ofY(z)/R(z) largely overlap with each other (Fig. 3).At the Nyquist frequency π rad (i.e., 13.2 kHz), wherez=ejπ,equalsfor NPZ-ignore and equalsfor ZPETC; that is to say, in log scale,in ZPETC ( ?1 4.72dB) has twice the magnitude ofin NPZ-ignore ( ?7 .36dB) (the top plot of Fig. 3). Moreover, in this HDD example, since the NMP zero is a real one at around ?2.5 andm=4, all threehave zero phase at the Nyquist frequency (the bottom plot of Fig. 3).

Fig. 3. Frequency responses of (indicating tracking performances) for different approximate model inversions used in the example of the HDD system in (9).

C. ILC-Based Model Inversion

1) Algorithm

ILC, originally developed for output tracking in repetitive tasks, can be extended to the field of model inversion[18]–[20]. Here, the inverse modelF(z) is constructed by designing its impulse responsef(k) as the feedforward signal in the following ILC

whereui(k) is the learned input at thei-th iteration

Here, the training referencer(k) is designed as the delta impulse δ(k). The ILC learning filterL(z) is built from the approximate model inversions (Section II-B) such that the stability conditionis satisfied. Withi→∞, from (10) and (11),that is,f(k) approximates the impulse response of the unstableG?1(z) . Recall thatf(k) is the impulse response ofF(z). Thus,we obtainF(z)≈G?1(z).

2) Frequency-Domain Analysis

In the ILC-based model inversion, the transfer function 1?L(z)G(z)determines not only the stability condition but also the convergence rate. Fig. 4 shows the frequency responses of (1?L(z)G(z))i, taking again the HDD system in(9) for example. Here,L(z) is built from ZPETC. With increasing iteration numberi, the magnitudes of(1?L(z)G(z))iat low frequencies start to converge to zero.Moreover, a largeriyields a wider low-frequency region with zero magnitude. Therefore, under finite implementation ofi,F(z) represents a low-pass approximation ofG?1(z) with a tunable bandwidth. One drawback, however, is that system hardware (or a very accurate modelG) is needed for iterative experiments to run.

Fig. 4. Frequency responses of (1?L(z)G(z))i for the example of the HDD system in (9), where L(z) is the learning filter built from ZPETC.

D. Summary of Literature Review and Motivations of This Paper

Table II summarizes the three model inversion strategies. It is noteworthy that these frequency-domain strategies can be implemented in both feedback and feedforward controls.Application of each method certainly depends on the specific problem at hand. Compared with the other two methods, theH∞-based model inversion can automatically identify the inverse model without knowing the exact NMP zeros, which particularly benefits unstable systems and high-order systems with complicated pole-zero distributions.

For inverse-based feedback control, all the surveyed algorithms have considered accurate model inversion but not robustness against model mismatch that is also crucial for closed-loop performance. In contrast, the algorithm to be proposed in Section IV enhances the system robustness by limiting the magnitude of the inverse model at frequency regions where large model mismatches exist. Before discussing the main algorithm, we provide in Section III some preparatory work on the effect of the NMP zeros.

III. Frequency-Domain Implications of Modulating NMP Zeros

This section studies the influence of modulating the NMP zeros (i.e., shifting the locations of the NMP zeros) on the frequency response of a system. For concreteness, we take the HDD system in [10] as an example, where model inversion underpins servo designs that control precisely the position of the read/write head to provide reliable storage.

The solid line in Fig. 5 shows the frequency response of an experimentally measured HDD system. The nominal model of the motors and actuators in the system is [10]

The ZOH equivalent ofGc(s) sampled at 26.4 kHz, namelyG(z), is expressed in (9) and has one unstable zero at around?2.5. As plotted in Fig. 5, the frequency response of the NMPG(z) matches well with the actual system dynamics (solid line).

TABLE II Overview of Frequency-Domain Inversion Strategies: Approximate [ 14] –[ 17] , ILC-Based [ 18] –[2 0] , and H∞-Based Methods[ 21] –[ 23]. DT and CT are Short for Discrete Time and Continuous Time, Respectively.

Fig. 5. Frequency responses of actual system dynamics from experiments and nominal system models in the HDD system.

We investigate next the frequency-domain implications of the NMP-zero locations by analyzing2.494311 in (9). Consider the rule of thumb that the closedloop bandwidthBpis around 10% of the Nyquist frequencyin this example,Bp=1300 Hz, and ωp=2π×1300/26400=17.72?. In other words, ω sweeps only a small arc on the unit circle from 0 to 17.72?in the main performance region, yielding mild changes to the vector ejω+2.494311, as shown in Fig. 6. Therefore,when shifting the NMP zero to a stable one, e.g., at ? 0.8 (Fig. 6),we can get a minimum-phase nominal modelthat has a stable inverse and largely maintains frequency response of the system in desired low-frequency regions

Fig. 6. Illustration of modulating the experimentally identified NMP zero in the HDD system.

In summary, a stable inverse is readily achievable through modulating the NMP zeros as long as the NMP zeros do not occur in the desired low-frequency regions. This result justifies the basic idea of theH∞-based optimal inversion,where the manual modulation is upgraded to an automatic and optimal search, as shall be proposed next.

IV. Proposed H∞-Based Optimal Inversion

Based on the frequency-domain analysis in Section III, this section develops anH∞-based optimal inversion. The design principle is to automatically search for the optimal inverse model to selectively fit different frequency regions. At frequencies where no NMP zeros exist and no large model uncertainties occur, we impose an accurate model matching between the minimum-phase modeland the original NMP model; at other frequencies, we limit the magnitude response of the inverse model to increase the system robustness. We explore the design procedures, case studies,and frequency-domain analyses of the proposed algorithm,first for NMP systems and then for unstable systems.

A. H∞-Based Optimal Inversion for NMP Systems

1) Algorithm

Let S denote the set of stable, proper, and rational discretetime transfer functions. We search among S to find the optimal inverse modelthat satisfies

a) F(z)is realizable/proper:This relates to thez?mterm inF(z). To minimize the delays,mcan be tuned and usually equals the relative degree ofG(z).

b) Model matching:Namely,we minimize the maximum magnitude of the model mismatchF(z)G(z)?z?mweighted byW1(z). The weightingW1(z)determines the frequency regions for accurate model matching. IfG?1(z) is stable, the direct solution isF(z)=z?mG?1(z).

c)Gain constraint:min||W2(z)F(z)G(z)||∞. Here, the magnitude ofF(z)G(z) is scaled by the weightingW2(z). For instance,W2(z) can be a high-pass filter to constrain noise amplification at high frequencies. The solution for this condition alone isF(z)=0, that is,F(z) does not amplify any input signals.

Integrating the above three goals yields the multi-objective optimization principle

The optimal inverse modelF(z) given by (14) preserves accurate model information in the frequency regions specified byW1(z) and, on the other hand, penalizes excessive high gains ofF(z) at frequencies determined byW2(z). Typically,W1(z) is a low-pass filter, andW2(z) is a high-pass one, as shown in the example of Fig. 7. For one system model, the weightings can be flexibly designed, yielding different inverse modelsF(z).

Fig. 7. Frequency responses of the weightings W 1 and W 2 in the active suspension system.

The optimization principle in (14) can be solved within the framework ofH∞controls.F(z) can be solved by thehinfsynfunction in the robust control toolbox of MATLAB and tuned for the target performance by changing the input argumentsgamTryandgamRangeof the function. Fig. 8 shows the block diagram realization of (14). Here, thehinfsynfunction minimizes the two error signalse1ande2. The solution ofF(z)exists as long asG(z),W1(z), andW2(z) are stable. After (14)is solved, a lower-orderF(z) can be reached by applying standard model-reduction techniques, if needed.

Fig. 8. Block diagram for the H∞-based optimal inverse design.

Remark:When the system model is subjected to perturbations, we can use a multiplicative uncertainty model to lump the various dynamic uncertainties.

where ∥?I∥∞≤1 [28]. Fig. 9 shows the block diagram of the proposedH∞-based optimal inverse with uncertainties taken into consideration. The problem now is to find a stabilizing inverse modelF(z) such that theH∞norm of the transfer functionbetweenrand [e1,e2]Tis less than 1 for all ?I, that is

which is no longer a standardH∞optimization but a robust performance problem. The μ-synthesis andDK-iteration procedures can be utilized to solve the problem [28], [23].

Fig. 9. Block diagram for the H∞-based optimal inverse design considering uncertainty.

2) Case Study With Frequency-Domain Analysis

This case study shows efficiency of the proposed algorithm for high-order NMP systems with complicated pole-zero distributions. We take for example the active suspension system in [29] that serves as a benchmark on adaptive regulation. The control goal there is to attenuate the vibrations transmitted to the base frame, and model inversion is critical for the best results achieved in the benchmark [30]. Although the system is open-loop stable, the existence of the NMP zeros challenges model inversion in general feedback and feedforward control.

Via standard system identification methods, the system modelG(z) is experimentally identified with a sampling rate of 800 Hz and has an order of 22. As shown in the pole-zero plot in Fig. 10, four NMP zeros show up inG(z). Furthermore,with the two weighting functions designed as in Fig. 7, we solve the optimization principle in (14) and obtain the optimal inverseF(z). After that, we reduce the order ofF(z) to 23 by applying the model-reduction functionreducein MATLAB.The pole-zero plot of the 23rd-orderF(z) is also shown in Fig. 10. Then the minimum-phase system model is secured by

Fig. 10. Pole-zero plot of the experimentally identified system model and its minimum-phase approximation of the active suspension system.

Fig. 11. Frequency responses of the experimentally identified system model G (z) and its minimum-phase approximations of the active suspension system. Models obtained from ZMETC and ZPETC, respectively, have the same magnitude and phase responses as the system model. Proposed H∞-based optimal inversion: red dashed line. Previous H∞-based method without gain constraint: magenta solid line.

B. H∞-Based Optimal Inversion for Unstable Systems

1) Algorithm

For unstableG(z), Fig. 8 and (14) are ill conditioned, and the MATLAB functionhinfsynreturns an empty solution ofF(z). The first intuition for applying theH∞-based optimal inversion is perhaps to ignore the unstable poles ofG(z) and take the remaining part as a fictitious system model. However,ignoring the unstable poles alters the relative degree of the system and may generate a non-causal system. Furthermore,numerical issues may arise after changing the magnitudes of the system. To overcome these difficulties, this section introduces an approach by using an all-pass factorization.

We first factor out the unstable poles ofG(z).

where |pi|>1 andG0(z) contains all the zeros and stable poles ofG(z).

Performing the all-pass factorization gives

whereis the complex conjugate ofpi. Here, the unstable poles inG(z) are replaced by their reciprocals inGs(z). The product termin (18) has unity magnitude,that is, the stableGs(z) has the same magnitude response as the unstableG(z). Then we can substituteG(z) withGs(z)when implementing the procedure proposed in Section IV-A.

For unstable systems, the design steps of theH∞-based optimal model inversion are modified as

a) Write the pole-zero representation ofG(z), determine the relative degreemofG(z), and then factor out the unstable poles as in (17);

b) Perform the all-pass factorization by transformingG(z) in(17) toGs(z) in (19);

d) Take into account the effect of the unstable poles in (18)byThe minimum-phase system model is then

2) Case Study With Frequency-Domain Analysis

In this case study, we show how to implement theH∞-based optimal inversion in unstable systems.

Consider a discrete-time transfer function

with a relative degree ofm=1 and a sampling rate of 26.4 kHz.G(z) contains an unstable pole 1.2 at low frequency and an unstable zero –1.5 at high frequency. Following the aforementioned design steps for unstable systems, we first get

The minimum-phase system model is therebyAs shown in Fig. 12,(dashed line)matches well withG(z) (solid line) particularly at frequencies below 5000 Hz, which is large enough for general feedback designs. Besides, compared with the existingH∞-based method (dotted line), near the Nyquist frequency, the high gain ofG?(z) from the proposed method (dashed line) indicates a small magnitude ofF(z), which matches with the gainconstraint design criterion in Section IV-A.

Fig. 12. Frequency responses of the system modelG(z)=z?1(z+1.5)/(z?1.2) and its minimum-phase approximations. Proposed H∞-based approach: red dashed line. Previous H∞-based method without gain constraint:magenta dotted line.

C. Feedback Applications of the Proposed Algorithms

Model inversion is fundamental to subsequent servo designs, such as Youla-Kucera parameterization and adaptive disturbance observers [8]–[12]. This section provides application examples that experimentally verify the preliminary NMP-zero modulation (Section III) and theH∞-based optimal inversion (Section IV).

In laser-based additive manufacturing, a galvo scanner system applies mirrors to reflect input laser beams to follow predefined trajectories. In [12], the authors first identify experimentally the NMP system model. After that, the minimum-phase model is obtained by moving the unstable zero from –4.419 to –0.6. Based on the minimum-phase model, [12] builds an outer-loop inverse-based Youla-Kucera parameterization scheme to reject single-frequency narrowband disturbances.

Reference [10] studies the track-following problem in a single-stage HDD system. The system model in (9) has one NMP zero, which is shifted inside the unit circle to make the inverse model strictly stable, as shown in Fig. 6. Then with the stable inverse model, [10] designs an adaptive disturbance observer based on the internal model principle to reject multiple narrow-band disturbances.

In the active suspension benchmark discussed in [30], the minimum-phase model (red dashed line in Fig. 11) is obtained by applying the proposedH∞-based optimal inversion. The model is then used to build an adaptive disturbance observer with an infinite impulse response structure to reject unknown or time-varying narrow-band vibrations.

V. Conclusion

In this paper, we discussed new frequency-domain analysis and design approaches to invert a nonminimum-phase (NMP)linear time-invariant system, with a focus on robustness and needed design constraints in feedback implementations. We reveal that among existing model inversion techniques, theH∞-based method stands out by automatically identifying the inverse model without knowing the exact NMP zeros.Furthermore, we illustrated that modulating the location of the NMP zero only changes the system response at selective frequency regions. Leveraging this fact, for general NMP systems, we propose a discrete-timeH∞-based optimal inversion to automatically design the inverse model for selective frequency regions defined by two weighting functions. Verifications in complex high-order systems and unstable systems show the strengths of the proposed algorithm.