Analysis and Comparison of Discrete-Time Feedforward Model-Inverse Control Techniques

Transcription

1 Analysis and Comparison of Discrete-Time Feedforward Model-Inverse Control Techniques Jeffrey A. Butterworth Graduate Research Assistant, Student Member of ASME Department of Electrical, Computer, and Energy Engineering University of Colorado at Boulder Boulder, Colorado Lucy Y. Pao Richard and Joy Dorf Professor Department of Electrical, Computer, and Energy Engineering University of Colorado at Boulder Boulder, Colorado Daniel Y. Abramovitch Senior Research Engineer Nanotechnology Group Agilent Laboratories 53 Stevens Creek Blvd., M/S:4U-SB Santa Clara, California Noncollocated sensors and actuators, and/or fast sample rates with plants having high relative degree, can lead to nonminimum-phase (NMP) discrete-time zero dynamics that complicate the control system design. In this paper, we examine three stable approximate model-inverse feedforward control techniques, the nonmimimum-phase zeros ignore (NPZ- Ignore), the zero-phase-error tracking controller () and the zero-magnitude-error tracking controller (), which have frequently been used for NMP systems. We analyze how the discrete-time NMP zero locations in the z-plane affect the success of the NPZ-Ignore,, and model-inverse techniques. We also examine the use of lowpass filters with the three model-inversions techniques. Experimental results on the x direction of an AFM piezoscanner are provided to support the discussions. Finally, tips on the use of these three model-inversion discrete-time feedforward methods are presented throughout the paper. Introduction Model-inverse control has made several appearances in the literature demonstrating its strength in improving tracking performance, settle time, and other performance metrics; a few examples include [ 8]. References [] and [] provide a comparison of two typical control architectures that may be used to implement model-inverse control. While our discussion here applies equally to both the closed-loopinjection architecture and the plant-injection architecture de- H CL x e x Σ C u d x F CL Fig.. P A block diagram of the closed-loop-injection architecture. When model-inverse control is used, F CL is designed to represent the inverse of the closed-loop system H CL consisting of the feedback controller C and the plant P. PC +PC scribed in [] and [], we will focus this paper on the closedloop-injection architecture seen in Fig.. When using the closed-loop-injection architecture for model-inverse control, we first design the feedback controller C of Fig. to maximize performance of the standalone closed-loop system H CL (z) =. Next, we set F CL equal (or approximately equal) to H CL (z) for even greater performance gains. The definition of performance depends on metrics defined by the designer, but in general the inclusion of F CL using model-inverse control methods will usually improve rise times, settle times, phase lag, tracking performance and more. Ideally, the design of F CL would yield a filter exactly equal to H CL (z). This, for example, would yield a transfer function from x d (t) to x(t) (in Fig. ) of unity, and allow a system s output x(t) to perfectly track a desired trajectory x d (t). Often the existence of nonminimum-phase (NMP) ze- x Corresponding Author

2 ros in the plant force a stable approximate inverse to be used in place of the exact inverse. This paper focuses on discrete-time single-input singleoutput (SISO) nonmimimum-phase systems. NMP zeros in a discrete-time model can result from sampling a continuoustime system to create the model [9]. Nonminimum-phase zeros may also be a result of sensors and actuators being physically noncollocated []. Due to both sources, the appearance of NMP zeros in discrete-time systems can be rather common. Various stable approximate model-inversion techniques exist, including Tomizuka s popular zero-phase-error tracking controller () [4]. A cousin of the is the comparatively named zero-magnitude-error tracking controller () that has appeared in [,,8,9,4,5,7,8]. Yet another approximation method is the use of the noncausal series expansion discussed in [6], [] and []. Using a zeroth-order series expansion is effectively the same as choosing to ignore the nonminimum-phase zeros (while accounting for the proper DC gain); approximating the inverse of a system in this way offers a more simplistic approach, but may not be as accurate [7, 8, 3]. In contrast, some have chosen to use the exact unstable inverse and maintain stability of the system by pre-loading initial conditions or using noncausal plant inputs [5,, 4]. In general, using an exact unstable inverse method or a higher-order noncausal series expansion method introduces additional complexity when the controllers are implemented. In contrast, the zeroth-order series or nonminimum-phase zeros ignore (NPZ-Ignore),, and approximation techniques are natural options for designers of modelinverse controllers. This is largely due to their documented effectiveness (especially in the case of and ) combined with their simplicity of design and implementation [7, 8]. However, the proper choice of the NPZ-Ignore,, or techniques for maximizing performance objectives depends largely on the system on which it will be applied. In this paper, we discuss how the positioning of a system s nonminimum-phase zeros might indicate to the designer the more effective approximate model-inverse technique when considering NPZ-Ignore,, and. The performances of these techniques vary considerably as a function of NMP zero location. In addition, we show how the use of low-pass filtering can improve design options and performance at the cost of added design and implementation complexity. This paper is organized as follows. In Section, we provide further motivation for this paper by discussing the use of the NPZ-Ignore,, and model inverse techniques on discrete models of an atomic force microscope and a hard disk drive. The basics of each model-inverse technique are described in Section 3. In Section 4, we perform an analysis on a first-order system to show the effect of nonminimum-phase zero location on the three model-inverse methods. An extension to the complex-conjugate, secondorder zeros case is presented in Section 5. Initial conclusions are drawn in Section 6 followed by a discussion of the use of.5.5 Fig.. (a) H CL:HDD (z) (b) H CL:AFM (z) The pole-zero maps of the closed-loop HDD and AFM stage discrete-time systems. The sample time of each is 68 µsec and 4 µsec, respectively. Both have an order of 7 with a relative degree of. a low-pass filter with these feedforward methods and presentation of some experimental results in Section 7. Finally in Section 8, we discuss some additional design considerations when using these discrete-time model-inversion feedforward techniques. Motivation When attempting to maximize performance, the correct choice of the NPZ-Ignore,, or techniques depends on the system on which the feedforward method will be applied. As an example, and for motivation of this discussion, we apply the three types of model-inverse control to two closed-loop mechatronic systems often investigated in the literature. The first system is a x-y piezoscanner that is used in the operation of an atomic force microscope (AFM). A closed-loop model (ĤCL) for the x direction of this device is Ĥ CL:AFM (z) =.9685(z + ) (z.89z ) (z.959z ) (z.78z )(z +.775) (z.33z +.368) (z.959z ). () The second system is from a hard disk drive (HDD) application. An expression for the complete closed-loop HDD system appears in [7] and is provided again in (). Ĥ CL:HDD (z) =.988(z ) (z.4z +.449) (z.874z +.887) (z.859z )(z.54) (z +.389z +.574) (z +.33z +.878). () The models for both systems were developed through system identification processes and the pole-zero maps for both are

3 shown in Fig.. The sample time for the HDD is slower than that of the AFM and as a result the two systems can not be directly compared. Regardless, the exact details of each are not critical to our discussion, as we are more interested in the relative location of the nonminimum-phase zeros of each system and how they affect the results of the NPZ-Ignore,, and techniques. From Fig., we see that both the HDD and the AFM stage have NMP zeros. The HDD has two unstable zeros in the left-half plane (LHP), while the AFM scanner has two in the right-half plane (RHP). Extensive work by Rigney et. al. in [, 6 8, 5] showed the use of the and NPZ- Ignore algorithms with some success in settle-time applications for HDDs. Noting this, we started our study of modelinverse based feedforward control for AFM scanning applications with the same algorithms. We soon discovered that the and NPZ-Ignore algorithms could not be safely applied to the AFM system as the resulting signal output from F CL is extremely large and ringy to such an extent that it cannot be implemented without risking damage to the system and/or catastrophic saturation. This was in contrast to the algorithm which was directly implementable on the AFM scanner. In comparison, one could safely implement any of the three algorithms on the HDD system. In Fig. 3, we provide several plots of simulation results for both systems under four different control scenarios. The input x d (dash-dot black) to both is a 5.4 Hz raster scan. (The raster scan is a common trajectory for scanners in AFM systems [6]). The corresponding outputs x of the HDD and the AFM stage are shown in solid blue and dashed red, respectively. The top plot is a simple feedback-only simulation where F CL = for both systems. The three lower plots display simulation results for both systems when using the NPZ-Ignore,, or techniques for the feedforward controller F CL, respectively. These shall be considered best-case scenario results as the input to the plants in these simulations was allowed to be as large as it needs to be (i.e. no saturation block). Despite this generous allowance, the faults of the AFM results for NMP-Ignore and are evident. Fig. 3(a) shows us that given feedback-only control, the HDD struggles to track the peaks of the 5.4 Hz raster scan while the AFM loses all high-frequency features in the scan. In Fig. 3(b), we introduce feedforward model-inverse control in the form of the NPZ-Ignore method. Here we see the HDD (solid blue) tracking the input rather well, but the AFM scanner (dashed red) experiences high-frequency ringing that causes the trajectory to bounce at the raster corners (containing the high-frequency components of the raster scan). Despite the corners, the rest of the AFM stage scan is tracked rather well. Using the method for both systems in Fig. 3(c), we see results similar to the NPZ-Ignore technique. The HDD continues to track well, but in this case, the AFM s trouble at the corners of the raster scan have been amplified considerably (so much that the trajectory travels well out of the frame of the plot numerous times). The method (in Fig. 3(d)) relieves the AFM stage of its ringing problems and forces a slightly phase-lagged tracking of the raster scan. Fig. 3. (a) Feedback Only (b) NPZ Ignore (c) (d) Time (msec) The simulation results for a 5.4 Hz raster scan input (dashdot black) for the HDD system (solid blue) and the AFM scanner system (dashed red) for (from top to bottom) feedback-only control and the NPZ-Ignore,, and combined feedforward/feedback techniques. The HDD continues to track the raster scan despite the fundamental change in the model-inverse procedure. In the next several sections, we discuss why the AFM scanner with RHP nonminimum-phase zeros struggles to track the raster scan using the NPZ-Ignore and techniques while the HDD with the LHP nonminimum-phase zeros tracks the scan well regardless of the feedforward method. In the last half of this paper, we will discuss a method for settling the feedforward signals of the NPZ- Ignore and techniques and provide some experimental results using that method. 3 Three Stable Approximate Model-Inversion Techniques Defining the closed-loop system in Fig. as H CL (z) = PC +PC and writing out the transfer function from x d(t) to x(t), we arrive at (3). X(z) X d (z) = H CL(z)F CL (z). (3) Assuming H CL (z) is exactly proper and all zeros are minimum phase, we could define F CL (z) = H CL (z), and achieve perfect tracking of an input signal. However, many systems are not exactly proper and further have nonmimimum-phase zeros that require a stable approximate

4 model inversion technique. The design of the three techniques of focus begin with the same basic structure. First, we write the dynamics of the system as in (4), partitioning B(z) into the polynomial B s (z) containing the stable (invertible) zeros and the polynomial B u (z) containing the unstable (noninvertible) zeros: Table. Approximate Inverses & Overall Transfer Functions for each Approximate Inverse Method. METHOD F CL (z) X(z) X d (z) H CL = B(z) A(z) = B s(z)b u (z). (4) A(z) NPZ-Ignore z q A(z) B s(z)b u() z q B u(z) B u() The polynomial A(z) contains all the poles of the closedloop system model, and B u (z) can be written in the form of the n th -order polynomial z (q+n) A(z)B f u (z) B s(z)(b u()) z (q+n) B u(z)b f u (z) (B u()) B u (z) = b un z n + b u(n ) z n + + b u, (5) z q A(z) B s(z)b f u(z) z q B u(z) B f u(z) where n is the number of NMP zeros. We can then write the basic model-inversion structure for NPZ-Ignore,, and as: H CL (z) = A(z) B s (z)bu(z), (6) where the indicates an approximate inverse of the system. Bu(z) is defined depending on the type of stable model-inverse technique to be used. The precise choice of Bu(z) will be described in detail in the following subsections. Since the number of roots of the polynomial A(z) do not always equal the number of roots of the polynomial (B s (z)bu(z)), (6) may not be causal. As a result, q units of delay in (7) may be required to ensure a causal implementation of F CL (z). F CL (z) = z q H CL (z). (7) 3. The NPZ-Ignore Technique When using the NPZ-Ignore technique, the designer ignores any nonminimum-phase zeros in the system model and makes the proper adjustments for how this might affect the DC gain of the overall system. In general, this is the least precise of all the approximate model-inverse techniques discussed here as there is no accounting for an entire portion of the system dynamics. Nevertheless, this is the simplest of the three and may have benefits if there are limited computational resources for implementing the controller. For a NPZ-Ignore design, B u(z) in (6) reduces to B u:ign(z) = B u (z) z= = B u (). (8) Here we are ignoring the nonminimum-phase zero dynamics and reducing B u(z) to a scalar that compensates for losses in the DC gain associated with not including the dynamics of B u (z). The resulting NPZ-Ignore design for F CL:Ign (z) appears in Table along with the overall transfer function that corresponds to (3). For F CL (z) to be causal, the delay q must be equal to the order of A(z) minus the order of B s (z), (q = O(A(z)) O(B s (z))). 3. The Technique The structure of the technique is very similar to the NPZ-Ignore method, but the is higher order because it retains the dynamics of the nonminimum-phase zeros. When using the method, B u(z) in (6) becomes: B u:zp(z) = where B f u(z) is defined by ( Bu (z) z= ) z n B f u(z) = (B u()) z n B f u(z) (9) B f u(z) = b u z n + b u z n + + b un. () Note that the difference between (5) and () is the flipping of the polynomial coefficients. This action is equivalent to reflecting the roots z i of B u (z) in (5) into the unit circle to /z i and then advancing by z n so that B f u(z) in () is also a polynomial in z. The design for F CL:ZP (z) appears in Table along with the corresponding overall transfer function. In order for the implementation of F CL (z) to be causal, the delay q = O(A(z)B f u(z)) O(z n B s (z)). The reader should note that for both NPZ-Ignore and, (3) is a finite-impulse-response (FIR) filter which can be advantageous for various applications [6 8]. 3.3 The Technique In contrast to the method that converts NMP zeros of the model into stable zeros of the approximate inverse, the technique transforms the nonminimumphase zeros of the model into stable poles of the approximate inverse. When using the method, B u(z) in (6) becomes: where B f u(z) is defined in (). B u:zm(z) = B f u(z), ()

5 Table. Approximate Inverses & Overall Transfer Functions for the st -Order Example for each Approximate Inverse Method. METHOD NPZ-Ignore F CL (z) (z p) ( a)z X(z) X d (z) (z a) ( a)z still influences the system s dynamics near the frequency specified by its location. Specifically, the LHP real zero in X(z)/X d (z) Ign appears at an extremely high frequency. In contrast, the RHP real zero in X(z)/X d (z) Ign appears at a frequency two orders of magnitude lower than that of its LHP counterpart. As a result, the RHP nonminimum-phase zero plays a much larger role in the dynamics of the system than the LHP zero, specifically in the form of amplification of high-frequency components of input signals (see Fig. 5). (z p)( az+) ( a) z (z a)( az+) ( a) z (z p) ( az+) (z a) ( az+) The expression for F CL:ZM (z) appears in Table along with the overall transfer function that corresponds to (3). In this case, q = O(A(z)) O(B s (z)b f u(z)). No compensation for changes in DC gain are required for the method because the DC gain of (3) remains at unity as per design. Unlike the NPZ-Ignore and methods, the technique leads to an overall transfer function that is an infinite-impulse-response (IIR) filter. 4 Analysis of a First-Order Example In this section, we will break down a simple first-order example for each model-inverse technique. To start, we assume a sampling time T s =. seconds and that the parameter representing the position of the zero a (, ) (, ) is real and outside the unit circle in the system (a) LHP Zero, X(z)/X d (z) Ign (c) LHP Zero, X(z)/X d (z) ZP (b) RHP Zero, X(z)/X d (z) Ign (d) RHP Zero, X(z)/X d (z) ZP.5 H CL = (z a) (z p). ().5.5 In this case, A(z), B s (z), B u (z), and B f u(z) are defined as: A(z) = (z p), B s (z) =, (3) B u (z) = (z a), and B f u(z) = ( az + ). Using each of the three techniques described in Section 3, we can define F CL (z) for each as shown in Table. Looking at the overall transfer functions in Table, we see that the pole p plays no role as it has been canceled out. Recall, that as per the definition of a (, ) (, ), the equations in Table are not defined when < a <. This is because when < a <, exact inverses can be used. If we assume p =.5 and a =. (to achieve LHP and RHP nonminimum-phase zeros, respectively), we can plot pole-zero maps for each model and model-inverse method (Fig. 4). As expected from Table, we see that in all cases the transfer function X(z)/X d (z) retains the NMP zero in the dynamics. As a result, the NMP zero (e) LHP Zero, X(z)/X d (z) ZM Fig. 4. (f) RHP Zero, X(z)/X d (z) ZM The pole-zero maps of X(z)/X d (z) for the simple storder example for each model-inverse method. The three plots in the left-hand column represent the systems with LHP nonminimumphase zeros whereas the right column of plots are the contrasting systems with RHP zeros. Here, we assume p =.5 and a =.. In Figs. 4 and 5, we see the problem with a RHP nonminimum-phase zero becomes worse when using the method. Here the low-frequency RHP nonminimum-phase zero gets reflected into the unit circle as an additional zero. This compounds the problems of the NPZ-Ignore technique as another zero (albeit stable) gets placed at the exact same frequency of the RHP nonminimum-phase zero. This additional zero counteracts

6 the phase drop associated with the RHP nonminimum-phase zero, but in the magnitude sense we get even more amplification of high-frequency components of input signals. In contrast, the method suffers no ill effects of the LHP nonminimum-phase zero as it and the additional stable zero occur at such extremely high frequencies that their influence on dynamics is limited to attenuating the extremely high frequency region. In contrast, we see in Figs. 4(e) and 4(f) that the method reflects NMP zeros into the unit circle as poles at the same frequency. This is how this method is able to attain zero-magnitude error at all frequencies. The increasing magnitude effects of any nonminimum-phase zero is canceled by a stable pole. One should note that in Fig. 5, the method for both LHP and RHP systems does not appear to have zero-phase error at all frequencies. This is due to the delay added to F CL to make it causal. The source of these high-frequency magnitude issues becomes clear if we look at the real and imaginary parts of X(e jω )/X d (e jω ) for each model-inverse method. Defining [ X(e jω ] [ ) X(e jω ] ) Γ = R X d (e jω and Φ = I ) X d (e jω ) for each model-inverse method, and utilizing some trigonometric identities, we can write Γ and Φ for each method (see Table 3). Looking at the magnitude for in Table 3, it again becomes clear why this method is called the zeromagnitude-error tracking controller as its magnitude remains at unity for all frequencies regardless of the value of a. In contrast, we see a very different behavior in the NPZ-Ignore and methods when the parameter a is varied. Specifically, we note that as ω approaches the Nyquist frequency (π radians/second in the case of T s = second), the imaginary term Φ Ign goes to zero regardless of the value of a. That leaves the real term Γ Ign to dominate the magnitude of X(z)/X d (z) Ign and lim Γ Ign (ω) = ω π + a a. (4) When <a<, we see that (4) is less than one and as a result we expect a rolling off of the magnitude near the Nyquist frequency. In contrast, when <a<, (4) achieves a value greater than one that suggests amplification of any high-frequency components of the input signal. A similar analysis on the method shows that as ω approaches π radians/second lim Γ ZP (ω) = lim Γ Ign (ω) = + a ω π ω π a, (5) which indicates the squaring effects of a RHP nonminimumphase zero when compared to the NPZ-Ignore method (which also can be seen in the magnitude column of Table 3). We can see this behavior graphically if we look at the positive-frequency portion of the Nyquist plots of both X(z)/X d (z) Ign and X(z)/X d (z) ZP for a =. in Fig. 6. Specifically, in Fig. 6(a) we see how the line representing the system with a RHP zero (dashed red) leaves the area near the origin and converges to a real value of as ω converges on π radians/second. In contrast, the line representing the system with a LHP zero (solid blue) never leaves a ball of radius one centered on the origin. When we turn to the method in Fig. 6(b), we see that the dashed red line achieves such a large value, that the solid blue line (which always has magnitude less than or equal to one) is so small and not visible on this scale. Fig. 7 is a plot of the limit of the magnitude as ω approaches the Nyquist frequency of X(z)/X d (z) in (3) for the closed-loop system in () for each control method as a is varied. Here the region where a value of a indicating a minimum-phase zero in () is marked in shaded yellow. This region is not valid for comparison of the three approximate model-inversion techniques as the exact inverse can be used. Again, we see that the method achieves unity gain for X(z)/X d (z) ZM as a is varied. For the NPZ- Ignore and methods, we see peaking magnitudes as a approaches from the right; this will serve to amplify any high-frequency components and noise in the input signal. Additionally, we see the rolling off of magnitude as a approaches from the left. As we will discover later in this paper, this rolling off may actually be a desired design feature of these methods. We should note that the rolling off of magnitude associated with LHP nonminimum-phase zeros in the NPZ-Ignore and methods does not generally affect tracking magnitudes as tracking at such highfrequencies is often beyond what the physical system is capable of doing. Clearly, the effect of these NMP zeros (regardless of their existence in the LHP or the RHP) becomes trivial as a becomes large. 5 Extension to Complex Zeros Extending to an example with complex-conjugate NMP zeros, we see the same trends of the first-order system. If we define r z > as the distance from the origin to the complex zeros and θ as the angle (measured from the positive-real axis in radians) defining the direction of r z, we can write H CL = (z r ze jθ )(z r z e jθ ) Ψ(z) = (z r z cos(θ)z + rz), (6) Ψ(z) where Ψ(z) represents a polynomial of stable poles of order two or more. As we discovered previously, the exact poles in the system are not of interest (assuming they are stable) as they will be canceled in any of the three model-inverse methods. The angle θ defines the LHP or RHP location of the complex zeros. Performing an analysis similar to that for the first-order system in Section 4, we can show that the conclusions in

7 Magnitude (db) 4 Magnitude (db) 4 Phase (deg) 5 NPZ Ignore 5 Frequency (rad/sec) Phase (deg) 5 5 Frequency (rad/sec) (a) LHP Zero, a =. (b) RHP Zero, a =. Fig. 5. The Bode plots of the transfer function X(z)/X d (z) of the st-order example for each model-inverse method with a LHP and RHP nonminimum-phase zero (a =. and a =., respectively). Table 3. Summary of X(e jω )/X d (e jω ) for the st-order Example for each Approximate Inverse Method. METHOD Γ = R [ X(e jω )/X d (e jω ) ] Φ = I [ X(e jω )/X d (e jω ) ] MAG ( Γ + Φ ) NPZ-Ignore Γ Ign = a cos(ω) a Φ Ign = a sin(ω) a +a a cos(ω) ( a) Γ ZP = cos(ω) ( a) ( + a acos(ω)) Φ ZP = sin(ω) ( a) ( + a acos(ω)) +a a cos(ω) ( a) Γ ZM = (+a ) cos(ω) a +a a cos(ω) Φ ZM = ( a ) sin(ω) +a a cos(ω) Imaginary Real (a) NPZ-Ignore Imaginary 5 a = Real (b) Fig. 6. The positive-frequency portion of the Nyquist plots of both X(z)/X d (z) Ign and X(z)/X d (z) ZP for a =. in (). Solid blue represents a =., while dashed red represents a =.. In (b), the solid blue line (a =.) is not visible due to the large scale of the dashed red line (a =.). Note that the axis scales of (a) and (b) are not the same. Section 4 extend to the case of complex-conjugate zeros by considering when ω approaches the Nyquist frequency similar to (4) and (5). We find that the imaginary terms Φ go to zero and the real terms dominate the magnitude: lim Γ Ign (ω) = ( + rz + r z cos(θ)) ω π ( + rz r z cos(θ)), lim Γ ZP (ω) = lim Γ Ign (ω), and ω π ω π

8 Mag. at Nyquist Frequency (db) Invalid Region NPZ Ignore Nonminimum Phase Zero Position, a Mag. at Nyquist Frequency (db) 5 5 NPZ Ignore Nonminimum Phase Zero Angle, θ (deg) Fig. 7. The limit of the magnitude of X(z)/X d (z) as ω approaches the Nyquist frequency for the st-order example for each control method as a is varied. The shaded yellow region indicates the area where the zero would become minimum phase and is not a valid area for comparison. lim Γ ZM (ω) =. (7) ω π These limits are only valid when r z > since exact inversion can be used when r z <. Fig. 8 shows the limit of the magnitude of X(z)/X d (z) as ω approaches the Nyquist frequency for constant r z =. and varying θ. Again, we see that the method achieves unity gain for X(z)/X d (z) ZM as θ is varied. For the NPZ-Ignore and methods, the effect of NMP zeros becomes trivial as the zeros move close to the imaginary axis. Similar to Fig. 7, in Fig. 8 we also see the rolling off effect of LHP zeros for the NPZ-Ignore and methods, and the amplification effect in the RHP. In support of Fig. 8, we provide Fig. 9 which presents the limit of the magnitude of X(z)/X d (z) as ω approaches the Nyquist frequency for each control method as r z is varied for three values of θ (,3, and 45 ). As expected, we see that the method achieves unity gain for X(z)/X d (z) ZM as r z is varied, and the effect of NMP zeros becomes trivial as r z grows toward ±. Similar to Fig. 7 and Fig. 8, in Fig. 8 we continue to see the expected rolling off effect of LHP zeros for the NPZ-Ignore and methods, and the amplification effect in the RHP. 6 Initial Conclusions Given the analysis thus far in this paper, we have shown that: (i) For systems with RHP NMP zeros near the unit circle, is the only viable choice of the three feedforward methods discussed here. (ii) When NMP zeros are far from the unit circle (in either the RHP or LHP), their affect on the feedforward methods is reduced. (iii) If the NMP zeros are near the unit circle and the real axis, the selection of a feedforward method should be based on the desired control objectives. Fig. 8. The limit of the magnitude of X(z)/X d (z) as ω approaches the Nyquist frequency for the complex-conjugate zeros example for each control method as θ is varied and r z is a constant.. θ is measured from the positive-real axis. Mag. at Nyquist Frequency (db) Fig Invalid Region NPZ Ignore 5 5 Nonminimum Phase Radius Length (r z ) The limit of the magnitude of X(z)/X d (z) as ω approaches the Nyquist frequency for the complex-conjugate zeros example for each control method as r z is varied for three values of θ (,3, and 45 ). θ is measured from the positive-real axis. The largest peaking curves of the NPZ-Ignore and methods are associated with θ = while the smallest peaking curves are from θ = 45. Regardless of the value of θ or r z, remains at unity. The shaded yellow region indicates the area where the complex zeros would become minimum phase and is not a valid area for comparison. Given our initial analysis, an inexperienced designer of approximate model-inverse controllers might make the conclusion that she/he will always use the method as it H CL guarantees unity gain of H CL (z) (z) for all ω frequencies and any LHP or RHP location of the nonminimum-phase zeros. For NMP zeros in the LHP or far outside the unit circle, other considerations may be important including signal saturation, noise in the system and more. When NMP zeros are far outside the unit circle (case (ii)), one should consider computational requirements and phase-lag issues. For any implementation that has limited computational resources, the NPZ-Ignore method is the simplest and may be the best option. In terms of phase-lag issues, depending on the system model, a different amount of delay (z q ) may be needed to make any of the three feedforward filters causal. This consideration could be crucial for

9 e x H CL x Σ C u d F Low Pass x x CL Filter P Fig.. A block diagram of the closed-loop-injection architecture with an added low-pass filter after the feedforward filter F CL. any time-critical applications [, 7, 8]. In contrast, some AFM scan applications are not phase critical [, ], so phase delays may not be problematic. When NMP zeros are near the unit circle and the real axis (case (iii)), one should consider the LHP or RHP location of these NMP zeros, computational requirements, phase lag issues, and any implementations requiring highfrequency attenuation. Recall from Section 4 that for LHP NMP zero locations, the or NPZ-Ignore methods tend to roll-off high-frequency regions providing potential attenuation to any high-frequency noise. Given a system like the HDD in which any of the three methods can be applied with some level of success, the designer can make a choice of the three based on their ultimate control goals. In general, might not be an ideal choice for the HDD if the provides sufficient results as the high-frequency attenuation of the method would likely be desired on an actual implementation. An HDD application might also have limited computational resources, so the NPZ-Ignore technique might be the best option under those circumstances. Unfortunately, the RHP zeros of the AFM piezoscanner limit it to only the method. 6. An Extension for RHP NMP Zeros We note that in [], the authors discuss the effectiveness of their higher-order noncausal-series approximation for model-inverse control on a system with RHP nonminimumphase zeros. Further, in the same paper, the authors preprocess the desired trajectory signal (x d (t)) with a low-pass filter in order to allow the use of the method with a system containing RHP nonminimum-phase zeros. The inclusion of low-pass filters (LPFs) in the feedforward design process for systems with RHP NMP zeros is an interesting extension of these model-inversion methods and will be the focus of the latter portion of this paper. 7 Filtering Feedforward Signals One conclusion of the discussions in the previous sections of this paper is that when a system has RHP NMP zeros near the unit circle, the only applicable design option of the three is. This is true if and NMP-Ignore are used in their strict form. By adding an additional design element in the form of a low-pass filter (LPF) (as suggested in [] and shown in Fig. ), we discover that these previously unusable techniques become of value again. Looking at the frequency response of the overall transfer Phase (deg) Magnitude (db) Fig NPZ Ignore 4 Frequency (Hz) The Bode plots of the transfer function X(z)/X d (z) of the AFM system in () for each model-inverse method. function X(z)/X d (z) of the AFM system in () in Fig., we note the same trend we saw in Fig. 5(b). Both the and the NPZ-Ignore techniques magnify the high-frequency regions, while the provides unity magnitude at all frequencies. By properly pairing a LPF with the and NPZ-Ignore filters, we can expect that we can alleviate this magnification at high-frequencies and return these magnitudes to the neighborhood of unity. A few drawbacks to adding a LPF to the system to make and NPZ-Ignore methods viable options are that (a) it is a further design requirement that adds complexity for implementation and (b) the performance is sensitive to the design of the low-pass filter. In the following subsections, we introduce the use of LPFs with these feedforward filters and discuss their use. Experimental results from the x-direction of an AFM piezoscanner are provided to support our conclusions. We begin by introducing some metrics we use to quantify the experimental tracking performance of our system. 7. Performance Metrics When discussing the performance of the tracking of a raster scan in AFMs, it is important to recall the overall goal of AFMs: to create a quality image in a timely manner [6]. But this goal requires a definition of a quality image when referring to an x-y raster scan. Focusing on the x direction, an ideal controller would cause the system output x(t) to track the desired raster pattern x d (t) flawlessly. This suggests that a mean-square-error metric might be informative in

10 Fig.. k* = x d (t) x d (t k T s) x(t) Time (sec) (a) FB Only: J e = and J m =.35 AFM scanner experimental results using only feedback control. Three curves appear in the figure: the 5.4 Hz raster input x d (t), the x d (t k T s ) used for the performance metrics J e and J m, and the x(t) output. In this case, k (displayed in the lower left corner of the plot) is defining the performance of a controller. However, for imaging, mean-square error might not give us the best measurement. This is because phase lag in the raster scan used for AFM imaging is not nearly as critical as consistently tracking the magnitude. Ultimately, this means that perfectly delayed tracking is better than imperfect timely tracking if we know the delay well. So, to identify that delay, we examine several periods (T ) of the recorded raster scan after a time t ss after which all transients (from initial conditions for example) have died out, and define the variable k as tss+nt k ( = arg min xd (t kt s ) x(t) ) dt. (8) k t ss Here, k is a variable defined on [, T T s ] where T s = 4 µsec is the controller sample period, and N is the number of recorded periods after transients have died out. N is always greater than or equal to, but can vary depending on the raster frequency and the size of the memory block to which the data is being written. The allowance of additional data through the use of a larger N provides more confidence in the determination of k. Specifically, k reflects the delay in the system which can be used to define two metrics that emphasize magnitude tracking over ten periods: J e = tss+t( xd (t k T s ) x(t) ) dt (9) t ss J m = ( max xd (t k T s ) x(t) ) t [t ss,t ss+t) () For further clarification, we provide Fig. which shows experimental results of a 5.4 Hz raster scan input Magnitude (db) Phase (deg) Fig NPZ Ignore 4 Frequency (Hz) The Bode plots of the transfer function X(z)/X d (z) of an AFM piezoscanner with each model-inverse method (NPZ-Ignore,, and ) when a LPF is included. The 5th-order Butterworth filter cut-off frequencies are 5, 75, and 5 Hz, respectively. into a feedback-only control loop. The feedback filter is a 5th-order H filter designed following techniques described in [7]. Design objectives include maximizing bandwidth, limiting saturation, and avoiding excitation of the plant s resonance frequency. We have chosen this feedback-only example for its clear delineation of each line. We see the actual 5.4 Hz raster scan input x d (t) in solid black, and the shifted 5.4 Hz input x d (t k T s ) in dashed black. From Fig., it is clear that the feedback-only controller lacks the bandwidth to be able to track the 5.4 Hz raster scan. Images created while using this raster scan would be highly distorted at the edges. The feedback-only experimental results in Fig. were recorded while using the same feedback controller as the AFM simulation results presented in Fig. 3(a). 7. LPF Design In systems with RHP NMP zeros near the unit circle, the and NPZ-Ignore methods can be implemented if low-pass filters are included in the feedforward path. The cost of such LPFs comes in the form of added complexity of design and implementation. Based on the discussion in previous sections of this paper, one might think that the best LPF would be the one that results in an overall transfer function for and NPZ- Ignore that mimics the system and retains approximately unity gain for all frequencies (see Fig. ). After several experiments on our AFM x-y scanner using low-pass filtered feedforwad signals on and NPZ-Ignore that were designed to achieve approximately unity gain for all frequencies for the overall transfer function, we discovered that this design approach is not ideal. By designing the LPFs such that the frequency response function (FRF) of the overall transfer function rolls-off at high frequencies (much like we see in Fig. 5(a)) for feedforward filter design for LHP NMP zeros), we note better experimental performance than when we design for unity

11 gain for all frequencies. This begs the question: although not necessary, what do designs have to gain when a LPF is included on RHP NMP zeros systems (like our AFM piezoscanner)? In our system, the advantage presents itself as improved raster tracking performance at speeds (5.4 Hz) greater than we had seen before. Prior to adding LPFs to the feedforward path, our RHP-NMP-zero AFM piezoscanner was limited to as the only viable option of the three modelinversion methods discussed here. The result of using this pure design is well-performing raster tracking [,3], but the raster speed was limited by saturation of the control signal u x. The filter F CL produces signals with amplified high-frequency components, and the result is that the control signal u x can easily grow beyond its physical limits as the desired speed of the raster input increases. This is because achieves unity gain of X(z)/X d (z) by applying an F CL filter that amplifies high frequencies in an effort to counteract the typical low-pass shape of H CL (z). Three Butterworth filters were used with varying cutoff frequencies for each of the three feedforward modelinversion techniques. For the sake of comparison, the order of these LPFs is kept at five. The choice of the exact cut-off frequency is made empirically for each feedforward method by selecting a frequency that limits signal excitement to sufficiently avoid control signal saturation. After including the LPFs into the overall transfer function, Fig. becomes Fig. 3. The NPZ-Ignore and methods are more sensitive to the choice of the LPF cut-off frequency. This is because in the case of these two techniques, the filter is playing the role of pulling-down the amplification at highfrequencies as well as rolling-off at extremely high frequencies. As a result, the cut-off frequency for these two methods must occur at a frequency much lower than that of the LPF used with the filter. Consequently, the LPFs for the NPZ-Ignore and methods will filter out more high-frequency components of the desired raster-pattern. Experimental results of the low-pass filtered use of the NPZ-Ignore and methods on our RHP-NMP-zero AFM piezoscanner are shown in Fig. 4. Here, each method uses the same feedback controller used in the feedback-only example of Fig.. If it were not for the use of the LPFs, we would not have been able to implement these feedforward design methods. Beyond that, the benefits of pairing a feedforward filter with a feedback controller becomes clear when visually comparing these results to those shown in the feedback-only example in Fig.. We are able to achieve a much higher tracking bandwidth when adding a LPF to the feedforward control. Experimental results of the low-pass filtered use of the method on our RHP-NMP-zero AFM piezoscanner are shown in Fig. 5. If it were not for the use of the LPF, we would not have been able to use as the feedforward filter design without experiencing saturation and negatively affecting the tracking performance. Note that the cutoff frequency for the LPF paired with is 5 Hz as compared to 5 and 75 Hz for NPZ-Ignore and, respectively. The optimization of the LPF cut-off frequencies for each feedforward method discussed in detail here would likely lead to further performance gains. At the moment, we are hesitant to spend much time on this task until we improve our system model and identify variations in the system that cause that model to deviate over time. 8 Discussion and Conclusions In this paper we have reviewed three well-known discrete-time model-inversion techniques and discussed their applications to systems with right and left-half plane NMP discrete-time zeros. When using these methods in the strictest sense, we note the conclusions summarized in Section 6. Further, as suggested in [], we combined the feedforward filters from each method with LPFs on a system with RHP discrete-time zeros near the unit circle. The lowpass filters are required for the NPZ-Ignore and to be physically implementable, and the method also benefits from the use of a LPF. This benefit comes in the form of reduced control signal energy which allows raster speeds to be increased without saturating the plant input signal u x. Given our experimental results and discussions, we can further conclude that: (a) For systems with RHP NMP discrete-time zeros near the unit circle, NPZ-Ignore and feedforward filters must be combined with a well-designed LPF in order to be viable feedforward design options. Further, combining an LPF with may not be required for implementation, but may allow better performance by reducing the likelihood of saturation. (b) For systems with LHP NMP discrete-time zeros near the unit circle, NPZ-Ignore,, and feedforward filters can be used directly without the use of a LPF. However, using a LPF filter with the design may allow better performance by reducing the likelihood of saturation. (c) The accuracy of the plant model is critical to performance when using these model-inversion techniques. In general, it is wise to consider the location of discretetime NMP zeros when using any of these three modelinversion feedforward methods. 9 Future Work Current and future work includes investigating the accuracy and variations of our model of our AFM system. We plan to identify and quantify these variations. Once properly understood, we plan to develop indirect adaptive methods to compensate for the slowly time-varying AFM system. Acknowledgments This work was supported in part by Agilent Technologies, Inc., the US National Science Foundation (NSF Grant CMMI-7877), and the Miller Institute for Basic Research

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