Optimal Selection of Basis Functions for Minimum-Effort Tracking Control of Nonminimum Phase Systems Using Filtered Basis Functions

Tracking control is a fundamental problem encountered in a wide range of application areas such as manufacturing, robotics, and aeronautics. The objective of tracking control is to force the output of the controlled system to follow a desired trajectory as closely as possible. It is also important that this objective is achieved with minimal control effort, e.g., due to power limits of actuators. Excellent tracking accuracy can be achieved using feedforward control by direct inversion of a sufficiently accurate model of a system (i.e., pole-zero cancellation) [1]. However, when applied to systems with nonminimum phase (NMP) zeros, direct model inversion gives rise to unstable or extremely high control inputs which are unacceptable [1]. There have been several feedforward tracking control methods reported in the literature that are applicable to linear systems with NMP zeros. These methods include NMP zero ignore (NPZ-ignore), zero-phase error tracking control (ZPETC) [1], zero magnitude error tracking control, extended bandwidth ZPETC [2], truncated series (TS) [3], direct inversion with bounded reference trajectories [4–8], approximate frequency-domain inversion [9], H_∞ matching [10,11], B-spline-based tracking with preview using iterative learning control (ILC) [12], spline filtering with feedback [13,14], etc. A major shortcoming of most of these methods is that they are not versatile in terms of the systems and/or the desired trajectories to which they are applicable (e.g., several of the methods cannot be applied to nonhyperbolic systems, i.e., systems with zeros on the unit circle)—see Ref. [15] for a more detailed discussion of this matter. Moreover, their tracking accuracy varies significantly depending on NMP zero location (in the complex plane) [15].

The filtered basis functions (FBF) approach has recently been gaining interest as an approach for tracking linear systems with NMP zeros [15–19]. The origin of the FBF approach can be traced back to the work of Frueh and Phan [20] on inverse linear quadratic learning in ILC. It expresses the control input as a linear combination of user-defined basis functions with unknown coefficients. The basis functions are forward filtered using the system dynamics, and their coefficients selected such that tracking error is minimized. Unlike most of the methods discussed above, the FBF method is effective in tracking any desired trajectory, irrespective of the location of NMP zeros in the z-plane (including nonhyperbolic systems [15,17,19]). Moreover, it has been observed by the authors that the FBF method maintains consistent tracking accuracy compared to popular linear time invariant (LTI) discrete-time tracking controllers irrespective of the location of the NMP zero in the z-plane [15,17,21].

There is a wide range of basis function available for use with the FBF approach. The choice of basis functions is entirely up to the control engineer. Prior work in the literature has explored cosine signals [20,22], reference trajectory-based basis functions [23,24], B-splines [12,17,21], Gaussian radial basis functions [18], etc. However, there has been no study on the optimal selection of basis functions to achieve a desired control objective using the FBF approach. This paper (and, in part, its preliminary version [25]) addresses the problem of optimal basis function selection by making the following original contributions to the literature:

1. It proposes the Frobenius norm of the lifted system representation (LSR) of dynamics as an excellent and appropriate metric for analyzing the performance of LTI and linear time varying (LTV) tracking controllers, independent of desired trajectory. The proposed metric is applicable to any discrete-time linear controller.

2. It demonstrates, using the metric that the tracking accuracy of the FBF approach is solely dependent on the number of basis functions used; it is independent of the type of basis functions and the plant dynamics (e.g., zero location), thus explaining the consistent tracking performance of the FBF approach relative to other tracking controllers.

3. It demonstrates that the metric for FBF's control effort dynamics is dependent on the system dynamics and the type of basis functions used. A methodology for determining the optimal set of basis functions for achieving a desired level of tracking accuracy with minimum control effort is derived analytically. These optimal basis functions are shown to be singular vectors (SV) of the lifted system representation of the plant dynamics.

Section 2 provides some background and motivation for the paper, and Sec. 3 presents the proposed metric, analyzes the FBF method using the metric and derives a methodology for determining the optimal set of basis functions. The discussion in Sec. 3 is validated in Sec. 4, using simulations and experiments. This is followed by conclusions and future work in Sec. 5.

The FBF approach relies on two assumptions:

1. the desired trajectory is known a priori; and

2. the control input u(k) is expressed as a linear combination of n + 1 user-defined basis functions φ_i(k); i.e.,

   $u(k)=∑i=0nγiφi(k)$

   (6)

Remark 1.C_FBF and E_ff,FBF both depend on the LSR of the plant, G, as well as the selected basis functions. Both matrices are, in general, non-Toeplitz and nontriangular implying that the FBF controller is, in general, LTV and noncausal [15].

Remark 2. Although, for simplicity, this paper describes the FBF approach in the context of LTI SISO systems, it is applicable to other types of linear systems such as LTV, linear parameter varying [18], and multi-input multi-output systems. Duan et al. [17] relaxed the assumption on a priori knowledge of the entire desired trajectory using the local property of B-splines as basis functions. Without loss of generality, this paper assumes that the initial conditions are zero. The authors' prior work [15, 17] can be consulted for approaches to incorporate effects of initial conditions into the FBF approach.

For simulations, the desired trajectory (y_d) is a zero-mean white noise signal, with variance equal to 1, M = 1000 and sampling frequency 10 kHz. As discussed in Sec. 1, there is a wide range of basis functions available for use with the FBF method. Here, three types of basis functions are used: (i) discrete cosine transform (DCT) [22], (ii) block pulse functions (BPF) [27], and (iii) B-splines [28]; their expressions are provided in Appendix  B. The DCT and BPF basis functions are rudimentary basis functions in frequency-domain and time-domain, respectively, whereas, B-splines are commonly used to parameterize commands sent to manufacturing machines and robots [29].

Figure 2(a) compares the normalized root-mean-square (RMS) tracking error e_RMS/y_d,RMS for the three basis functions (n = 990) for various values of a. It must be pointed out that approximate inversion is not generally used for tracking control in the minimum phase (MP) region because C = G⁻¹ can be employed (provided a is not poorly damped [1]). However, the MP region is included in this paper for the sake of completeness. The FBF approach demonstrates consistent tracking accuracy, as compared to popular methods in the literature (see Fig. 12 in Appendix  C for a comparison with two such popular methods, viz., ZPETC and TS), irrespective of zero location and type of basis functions [15,17,21]. Similar observations have been made in prior work [15,17] and contrasted with other tracking controllers whose tracking accuracy typically varies significantly depending on NMP zero location [3,26]. A theoretical justification for this observation was preliminarily explored in Ref. [25], and is further discussed in Sec. 3 below.

Figure 2(b) compares the normalized RMS control effort of the various basis functions applied to the FBF approach. Notice that there is significant variation in control effort for various basis functions, even when tracking accuracy is similar. For instance, at a = 1.02, all the basis functions achieve similar levels of tracking accuracy, but the control effort required by DCT is 370 times that of BPF, whereas, the control effort required by B-splines is 11,800 times that required by DCT. In contrast, all the basis functions have very similar values of e_RMS and u_RMS for −1 ≤ a ≤ 1. This suggests that the system dynamics and choice of basis functions play a significant role in the control effort required to achieve a desired level of tracking accuracy using the FBF approach. Hence, a methodology for determining the best set of basis functions for a given plant and desired level of tracking accuracy is needed.

A suitable metric is needed in order to probe deeper into the observations made in Sec. 2.3 above and provide a methodology for selecting an optimal set of basis functions. While Bode diagrams [3] and magnitude at Nyquist frequency [26] have been used as metrics to analyze LTI tracking controllers, they are not applicable to LTV controller like FBF. In this section, we propose a metric that is suitable for analyzing LTV controllers like FBF, independent of trajectory to be tracked, and utilize the metric to develop a methodology for optimal selection of basis functions for minimum control effort.

In other words, J_e approaches the error 2-norm criterion (sometimes used in the design and analysis of tracking controllers [3]). The singular values of E_ff approximate the magnitude of the frequency response of E_ff(q) [31]; and ǁE_ff(q)ǁ_∞ is the maximum gain of the system E_ff(q) which is approximated by ǁE_ffǁ₂ (because the 2-norm of a matrix is its maximum singular value), and this approximation is more accurate as M → ∞.

The implication is that J_e is an upper bound on the RMS tracking error (e_RMS).

Remark 3. The LSR is employed for the proposed metric because it applies to both LTI and LTV controllers [32]. Moreover, the LSR is applicable to feedforward as well as feedback controllers, SISO as well as multi-input multi-output controllers. Thus, the proposed metric is broadly applicable to any linear discrete-time tracking controller. It can also be used for evaluating other performance criteria beyond tracking accuracy and control effort.

▪

Remark 4. Note that J_e,FBF is independent of G(q) and the type of basis functions employed. It depends only on the number of basis functions (relative to the number of discrete points in the trajectory). As discussed in Ref. [25], the independence of J_e from G(q) cannot be taken for granted with other tracking controllers [26]. The consistent tracking accuracy of FBF stems from the unique structure of E_ff,FBF, and it provides an analytical explanation of the relative independence of the FBF method's tracking accuracy from G(q) observed in prior work [15,17,21], and demonstrated in Sec. 2.3. Also, the result of Proposition 1 holds for any linear plant dynamics, i.e., it also applies to LTV and linear parameter varying plants.

Remark 5. Note that, ǁE_ff,FBFǁ₂ is equal to 1, irrespective of the number of basis functions used, which is not a reasonable representation of the tracking accuracy of the FBF method, which varies significantly with n [15,17,21]. Hence, the proposed metric is more appropriate compared to 2-norm metrics like those used in convergence analysis in ILC [30].

Remark 6. J_e = 0 implies C = G⁻¹ which might be undesirable if G(q) contains uncancellable zero(s) because such zeros result in very small singular values of G, and large control signals [33]. The FBF approach is rank constrained minimization of the metric applied to error dynamics [25]. A rank constraint, which in the LSR implies a restricted space of input and output [15], is used to avoid inversion of the full G, while also reducing the computational demands of the control problem [22,34,35]. However, the rank constraint does not necessarily result in minimization of control input and hence, analysis of FBF controller dynamics (discussed in the remainder of this section) and selection of an optimal set of basis functions (presented in Sec. 3.3) is necessary.

Without the loss of generality, this paper assumes that G has distinct singular values.

The squared Frobenius norm of FBF controller dynamics $‖CFBF‖F2$ is a linear combination of the inverse of squared singular values of G, i.e., ${1/σi2}i=1M+1$ with coefficients of the linear combination $∑j=0nξ̃ij2$ determined by interaction between filtered basis functions $ψ̃j$ and system dynamics G. The coefficients are bounded between 0 and 1 and need to satisfy orthogonality condition (see Eq. (20)).

It has been shown, so far, that the tracking accuracy of the FBF method (measured by J_e) is always fixed for a given number of basis functions, irrespective of the type of basis functions or the plant dynamics. Also, for a given number of basis functions, the control effort of the FBF method (measured by J_c) is dependent on the plant dynamics and type of basis functions. In this section, the optimal set of basis functions that minimizes J_c for a given J_e are presented—and the resulting controller is called the optimal FBF controller (i.e., optimal in terms of minimizing control effort).

Proof. Proposition 2 is substantiated by

• proving by contradiction that there is no $ξ̃ij$ such that

  $‖CFBF‖F2<∑i=1n+11σi2$

  (31)
• showing there exists $ξ̃ij$ such that

  $‖CFBF‖F2=∑i=1n+11σi2$

  (32)

▪

However, when n = M, the control input is undesirably high if system G contains NMP zero(s) [16,33].

Remark 7. The SV-based basis functions (given by Eq. (43)) are the optimal set of basis functions which result in minimum J_c for given J_e. Note that these optimal basis functions are system dependent. Inversion of the LSR, G⁻¹ for NMP systems can be realized, approximately, by truncating the smallest singular values from the SVD of G [16,33]. The truncated SVD-based approximation of G⁻¹ is a special case of the optimal FBF controller (i.e., it uses the SVs of G as basis functions with n = M − r, where r is the number of NMP zero(s) of system G).

Remark 8. For FBF, the fact that J_e is independent of plant dynamics and basis functions, whereas, J_c is dependent on the plant dynamics and basis functions permits a sequential two-stage design procedure for achieving the optimal FBF controller. In the first stage, the user selects the number of basis functions (n + 1) to achieve a desired level of tracking accuracy (J_e). Then, in the second stage, the control effort (J_c) is minimized by selecting n + 1 of the highest M + 1−r SV components of G as the optimal set of basis functions.

Remark 9. While the proposed two-step methodology is general in that it is independent of desired trajectory, it can accommodate special cases where information (e.g., frequency spectrum) of a specific desired trajectory to be tracked is available. For instance, the dynamics can be premultiplied with a weighting filter that emphasizes the frequency content of the desired trajectory and the proposed methodology can be applied to the augmented dynamics.

Remark 10. The optimization-based methodology discussed in this section can easily be extended to other control objectives (e.g., basis functions selection in the presence of uncertainty) and other applications (e.g., basis functions selection for ILC).

Section 2.3 motivated Sec. 3 using a first-order plant and three different basis functions, viz., DCT, BPF, and B-splines. This section continues with the same example and compares the three basis functions mentioned above, with the optimal basis functions proposed in Sec. 3.3. The desired trajectory and other parameters (M = 1000, n = 990) are same as Sec. 2.3.

Figure 3 plots J_e and J_c for different basis functions and Fig. 4 plots the normalized tracking error e_RMS/y_d,RMS and normalized control input u_RMS/y_d,RMS. Note that Figs. 3 and 4 validate the discussion in Sec. 3. The trend for e_RMS/y_d,RMS and u_RMS/y_d,RMS in Fig. 4 are quite similar to those of J_e and J_c, respectively, in Fig. 3, which demonstrates the effectiveness of the proposed metric. Note that there might be instances when performance trends may not exactly follow the predictions of J_e or J_c. For example, FBF might have much better tracking accuracy than predicted by J_e or smaller control effort than predicted by J_c if one purposely (or accidentally) uses filtered basis functions that span the desired trajectory (y_d). However, in general, the proposed metric provides good insights on the upper limits of RMS tracking performance and control effort. As discussed in Sec. 3.2, all the four basis functions have similar tracking error (see Fig. 4(a)) but different control efforts (see Fig. 4(b)). Table 1 shows the mean values of e_RMS/y_d,RMS and u_RMS/y_d,RMS over all a. For e_RMS/y_d,RMS, the values of the mean for different basis functions are of the same order of magnitude. This validates the discussion about consistent tracking accuracy of the FBF approach, as compared to popular methods in the literature (see Figs. 12 and 13 in Appendix  C for a comparison with two such popular methods, viz. ZPETC and TS), for different types of basis functions. However, when it comes to u_RMS/y_d,RMS, the value of the mean for B-splines is four orders of magnitude higher than the value of the mean for DCT, and the value of the mean for DCT is two orders of magnitude higher than BPF and optimal basis functions. Although BPF and optimal basis functions have similar order of magnitude, the value for optimal is 10% lower than the value for BPF. The example demonstrates the effectiveness of the optimal basis functions, proposed in Sec. 3.3, in tracking the desired trajectory with minimal control effort.

This section demonstrates the practical benefits of optimal basis function selection in experiments. The biaxial (X–Y) linear motor driven stage (Aerotech ALS 25010), shown in Fig. 5, is used for the experiments. The stage is controlled using a P/PI feedback controller, augmented with velocity and acceleration feedforward [37] (see Fig. 6). The controller is implemented on a dSPACE 1202 real-time control board with 10 kHz sampling frequency. A flexible fixture consisting of a block mounted on a slender rod is attached to the stage. The block is assumed to represent an apparatus, for example, a tool, a workpiece, or a measurement device whose position needs to be tracked accurately despite its flexible structure. The FBF approach is used as a feedforward tracking controller as shown in the block diagram of Fig. 6. The FBF approach takes in the desired position commands y_d for each axis and generates modified position commands u that are sent to the stage to ensure that the actual position of the block y follows y_d accurately, in spite of its inherent structural flexibilities. The actual position of the block is observed from its accelerations measured using two unidirectional accelerometers (PCB piezotronics 393B05).

Figure 7 shows the frequency response function (FRF) of the dynamics of each axis of the stage, generated by applying swept sine acceleration inputs to the stage and measuring the corresponding accelerations of the block using the accelerometers. Each axis has four modes (two dominant and two less dominant), and hence, the plant dynamics is eighth-order. Prior work of the authors [38] provides more details about a continuous-time model for the system, which indicates the presence of one NMP zero in the dynamics of each axis. The Markov parameters of the dynamics are obtained from the continuous-time model and are used to construct a finite impulse response (FIR) representation of the dynamics along each axis. Figure 7 shows a good match between the measured FRF and the FRF generated using the FIR representations (modeled). Based on the methodology discussed in Appendix  A, the LSR of the dynamics of each axis is generated using the FIR representation. Singular values of the LSRs (for M = 10,000) are shown in Fig. 8. Note that each axis has two very small singular values which deviate from the cluster, and these singular values result in large control inputs, if the basis functions are not properly selected. One of the two singular values along each axis arises from the NMP zero, whereas, the other small singular value is a result of relative degree of one (the first Markov parameter is zero). Figure 9 shows the desired butterfly shaped path, whereas Fig. 10 show the desired paths position along the X- and Y-axes (for more details see Ref. [38]). The duration of the trajectory is 1 s (i.e., M = 10,000, based on 10 kHz sampling frequency).

For experiments, the optimal basis functions proposed in the paper are compared with B-splines, because they are often the basis functions of choice for manufacturing and robotics applications [12,17,21,28,38]. Figure 11 shows the control input (i.e., modified position commands) sent to the X- and Y-axes for the two sets of basis functions (for n = 600). Also, shown are the resultant tracking errors, which are based on position signals derived from measured acceleration signals using an observer. Note that the B-spline-based control input show rapid growth in magnitude toward the end of the signal because of the small singular values of the LSR corresponding to the NMP zero and the relative degree. For safety reasons, a limit ±10 mm is placed on the position commands for both axes, as shown in Fig. 11; the B-spline control input saturates at the limit. Notice that, before saturation, the control inputs for the proposed optimal basis functions and B-splines are quite similar which results in similar tracking errors. In the time interval between 0 and 0.99 s, the RMS tracking errors for optimal basis functions and B-splines are 263.31 μm and 250.85 μm, respectively, for the X-axis and between 0 and 0.96 s, 171.29 μm and 186.12 μm, respectively, for the Y-axis. However, because of saturation, the B-spline-based commands generate large tracking errors, as shown in Fig. 11. Consequently, the overall RMS tracking error for B-splines, along the X- and Y-axes, are, respectively, three and 19 times the RMS tracking error for the optimal basis functions (see Table 2). Moreover, the optimal basis functions require 3% and 13% lower control effort than B-splines for X-axis and Y-axis, respectively. The proposed optimal basis functions track much better than B-splines and require less control effort.

Remark 11. The violation of actuator limits by B-splines observed in the results of Fig. 11 can be mitigated by formulating the FBF approach as a constrained optimization problem using the constraint handling capabilities of B-splines, as done in prior work of the authors [38]. However, all things being equal, it is theoretically and practically preferable to avoid large control signals altogether than to contain them via constraints.

An appealing feature of the FBF approach is that it provides a control engineer with a wide variety of basis functions for use in tracking control (of NMP systems). However, to date, there has been no work on how to determine the best set of basis functions to achieve a given tracking control objective. To address this shortcoming, this paper has proposed a methodology for optimal selection of basis functions that minimize control effort of the FBF controller without sacrificing its tracking accuracy.

To facilitate optimal basis functions selection, a new metric based on the Frobenius norm of the LSR of system dynamics is proposed. The proposed metric is versatile. It is useful for analyzing and evaluating the performance of any linear discrete-time controller (including LTV controllers like FBF) independent of the inputs to the controller, in a manner akin to the use of Bode plots for LTI systems. It is shown that, for FBF, the proposed metric applied to tracking error dynamics is independent of the plant dynamics and type of basis functions; it depends only on the number of basis functions, relative to the length of the trajectory to be tracked. This finding is remarkable because it provides a theoretical justification for observations made by the authors in prior work about the unusual consistency of the tracking accuracy of the FBF approach irrespective of system dynamics and basis functions. Conversely, the metric shows that the control effort of the FBF approach depends on basis functions and plant dynamics.

Leveraging the analysis, a two-step process for selecting optimal basis functions that minimize control effort for a specified tracking accuracy is proposed. In the first step, the number of basis functions is selected to satisfy a desired level of tracking accuracy regardless of the type of basis functions; in the second step, the optimal set of basis functions—which are related to the singular vectors of the controlled system—are determined for minimum control effort. Simulations and experiments are used to demonstrate the effectiveness of the proposed optimal basis functions. Simple first-order plants with varying zero locations in the z-plane are used in simulations, and the proposed basis functions are compared with three other commonly used basis functions (DCT, BPF, and B-splines). The results demonstrate the effectiveness of the proposed metric as well as the superiority of the optimal basis functions. Experiments on an Aerotech linear motor driven stage, with a flexible structure, are used to show that the proposed basis functions can effectively track a desired trajectory with minimal effort, as compared to B-splines which require much higher control effort, resulting in control saturation and degradation of tracking accuracy.

This study demonstrates the importance of a systematic approach for optimally selecting basis functions not only for the FBF approach but also for related control techniques, like iterative learning control, where basis functions are commonly used and often selected in an ad hoc manner. Ongoing work has focused on the design of robust FBF controllers [39,40]. Future work will investigate the use of the proposed metric to study selection of basis functions to enhance the robustness of the FBF approach in presence of model uncertainties. The use of the proposed metric as a tool to analyze and design other optimal linear discrete-time controllers beyond the FBF approach will also be explored.

• The National Science Foundation—CMMI 1825133: Boosting the Speed and Accuracy of Vibration-Prone Manufacturing Machines at Low Cost through Software (Funder ID: 10.13039/100000001).

Similarly, overall dynamics L and error dynamics E_ff can be expressed in LSR as L and E_ff. For LTI systems, the LSR is Toeplitz. For LTV systems or controllers, the construction of the LSR for L and E_ff follows a similar process but the resulting matrices are not Toeplitz [32].

The BPF expressed in Eq. (B2) seeks to divide the time interval from 0 to M among n + 1 basis functions in a quasi-uniform manner.

This Appendix compares the tracking accuracy of FBF with that of ZPETC and TS, using the normalized RMS tracking error e_RMS/y_d,RMS (Fig. 12) and the proposed Frobenius norm metric J_e (Fig. 13), for various zero locations a ϵ [−10, 10]. The trend for J_e and e_RMS/y_d,RMS are quite similar and agree with observations made in the literature [3,15,17,21,26]. The results show that the tracking performance of FBF is consistent as compared to popular methods in the literature, viz., ZPETC and TS. For more details, interested readers can see prior work of the authors [25].