Tracking Control for Non-Minimum Phase System and Brain Computer Interface

For generations, humans dreamed about the ability to communicate and interact with machines through thought alone or to create devices that can peer into a person’s mind and thoughts. Researchers have developed new technologies to create brain computer interfaces (BCIs), communication systems that do not depend on the brain’s normal output pathways of peripheral nerves and muscles. The objective of the first part of this thesis is to develop a new BCI based on electroencephalography (EEG) to move a computer cursor over a short training period in real time. The work motivations of this part are to increase: speed and accuracy, as in BCI settings, subject has a few seconds to make a selection with a relatively high accuracy. Recently, improvements have been developed to make EEG more accurate by increasing the spatial resolution. One such improvement is the application of the surface Laplacian to the EEG, the second spatial derivative. Tripolar concentric ring electrodes (TCREs) automatically perform the Laplacian on the surface potentials and provide better spatial selectivity and signal-to-noise ratio than conventional EEG that is recorded with conventional disc electrodes. Another important feature using TCRE is the capability to record the EEG and the TCRE EEG (tEEG) signals concurrently from the same location on the scalp for the same electrical activity coming from the brain. In this part we also demonstrate that tEEG signals can enable users to control a computer cursor rapidly in different directions with significantly higher accuracy during their first session of training for 1D and 2D cursor control. Output tracking control of non-minimum phase systems is a highly challenging problem encountered in many practical engineering applications. Classical inversion techniques provide exact output tracking but lead to internal instability, whereas modern inversion methods provide stable asymptotic tracking but produce large transient errors. Both methods provide an approximation of feedback control, which leads to non robust systems, very sensitive to noise, considerable tracking errors and a significant singularity problem. Aiming at the problem of system inversion to the true system, the objective of the second part of this thesis is to develop a new method based on true inversion for minimum phase system and approximate inversion for non-minimum phase systems. The proposed algorithm is automatic and has minimal computational complexities which make it suitable for real-time control. The process to develop the proposed algorithm is partitioned into (1) minimum phase feedforward inverse filter, and (2) non-minimum phase inversion. In a minimum phase inversion, we consider the design of a feedforward controller to invert the response of a feedback loop that has stable zero locations. The complete control system consists of a feedforward controller cascaded with a closed-loop system. The outputs of the resulting inverse filter are delayed versions of the corresponding reference input signals, and delays are given by the vector relative degree of the closed-loop.


INTRODUCTION
Brain activity produces electrical signals that can be detected from the scalp, from the cortical surface, or within the brain [1]. Brain-computer interfaces (BCIs) use these signals to communicate between the brain and the outside world. BCIs enable users to control devices with direct brain communication using electroencephalographic (EEG) activity recorded from electrodes placed on the scalp (noninvasive BCI) [1,2] or with activity recorded from on or within the brain (invasive BCI) [3,4]. The EEG is used to monitor brain electrical activity; however EEG signals have low signal to noise ratio (SNR), low spatial resolution, and are contaminated by various artifacts from other sources. These characteristics limit measuring the spatial distribution of brain electrical activity and thus necessitate significant preprocessing [1]. Recently, improvements have been developed to make EEG more accurate by increasing the spatial resolution. One such improvement is the application of the surface Laplacian to the EEG, the second spatial derivative. Tripolar concentric ring electrodes (TCREs) [6,7] automatically perform the Laplacian on the surface potentials. Previously we have shown that TCRE (Laplacian EEG or tEEG) has significantly better spatial selectivity, SNR, localization, approximation of the analytical Laplacian, and mutual information than conventional EEG with disc electrodes [6,7]. In manuscript 1, the signal from the outer ring of the TCRE was used as an emulation (eEEG) of EEG recorded using conventional disc electrodes. This allows to record EEG emulation from the exact same locations at the exact same time as the tEEG using a single recording system. Time domain neuronal signal synchrony was measured using cross-correlation suggesting the potential of eEEG as an emulation of EEG (r ≥ 0.99).
An important concern in BCI research is to control the computer cursor. Cursor control requires the subject to learn how to adapt their brain signals by using different thought for different tasks [8]. Studies in recent years have shown that EEG-based BCI has great potential in achieving one-dimension cursor control [9][10][11]. However, these systems usually require long-term training in regulating brain signals and the performance in long-term use is often not robust [12]. For example, Wolpaw et al [1] performed over twenty sessions per subject, at a rate of two to four per week to develop high-accuracy cursor control (i.e., >90%) [1]. Slow training of subjects and low spatio-temporal resolution is still a serious problem [8]. In order to make future BCI convenient, the training time must be reduced without loss of accuracy. The objective of manuscript 2 was to compare the accuracy of online 1-Dimensional (1-D) (left-right) computer cursor control using two different types of electrodes, the traditional disc electrode (EEG) and a new tripolar concentric ring electrode (tEEG) [6,7].
Compared with 1-D cursor control, multidimensional cursor control, such as 2- , provides a much wider range of applications. Most of the multidimensional cursor controls BCIs have been invasive [13]. On the other hand, the development of noninvasive EEG-based 2-D control BCI is delayed by the problem of noisy EEG and the time consumed for subject training. In manuscript 3 we conduct a 2-D (left, right, up and down) computer cursor control using the tEEG signal recorded from subjects performing just one training session.
Although noninvasive EEG-based BCI have been studied increasingly over the recent decades, their performance is still limited. Thus, improving EEG-based BCI performance is still a challenge and the search for paradigms that can detect BCI commands with high temporal resolution is still active. Another noninvasive signal used in the area of BCI is called near infrared spectroscopy (NIRS) [5]. NIRS measures the concentration changes of oxygenated hemoglobins (HbO) and deoxygenated hemoglobins (HbR) in the superficial layers of the human cortex by means of distinct absorption spectra in the near-infrared range [5]. The advantages of NIRS are that it is impervious to the widely-spread environmental electrical noise and much less sensitive to EMG (muscle) artifacts than EEG [5]. NIRS measures oxygenated hemoglobin (Hb) and deoxygenated Hb concentrations. Blood Hb concentration changes have been clearly shown to be related to the presence and absence of the stimulation [5].
The feasibility of using EEG-based motor imagery BCI and NIRS-based motor imagery BCI has been demonstrated in different studies. For example, in [14] they tried to detect motor imagery with online feedback in a NIRS-based motor imagery BCI system. Study [15] introduced a multimodal NIRS and EEG-based system and used NIRS as a predictor for EEG-based BCI performance. In studies [16] and [17], the process was based on the EEG signal to classify the motor imagery based BCI. In manuscript 3, the subjects were asked to use their motor imagery to think about moving their left hands, right hands, both feet or both hands during the appearance of an arrow to the left, right, up, or down on the computer screen. The mean and the median of the recorded data were used as features to classify the recorded data from tEEG and NIRS into different tasks (left, right, up or down). Two different classifiers were used; the support vector machine (SVM) [18] and the k-nearest neighbor (kNN) [19].
This research also considers the design of digital tracking control systems for minimum phase and non-minimum phase feedback control loops. For a minimum phase system the zeros of the closed-loop system are located inside the unit circle. In the other hand, a non-minimum phase system zeros are locate outside the unit circle.
The complete control system consists of closed-loop control system cascaded with a feedforward inverse system called (FIF). The outputs are delayed versions of the corresponding reference input signals as discussed in manuscript 4. By accounting for delays, theoretically perfect tracking is achieved for minimum phase system. Note that the resulting delay is the relative degree of the closed-loop [20][21][22]. However, the total delay for a non-minimum phase system is a very challenging process due to the zero locations (zeros outside the unit circle), which require a number of delays larger than the relative degree to do approximate inversion. First, we calculate an exact inverse of the closed-loop system. At this point, the inverse calculated is not stable because the zeros located outside the unit circle becomes poles. Second, we approximate the inverse system by adding more delays until we reach stability. The total delay is the sum of the relative degree with the delay added to reach stability.
Manuscript 5 discussed the tracking control of a fully-coupled Multi-Input Multi-Output (MIMO) plant. We assumed that a 2-degree-of-freedom feedback control system has been designed to get a stable closed-loop system having zero steady-state error to step inputs. Such a system has an inner feedback loop that stabilizes the plant.
In order to have precision tracking for other types of inputs, some type of feedforward control is needed. The main contribution of this manuscript is an algorithm for calculating an inverse filter for the stabilized plant, which is an exact or approximate inverse for the closed-loop system. If the closed-loop system is minimum phase, the result is a decoupled system of delays. If the closed-loop system is non-minimum phase, the result is approximately a decoupled system of delays over a certain bandwidth. The calculation of the inverse filter and resulting tracking performance is demonstrated on an experimental belt-driven H-frame XY table, for which the system to be inverted is fully coupled and non-minimum phase.
The most important advantage of EEG signal over other types of control systems, such as body-powered mechanical systems, is its hands-free control a user's intention.
Many real-world applications operated through EEG have been reported, including intelligent wheelchairs, video games, grasping control, television, robotic arm, virtual keyboards, or a neuro-prosthesis that enables the multidimensional movements of a paralyzed limb, etc. [23,24]. In our future work we will develop an EEG control system to move and control a robotic arm using different mental (motor imagery)

Abstract
Conventional electroencephalography (EEG) with disc electrodes has major drawbacks including poor spatial resolution, selectivity and low signal-to-noise ratio that critically limit its use. Concentric ring electrodes are a promising alternative with potential to improve all of the aforementioned aspects significantly. In our previous work, the tripolar concentric ring electrode (TCRE) was successfully used in a wide range of applications demonstrating its superiority to conventional disc electrodes, in particular, in accuracy of Laplacian estimation (tEEG). For applications that may benefit from simultaneous recording of EEG and tEEG, in this paper we propose to use the signal from the outer ring of the TCRE as an emulation (eEEG) of EEG recorded using conventional disc electrodes. This will allow us to record EEG emulation from the exact same locations at the exact same time as the tEEG using a single recording system. Time domain neuronal signal synchrony was measured using cross-correlation in phantom and human experiments suggesting the potential of eEEG as an emulation of EEG (r ≥ 0.99).

Introduction
Electroencephalography (EEG) is an essential tool for brain and behavioral research and is used extensively in neuroscience, cognitive science, cognitive psychology and psychophysiology. EEG is also one of the mainstays of hospital diagnostic procedures and pre-surgical planning. Despite scalp EEG's many advantages end users struggle with its poor spatial resolution, selectivity and low signal-to-noise ratio, which are EEG's biggest drawbacks and major hindrances in its effectiveness critically limiting the research discovery and diagnosis [1]- [3].
EEG's poor spatial resolution is primarily due to (1) the blurring effects of the volume conductor with disc electrodes; and (2) EEG signals having reference electrode problems as idealized references are not available with EEG [2]. Interference on the reference electrode contaminates all other electrode signals [2]. The application of the surface Laplacian (the second spatial derivative of the potentials on the body surface) to EEG has been shown to alleviate the blurring effects enhancing the spatial resolution and selectivity [4]- [6], and reduce the reference problem.
While several methods were proposed for estimation of the surface Laplacian through interpolation of potentials on a surface and then estimating the Laplacian from an array of disc electrodes [5]- [9], concentric ring electrodes (CRE) have shown more promise. The CREs can resolve the reference electrode problems since they act like closely spaced bipolar recordings [2]. Moreover, CREs are symmetrical alleviating electrode orientation problems [10]. They also act as spatial filters enhancing the high spatial frequencies [10], [11]. Finally, bipolar CREs, consisting of just two elements including a single ring and the central disc, improve the radial attenuation of the conventional disc electrode from 1/r 3 to 1/r 4 with higher numbers of poles having the potential to enhance radial attenuation even further [12]. an extension of the five-point method used for bipolar CREs, significantly better than other electrode systems including bipolar and quasi-bipolar CREs [13], [14]. The Laplacian algorithm is two-dimensional and weights the middle ring and central disc signal difference sixteen times greater than the outer ring and central disc signal difference [13], [14]. Compared to EEG with conventional disc electrodes, Laplacian EEG using TCREs (tEEG) have been shown to have significantly better spatial selectivity (approximately 2.5 times higher), signal-to-noise ratio (approximately 3.7 times higher), and mutual information (approximately 12 times lower) [15]. TCREs also have very high common mode noise rejection providing automatic artifact attenuation, -100 dB one radius from the electrode [14]. Because of such unique capabilities TCREs have found numerous applications in a wide range of areas including brain-computer interface [16], seizure attenuation using transcranial focal stimulation applied via TCREs [17]- [20], seizure onset detection in animal models [21], [22] and, most recently, humans [23], etc.
Some current and future applications may require simultaneous recording of EEG using conventional disc electrodes and tEEG. For example, in [23] EEG and tEEG were recorded simultaneously from human patients with epilepsy to allow a direct comparison of seizure onset detection results for two sensor modalities. In [23] EEG and tEEG data were recorded by placing a set of TCREs directly behind the conventional disc electrodes that were in the standard 10-20 system locations but this approach has two disadvantages. First, EEG and tEEG are not being recorded at exactly the same locations. Second, this approach may require additional hardware as two recording systems may have to be used at the same time for EEG and tEEG data respectively (as was done in [23]) resulting in imperfect synchronization in time between EEG and tEEG.
In this preliminary study, we propose to use the signal from the outer ring of TCRE as an emulation (eEEG) of EEG recorded using conventional disc electrodes.
This will allow us to record eEEG from the exact same locations at the exact same time as the tEEG using a single recording system. Time domain neuronal signal synchrony was measured using cross-correlation in phantom and human experiments to assess the potential of eEEG as an emulation of EEG. Moreover, in the phantom experiments, a shorted TCRE was also assessed as an alternative potential emulation of EEG. TCREs with diameter of 1.0 cm (Figure 1.1, B) were used in all of the experiments.

Phantom Experiments
A diagram of the setup used for the phantom data collection is presented in Figure   1 Anode and cathode corrosion were cleaned after each series of recordings.

Human Experiments
The human data were collected from six healthy subjects (1-6, ages 24-40, one female). Baseline brain activity was recorded with the subjects seated in a chair and asked to remain motionless during the recording process to reduce movement induced artifacts. Durations of individual recordings ranged from 110s to 550s for a total duration of 1730s for 6 subjects which, when subdivided into non-overlapping segments of 10s resulted in 173 segments total for this part of the study. The conventional disc electrode and a TCRE, recording both outer ring signal and Laplacian tEEG, were side-by-side at location P4 of the standard 10-20 system with reference and ground located on the right mastoid process. Skin-to-electrode impedances were maintained below 5kΩ. Signals from the TCRE were preamplified using a custom preamplifier with a gain of 6 after which both TCRE and conventional EEG signals were band pass filtered (0.1-100Hz) and digitized at 1200Hz using a gUSB amplifier with normalized unit gain (g.tec medical engineering GmbH, Schiedlberg, Austria).

Signal Processing and Synchrony Measure
All the signal processing was performed using Matlab (Mathworks, Natick, MA).
Sixty 30s recording segments for the phantom data part of this study and 173 10s recording segments from 6 subjects for the human data part of this study were digitally filtered (zero-phase fifth-order Butterworth) with band pass of 1-100Hz and 60Hz notch filter active since this frequency range is the current clinical standard for EEG recording and, therefore, is the primary goal for EEG emulation. Next, crosscorrelation, a widely used linear measure of neuronal signal synchrony in the time domain was applied to all the respective pairs of signals from different electrode modalities [24]. For cross-correlation the signals were normalized to zero mean and unit variance. We calculated both the correlation coefficient at lag zero as well as the maximum correlation coefficient value corresponding to the optimal lag to account for possible time delay between the acquired signals.

Statistical Analysis
For the phantom data, statistical tests were used to assess significance of difference between the two proposed EEG emulation options: eEEG via the TCRE outer ring and shorted TCRE signals. First, we calculated average cross-correlation coefficients for each of 10 series of recordings averaging together coefficients for 6 recordings that comprised each series. Next, we applied unpaired or "independent samples" tests to samples of series cross-correlation coefficients (n = 10) between EEG vs. eEEG and EEG vs. shorted TCRE signal respectively: parametric two-sample Student's t-test (alternative hypothesis of sample means being not equal) and nonparametric Mann-Whitney test (alternative hypothesis of sample medians being not equal) [25]. The Ryan-Joiner (similar to Shapiro-Wilk) normality test was used for all the samples compared [26]. A parametric test was used only when both samples to be compared were normally distributed. Otherwise, a nonparametric test was used.

Phantom Data
At lag zero the following cross-correlation coefficients (r) were obtained (average ± standard deviation) for 10 series of recordings: for EEG vs. eEEG, r = 0.9744 ± 0.0121; for EEG vs. shorted TCRE signal, r = 0.9445 ± 0.0281. There was a statistically significant difference between the two (p = 0.009).
Individual optimal lags varied between the recordings. Group average optimal lag was equal to one both for EEG vs. eEEG and for EEG vs. shorted TCRE signal. The maximum cross-correlation (rmax) corresponding to the unit optimal lag was higher than cross-correlation at lag zero (r) both for EEG vs. eEEG with rmax = 0.9841 ± 0.012 (p = 0.045) and for EEG vs. shorted TCRE signal with rmax = 0.9766 ± 0.0195 (p = 0.013). There was no statistically significant difference between the two maximum cross-correlations (p = 0.385).

Human Data
At lag zero the following cross-correlation coefficients were obtained (average ± standard deviation) for EEG vs. eEEG on data from 6 subjects r = 0.9905 ± 0.0065. At optimal lag the maximum cross-correlation coefficients were equal to the zero lag coefficients suggesting that there was no time delay between the acquired signals.

Discussion
Significance of difference between zero lag (r) and optimal lag cross-correlation devices (e.g., robot arm, artificial limb or computer cursor). The most important clinical applications of BCI systems include brain-derived communication in paralyzed and locked-in patients [2,3] and restoration of motor function in patients with spinal cord injuries [4].
BCIs enable users to control devices with direct brain communication using electroencephalographic (EEG) activity recorded from electrodes placed on the scalp (noninvasive BCI) [5,6] or with activity recorded from on or within the brain (Invasive BCI) [7,8]. The EEG is a noninvasive method to monitor brain electrical activity; however EEG signals have low signal to noise ratio (SNR), low spatial resolution, and are contaminated by various artifacts from other sources. These characteristics limit measuring the spatial distribution of brain electrical activity and thus necessitate significant preprocessing [6]. Invasive BCIs face substantial technical difficulties and clinical risks as they require that recording electrodes be implanted in or on the cortex and function well for long periods, with risks of infection and other damages [8].
Recently, improvements have been developed to make EEG more accurate by increasing the spatial resolution. One such improvement is the application of the surface Laplacian to the EEG, the second spatial derivative. Tripolar concentric ring electrodes (TCREs) (see figure 2.1(B)) automatically perform the Laplacian on the surface potentials. Previously we have shown that TCRE EEG (tEEG) has significantly better spatial selectivity, SNR, localization, approximation of the analytical Laplacian, and mutual information than conventional EEG with disc electrodes [9,10]. These findings suggest that tEEG may be beneficial for neurological disorders analysis like seizure detection [11,18].
An important concern in BCI research is to control the computer cursor such as able-bodied persons do. Cursor control requires the subject to learn how to adapt their brain signals by using different thought patterns for different tasks [12]. Studies in recent years have shown that EEG-based BCI has great potential in achieving onedimension cursor control [13][14][15]. However, these systems usually require long-term training in regulating brain signals and the performance in long-term use is often not robust [16]. For example, Wolpaw et al [6] performed over twenty sessions per subject, at a rate of two to four per week to develop high-accuracy cursor control (i.e., >90%) [6]. Slow training of subjects and low spatio-temporal resolution is still a serious problem [12]. In order to make future BCI convenient, the training time must be reduced without loss of accuracy.
The objective of this research project was to compare the accuracy of onedimensional ('left'-'right') cursor control between EEG and tEEG. A secondary objective was to determine if high accuracies could be accomplished using only a single training session. We demonstrate that tEEG signals can enable users to learn how to control a one-dimensional computer cursor rapidly and accurately during their first session with significantly higher accuracy when compared to the EEG signals.
The rest of the paper is structured as follows: section 2 presents a general description of the dataset used in this work and the procedure employed is explained in detail. Section 3 presents the obtained results with a general discussion. Finally, some conclusions with future work are offered.

Laplacian Electroencephalography (tEEG)
The scalp surface Laplacian is an alternative method for presenting EEG data with higher spatial resolution. It has been shown that the surface Laplacian is proportional to the cortical potentials and improves the high spatial frequency components of the brain activity near the electrode [19]. To obtain the Laplacian, we take a new approach by using unique sensors and instrumentation for recording the signal [9,20]. The unique sensor configuration which measures the Laplacian potential directly is the TCRE (Figure 2.1, panel B) [9,10]. Two differential signals from each electrode were combined algorithmically for a tEEG derivation of the signal as reported previously in [9]. Briefly, the algorithm is two-dimensional and weights the middle ring and central disc difference sixteen times greater than the outer ring and central disc difference, 16 * (middle-disc) -(outer-disc), where disc is the central disc, middle is the middle ring, and outer is the outer ring of the TCRE. The outer ring was used as the disc electrode ( Figure 2.1, panel B). In our previous work [22] we have shown that the outer ring signal has a 0.99 correlation to disc electrode signals. All the signal processing was performed using MATLAB (Mathworks, Natick, MA, USA).

Data Recording
The brain signals were gathered from eight healthy subjects (1-8, ages 24-40, two female). The subjects were naive to neuro-feedback training and their task was to use their thoughts to move a cursor, in the form of a ball, from the center of a computer screen to a target. The target was a pink rectangle which appeared in the 'left' or the 'right' of the periphery of the screen. The cursor width was 10% of the screen width.  During BCI operation, subjects were seated in a chair, facing a computer screen which was placed about 1.5 min front of the subject. The subjects were asked to remain motionless during the recording process to reduce the introduction of artifacts.
The BCI2000 [16] software application was used to acquire and process in real-time signals recorded from eight scalp surface electrodes (C3, C1, Cz, C2, C4, FC1, FCz, FC2) according to the international 10-20 system, with reference and ground from the right mastoid process (see Figure 2.1 panel A). Skin-to-electrode impedances were maintained below 5 kΩ. Signals from all the channels were first pre-amplified with a gain of 6, then amplified, filtered (0.1-100 Hz) and digitized (sampling frequency was 256 Hz) with a gUSB amplifier (g.tec GmgH, Schiedlberg, Austria). For a direct comparison between EEG and tEEG cursor accuracy TCREs were placed once and used for both experiments. The subjects were unaware as to whether EEG or tEEG was used for BCI.

Procedure
In this paper, for achieving one-dimension cursor control two stages were used: (1) features identification during an offline analysis, and (2) online one-dimensional BCI cursor control (real-time one-dimensional cursor control). The two stages are described in the following section (see Figure 2.3).

Offline Analysis for features identification
The offline analysis was performed to determine which components and features of the signal the user could most easily modulate for BCI control. First we determined which EEG and tEEG features (i.e., signal amplitudes at particular frequency bands and particular electrode locations) were correlated with a particular motor imagery task, and might thus be the basis for BCI experiments.
To measure and characterize responses to motor imagery, each subject had to imagine 'left' and 'right' hand movements, following a fixed repetitive time scheme of an arrow pointing 'left' or 'right' for 2 s, and to rest while the screen was blank for 1 s.
The appearance of the arrow was random and each 'left' or 'right' task was repeated ten times. Moreover, the data were collected just for one session (to assure that the subject was a first-time BCI user). The session consisted of five runs, and each run consisted of 20 trials for 100 trials total.
For these analyses, we converted the time-series EEG and tEEG data into the frequency domain, with 2 Hz wide bins from 0 to 140 Hz, using an autoregressive model of order 20 (BCI200 function) to produce a set of frequency spectra for each location and for each task and rest. We then calculated the statistical difference for the distribution of frequency magnitude at each electrode location and frequency (i.e., values of r 2 , which indicated what fraction of the signal variance at that location and frequency was due to the condition of task and rest). This procedure was performed to identify features that could be modulated by the subject using imagined tasks.
Electrodes for each subject were identified for a selected frequency band that exhibited a high correlation with the imagined right hand movement task (i.e. the largest value of r 2 ) for both EEG and tEEG signals (see Table 2

Online Testing
The order of the EEG or tEEG closed-loop control was randomized without the subject knowing which they were performing. The closed-loop BCI experiments, the real time testing, were performed with the subject receiving online feedback. The feedback consisted of one-dimensional cursor imaginary movement controlled by the EEG or tEEG features. Moreover, the subjects received feedback that was proportional to the extracted features identified in the methods of section 3.1. To translate the extracted features into a set of signals that moved the cursor toward the target during the online feedback, the amplitudes of an identified frequency band were summed linearly for the identified electrodes [23]. The weights were chosen so that the cursor moved to the 'right' with task performance and to the 'left' during rest. The weights were selected manually to set this configuration. Initially, we performed a trial task for each subject using EEG and an initial weight value of (1). We monitored the speed of the ball and adjusted the weight so that the cursor moved from the center to the target within 2 s for each subject. The same weights determined for the EEG were also used for the tEEG. The EEG and tEEG features were integrated over time to yield the current cursor position.
The subject's goal was to move the cursor horizontally so that it hit the appropriate (i.e., 'left' or 'right') target. Data were collected from each subject for 10 runs, each run comprised 20 trials. The runs were separated with short breaks. The random 'left' or 'right' target appeared 2 s prior to the cursor (a ball) which appeared in the center of the screen. The trial ended 1 s after a hit, miss, or abort with a total run duration of approximately 8 s. The feedback duration was 2 s with a maximum duration of 4 s.
One session comprised ten runs for approximately 2.7 min per run and approximately 27 min for a session. Data were collected for only one session from each subject. The cursor was visible and controllable throughout the whole run. Once the cursor hit the target, the color changed from pink to green to show the successful performance.
Since there were two possible outcomes in each trial, the expected probability in the absence of any control was 0.5.

Results and Discussion
In this study, we report the first tEEG-based BCI system for one-dimensional realtime cursor movements. Furthermore, we emphasize that the system requires only one training session of an offline analysis for features identification (~27 min) and online testing (~27 min) resulting in proficient operation. The procedure for achieving onedimensional cursor control had two stages. In the first stage, the data were collected from naıve subjects and analyzed using an offline process to determine which features could be used for BCI control. In the second stage, the subjects attempted to control of the computer cursor.
During offline analysis, the signal amplitudes related to the electrodes location and frequency bands with the most significant task (i.e., the highest values of r 2 ) were identified as features to be used to control cursor movement in the subsequent online BCI experiments. The r 2 is a measure of how relevant a signal is to the presented target [21]. An analysis example comparing tEEG and EEG signals for the right-hand imaginary and rest is shown in Figure 2 In the online analysis, the subjects controlled a cursor (moving the cursor to the left or right edge of the screen to hit the target) with their EEG or tEEG signals. The obtained accuracies are comparable to other noninvasive and invasive BCI studies aimed at one-dimensional cursor control [6, 8, 17 and 21]. For examples in [6], human subjects try to control at first a one-dimensional movement and then moved to a two dimensional movement using a noninvasive BCI. The tasks were performed from 2-4 times a week with a 92% hit rate achieved. In [17], the subjects controlled a cursor in one dimensional direction by imagined movements. All data of each subject were recorded on the same day without subject training. The average accuracy of all subjects obtained online in the feedback application was 90.5%.
Over short training periods (~27 min), all eight users achieved significant control of the computer cursor. Figure 2.5 shows the average group accuracy during online cursor control obtained from each user for the period of ten runs. The average rates of online experimental results for each subject using both disc and tripolar electrodes are shown in  (Figure 2.5). One-way analysis of variance (ANOVA) with blocking factor levels corresponding to eight human subjects was used to confirm that there was a significant difference in the mean BCI hit rate between data corresponding to conventional disc EEG and tripolar concentric ring Laplacian tEEG (p < 0.0001).
These results show that there is a significant difference in accuracy of the tEEG to disc EEG for new users in real-time one-dimensional cursor control.
Comparing the obtained results from EEG and tEEG signals, the cursor control accuracy using tEEG, on average, was significantly higher than for the disc EEG.
When moving the cursor in one dimension, two of the eight subjects were able to hit 100% of the targets within one session (~27 min) using tEEG. We believe that this increased accuracy in a shortened time will help users train on the BCI quicker.
We always used the features that provided the highest r 2 for each electrode type.
Many times the electrode locations used for EEG and tEEG overlapped and sometimes the frequency bands also overlapped. Although, the performance of tEEG over EEG may be partially due to different selection of features, we believe this improvement is also due to tEEG having higher spatial resolution, lower mutual information, and better SNR than EEG which provides more specific features.

Conclusion
In this paper, we have shown a significant improvement in accuracy for onedimensional BCI real-time center out cursor control using tEEG compared to EEG in humans after minimal training. The obtained results indicate that tEEG-based BCI could provide one-dimensional control that is more accurate with shorter training time requirements than EEG-based BCI. All eight subjects demonstrated reliable control achieving an average of 70.2% using tEEG-based BCI and an average of 59.1% using EEG-based BCI after only one offline training session. As a result, the average improvement of the tEEG-based BCI over the EEG-based BCI was more than 15%.
Using TCREs allowed us to collect the tEEG and EEG data from the same locations at the same time for comparison.
Our future studies will focus on two-dimensional cursor control to determine the hit accuracies for tEEG and EEG signals. and NIRS based BCIs. To this end, we recorded the tEEG signal and the event-related hemodynamic responses from nine participants while performing these imagery tasks.

MANUSCRIPT 3 Imagery Tasks Classification using Laplacian EEG and NIRS based BCI
Classification accuracies were then estimated comparing these four mental tasks with rest using two different classifiers, the support vector machine (SVM) and the knearest neighbor (k-NN) algorithm.

Introduction
A brain-computer interface (BCI) translates electrical signals detected from the brain to control an output device which can help those suffering from neuromuscular impairments to interact and communicate with their surrounding environment [1].
BCIs can be invasive or non-invasive. Invasive BCIs read the user's intent from neuronal action using electrodes embedded on or directly within the brain. Noninvasive BCIs derive the user's intent from neuronal action recorded from the scalp.
EEG is the measurement and recording of the brain electrical activity using sensors arrayed across the scalp. Surface EEG has many advantages. For example, it is non-invasive, has a good temporal resolution and available at low cost. However, EEG has low signal-to-noise ratio and low spatial resolution that does not allow an accurate localization [11]. Recently, improvements have been applied to EEG recording techniques making it more accurate by increasing the spatial resolution. One such improvement is the application of the surface Laplacian to the EEG [12,13]. The tripolar concentric ring electrodes (TCREs), (see Figure 3.3 panel B) were used to record the tEEG. It has been shown that tEEG has significantly better spatial resolution and better signal-to-noise ratio than EEG recorded using the traditional disc electrodes [12,13]. In our previous work [22], we found a significant improvement in detection of single event imagined movements with tEEG compared to EEG. Recently we found a significant improvement in real-time 1D cursor control accuracy using tEEG compared to EEG [14].
Near-infrared spectroscopy (NIRS) is a non-invasive optical imaging technique used to acquire brain signals by transmitting near-infrared (650 nm -950 nm wavelengths) electromagnetic radiation through the skull and comparing the intensities of the returning and incident light. During mental activity, an increase in the blood flow to pre-defined regions of the brain is observed, e.g., motor cortex during motor imagery tasks [15,16]. The increase in blood flow causes changes to the regional concentrations of oxygenated and deoxygenated hemoglobin [8]. Near-infrared spectroscopy provides researchers with a variety of advantages such as flexibility of use, good spatial and temporal resolution, localized information, affordability and high sensitivity in detecting small substance concentration [11]. Moreover, NIRS is much less sensitive to EMG artifacts than EEG. It has also been shown that NIRS has some disadvantages such as slow operation due to the inherent latency of the hemodynamic response and weak signal due to hair on the head which may introduce artifacts. Coyle et al. 2004 [8] were one of the first to explore the feasibility of NIRS for a BCI system. Recently, NIRS-based brain-computer interfaces were introduced for discriminating motor tasks imagery by different studies [8,9,[19][20][21].
The goal of our study was to investigate the potential of classifying different imagery tasks using tEEG and NIRS based BCI. The imagery tasks were, left-hand motor imagery, right-hand motor imagery, both hands-motor imagery and both-feet motor imagery. During offline analysis, the extracted mean and the median values of tEEG and NIRS signals during task were used as input to two different classifiers, the support vector machine (SVM) [27] and k-nearest neighbor (k-NN) [28]. The block diagram describing the procedure of data acquisition and the control command process is shown in Figure 3.1. In this study we will also discuss online classification towards developing a tEEG-based BCI.

Subjects and Experimental Paradigm
Nine healthy subjects (20 to 36 years old) voluntarily participated in this study. All

Data Acquisition
Simultaneous measurements of the brain activities related to mental imagery tasks were performed from tEEG and NIRS recorded data.
The BCI2000 [24] software application was used to acquire and process signals in real-time.
To record the NIRS data, the NIRS-System (NIR_Scout, NIRX Medical Technologies, USA) was used in this experiment with wavelength of 760 and 850 nm.

Signal Processing
Further analysis of the NIRS and tEEG offline data were performed as follows.
The session was constructed from 10 runs, each run was formed from 20 trials, and the length of each trial was 20 s representing one of the four tasks: (LI), (RI), (BHI), (BFI) and rest that were recorded using both NIRS and tEEG data.
The recorded NIRS signals were converted into oxygenated hemoglobin (HbO2) and deoxygenated hemoglobin (Hb) concentration using the modified Beer-Lambert law [25]. In order to remove the high frequency noise due to respiration and heartbeat, the signals were low-pass filtered using a 3 rd -order Butterworth filter of cut-off frequency 0.2Hz. In this study, the averaged oxygenated hemoglobin (HBO) signal for each task over each run was considered for classification. The nirsLAB (v2014.05 by NITRC) was used to preprocess the NIRS measurements. A total of 200 segments of 10 s were associated for each task.
The same process was performed for tEEG data. Each run was decomposed into segments of 10 s corresponding to each task (left, right, up and down) and rest. The average of these tasks over each run was calculated. Thus, for each subject there are a total of 80 segments of 10 s corresponding to each of the four tasks and for rest for the 8 electrodes used to record the tEEG signals. The tEEG segments were filtered using an 8 th order low pass Butterworth filter with 30 Hz cut-off frequency.

Motor Imagery Detection based on tEEG and NIRS System
The aim of BCI is to translate the brain activities into commands for a special device such as computer. To achieve this goal, classification algorithms can be used.
In this paper, an offline classification of the tEEG and NIRS signals recorded during motor task imagery were performed. The classification accuracy depends on both the features and the classification algorithm used. These two steps are highlighted in this section.

Feature Extraction
As already mentioned in the above section, the duration of each task is 10 s. To improve the classification accuracy, the window sizes (1s, 2s, 3s 4s and 5s) were tested for feature extraction. One second sliding Hamming window was adopted since it gives the best accuracy among all the tested window sizes from both tEEG and NIRS. Then the signal mean and the signal median (MAD) for each one second segment were computed. For the dataset: , the MAD is defined as the median of the absolute deviations from the data's median [26]: Therefore, for each 10 s segment, there were 10 two dimensional data points for mean and median values of windowed segments. For NIRS data the feature vector was calculated from 200 segments of 10 s each for each task (20 channels for 10 runs). For tEEG data, the feature vector was calculated from 80 segments of 10 s each for each task (8 electrodes for 10 runs).

Classification
The classification was performed using two different classifiers, the support vector machine (SVM) [27] classifier and the k-nearest neighbor (k-NN) [28] classifier.

Support Vector Machine (SVM)
The SVM classifier developed by V. Vapnik [27] has shown to perform well in real world problems, including BCI. The idea of SVM is to separate the data n R x ∈ into two classes by finding a weight vector n R w ∈ and an offset R b ∈ of a hyperplane H : The classification is done by constructing an n-dimensional hyper-plane that optimally separates the data into two categories by maximizing the distance from the decision boundary to the nearest data-points (called support vectors) in the training data. In this study, we generate non-linear decision boundaries using Gaussian radial basis function (RBF) kernel [29]. This kernel was selected due to the highest classification results obtained were better when compared to other SVM kernels (linear, quadratic, polynomial) that were tested.

k-Nearest Neighbor (k-NN)
k-NN is a non-parametric and nonlinear classifier suitable for n-dimensional, multi-modal problems [28]. The k-NN approach has recently been recognized as a very important algorithm, due to its high classification accuracy in problems with unknown and abnormal distributions [28]. The k-NN classification finds a group of k objects in the training set that are closest to the test object, and assigns a label on the predominance of a particular class in this neighborhood [28]. Using the k-NN classifier, three important components must be identified: a set of labeled objects, a distance or similarity metric to compute distance between objects, and the value of k, the number of nearest neighbors. In this paper, the linear distance measure was selected and the number of nearest neighbor k was set to be one, which provided highest detection accuracy when compared to other values of k (k=2, 3, 4,..).

Data Training and Performance Measurement
The classification is based on two separate sets of training and testing data containing the extracted feature from each task. In this study, we performed a ten-fold cross-validation to determine the classification accuracy. In ten-fold cross validation, the data set is split into 10 equal sub-set partitions. In each iteration, one of the 10 subsets is used for testing whereas the other 9 subsets are used for training the dataset.
The total data were randomly split into ten subsets. The whole procedure is repeated ten times. The final result is the average of all 10 repetitions. One important advantage of the cross-validation technique is the avoidance of the over-fitting problem [32].
In order to evaluate the performance of the proposed algorithm for imagery task classification, accuracy was calculated as follows [30]:

Online tEEG Control of Cursor Movement
In this part, we conducted an online experiment involving all 9 subjects. The subjects performed one session of feedback training to perform a two dimensional cursor control using their motor imagery. The tEEG recording methods and processing in BCI2000 were the same as in the offline analysis. The signal was recorded from the same 8 locations (C3, C1, Cz, C2, C4, FC1, FCz, FC2) using the TCREs according to the international 10-20 system.
The subject was asked to move a computer cursor from the center of the screen and hit a target that appeared in one of four locations of the screen: up, down, right, or left. Subjects imagined moving their right hand to move the cursor to the right, their left hand to go left, both hands for up, and both feet for down. The trial sequence used for two-dimensional cursor movement is shown in Figure 3

Results and Discussion
The objective of this study was to detect four different imagery tasks using two different modes of BCI: (1) Laplacian EEG (tEEG) based BCI and (2)       The single event for tEEG and NIRS were also tested. The individual offline classification accuracies ranged from 50.65% to 92.27% using the tEEG signals and from 50.12% to 80.55% using the NIRS signals. Table 3.3 and Table 3 In this study the subjects performed one session in the online target acquisition phase. The subjects were trained to move a cursor with their tEEG signals to hit targets that appeared randomly one at a time, left, right, top, or bottom. The subjects were given 10 initial runs which were used for offline analysis.

Accuracy (%) S u b je c t 1 S u b je c t 2 S u b je c t 3 S u b je c t 4 S u b je c t 5 S u b je c t 6 S u b je c t 7 S u b je c t 8 S u b je c t 9
Figure 3.8 The average accuracy of k-NN and SVM of each offline task for each subject using the single event NIRS signals.
As seen in Table 3  One positive outcome of this study is that only one session for each subject was needed to accurately classify the four imagery tasks. Moreover, a simple set of features was used, the mean and the median, therefore it is expected that the processing could be performed in near real-time. Moreover, it is recommended to test the method on subjects with significant motor impairment.

Conclusion
The present study addresses the usefulness of tEEG and NIRS based BCI for imagery task classification. The tripolar EEG and NIRS signals corresponding to lefthand motor imagery, right-hand motor imagery, both-hand motor imagery and bothfeet motor imagery were acquired simultaneously from nine healthy subjects. The signal mean and signal median calculated during one second duration sliding time windows were used as the inputs to two different classifiers, SVM and k-NN, to evaluate the performance of the proposed method. The results demonstrated there was a significant difference in accuracy of 2D motor imagery with tEEG compared to NIRS-based BCI.
. [4] F. Lotte  [20] S. D. Power, T. H. Falk and T. Chau, "Classification of prefrontal activity due to mental arithmetic and music imagery using hidden Markov models and

Introduction
The objective of tracking control is to obtain a small path-following error for a given reference input or desired path. e.g., automated arc welding (certain trajectory) and servo turning table (uncertain trajectory). One way to achieve this is to design a stable feedback control system and then calculate a feedforward controller consisting of the inverse of this closed-loop system. Early work on system inversion was done by Massey et al. [1] who stated the necessary and sufficient conditions for the existence of a feedforward linear inverse system. Vito et al. [2] has addressed the problem of designing a feedforward block of a two degree of freedom controller for a discrete single input single output linear system with model uncertainty [3]. Fabrizio et al. [4] proposed a solution for the synthesis of a feedforward control action for a fractional control synthesis; this research has received a great attention from academia and industry because of the need to meet more stringent tracking system requirement.
Two types of architectures for inversion exist: plant inversion and closed-loop inversion. When the closed-loop system is non-minimum phase (i.e. has zeros outside the unit circle), the inverse system will be unstable. There are several partial solutions to this problem: the zero phase error tracking controller (ZPETC) [5,6,7,8], the zero magnitude error tracking controller (ZMETC) [9,10,11], non-causal inverse [12,13], the Truncated Series Approximation scheme (TSA) [14] and the non-causal Taylor series approximation [15].
ZPTEC was proposed in [8] and developed in [16,17]. This algorithm is considered as one of the most successful feedforward controller designs and has attracted the attention of many researchers as an effective and simple method for approximately inverting non-minimum phase systems (e.g. [18,19]). Using ZPETC, the poles that arise from inverting non-minimum phase zeros are approximated using stable poles and an appropriate gain factor; hence, the cascade of ZPETC and a given closed-loop system is approximately a pure delay. The use of an inverse system such as ZPETC allows the tracking system to follow time-varying reference inputs with small error. However, ZPETC has some shortcomings, such as gain errors and its inability to handle systems with a zero inside the unit circle, located on the negative real axis, close to -1 and at z = 1. This has motivated additional research [20][21][22].
Many applications related to this work, electro-hydraulic actuator [23,24], NC machine tools [25], industrial motion control [26], ZPETC Path-Tracking gainscheduling design and real-time multi-task flight simulation for the automatic transition of tilt-rotor aircraft [27], method for precise time synchronization based on FPGA [28] and so forth… In this paper, we consider the above limitations of ZPTEC algorithm, and we introduce a state-space algorithm for computing the inverse of a given closed loop control system. We advocate the use of a feedforward system that inverts the closedloop system model rather than the plant model. The reason is that the plant model is inaccurate while the modeling errors for the closed-loop system are reduced by the effects of feedback. The cascade of the inverse system, which we call the feedforward inverse filter (FIF), and the given closed-loop system, is a pure delay. Thus, the plant output will be an exact, delayed replica of the reference input. The proposed approach for computing the FIF works even when the plant has zeros arbitrarily close to the unit circle. Two inversion methods, ZPETC and FIF, will be used with two different closed-loop control systems: a proportional integral derivative (PID) controller and a standard state-space tracking system. The best results, in both simulations and hardware experiment, are obtained by combining FIF with the standard state-space tracking system.
The remainder of this paper is organized as follows; Section II introduces a standard PID controller, a standard state-space tracking system, as well as the algorithm for computing the FIF. Section III presents simulation and experimental results, with a motor positioning system. Section IV contains conclusions and recommendations for future work.

Digital Closed-Loop system design
Consider an n th -order single-input, single-output plant with state-space model: and n R C × ∈ 1 , where u(t) is the input of the system, y(t) the output and x(t) the state vector. In order to design a digital closed-loop control system for this plant, a discrete-time zero-order hold (ZOH) equivalent model that includes the effects of the D/A converter as well as the A/D converter sampling y(t) and x(t), is needed. The D/A and A/D converters operate with a sampling interval T seconds. The ZOH equivalent model is given by:  Notice that the derivative term is combined with a low-pass filter N s N + to limit its gain at high frequencies:

PID controller system
where N is the filter coefficient. The derivative term can be written in the form shown  Letting 1 x be the state variable for the integral term and 2 x be the state variable for the derivative term, the following state-space model for the digital PID controller is obtained: where, in Figure 4.1 is described by the following state-space model:

Digital state-feedback tracking system
In order to design a state-space feedback control system in which ] [k y tracks a class of reference inputs ] [k w with zero steady-state error one must use additional dynamics (see Figure 4.2).
Where the eigenvalues of a Φ include the poles of the z-transform of ] [k w [15]. For the class of step inputs, only one eigenvalue is required at z = 1, which gives digital integral control. That is, for step input tracking, the additional dynamics are given by [29]: In order to compute the ( n × 1 ) state-feedback gain vector L1 and the integrator gain 2 l , a design model consisting of the cascade of the plant followed by the additional dynamics is needed. This design model is given by [29,30]: Given a desired vector of n+1 closed-loop poles, call it zpoles, a ) 1 ( 1 + × n gain vector L is computed (e.g. using the MATLAB place command) to obtain:

Feedforward inverse filter
This subsection shows how to find a state-space model ) , , ( , Let the degree of the numerator polynomial be called m, and the degree of the denominator polynomial a(z) be called n. A formal geometric series expansion for the matrix inverse in Equation (4.17) is: (4.18) .
Dividing an m th degree polynomial b(z) by an n th degree polynomial a(z) yields a series in inverse powers of z with coefficients denoted by k α . The first non-zero term in the series is the n m z − term: Comparing Equation (4.20) with equation (4.19) it can be seen that: where d is the relative degree of the system, the number of poles minus the number of zeros. Once the FIF is calculated, it can be used in the overall system shown in Figure   4.3 to achieve perfect tracking with delay; that is, The FIF is derived by considering shifts of the outputs for the system shown in Equation (4.15), for example: .
. A particular shaped reference input, shown in the next section, is used in this paper.

Simulation and experimental results
To illustrate the performances of the PID and the ADD control systems, with or without an inverse filter obtained from ZPETC or FIF, we considered the control of a

Open-Loop modeling process
A mathematical model for the first order motor system with velocity as the output signal was obtained from measured step response data for the actual motor system.
Step inputs of different amplitudes were sent to the motor and the motor velocity was  Table 4.2.
In Table 4.2, we also included a coulomb friction torque estimate for this system, where the friction torque is given by Equation (4.26). The non-linear friction was included in the simulation results that follow, and is present in the hardware system. However, only the linear system model, which ignored the Coulomb friction, was used to design the control systems. This linear system model, using the numbers in Table 4 with: T=0.001s, the resulting ZOH equivalent:

Closed-Loop tracking systems
After obtaining a model for the plant, the next step is to design a stable closed-loop system. We consider two different control systems, PID and ADD, shown in Then the desired discrete time poles used in (4.14) are:    give a bandwidth of about 16Hz for the low-pass filter, which was found to adequately limit the derivative noise in the hardware system.
The poles and zeros of the closed-loop PID and ADD systems and the corresponding stability margins are shown in Table 4.3.

Shaped input
When an inverse system such as ZPETC or FIF is used in conjunction with a closed-loop control system such as PID or ADD, the reference must be suitably band limited to prevent the inverse system from generating an excessively large output that would saturate the input to the hardware plant. In this paper, we calculate a shaped input signal by starting with the following sinusoidal acceleration signal, with amplitude A 2 / s rad and period D seconds:  (4.31) This equation may be integrated twice to obtain velocity and position signals, respectively. The constants of integration are chosen to make the initial values of velocity and position both zero. The resulting position signal is: In order to obtain a position of 1 radian at t = D seconds, the following relationship between A and D must be satisfied: In what follows, we show results for 4D = 0:2 seconds. Figure 6 shows the simulated and experimental output for the PID controller without any inverse system.
There is a significant phase lag for the PID controller and similarly for the ADD controller. However, the phase lag is eliminated from both controllers by using an inverse system, as shown next.

Inverse systems
In principal, either ZPETC or FIF could be used to obtain the inverse system for the PID or ADD control systems. However, as shown previously in Table 4.3, the PID control system has a pole-zero cancelation at z = 0.9999. If the PID system is given to the ZPETC as is, the ZPETC algorithm fails due to the zero location at z = 0.9999. In order to use ZPETC for the PID system one would have to give it the reduced-order model obtained by cancelling the pole and zero near 1. The FIF algorithm works with the given PID system as is. Note, that when the ADD has a zero at z = -0.9723, the ZPETC algorithm fails when a zero is located inside the unit circle, laying on the negative real axis and close to -1. As mentioned before, this behavior is documented in [1] and was confirmed in our study. For this reason the zero located at z = -0.9723, after inversion, will be considered as non stable pole. However, the FIF work as is.
In order to see the need for the FIF, the closed-loop PID system is used by itself to track a reference input that has two start-stop segments in the positive direction followed by two start-stop segments in the negative direction. The reference input has 95% of its energy in the frequency range from 0 to 5.5 Hz. The tracking results are shown in Figure 4.6, and it is clear that the closed-loop system has significant phase lag. Similar results are obtained with the ADD closed-loop system. The FIF may be used with either the PID or ADD closed-loop systems to eliminate the phase lag.   Figure 4.10 shows the tracking errors for the PID system, as well as for the PID+FIF, ADD+ZPETC and ADD+FIF systems. It can be seen that the FIF with either the PID or ADD closed-loop systems provides nearly perfect tracking of the reference input. Now suppose the reference input were scaled to a shorter duration, thus increasing its frequency content. Could the hardware system track this faster reference input? An answer to this question is found in Figure 4.11, which shows the plant-input voltages for the various tracking systems with the original reference input.
The peak plant-input is about 6 volts, and the hardware has a 10-volt limit. Thus, the reference input used is within a factor of two of the tracking capability of the hardware.
Due to nonlinearities in the hardware system (e.g. nonlinear friction), the response to a reference input is not exactly repeatable. For this reason, we use a periodic reference input consisting of ten periods with a total duration of 2 second. The average squared error over the ten periods was computed as follows: The maximum absolute value of the error over the 2 second duration was also recorded. The values of these error measures for the various tracking systems are shown in Table 4

Conclusion
A digital tracking control system for minimum-phase plants is presented in this paper. This controller is developed to reduce the phase lag caused by the feedback controller during the tracking control. The control system consists of two parts: a feedback control loop and a feedforward inverse system. For the feedback loop we used either a standard PID loop or a state-space tracking system with additional dynamics, which we denote by ADD. For the feedforward system we used either the ZPETC filter or a feedforward inverse filter, which we denote FIF. The FIF is calculated using the state-space formula shown in Table 4.1.
The main result in this paper is the state-space formula for the FIF. Because the FIF is outside of the feedback loop, it does not affect the stability of the closed-loop system or its robustness to modeling errors or disturbances. These depend only on the feedback loop. In this paper, the standard state-space and PID feedback loops both had good stability robustness as indicated by the gain and phase margins shown in Table   4.3.
As with any inverse method, one must question the robustness of the tracking error with respect to plant modeling errors. There will be errors in the model of any physical system, as shown for example in Figure 4

Abstract
The problem considered is tracking control of a fully-coupled Multi-Input Multi-Output (MIMO) plant. We assume that a 2-degree-of-freedom feedback control system has been designed to get a stable closed-loop system having zero steady-state error to step inputs. Such a system has an inner feedback loop that stabilizes the plant.
In order to have precision tracking for other types of inputs, some type of feedforward control is needed. The main contribution of this paper is an algorithm for calculating an inverse filter for the stabilized plant, which is an exact or approximate inverse for the closed-loop system. If the closed-loop system is minimum phase, the result is a decoupled system of delays. If the closed-loop system is non-minimum phase, the result is approximately a decoupled system of delays over a certain bandwidth. The calculation of the inverse filter and resulting tracking performance is demonstrated on an experimental belt-driven H-frame XY table, for which the system to be inverted is fully coupled and non-minimum phase.

Introduction
Feedforward filters from the reference signal to the plant input have been used to improve the performance of Single-Input Single-Output (SISO) tracking systems for many years. Typically, the feedforward filter is the inverse of the plant model. If the plant model is non-minimum phase the inverse model will be unstable, and some type of approximation is necessary. When the tracking system is implemented in discretetime, non-minimum phase systems can be approximately inverted with some delay.
Methods for using feedforward filters in MIMO tracking systems are less well developed, especially for non-minimum phase systems. One recent contribution for MIMO systems is [1]. Instead of using one feedforward inverse system, two FIR blocks are used in a novel feedforward architecture. The resulting control system provides theoretically perfect tracking with delay. However, the development in [1] relies on the MIMO plant not being fully coupled in the sense that the method works only for plants having lower triangular transfer function matrices.
In this paper, we consider fully coupled MIMO systems and use a standard architecture in which there is a single feedforward filter from the reference signal to the plant input. We present a new method for approximately inverting a non-minimum phase stable MIMO system. The method may be used for plants that are not stable, provided a 2 degree-of-freedom control system is used which has an inner stabilizing feedback loop around the plant. There is a standard state-space tracking system architecture that has this required characteristic. The feedforward filter designed by the proposed approach approximately inverts the stable combination of the plant and inner feedback loop.
The outline of the paper is as follows. Section II reviews the development of a standard 2 degree-of freedom feedback (DoF) control system. Section III deals with MIMO system inversion. For stable non-minimum phase systems, an approximation is made leading to an approximate inverse system which is stable. Section IV provides simulation and experimental results on tracking control using the proposed method for a belt-driven H-frame XY positioning system. Conclusions are given in Section V.

2-DOF Tracking System
In order to use the MIMO inversion algorithm derived in Section III, it is first necessary to design a feedback control system to achieve a desired bandwidth with good stability robustness. As this is not a trivial task, we review in this section a specific methodology for obtaining a 2-degree of freedom (DoF) feedback control system.
Consider an nth-order linear, m-input, p-output plant with state-space model: A digital state-feedback tracking system is shown in Figure 5.1. In order to compute the n m × state-feedback gain matrix K1 and the integrator gain matrix K2, a design model consisting of the cascade of the plant followed by the additional dynamics is needed [3]. This design model is given by: Given a desired set of n +1  algorithm [4], which uses a parameterization of the set of Kd matrices to find the one which maximizes the input-multiplicative stability robustness bound (see [5] for a definition of this bound).

Stable MIMO System Inversion
The system shown in Figure 5.1 is a 2 DoF control system having an inner feedback loop around the plant. The model for the system from v[k] to y[k] will be called the modified plant, and is given as follows: This modified plant is stable due to the feedback through K1. The transmission zeros of the modified plant might all be inside the unit circle (minimum phase) or some may be outside the unit circle (non-minimum phase). In this subsection we derive a statespace algorithm for inverting a minimum phase system. The following subsection deals with approximately inverting a stable non-minimum phase system. By inverting the modified plant we obtain a feedforward filter that may be used with the control system shown in Figure 5.1 to obtain a precision tracking system.
In order to derive the inverse model for the modified plant where Df is given by (5.12) and  The preceding development is useful only if the IMP is stable. The poles of the IMP will include the transmission zeros of the modified plant. If the plant itself is non-minimum phase, some of the poles of the IMP will be outside the unit circle. Even if the continuous-time plant is minimum phase, some of the eigenvalues of the IMP may be outside the unit circle due to the presence of "sampling zeros." Whatever the cause of unstable eigenvalues of the IMP, additional calculations are needed to obtain a stable approximate inverse for the modified plant, as shown next.

Approximate Inverse for Stable Nonminimum Phase System
Consider adding s more advances to the r advances shown in (5.10). The result is, with d = r + s,

Simulation and Experimental Results
In this section we carry out the design and implementation of a tracking control system for an H-Frame XY positioning system. This system consists of two stationary motors, eight pulleys, and a single drive belt, as shown in Figure 5.3. A detailed description of the derivation of an 8th-order state space model is given in [6]. That This choice is influenced by the locations of the poles and zeros of the continuoustime plant [4,7]. This plant has no finite transmission zeros, and has the following poles: . Using the rules given in [4] four of the closed-loop poles are selected by keeping the imaginary parts of the complex plant poles and replacing their real parts with a value determined by the desired settling time, call it TS seconds. The value that gives the desired damping is s1/Ts, where s1 is the first order normalized Bessel pole, shown in  Table 5.1). With this selection of closed-loop poles, the algorithm described in [4] was used to calculate the gain matrices K1 and K2. The input multiplicative stability robustness bound for the resulting tracking system was 71 . 0 = δ . If the gain matrices are calculated using Matlab's place command, the stability robustness bound is 0.36.
The robustness bound for state feedback gains computed via an optimal control linearquadratic regulator (LQR) formulation is guaranteed to be at least 0.5 [5]. The place command gives much less robustness than LQR for this plant, but the new formulation [V] gives robustness that is competitive with LQR with the ease of a pole-placement design. With the gain matrices K1 and K2 calculated, the modified plant (5.8) is formed. The poles and zeros of the modified plant are: This modified plant can be inverted exactly with a delay of r = 1 sample. However, the two zeros outside the unit circle will be unstable poles of the exact inverse model. Furthermore, an exact inverse would have poles at the locations of the zeros of the modified plant on the negative real axis, resulting in a highly oscillatory output signal from the inverse model. Using the procedure described in Section IIB, it is found that only one additional delay, s = 1, is needed to make the IMP stable. With d = r + s = 2 delays, the poles and zeros of the IMP are:    6 Experimental x and y reference (red) and achieved (blue) trajectories for the H-frame system using the tracking system shown in Figure  5.2 without the Inverse Modified Plant filter.

Conclusion
A method of obtaining an approximate inverse of a non-minimum phase MIMO system was presented and confirmed with simulations and experimental results. The example used a relatively low frequency reference signal. In the hardware system, attempts at higher speed trajectories resulting in chattering, which is thought to be caused by the nonlinear friction that is present. The proposed control techniques used in this paper are for linear systems. A friction-compensation scheme such as shown in [8] might be used to make the hardware system behave more like a linear system, allowing higher tracking speeds.