Download Closed-loop Neural Network Controlled Accelerometer

Closed-loop neural network controlled accelerometer
E.I. Gaura, R.J.Rider, N. Steele,
Nonlinear Systems Design Group, Coventry University,
Priory Street, Coventry, UK
tel: +44-(0)1203-838825
fax: +44-(0)1203-838949
e-mail: [email protected]
SUMMARY
The purpose of this paper is to present aspects of an integrated micromachined sensor/neural
network transducer development. Micromachined sensors exhibit particular problems such as
nonlinear characteristics, manufacturing tolerances and the need for complex electronic
circuitry. The novel transducer design described here, based on a mathematical model of the
micromachined sensor, is aimed at improving in-service performance, and facilitating design
and manufacture over conventional transducers. The proposed closed-loop transducer
structure incorporates two modular artificial neural networks: a Compensating Neural
Network (CNN), which performs a static mapping and a Feedback Neural Network (FNN)
which both linearises and demodulates the feedback signal. Simulation results to date show an
excellent linearity, wide dynamic range and robustness to shocks for the proposed system. The
design was approached from a control engineering perspective due to the closed-loop structure
of the transducer.
Keywords:
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accelerometer, mechatronic, micromechanical devices, micromachined sensor, multilayer
perceptron, neural network.
List of notations:
a
acceleration
b(x)
damping term
C1(x) capacitance between seismic mass and top electrode
C2(x) capacitance between seismic mass and bottom electrode
g
gravitational acceleration
k
spring constant
m
seismic mass
Vexc high frequency excitation voltage
x
displacement of seismic mass
1. Introduction
Transducers employing micromachined sensors are suited to a large variety of applications
(including automotive, aerospace and defence), due to their small size, low weight, low power
consumption and compatibility with VLSI technology [1, 2, 3]. However, the small dimensions
cause unique problems in the integration of microengineered devices into a full system.
This paper considers a transducer which is the subject of considerable research and
development, namely a micromachined accelerometer with a capacitive type of pick-off. The
particular class of devices investigated has interesting nonlinear properties from both a static
and a dynamic analysis point of view. Typical non-linear effects which must be addressed are:-
1. the nature of the electrostatic forces (proportional to square of voltage),
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2. "squeeze film" damping phenomena,
3. the existence of production tolerances of the micromachined device leading to the
requirement of calibration at the manufacturing stage.
Also, due to the method of pick-off used to detect the position of the seismic mass, these
micromachined acceleration sensors have an inherently narrow stability margin [4] and a
reduced dynamic range. For static/low frequency applications, which do not require a high
degree of accuracy, they can be used in open-loop (compensated or uncompensated) but for
higher precision applications, negative feedback must be used to increase linearity, bandwidth
and dynamic range [1, 2, 5, 6]. Conventional closed-loop designs use a simple, linear control
strategy based on a proportional and integral (PI) control action [1]. However, the systems
obtained by using this approach are only conditionally stable (i.e. stability is guaranteed only
over the dynamic range of the transducer). Such behaviour is unacceptable in most application
areas. In contrast, the neural network approach proposed here offers the prospect of
overcoming this crucial deficiency by providing stability over double the dynamic range. It is
expected that this novel control approach will reduce the mathematical problems associated with
the modelling and design of nonlinear micromachined accelerometers, lead to a simplified
electronic structure and will also address the manufacturing problems.
The concept of using an artificial neural network (ANN) as a potential solution strategy for
problems which require complex data analysis is not new [7, 8]. To date, over 100 alternative
ANN structures have been reported in the literature. However, most of them are highly
application-specific. The basic feedforward multilayer perceptron (MLP) architecture remains
the most widely applied and analysed [7]. The capability of MLP’s, trained by error-backpropagation, to approximate nonlinear functions with any degree of accuracy, has already been
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shown in dozens of successful applications in the fields of pattern recognition/classification
and image processing [7, 9, 10]. The MLP has been selected for this study due to its simplicity
and ease of training for small-scale problems.
The following sections describe the various stages of modelling, design, testing and validation
undertaken in developing a novel micromachined transducer based on neural networks.
2. The mathematical model of the micromachined accelerometer
The basic acceleration sensor considered consists of a proof mass suspended between two
plates which form the covers for the device (Figure 1); the shape of the mass may vary
depending on the design. Typically, the mass is symmetrically suspended on eight cantilever
beams. The form of suspension is arranged to ensure both that the mass displacement is linear
with force and that torsional effects are minimised.
The motion of the mass is detected by the use of sensing electrodes on the inside surfaces of
the top and bottom covers and the proof mass itself. These electrodes enable the position of
the mass to be inferred using capacitive bridge techniques [2].
Some typical dimensions for this class of devices would be:
 gap from the seismic mass to either electrode - 10 m;
 area of the seismic mass - 2.4 mm x 2.4 mm;
 thickness of the seismic mass - 0.35 mm;
 mass of the seismic mass - 6.2 mg.
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The measurement of the position of the seismic mass (performed by a pick-off circuit) is
realised by applying a high-frequency sinusoidal signal to the top electrode and the same
signal in antiphase to the bottom electrode. The signal, coupled through the seismic mass, is
passed to a charge amplifier. In order to extract the direction of the mass deflection it is
necessary to perform phase-sensitive demodulation and filter the output voltage of the charge
amplifier. A model-based strategy has been adopted for the design of the closed-loop
transducer system, therefore a mathematical model has had to be derived to represent the
physical system. The corresponding block diagram is shown in Figure 2.
This model is a true mechatronic system, consisting of a mechanical component (the
micromachined sensing element) and an electronic part (the pick-off and conditioning
circuitry) [11]. The input to the system is the acceleration force acting on the proof mass,
causing it to deflect from the rest position; the output signal is a measure of the position of the
mass. The sensing element is basically a second order system with a mass, spring and a form
of nonlinear (squeeze-film) damping caused by the motion of the proof mass in a gaseous
medium. The generated damping force is a function of the velocity of the mass and its
displacement from the midpoint between the plates. The sensing element itself is a closedloop system in its own right, with a spring force and a damping force providing negative
feedback. The pick-off circuit can be modelled as a proportional gain factor which converts
the displacement of the proof mass into a voltage. Apart from the nonlinearities caused by
damping, another important nonlinear effect is introduced by the method of sensing: the
relationship between the electrostatic force on the mass and applied voltage to the plates is
proportional to the square of the voltage and inversely proportional to the square of the
distance between the electrode and the mass.
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This mathematical model was simulated in the SPICE environment (standard MicroSim
simulation package) using the behavioural modelling library. Since, as previously stated, this
micromachined accelerometer with capacitive pick-off has inherent nonlinear properties in
open-loop operation, an obvious solution is to apply some form of feedback. Due to the small
physical dimensions of the sensing element, this can be achieved by balancing the inertial
force acting on the proof mass with electrostatic actuation. Conventional transducers use a
simple linear PI controller, cascaded with the pick-off circuit, to provide the feedback. Under
normal operating conditions (input accelerations in the range 4g), a stable linear relationship
between the controller output and the input acceleration is obtained. However, certain
conditions (such as large input accelerations, shocks in acceleration, unknown mass position
at power up, etc.) may arise in which stable linear behaviour is lost. This leads to an
irreversible electrostatic lock-up of the seismic mass to one electrode which can only be
rectified by switching-off the power supply to the sensor. Such behaviour is unacceptable for
many applications (e.g. inertial navigation). Despite this problem, the PI approach has been
used in many devices described in the literature [2,3] since it nevertheless improves the sensor
performance compared to open-loop operation.
In contrast to the above linear control approach, the novel transducer design uses the nonlinear
mapping capabilities of neural networks for both controlling the sensing element and
linearising the electrostatic forces. A block diagram representation of the proposed system is
given in Figure 3.
The compensating neural network (CNN) performs a static mapping acting as a nonlinear gain
controller. The feedback neural network (FNN) has two functions. Firstly, it calculates the
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square root of the output voltage, providing a linear feedback relationship between the system
output and the electrostatic forces acting on the electrodes. Secondly, the network
demodulates the output signal in order to apply the feedback to only one electrode at a time:
the bottom electrode will be activated if the proof mass has moved towards the top electrode
and vice-versa. Details of the design of both networks are given in the following sections.
3. The feedback linearisation procedure
As stated above, the FNN has the role of providing a linear relationship between the output
voltage and the feedback electrostatic forces acting on the seismic mass. Moreover, the FNN
acts as a demodulator for the feedback signal, by applying it in a discontinuous manner, to
only one electrode at a time. This novel reset concept ensures that the feedback is always
negative and therefore, even under adverse operating conditions, the seismic mass cannot
lock-up.
The network used for this purpose has one input (the output voltage of the transducer), one
hidden layer and two outputs, connected to the outer electrodes of the sensing element. Both
the hidden and the output neurons are governed by a sigmoid-type transfer function, with bias.
The network has been trained to approximate the following input - output function:
 input if input 0
output1  
0 if input 0
(1)
  input if input 0
output 2  
0 if input 0
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The training set contained 200 examples, with the input samples evenly distributed in the
range [-1 ; +1]. A dynamic version of the backerror propagation algorithm was used for the
network training [9, 12]. This included both a variable learning rate (lr) and a momentum term
[13]. The input and desired outputs of the network are presented in Figure 4a. The
approximation provided by a six hidden neuron net is presented in Figure 4b. 20000 epochs of
training (using MATLAB) were necessary for the network to reach a preset sum-square error
(of the difference between the desired and actual outputs) of 0.05/200 samples. It should be
noted that determining the size and structure of a network suitable for a particular problem is a
major issue. Here, a simple trial and error approach proved adequate and resulted in the small
sized networks used throughout.
Some scaling was necessary in order to integrate the trained network into the closed-loop
transducer structure: the output of the transducer (which is the input to the feedback network)
was divided by 100 and the outputs of the net were multiplied by 10 (in view of the required
square root law). The testing and validation of the trained net was done by describing it
behaviourally in the SPICE environment, subjecting it to previously unseen input stimuli and
assessing its performance in terms of the sum-square error [7]. Figure 5 shows the SPICE
behavioural description of the trained neural network, with the “Function” blocks obeying
equation (2) and the fixed weights being represented by “gain blocks”.


2
Function _ out  
 1
 1  exp( 2( Function _ in)  bias) 
4 The Neural Network nonlinear gain controller
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(2)
The Compensating Neural Network (CNN) - acting as a nonlinear gain feedback neural
controller - performs a static mapping. The same architecture and training algorithm have
been used as in section 3, for the CNN design. The training set was formed by subjecting a
prototype PI transducer [4] to a ramp type of acceleration rising from -6g to +6g (which is the
full working range of the conventional accelerometer, before lock-up of the seismic mass
takes place). The resultant mapping (after suitable scaling) required to be performed by the
network is presented in Figure 6. The CNN transfer characteristic contains a linear region,
(equivalent to a purely proportional gain) corresponding to small input signals, and softlimiting transitions to saturated regions, corresponding to larger input signals. A 1-6-1 MLP
was successfully trained to perform this mapping, reaching in roughly 50000 epochs a sumsquare error of 0.1/400 samples. As for the FNN, the trained CNN was modelled and tested in
SPICE.
5. Design and performance of the closed-loop neural transducer
In the preceding sections, a feedback signal linearisation procedure and a nonlinear gain
controller design have been proposed. It has been shown [14] that both of the resulting neural
networks produced improved performance, when used separately to replace the corresponding
block in the prototype PI transducer. The novel transducer design described in this section
differs, in that it incorporates both the FNN and the CNN with the aim of combining the
advantages of linear feedback electrostatic forces, time domain separation of feedback/sensing
phases (FNN) and soft-limiting nonlinear gain control (CNN). In the following, the
functionality of the system is studied by subjecting the transducer to a wide variety of stimuli
and establishing both its advantages and limitations.
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The nonlinear control action imposed by the CNN is evident from the simulation results
presented in Figure 7, where the neural transducer was excited with a sine wave of 6|g|
amplitude and 1Hz frequency. Although the mass displacement is highly nonlinear, the
transducer output closely resembles the form of the input. Additionally, the saturating
properties of the CNN and the demodulating action of the FNN are revealed when a 12g input
acceleration is imposed on the system (Figure 8). In this case, the seismic mass is deflected to
a maximum of 8m, followed by a sharp return to the rest position, midway between the top
and bottom electrodes. The output voltage is therefore reaching its limit values of 20V
without the system becoming unstable.
Shocks in acceleration of up to 25g can be withstood by the transducer, without irreversible
displacements of the mass, as indicated in Figure 9, provided that the duration of such a shock
is less than 30ms (amplitudes of this level and duration greater than 30ms cannot be classed as
shocks and therefore fall outside of the dynamic range specification).
The calibration curve for the new system is presented in Figure 10. The system exhibits a
maximum departure from linearity of 3.8% over the range 6g, maximum hysteresis of 5%
between 6g and 8g, followed by saturation for acceleration magnitudes in excess of |8|g.
This performance compares with a departure from linearity of 8% for the conventional PI
transducer, over its entire dynamic range of 4g.
According to the application requirements for the acceleration sensor, the design can be easily
altered. Improved linearity can be obtained for a restricted range (precision applications), or
the whole dynamic range extended, by reducing accuracy in linearity. This can be done by
modifying the CNN scaling factors, which currently have the following values:
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 CNN input gain - 0.1
 CNN output gain - 20
Two examples of transfer characteristics for modified designs are presented in Figure 11 with
the corresponding scaling factors given in Table 1.
In order for the analysis of the transducer to be complete, the frequency behaviour of the
system must be considered. Unfortunately, for this system, such a study is difficult to be
performed, as the simple AC sweep option available in SPICE cannot be used, due to the
nonlinear nature of the system. However, parametric analysis can be performed for such a
system, for a given magnitude of the input signal, at several frequencies. Bode diagrams can
then be drawn for the fundamental component of the transducer output. Figure 12 presents the
results of such a frequency analysis, for a sinusoidal input acceleration of 4|g| magnitude and
frequency in the range [1 - 1000Hz]. It can be observed that the transducer has a flat frequency
response up to approximately 300Hz, but the phase shift reaches quite large values at
frequencies above about 30Hz. For phase sensitive applications, phase compensation is
therefore necessary if the transducer is to be used outside the [0 - 30Hz] range.
The frequency response of the circuit for several other input acceleration magnitudes has been
studied. As expected from the slightly nonlinear static transfer characteristic of the novel
transducer, some gain changes take place as the magnitude of the input acceleration is
increased. The gain and bandwidth of the transducer for the [1|g| ; 6|g|]: magnitude range are
given in Table 2.
A major decrease in the bandwidth takes place for sinusoidal accelerations around 6|g| for this
particular design. However, the general performance of the novel transducer show
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considerable improvement over that of the prototype PI transducer where the gain variation
for the [1|g| - 4|g|] magnitude range was 6dB (compared to less than 3 dB for the novel
design).
6. Conclusions
A neural, closed-loop accelerometer has been developed based on the mathematical model of
the sensing element. The aim of the design work was to improve the functionality of a
prototype PI controlled accelerometer previously developed at Coventry University. A
nonlinear control strategy was developed, based on MLP type neural networks, as opposed to
the linear approach previously adopted.
Electrostatic forces are employed to provide the feedback action for the closed loop system. A
linear, negative feedback relationship is ensured in the new design by inserting a neural
network in the feedback path of the system. A novel concept for the feedback action has been
used: the feedback electrostatic force is applied in a discontinuous manner, only one electrode
being activated at a time. It should be mentioned here that, although other ways of
implementing this feedback arrangement exist - for example by using dedicated integrated
circuits to perform the square root function and using logic gates for the feedback signal
demodulation - the neural network implementation was preferred due to its simplicity both in
the design phase and from an integration point of view.
The performance of the transducer in terms of bandwidth and linearity was further improved
by replacing the PI controller by a neural network nonlinear feedback gain controller.
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The SPICE environment was successfully used to simulate the model of the closed-loop
accelerometer at different stages during the design process and to observe its stable and
unstable behaviour.
It was shown that the novel transducer proposed has a stable region extended by 200%
compared to the prototype PI transducer, a dynamic range increased by 70% and a bandwidth
of around 300Hz. Possible application oriented changes to the novel design are suggested in
order to increase the dynamic range or to increase accuracy, as required.
As far as the hardware implementation of the transducer is concerned, the new methods of
feedback and control could significantly reduce the complexity of the electronics associated
with the sensing element. A first step forward has been made towards simpler integration of
the transducer, provided that appropriate hardware is chosen for the implementation of the
two modular neural networks. Hardware devices for implementing neural usually support a
large number neurons, therefore, both the FNN and the CNN may be physically realised using
a single chip.
Acknowledgements
The work presented here is based on the modelling and design work conducted by members of
the Nonlinear Systems Design Group at Coventry University who have developed both an
analogue and a digital closed loop accelerometer, using conventional control techniques [1, 4].
References
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Proceedings - Circuits, Devices and Systems, Oct 1998, Vol. 145, No. 5, pp. 325 – 331.
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integrated force balanced capacitive accelerometer for low-g applications. Sensors and
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1997, Coventry University.
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Electronic Conf., 1991, Firenze, Vol. 3. pp. 488-490.
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Inst. Meas. and Control, to be published 1999.
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Control Engineering Series 53, 1995, Short Run Press Ltd., UK.
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11.Kraft, M., Lewis, C.P. System level simulation of a digital accelerometer, Proc. 1st Int.
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Tables
Table 1: Scaling factors of modified designs
Transducer range
Low
CNN input gain
0.04
CNN output gain
25
Wide
0.05
40
Table 2: Transducer gain and bandwidth for different input acceleration magnitudes
Acceleration
magnitude
1g
Gain
[dB]
11.74
Bandwidth
[Hz]
350
2g
10.88
350
4g
9.76
300
6g
8.99
60
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List of Captions
Fig. 1: The sensing element
Fig. 2: Mathematical model of sensing element and pick-off circuit
Fig. 3: Block diagram of the closed-loop neural transducer
Fig. 4a: FNN inputs and desired output
Fig. 4b: Approximation provided by the trained FNN
Fig. 5: Neural network behavioural description in SPICE
Fig. 6: CNN input and desired output
Fig. 7: Mass displacement (1V  1m) and output voltage for the fully neural transducer for
a 6g input acceleration (1g  10V)
Fig. 8: Mass displacement (1V1m) and output voltage for the fully neural transducer for a
12g input acceleration (1g  10V)
Fig. 9: Displacement (1V  1m) and output voltage of the fully neural transducer for shock
in acceleration of 25g (1g  10V)
Fig. 10: Transfer characteristics of the fully neural transducer
Fig. 11: Transfer characteristics of a low range and a wide range transducer
Fig. 12: Gain and phase plots for the fully neural transducer excited by a 4g acceleration
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Fig. 1
Sensing Element
Accelerati
on
m
-Vexc

1/m
- -
Integrat
or
b(x
)
Nonlinear
damping
k
Signal Pick-off
x

Pick-off
circuit
C1(x)
Integrat Saturatio
or
n
C2(x
)
Vexc
Output
voltage
Spring
constant
Low pass
filter
Fig. 2
Input
acceleration
Sensor and pick-off
circuitry
CNN
Transducer output
voltage
FNN
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Fig. 3
Signal Level
1
output2
output1
0.5
0
output1
output2
input
-0.5
-1
0
50
100
150
Sample Number
Fig. 4a
Signal Level
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200
250
1.2
1
output2
output1
0.8
0.6
0.4
0.2
0
output2
output1
-0.2
0
50
100
150
Sample Number
Fig. 4b
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200
250
Fig. 5
Signal Level
1
input
0.5
0
-0.5
output
-1
0
100
300
200
Sample Number
Fig. 6
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400
500
Fig. 7
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Fig. 8
Fig. 9
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input acceleration (1g10V)
Fig. 10
input acceleration (1g10V)
Fig. 11
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Fig. 12
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