distortions being reduced (Pintelon and Schoukens, 2001; Tan, Godfrey and Barker, 2005). In the training set,Multisine Awas applied toSubsystem 1 andMultisine B
was applied to Subsystem 2, this order wasreversed in the validation set.
From the information given in Cham, Tan and Tan (2010) (hereinafter re- ferred as ‘the Benchmark’), Subsystem 1 contained a highly nonlinear element (a stepper motor). However, the step time of the motor was only 0.05s, so that there are 20 such steps within each sampling interval of 1s. Further, with a movement of 7.5° per step, the motor can move150° within each sampling interval, and the range of operation was confined to be from 75° to 225°. The relationship between angle and flow rate, given in Cham, Tan and Tan (2010, fig. 3) was nonlinear, but over the operating range was almost linear. Again from the information of the Benchmark,
Subsystem 2 was very nearly linear.
In the figures given in theBenchmark, there is almost no spectral content in the output at any harmonic other than those of either of the input signals, partly as a result of the harmonic suppression in the inputs, but also suggesting that the overall system was very nearly linear.
8.1
System modelling
In system identification, the term whitebox modelling refers to the process of mod- elling a system through first principles, laws of physics and explicit assumed re- lationships between the input and output through prior knowledge of the system, resulting in amechanisticmodel. The termblackboxmodelling refers to the process of modelling a system through non-parametric techniques, without knowledge of physical inner workings of the system, resulting in an empirical model. A ‘greybox’ modelling approach is a mixture betweenwhitebox and blackboxmodelling.
Agreybox approach was taken to model each of the three subsystems separ- ately. Detailed information given in theBenchmark readily allowedSubsystem 1 to be modelled aswhitebox. Subsystem 2was modelled asblackboxwith the knowledge given in the Benchmark of it being very linear in its characteristics. A trial-and-error approach was then taken to modelSubsystem 3 to condition the outputs fromSub- systems 1 and 2 to give the output frequency spectrum and time domain response as close as possible to those given in the Benchmark.
A block diagram of the overall system is shown in Fig. 8.1. The outputs of thesingle-input single-output (Siso) Subsystems 1and2 are combined to form the inputs to the dual-input single-output Subsystem 3.
8.1. System modelling
Subsystem 3: Heat exchange unit
Air inducer &
Cooling chamber Subsystem 2:
Peltier system
PWM & Peltier cooler
Subsystem 1: Flow control system
Stepper motor & Air valve Duty ratio Valve angle Chamber temperature Peltier temperature
Figure 8.1: Overview of the Peltier cooling system
8.1.1 Flow control system
Subsystem 1 consisted of an air flow valve controlled by a stepper motor, with the input signal given to the controller of the motor being the desired valve angle. The input multisine signals used were scaled to span the input amplitude range, from 75° to225°.
AMatlabSimulinkmodel utilising a rate limiter, quantisation nonlinearity, saturation nonlinearity and a lookup table translating valve angle to air flow rate was developed with aid of the steady-state gain characteristic (Cham, Tan & Tan, 2010, fig. 3) and specified stepper motor characteristics. This is shown in Fig. 8.2a.
Saturation 75°−225° Rate limiter
150°/s Quantiser
7.5°/step Lookup table:
Valve angle
(a) Subsystem 1: Flow control system
Peltier temperature F1: 2nd order low-pass filter
−1
1−1.4894z−1 +0.5233z−2
Duty ratio
(b)Subsystem 2: Peltier system
Figure 8.2: Block diagrams ofSubsystems 1 and 2
8.1.2 Switching Peltier cooling system
Subsystem 2 was a Peltier cooler with the duty ratio of a pulse-width modulation (Pwm) waveform used to vary the power supplied to a120W Peltier module. The input signal was scaled to span the input amplitude range from 0 to 1, which rep- resents the duty ratio. The characteristics given in the Benchmark suggests that
8.1. System modelling
this subsystem was very nearly linear and that it had much slower dynamics than
Subsystem 1. The system dynamics were modelled by a simple second order digital low pass filterF1, as shown in Fig. 8.2b, to approximate the very nearly linear and low pass characteristics. The order of the filter was decided on by observing the sim- plicity of the transfer characteristics from the output ofSubsystem 2 to the overall system output with respect to excited frequency lines of input Multisine B. The gain of this subsystem was negative, since an increase in the duty ratio results in a decrease of temperature at the output. A guess was made to the filter coefficients by observing the harmonic components of the master output incorporating both
Subsystem 3 Path 2 and Subsystem 2 initially. The coefficients were subsequently modified through a trial-and error approach to match the output spectrum of the intermediate signal for Subsystem 2.
8.1.3 Heat exchange unit
Subsystem 3 physically comprised a cool air inducer attached to a cooling chamber. From a modelling perspective the subsystem conditions and combines the signals from Subsystems 1 and 2. This system was modelled as a blackbox, and because of a lack of physical knowledge of Subsystem 3, the model structure had redundant gain blocks to allow for flexibility in the trial-and-error modelling process.
The model shown in Fig.8.3was constructed by comparing the spectrum of the overall system output (from Subsystem 3) in the training data with the input spectra of Multisine A for the Subsystem 1 signal path and Multisine B for the
Subsystem 2 signal path to deduce appropriate filters and relative contributions between these two paths. All filter frequencies specified from this point onwards are normalised in the range of 0 to 1.
For the Subsystem 1 pathway, the resulting structure was a combination of a fourth order finite impulse response (Fir) band stop filter F2 with stop at about 200/600 and a second order high pass filter F3 (cascaded by two first order Butterworth filters in series) with a low cut-off frequency of1/600. F2 was designed using arbitrary magnitude fitting function “arbmag” in “fdesign”, and the GUI “fdatool” of the Signal Processing Toolbox of Simulink (Krauss et al., 1994) by observing the transfer characteristics from the output of Subsystem 1to the overall system output with respect to excited frequency lines of input Multisine A(Cham, Tan & Tan, 2010, eq. 1). The filter order of four was chosen on the criteria of simplicity and sufficiency, based on previews of frequency responses in the Toolbox.
F3, designed using the same Toolbox with standard Butterworth filter coefficients, gave the appropriate attenuation seen on the first harmonic number of the training
8.1. System modelling Subsystem 3: Chamber temperature Transport delay 5s Offset To K1 K2 0.113 0.11 0.24
F5: Two cascaded 1st order low-pass Butterworth filters cut-off = 3 Hz, fs = 600 Hz
F4: Single 1st order low-pass Butterworth filter cut-off = 7 Hz, fs = 600 Hz
F3: Two cascaded 1st order high-pass Butterworth filters
cut-off = 1 Hz, fs = 600 Hz F2: 4th order ~200 Hz FIR band-stop filter Subsystem 2: Peltier temperature Subsystem 1: Air flow Path 2 Path 1 Path 1 Path 2 0.1065 + 0.2250z−1 +0.3324z−2+ 0.2250z−3 +0.1065z−4 z 1 − 0.9896 0.9947 z 1 − z 1 − 0.9293 0.03537 z 1 + z 1 − 0.9691 0.01547 z 1 + output
Figure 8.3: Overview of the Peltier cooling system
data output frequency spectrum in theBenchmark. For the Subsystem 2 pathway, the signal was filtered by a first-order Butterworth low pass filter F4 with cut-off frequency of 7/600, in parallel with a second order low pass filter F5 cascaded by two first order Butterworth filters in series, with cut-off frequencies of 3/600. The signal itself was then added to its two filtered counterparts with different weightings, as shown in Fig. 8.3. A transport delay (pure time delay) was also incorporated in this pathway (see Section 8.3.2).
After combining the contributions from the two pathways, a constant offset 𝑇o was added. This was based on comparing the simulated output with the mean value of the training data time domain response. It is reasonable to assume the offset temperature was related to the ambient temperature when the measurement took place (Cham, Tan & Tan,2010).