From ControlTheoryPro.com

1 Simple Mirror Model
Modeling of even simple systems, such as a Fast Steering Mirror (FSM), can appear complex to those who have not done much modeling. This section will include a simple model of the system. Being a simple model the following items will be ignored:
 Noise
 Conversion factors (Amps to torque for driving circuits, Volts to radians for position sensors, etc.)
The simple example will focus on the creation of 2^{nd} order system models for each piece of the system. Those pieces are:
 mirror
 driver circuit
 position
 controller
At the end of the simple example the complexities of a true to hardware model will be addressed.
1.1 Mirror
The chosen mirror is by Axsys Technologies Imaging Systems and it is described in this whitepaper (used to be free in a couple of places but I can only find it as a paid item anymore).
Keep Figure 1 in mind as the example progresses. is the plant, in this case the physical dynamics of the mirror and its flexures. is the mirror's driver circuit. is the feedback sensor, in this example a position sensor built by Kaman. is a prefilter that could be designed but is unnecessary in this example so . The reference command is typically a step command provided by another sensor somewhere else in the parent system. This example will follow along the Axsys whitepaper's laser communications example. For laser communications the outside sensor providing the reference command would typically be an optical feedback sensor which returned the position of the communication target relative to the platform this mirror resided on.
is what needs to be designed.
1.1.1 Mirror Parameters
The whitepaper suggests that the mirror can be accurately described by a 2^{nd} order system. Therefore the governing transfer function is
Generic 2^{nd} Order System 
 where
 is the system gain
 is the system's natural frequency
 is the system's damping ratio.
For the Fast Sterring Mirror described in the whitepaper:
 Hz
Since the mirror is our plant and the plant transfer function is denoted by in Figure 1 replace with .
Figure 2 shows the frequency response of the Axsys FSM. The whitepaper provides only 1 value for . So only a single axis is modeled. The 2^{nd} axis could be modeled identically, at least in early design stages.
1.1.2 Driver Circuit
Figure 9 of the whitepaper includes a circuit which converts voltage commands from the controller to amps used by the mirror motor.
The driver circuit is not described in detail. However, the proposed closed loop bandwidth is between 500 and 1000 Hz. This requires that the driver circuit have a bandwidth of at least 10x the maximum desired closed loop bandwidth. This means the driver circuit has a minimum bandwidth of 10 kHz. The driver circuit will be modeled with a 2^{nd} order system with the following parameters:
 kHz
In Figure 1 the driver circuit is denoted by . With the bandwidth of the driver circuit at only 10 kHz it will effect the phase margin of any system with a closed loop bandwidth of near 1 kHz. If the phase margin was not going to be effected we could model the driver circuit as a unity gain.
1.1.3 Sensor Parameters
Figure 10 of the whitepaper includes a position sensor. The position sensor is not described but a common sensor for FSMs is a differential impedence transducer.
A position sensor for mirror feedback is required when the system is meant to follow step commands. The Kaman DIT 5200 is described in this datasheet. The datasheet provides the following parameters
 kHz
The sensor is traditionally denoted with . With the bandwidth of the sensor at only 20 kHz it will also effect the phase margin of any system with a closed loop bandwidth of near 1 kHz.
1.2 Controller
The Figure 9 of whitepaper also inlcudes reference to a PID controller with a parallel Integral Control and Lead Lag Controller network. The authors present a circuit drawing of their controller. They show a bandwidth of 560 Hz with closed loop peaking of 8 dB.
1.2.1 Plant and Sensor System
Combining the Mirror, Driver Circuit, and Sensor transfer functions we get an overall transfer function.
Below are controller examples.
1.2.2 Axsys Whitepaper Author's Controller
The author's of the Axsys FSM whitepaper presented what they called a PID controller. However, the form of the controller is a Lead/Lag network in parallel with an integrator.
Below are the system metrics of for the closed loop system
System Metrics  
Phase Margin  = 24.03  degrees 
Gain Margin  = 15.13  dB 
Peak Closed Loop Magnitude  = 7.62  dB 
Open Loop Crossover Frequency at 0 dB  = 413.43  Hz 
Closed Loop Bandwidth Frequency at 3 dB  = 677.17  Hz 
Disturbance Rejection Bandwidth Frequency at 3 dB  = 256.66  Hz 
A phase margin of 24 degrees is not very good. The whitepaper reports a closed loop bandwidth of 560 Hz and a phase margin of 35 degrees. A phase margin of 35 degrees is just barely good enough. The 8 dB of closed loop peaking is related to the low phase margin. The poor phase margin in both closed loop systems leads to excessive closed loop peaking. The closed loop peaking is related the system overshoot as well. So the lower the phase margin, the higher the closed loop peaking, the more the system rings in reponse to a step input. Figure 7 shows the Simulink model built to test this controller. Figure 9 shows the system's reponse.
1.2.3 PI with Lead/Lag Network
With a little tuning the controller can be made significantly better.
Notice that the phase margin is larger and the peak closed loop magnitude is smaller. This controller's better response is largely due to more phase gain from the lead/lag network. The phase gain was achieved by a larger pole/zero separation.
Today analog lead/lag circuits can achieve this much pole/zero separation and as such this much phase gain. However, some years ago this controller was not achievable with an analog controller. It is easily accomplished with a digital controller.
System Metrics  
Phase Margin  = 57.03  degrees 
Gain Margin  = 21.23  dB 
Peak Closed Loop Magnitude  = 2.08  dB 
Open Loop Crossover Frequency at 0 dB  = 413.43  Hz 
Closed Loop Bandwidth Frequency at 3 dB  = 684.86  Hz 
Disturbance Rejection Bandwidth Frequency at 3 dB  = 251.60  Hz 
Figure 11 shows the step response of the retuned controller. Note the smaller overshoot and shorter settling time.
Designing and tuning controllers is a broad topic beyond the scope of this example.
2 True to Hardware Mirror Model
The most important difference between a simple model used in early design phases and a mature model used for accurate system performance is an accurate accounting of the noise. However, there are also important factors such as conversion factors and quantization error (for digital systems).
2.1 Noise
Digital System Note 
There are several types of noise to be considered
 bias
 white noise
 shaped noise
 quantization error
For this example we are assuming all noises are either white noise or quantization error. In Simulink white noise is easily modeled using the BandLimited White Noise block. Simulink also includes a block specific to quantization errors called the quantizer.
2.1.1 Driver Circuit
Depending on the circuit components and the length of cabling used to connect the circuit to the rest of the system, electrical circuits can pick up noise from radio stations, cell phones, the electrical power system of the building and any number of other sources. Electrical noise is usually modeled very well by white noise. See white noise for more information.
2.1.2 Position Sensor
Position sensor noise is complicated by sensor misalignment, linearity, proximity, and temperature. However, systems design will take care of these items. In other words, buy a sensor with small (by comparison to required error) misalignment, good linearity across the entire operating range, etc. If a sensor with all these characteristics is not available then really good calibration is required so the negative impacts can be minimized.
Assuming a good sensor can be procured, the noise source that eliminated or minimized is the electrical noise of the sensor. This is usually white noise.
2.2 Conversion Factors
Conversion factors are typically gains. This means they are easily handled once a plant transfer function is measured. However, sometimes it is preferrable to include these conversion factors in the model so that it is as accurate as possible.
2.2.1 Mirror Gain
The Axsys whitepaper states that the conversion factors are
 0.365 Amps/Volt
 the second factor radians/amp is provided but this author does not know what stands for, it assumed to be related to inertia
In reality the mirror transfer function will not have a but instead will have a nonunity DC gain. The gain will be a function of the mirror's inertia.
The mirror gain does not necessarily need to be modelled accurately. The change to the controller is only the inverse of the mirror's gain (K). An accurately modeled mirror gain is only necessary for the following reasons:
 driver circuit saturation is a concern
 hardware is to be tested
In the case of driver circuit saturation the circuit needs to be capable of generating enough amps (torque) to achieve the necessary response. If it cannot do this without saturating then something in the loop needs redesign.
In the case of hardware being built models of the mirror are not likely to be quite accurate enough. Therefore the mirror plant transfer function needs to be measured. The DC gain from this measurement can then be used to update the model. The controller gain will need to be updated to reflect this new gain in order to maintain the same open loop crossover and closed loop bandwidth.
2.2.2 Driver Circuit  Amps to Torque
The driver circuit conversion factor was not specified in the whitepaper. There is an Amps/Volt factor and radians/Amp factor. In reality the driver circuit will produce a current. The voice coils will produce a force based on this current. The force acts via a lever arm to produce a torque about the mirror's center of rotation.
Ultimately the conversion factor will be a combination of all the factors to create a Torque/Amp or Torque/Volt.
2.2.3 Position Sensors  Volts to Angular Position (radians)
The Axsys whitepaper specifies a position sensor conversion factor of
 100 Volts/radian
2.2.4 Controller Adjustment
If the controller works well when all the 2^{nd} order systems have a gain of 1 then it will work fine with only a magnitude adjustment. The adjustment should be equivalent to the inverse of all the conversion factors. The easy way to adjust the magnitude is to form the system
(see Figure 1) 
 where
 is the driver circuit transfer function
 is the mirror plant transfer function
 is the position sensor transfer function
 is the controller transfer function
Then adjust the magnitude in order to get the same open loop crossover frequency that the system had before the conversion factors.
2.3 Digital Considerations
The Axsys whitepaper also shows that the controller is digital. There is a 12 bit DAC which turns the digital controller command into an analog voltage. The analog output from the position sensor is passed through a 12 bit ADC to turn the sensor output into a digital value for the controller. The analog/digital conversion introduce a courseness to the signals which introduces yet another error source.
2.3.1 ADC
Modelling an ADC in the frequency domain is not possible. In order to model an ADC in the time domain (Simulink) use the quantizer block.
2.3.2 DAC
Modelling a DAC is a simple matter of using a gain equal to the Least Significant Bit (LSB) which will be the DAC range divided by 2^(number of effective bits).
3 Final Mirror Model
The Kaman 5200 DITs specificaton state that the 15N sensor head has 14 nanoinches of error RMS. If we assume that the mirror is a square mirror, 4 inches on a side then the position error is 7 nanoradians .
The Axsys whitepaper states that there is a driver circuit but it does not specify the RMS noise for that circuit. It also specifies a mechnical range of +/3 degrees (52.4 mrad). The table below show the results of the simulation in Figure 12 as various parameters are changed.
Simulation Results  

Sample Rate (kHz)  Bits  Mechanical Range (deg)  Residual RMS (rad) 
10  12  3  1.60E5 
10  12  0.3  2.88E5 
10  14  3  2.70E5 
20  12  3  1.28E5 
20  12  0.3  8.05E6 
20  14  3  8.00E6 
30  12  3  6.23E6 
30  12  0.3  4.07E6 
1000  12  3  4.28E6 
1000  12  0.3  3.29E6 