Update Content - 2021-05-25

content/book/schmidt20_desig_high_perfor_mechat_third_revis_edition.md

@@ -8,7 +8,7 @@ Tags
 : [Reference Books]({{< relref "reference_books" >}}), [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
 
 Reference
-: ([Schmidt, Schitter, and Rankers 2020](#orgb603526))
+: ([Schmidt, Schitter, and Rankers 2020](#org343c570))
 
 Author(s)
 : Schmidt, R. M., Schitter, G., & Rankers, A.
@@ -64,7 +64,7 @@ Year
 
 #### Electric Field {#electric-field}
 
-<a id="org68292ec"></a>
+<a id="org8b0a7f3"></a>
 
 {{< figure src="/ox-hugo/schmidt20_electrical_field.svg" caption="Figure 1: Charges have an electric field" >}}
 
@@ -397,7 +397,7 @@ Finally it can be concluded, that these insights help in designing actively cont
 
 ### A Walk around the Control Loop {#a-walk-around-the-control-loop}
 
-Figure [2](#orgc838464) shows a basic control loop of a positioning system.
+Figure [2](#orgc78b355) shows a basic control loop of a positioning system.
 First, the A/D and D/A converters are used to translate analog signals into time-discrete digital signals and vice versa.
 Secondly, the impact locations of several disturbances are shown, which play a large role in determining what reqwuirements the controller needs to fulfil.
 The core of the control system is the _plant_, which is the physical system that needs to be controlled.
@@ -405,7 +405,7 @@ The core of the control system is the _plant_, which is the physical system that
 <a id="table--tab:walk-control-loop"></a>
 <div class="table-caption">
   <span class="table-number"><a href="#table--tab:walk-control-loop">Table 1</a></span>:
-  Symbols used in Figure <a href="#orgc838464">2</a>
+  Symbols used in Figure <a href="#orgc78b355">2</a>
 </div>
 
 | Symbol     | Meaning                              | Unit |
@@ -419,14 +419,14 @@ The core of the control system is the _plant_, which is the physical system that
 | \\(y\\)    | Measured output motion               | [m]  |
 | \\(y\_m\\) | Measurement value                    | [m]  |
 
-<a id="orgc838464"></a>
+<a id="orgc78b355"></a>
 
 {{< figure src="/ox-hugo/schmidt20_walk_control_loop.svg" caption="Figure 2: Block diagram of a motion control system, including feedforward and feedback control." >}}
 
-The plant combines the mechanical structure, amplifiers and actuators, as they all deal with energy conversion in close interaction (Figure [3](#org969d1d3)).
+The plant combines the mechanical structure, amplifiers and actuators, as they all deal with energy conversion in close interaction (Figure [3](#orge91544d)).
 They interact in both directions in such a way that each element not only determines the input of the next element, but also influences the previous element by its dynamic load.
 
-<a id="org969d1d3"></a>
+<a id="orge91544d"></a>
 
 {{< figure src="/ox-hugo/schmidt20_energy_actuator_system.svg" caption="Figure 3: The energy converting part of a mechatronic system consists of a the amplifier, the actuator and the mechanical structure." >}}
 
@@ -457,7 +457,7 @@ Fortunately the effect is mostly so small that it can be neglected.
 
 #### Overview Feedforward Control {#overview-feedforward-control}
 
-Figure [4](#org3b63c15) shows the typical basic configuration for feedforward control, which is also called _open-loop control_ as it is equal to a situation where the measured output is not connected to the input for feedback.
+Figure [4](#org0284035) shows the typical basic configuration for feedforward control, which is also called _open-loop control_ as it is equal to a situation where the measured output is not connected to the input for feedback.
 
 The reference signal \\(r\\) [m] is applied to the controller, which as a reference transfer function \\(C\_{ff}(s)\\) in [N/m].
 The output \\(u\\) in [N] of the controller is connected to the input of the motion system, which has a transfer function \\(G(s)\\) in [m/N] giving the output \\(x\\) in [m].
@@ -469,7 +469,7 @@ If one would like to achieve perfect control, which means that there is no diffe
 G\_{t,ff}(s) = \frac{x}{r} = C\_{ff}(s)G(s) = 1 \quad \Longrightarrow \quad C\_{ff}(s) = G^{-1}(s)
 \end{equation}
 
-<a id="org3b63c15"></a>
+<a id="org0284035"></a>
 
 {{< figure src="/ox-hugo/schmidt20_feedforward_control_diagram.svg" caption="Figure 4: Block diagram of a feedforward controller motion system with one input and output (SISO)." >}}
 
@@ -493,7 +493,7 @@ The drawbacks and limitations of feedforward control are:
 
 In feedback control the actuator status of the motion system is monitored by a sensor and the controller generates a control action based on the difference between the desired motion (reference signal) and the actuator system status (sensor signal).
 
-The block diagram of Figure [5](#org2d7cf84) shows a SISO feedback loop for a motion system without the A/D and D/A converters.
+The block diagram of Figure [5](#orgf739dfa) shows a SISO feedback loop for a motion system without the A/D and D/A converters.
 The output \\(x\\) in [m] is the total motion of the plant on all its parts and details, while \\(y\\) is the measured motion with a measured value \\(y\_m\\) measured on a selected location in the plant.
 This measured is compared with \\(r\_f\\), which is the reference \\(r\\) after filtering.
 The result of this comparison is used as input for the feedback controller.
@@ -505,7 +505,7 @@ The transfer function of any input to any output in a closed-loop feedback contr
 
 </div>
 
-<a id="org2d7cf84"></a>
+<a id="orgf739dfa"></a>
 
 {{< figure src="/ox-hugo/schmidt20_feedback_control_diagram.svg" caption="Figure 5: Block diagram of a SISO feedback controlled motion system." >}}
 
@@ -560,9 +560,9 @@ Also, some pitfalls have to be dealt with:
 #### Model-Based Feedforward Control {#model-based-feedforward-control}
 
 In the following an example of a model-based feedforward controller is introduced.
-The measured frequency-response of the scanning unit taken as as an example is shown in Figure [6](#org5b8e277).
+The measured frequency-response of the scanning unit taken as as an example is shown in Figure [6](#org70c7ce4).
 
-<a id="org5b8e277"></a>
+<a id="org70c7ce4"></a>
 
 {{< figure src="/ox-hugo/schmidt20_bode_plot_scanning.svg" caption="Figure 6: Bode plot of a piezoelectric-actuator based scanning unit for nanometer resolution positioning. It shows the measured response (solid line) and the second order model, which is fitted for the low-frewquency system behaviour (dashed line)." >}}
 
@@ -607,10 +607,10 @@ G\_{t,ff}(s) &= G(s)G\_{ff}(s) \\\\\\
 \end{align}
 \end{equation}
 
-The bode plot of the resulting dynamics is shown in Figure [7](#orga61b541).
+The bode plot of the resulting dynamics is shown in Figure [7](#org6f65baa).
 The controlled system has low-pass characteristics, rolling of at the scanner's natural frequency.
 
-<a id="orga61b541"></a>
+<a id="org6f65baa"></a>
 
 {{< figure src="/ox-hugo/schmidt20_bode_plot_feedfoward_example.svg" caption="Figure 7: Bode plot of the feedforward-controlled scanning unit" >}}
 
@@ -628,9 +628,9 @@ The oscillation caused by each individual step are 180 degrees out of phase and
 This method is clearly very different form pole-zero cancellation.
 In the frequency domain, these sampled adaptations to the input create a frequency spectrum with a multiple of notch filters at the harmonic of the frequency where these adaptations are applied.
 
-Applying input-shaping to the triangular scanning signal results in the introduction of a plateau instead of the sharp peak, where the width of the plateau corresponds to half the period of the scanner's resonance as can be seen in Figure [8](#orgfe06e66).
+Applying input-shaping to the triangular scanning signal results in the introduction of a plateau instead of the sharp peak, where the width of the plateau corresponds to half the period of the scanner's resonance as can be seen in Figure [8](#org243dcf9).
 
-<a id="orgfe06e66"></a>
+<a id="org243dcf9"></a>
 
 {{< figure src="/ox-hugo/schmidt20_input_shaping_example.svg" caption="Figure 8: Input-shaping control of the triangular scanning signal in a scanning probe microscope." >}}
 
@@ -658,11 +658,11 @@ The limitations of the actuators and electronics in a controlled motion system a
 Of at least the levels of Jerk and preferable also Snap should be limited.
 The standard method to cope with these limitations involves shaping the input of a mechatronic motion system by means of _trajectory profile generation_ or _path-planning_.
 
-Figure [9](#org68ee0e9) shows a fourth order trajectory profile of a displacement, which means that all derivatives including the fourth derivative are defined in the path planning.
+Figure [9](#org1bf0f60) shows a fourth order trajectory profile of a displacement, which means that all derivatives including the fourth derivative are defined in the path planning.
 A third order trajectory would show a square profile for the jerk indicating an infinite Snap and the round of the acceleration would be gone.
 A second order trajectory would show a square acceleration profile with infinite Jerk and sharp edges on the velocity.
 
-<a id="org68ee0e9"></a>
+<a id="org1bf0f60"></a>
 
 {{< figure src="/ox-hugo/schmidt20_trajectory_profile.svg" caption="Figure 9: Figure caption" >}}
 
@@ -682,12 +682,12 @@ Feedback control is more complex and critical to design than feedforward control
 In general, a feedback controlled motion system is to perform a certain predetermined motion task defined by the reference input \\(r\\), while reducing the effects of other inputs like external vibrations and noise from the electronics.
 All these input signals, whether desired of undesired, are treated by the feedback loop as disturbances and it is the sensitivity of the desired output signal to all input signals that determine the performance of the feedback controller.
 
-<a id="org2c29057"></a>
+<a id="orgdddc940"></a>
 
 {{< figure src="/ox-hugo/schmidt20_feedback_full_simplified.svg" caption="Figure 10: Full and simplified representation of a feedback loop in order to determine the influence of the reference signal and most important disturbance sources on real motion output of the plant \\(x\\), the feedback controller output \\(u\\) and the measured motion output \\(y\\). \\(y\_m = y\\) when the measurement system is set at unity gain and the sensor disturbance is included in the output disturbance." >}}
 
 Several standard sensitivity functions have been defined to quantify the performance of feedback controlled dynamic systems.
-There are derived from a simplified version of the generic feedback loop as shown in Figure [10](#org2c29057).
+There are derived from a simplified version of the generic feedback loop as shown in Figure [10](#orgdddc940).
 The first simplification is made by approximating the measurement system to have a unity-gain transfer function.
 For further simplification the sensor disturbance in the measurement system is included in the output disturbance \\(n\\), thereby defining the output of the system \\(y\\) as the measured output.
 With this simplified model, the transfer functions of the different inputs of the system to three relevant output variables in the loop are written down in a set of equations.
@@ -729,6 +729,8 @@ T(s) = \frac{GC}{1 + GC}
 \end{equation}
 
 represents the ability of the system to follow a given reference position signal.
+For motion systems, the complementary sensitivity function is less important, because using model-based feedforward control for handling known inputs, feedback should only be applied for correction of unknown factors caused by disturbing inputs.
+For that reason the most relevant motion system performance criteria are the Sensitivity \\(S(s)\\) and the Process Sensitivity \\(GS(s)\\).
 
 <a id="table--tab:gang-of-four"></a>
 <div class="table-caption">
@@ -740,9 +742,64 @@ represents the ability of the system to follow a given reference position signal
 |-----------------------------------------------------------------------------------|----------------------------------------------------|
 | \\(\frac{u}{r} = \frac{u}{n} = \frac{C}{1 + GC}\\)                                | \\(\frac{y}{n} = \frac{1}{1 + GC}\\)               |
 
+A common factor in all the sensitivity functions is the _Feedback-loop Transfer Function_:
+
+\begin{equation}
+L(s) = G(s) C\_{fb}(s)
+\end{equation}
+
+The magnitude of the _Feedback-loop response_ \\(L(j\omega)\\) is called the _loop-gain_.
+
+The sensitivity functions were defined as the impact of any of the inputs to the measured output \\(y\\).
+In reality, we would be more interested in errors at the real position \\(x\\).
+We can compute the error \\(e\_{\text{real}} = r - x\\) and find:
+
+\begin{equation}
+e\_{\text{real}} = r - x = \sqrt{(S(s)r)^ 2 + (GS(s)d)^2 + (T(s)n)^2} \quad \text{[m]}
+\end{equation}
+
+This clearly points out that a high control gain \\(C(s)\\) only reduces errors related to the reference and the process disturbance but not the errors included in \\(n\\) that originate in the measurement system, because \\(T(s)\\) will approach unity with a high control gain.
+The quality of the sensor therefore determines the _minimal_ achievable error.
+
 
 #### Stability and Robustness in Feedback Control {#stability-and-robustness-in-feedback-control}
 
+To achieve sufficient robustness against instability in closed-loop feedback control of a motion system, several margins are defined that are applied in the analysis of the transfer function of the feedback loop.
+
+The condition for robustness of closed-loop stability is that the total phase-lag of the **total feedback-loop**, consisting of the feedback controller in series with the mechatronic system, must be less than 180 degrees in the frequency region of the _unity-gain cross-over frequency_.
+
+The Nyquist plot of the feedback loop, like the example shown in Figure [11](#org2768f88), is most appropriate to analyze the robustness on stability of a feedback system.
+It is an analysis tool that shows the frequency response of the **feedback-loop** combining magnitude and phase in one plot.
+In this figure, two graphs are shown, designed for a different purpose.
+The first graph from the left shows margin circles related to the capability of the closed-loop feedback controlled system to follow a reference according to the complementary sensitivity.
+The second graph shows a margin circle related to the capability of the closed-loop feedback controlled system to suppress disturbances according to the sensitivity function.
+
+<a id="org2768f88"></a>
+
+{{< figure src="/ox-hugo/schmidt20_nyquist_plot_stable.svg" caption="Figure 11: Nyquist plot of the feedback-loop response of a stable feedback controlled motion system. Stability is guaranteed as the \\(-1\\) point is kept at the left hand side of the feedback loop repsonse line upon passing with increased frequency, even though the phase-lag is more than 180 degrees at low frequencies." >}}
+
+Three values are shown in Figure [11](#org2768f88) related to the robustness of the closed-loop feedback system:
+
+-   **The gain margin** determines by which factor the feedback loop gain additionally can increase before the closed-loop goes unstable.
+-   **The phase margin** determines how much additional phase-lab at the unity-gain cross-over frequency is acceptable before the closed-loop system becomes unstable.
+-   **The modulus margin** is defined by the closest distance in a Nyquist plot between the graph and the \\(-1\\) point. It determines the highest level of the Sensitivity \\(S(s)\\) in the frequency range where the error is increased by feedback.
+
+The robustness margins determine how much the properties of the system are allowed to change before the system becomes unstable.
+With smaller margins, the system will behave more nervous, less damped.
+Higher margins corresponds to a higher level of damping.
+
+The Nyquist plot has one significant disadvantage as it does not show directly the frequency along the plot.
+For that reason many designers prefer to use the Bode plot.
+
+Fortunately it is also possible to indicate the phase and gain margin in the Bode plot as is shown in Figure [12](#org065335c).
+
+In many not too complicated cases, these two margins are sufficient to tune a feedback motion controller.
+In more complicated control systems, it remains useful to also use the Nyquist plot as it also gives the Modulus margin.
+
+<a id="org065335c"></a>
+
+{{< figure src="/ox-hugo/schmidt20_phase_gain_margin_bode.svg" caption="Figure 12: The gain and phase margin in the Bode plot" >}}
+
 
 ### PID Feedback Control {#pid-feedback-control}
 
@@ -842,12 +899,37 @@ represents the ability of the system to follow a given reference position signal
 
 ### Digital Signal Processing - The Z-Domain {#digital-signal-processing-the-z-domain}
 
+Most modern controllers operate with digital processors in the discrete time domain, also called the _Z-domain_, which allows to create filters by combining scaled sampled data at fixed intervals.
+In many situations it is sufficient to design the system by creating the control filters in continuous-time frequency responses and use a corresponding transformation that translates the filters into their discrete-time counterparts.
+In the transition to a digital and discrete time implementation the phase lag introduced by delay and latency of the sampling, calculation and communication process needs to be considered.
+This is mostly fine when the sample frequency is very high (\\(\gg 10\\) times) in respect to the bandwidth of the controlled system.
+
 
 #### Continuous Time versus Discrete Time {#continuous-time-versus-discrete-time}
 
-<a id="orgffe316d"></a>
+In the frequency-domain a position controller acts as a filter, which attenuates or amplifies the system response and shapes the phase response of the system in certain frequency areas.
+In principle these filters, all representing a differential equation or a transfer function with poles and zeros, can be implemented in analogue electronics or as a digital filter that runs on a digital real-time hardware computing platform.
 
-{{< figure src="/ox-hugo/schmidt20_digital_implementation.svg" caption="Figure 11: Overview of a digital implementation of a feedback controller, emphasising the analog-to-digital and digital-to-analog converters with their required analogue filters" >}}
+The implementation of a filter in analogue electronics, allows the realization of very high bandwidth frequencies at low cost.
+However, analogue controllers have three important disadvantages:
+
+-   its properties can be less consistent over time and also sensitive to changing environmental conditions such as temperature
+-   each filter parameter has to be adjusted in hardware by selecting discrete passive components, which cannot be adapted easily during operation
+-   the noise of the filter and its propagation has to be considered in the design process
+
+The digital implementation of filters overcome these problems as well as allows more complex algorithm such as adaptive control, real-time optimization, nonlinear control and learning control methods.
+
+In Figure [13](#org144efa1) two elements were introduced, the _analogue-to-digital converter_ (ADC) and the _digital-to-analogue converter_ (DAC), which together transfer the signals between the analogue and the digital domain.
+
+<a id="org144efa1"></a>
+
+{{< figure src="/ox-hugo/schmidt20_digital_implementation.svg" caption="Figure 13: Overview of a digital implementation of a feedback controller, emphasising the analog-to-digital and digital-to-analog converters with their required analogue filters" >}}
+
+Anti-aliasing filter is needed at the input of the ADC to limit the frequency range at the input to less than half the sampling frequency, according to the Nyquist-Shannon sampling theorem.
+
+A DAC needs a filter (called a reconstruction of _anti-imaging_ filter) at its analogue output, because the digital data are provided at fixed intervals after which they are kept constant between each sampling moments by means of a zero-order hold element.
+
+The selection of the resolution of ADCs and DACs determines the resolution and achievable precision of the controlled motion system, while the choice of the sampling time determines the maximum frequency that the filter can correctly handles.
 
 
 #### Sampling of Continuous Signals {#sampling-of-continuous-signals}
@@ -855,20 +937,81 @@ represents the ability of the system to follow a given reference position signal
 
 #### Digital Number Representation {#digital-number-representation}
 
-{{< figure src="/ox-hugo/schmidt20_digital_number_representation.svg" >}}
+For the implementation of digital algorithms on digital computing platforms, two representation forms of digital number exist, floating point or fixed point.
+
+Fixed point arithmetic has been favored in the past, because of the less complex DSP processor structure.
+A main drawback is, that the developer must pay attention to truncation, overflow, underflow and round-off errors that occur during mathematical operations.
+Fixed points numbers are equally spaced over the whole range, separated by the gap which is denoted by the least significant bit.
+The two's complement is the most used format for representing positive and negative numbers.
+For representing a fixed point fractional number of two's complement notation, the so called \\(Q\_{m,n}\\) format is often used (see Figure [14](#orgda6fa80)).
+\\(m\\) denotes the number of integer bits and \\(n\\) denotes the number of fractional bits.
+\\(m+n+1=N\\) bits are necessary to store a signed \\(Q\_{m,n}\\) number.
+If the binary representation is given, the decimal value can be calculated to:
+
+\begin{equation}
+x = \frac{1}{2^n} \left( -2^{N-1 }b\_{N-1} + \sum\_{i=0}^{N-2} 2^i b\_i \right)
+\end{equation}
+
+where \\(b\\) indicate the bit position, starting with \\(b\_0\\) from the right in Figure [14](#orgda6fa80).
+
+<a id="orgda6fa80"></a>
+
+{{< figure src="/ox-hugo/schmidt20_digital_number_representation.svg" caption="Figure 14: Example of a \\(Q\_{m.n}\\) fixed point number representation and a single precision floating point number" >}}
+
+Floating point arithmetic has a higher dynamic range than fixed point arithmetic, given by the largest and smallest number that can be represented, has a higher precision due to the smaller gaps between adjacent numbers, less quantization noise, and it is easier to handle in terms of programming.
+A floating point number is represented by a multiplication of a _mantissa_ \\(M\\) with a _base_ \\(b\\) to the power of the _exponent_ \\(q\\):
+
+\begin{equation}
+x = -1^i M b^q
+\end{equation}
+
+A common used standard for representing floating point numbers defines basic formats, single precision (32 bit wide format) and double precision (64 bit wide format).
+The single precision format with a base of 2 is chosen as an example.
+The decimal value can be calculated by:
+
+\begin{equation}
+x = -1^i M 2^{E-127}
+\end{equation}
+
+The term \\(E\\) in the exponent is stored as a positive number ranging from \\(0 \le E < 256\\) with 8 bits.
+An offset of \\(-127\\) is added in order to allow very small to very large numbers.
+The decimal value is normalized, meaning that only one nonzero digit is noted at the left of the decimal point.
+The storage register is divided into three groups, as shown in Figure [14](#orgda6fa80).
+1 bit represents the sign, the exponent term \\(E\\) is represented by 8 bits, and the mantissa is stored in 23 bits.
 
 
 #### Digital Filter Theory {#digital-filter-theory}
 
+<a id="org9922abe"></a>
 
-##### Z-Transform and Difference Equations {#z-transform-and-difference-equations}
+{{< figure src="/ox-hugo/schmidt20_s_z_planes.svg" caption="Figure 15: Corresponding points and area in s and z planes" >}}
 
 
 #### Finite Impulse Response (FIR) Filter {#finite-impulse-response--fir--filter}
 
+<a id="org4d4c293"></a>
+
+{{< figure src="/ox-hugo/schmidt20_transversal_filter_structure.svg" caption="Figure 16: Transversal filter structure of a FIR filter. The term \\(z^{-1}\\) each represent a sampling period which means that \\(b\_0\\) is the gain of the last sample, \\(b\_1\\) is the gain of the precious sample etcetera." >}}
+
+<a id="orgbd6bb94"></a>
+
+{{< figure src="/ox-hugo/schmidt20_optimized_fir_filter_structure.svg" caption="Figure 17: Optimized FIR filter structure with symmetric filter coefficients" >}}
+
+<a id="org670458c"></a>
+
+{{< figure src="/ox-hugo/schmidt20_dir_filter_cascaded_sos.svg" caption="Figure 18: Higher-order FIR filter realization with cascade SOS filter structures" >}}
+
 
 #### Infinite Impulse Response (IIR) Filter {#infinite-impulse-response--iir--filter}
 
+<a id="orgdfc31d4"></a>
+
+{{< figure src="/ox-hugo/schmidt20_irr_structure.svg" caption="Figure 19: (a:) IIR structure in DF-1 realization and (b:) IIR structure in DF-2 realization" >}}
+
+<a id="orgb617097"></a>
+
+{{< figure src="/ox-hugo/schmidt20_irr_sos_structure.svg" caption="Figure 20: IIR SOS structure in DF-2 realization" >}}
+
 
 #### Converting Continuous to Discrete-Time Filters {#converting-continuous-to-discrete-time-filters}
 
@@ -1847,4 +1990,4 @@ Motion control is essential for Precision Mechatronic Systems and consists of tw
 
 ## Bibliography {#bibliography}
 
-<a id="orgb603526"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2020. _The Design of High Performance Mechatronics - Third Revised Edition_. Ios Press.
+<a id="org343c570"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2020. _The Design of High Performance Mechatronics - Third Revised Edition_. Ios Press.