Digital Control Systems Chapter 3 Solution Fadali

Digital Control System

In digital control systems, the variable resistance is often translated to a temperature signal by using a software look-up table that maps the temperature corresponding to the measured resistance, or by solving an exponential equation using exponents and coefficients provided by the thermistor manufacturer.

From: Fundamentals of HVAC Control Systems , 2008

Robustness of feedback control systems

Mark A. Haidekker , in Linear Feedback Controls (Second Edition), 2020

13.4 Robustness of digital control systems

Digital control systems are subject to the same effects of coefficient variability and coefficient uncertainty as time-continuous controllers. Two additional effects deserve consideration for the design of robust digital control systems:

•

Aliasing: Noise was identified as a broadband signal, which implies that the sensor noise, unless filtered, exceeds the Nyquist frequency. The filtering requirement becomes more stringent in digital control systems.

•

Control period T: In Chapter 4, T was identified generally as the sampling period. Any digital control system requires computation time to prepare the value for the corrective action and apply it to the output. More specifically, therefore T is the period between two complete control actions (sensor readout, computation, update of the output). The control period is fundamental to the transfer function of a digital control system, and any variability of T affects the closed-loop transfer function.

Noise and its influence on a time-continuous control system was discussed in Section 13.3. In the example in Fig. 13.7, a first-order lowpass filter was proposed to reduce the spectral energy of the sensor noise. A first-order lowpass attenuates frequency components by a factor of 10 for a 10-fold increase of frequency. In the specific example, a lowpass of $f_c=8$   Hz ( ${\omega }_c=50{\mathrm{\text{s}}}^{-1}$ ) was used. If the sampling period of a hypothetical control system is $T=6.25$   ms ( $f_s=160$   Hz), then the noise magnitude at the Nyquist frequency is merely attenuated to 10% of the average noise spectral energy, and the entire spectral energy above $f_N=f_s/2$ is mirrored (aliased) into the frequency band below $f_N$ . The aliased noise spectrum adds to the passband noise spectrum and leads to a deterioration of SNR at the controller input. For this example with a factor of ten between filter cutoff frequency and Nyquist frequency, the signal with frequency components between $f_c$ and $f_N$ is more than two times larger than passband component with frequencies below $f_c$ , although this part of the spectrum is not subject to aliasing. The remaining spectral components, under the assumption of reasonable op-amp cutoffs, are roughly six times larger than the passband component, and this part of the spectrum is subject to aliasing.

Since the aliased spectrum appears to the digital control system as a low-frequency spectrum, digital filtering is ineffective. Consequently, digital systems require stronger analog filtering than the corresponding time-continuous systems. Sometimes it is possible to raise the sampling frequency. If the sampling period in the preceding example can be reduced, for example, from $T=6.25$   ms to $T=625$   μs, the spectral component that is subject to aliasing is roughly 50% lower than that with the longer sampling period. To reduce the control period, however, faster analog-to-digital conversion and faster computation are needed.

Higher-order analog filters are effective in reducing frequencies above the Nyquist frequency. Even a second-order filter in place of $G_s(s)$ (Fig. 13.6) reduces the spectral energy between $f_c$ and $f_N$ to about the same level as the energy below $f_c$ , and the energy above $f_N$ to a fraction around 10% of the energy below $f_c$ . The effect of a third-order filter is even more dramatic, and such higher-order filters allow more design freedom to place the cutoff frequency closer to the sampling frequency. However, any filter becomes part of the transfer function, and the filter cutoff frequency is ideally placed far higher than the system response so that the filter poles can be considered nondominant. This requirement again calls for an appropriately fast control system to still effectively reduce aliasing.

The control period T has no correspondence in time-continuous systems, and it is an integral part of time-discrete transfer functions. Consider, for example, the transfer function of a time-discrete PID controller,

(13.24) $H_{\mathrm{PID}}(z)=\frac{(k_p+\frac{k_{I}T}{2}+\frac{k_D}{T})z+(\frac{k_{I}T}{2}-\frac{k_D}{T})}{z-k_I}.$

The controller contributes one open-loop pole ( $z=+k_I$ ) and a zero whose location in the z-plane depends on T. Any Laplace-domain transfer function requires introducing terms with T or $e^{-aT}$ as shown in Section 4.3. For example, a simple first-order system with an exponentially decaying impulse response relates to its z-transform through

(13.25) $\mathcal{Z}\left\{\frac{a}{s+a}\right\}=\frac{az}{z-e^{-aT}}.$

The underlying assumption for the z-domain model is a fixed sampling rate T. If the first-order system in Eq. (13.25) has a decay constant $a=2.5$   s, and the sampling rate is $T=50$   ms, the z-domain transfer function is $2.5z/(z-0.88)$ . The model with the assumed fixed sampling rate of $T=50$   ms would show the pole at $z=0.88$ , but if the controller exhibits variability of T, then the actual pole location would fluctuate as well. Extreme cases where a minor jitter of the sampling rate would move a pole in and out of the unit circle can easily be found.

Many software operations, notably floating-point arithmetic, have data-dependent execution times. A simple loop that continuously repeats reading sensor data, computing a control action, and writing the control action to an output can exhibit significant jitter of the loop execution time. This behavior is particularly detrimental when the control system has a tightly tuned dynamic response. Fortunately, this situation can be easily avoided. Most modern microcontrollers have independent hardware timers, which can be used to control the sampling rate T. The control action would be executed only once when the timer elapses, and the timer period would be selected large enough so that the execution time of the control loop is always smaller than T.

Nondedicated computers (e.g., desktop computers, but also card-sized PCs, such as the Raspberry Pi, which are popular for control applications) run the control software under the management of an operating system. A defined response time cannot normally be guaranteed. High-end data acquisition hardware often has built-in timers that enable the control software to circumvent variability of the operating system response time. In addition, some operating systems, such as Linux-based flavors, have optional real-time kernels that guarantee a maximum response time. Typically, the aforementioned card-size PCs already have real-time kernels in their operating software.

For both microcontroller and microcomputer, it is the programmer's responsibility to ensure accurate timing of the control action.

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Instrumentation and control

In The Efficient Use of Energy (Second Edition), 1982

Digital System Applications

Digital-control systems, by definition, depend on translation of primary measurements into a digital code that enables an increment of adjustment to be made; the magnitude of the adjustment is determined by:

1.

The resolution or minimum 'bit level' of the primary data, and

2.

The corresponding response or step change of the output actuator.

Digital-control systems may range in complexity from the contact-operated motor described above to self-optimizing systems. In the latter, calculations are repeatedly made on the primary physical variables associated with a process, and corrections thereby computed to achieve the criteria defining a pre-programmed 'optimum' result.

Having due regard for the 'analogue' nature of the majority of physical process variables and the exponential nature of the control equations to be satisfied for any control system to stabilize, it is seldom practicable to rely on a system that is literally 100% digital, although it may be expedient for certain elements (particularly those involving calculations or requiring adjustment to some preset datum) to be instrumented digitally. This point is discussed in more detail in the context of digital control with analogue back-up facilities.

It is apposite to review digital-control techniques by first reviewing analogue-control fundamentals to indicate the problems associated with the substitution of digital control terms, and to follow with examples of digital-control systems developed from step-by-step replacement of analogue-control elements by their equivalent digital counterparts. It will be shown that within certain limitations, the datum-holding capability of a typical control system may be improved by one or more orders of accuracy by the addition of digital measuring and control techniques.

The first point to note is that by the very nature of the time-dependent variables in any control problem, digital techniques (i.e. the use of incremental or discontinuous measuring and control processes) can only be effective if the time constant of the system is long compared with the digital sampling time. For this reason the use of digital techniques tends to be confined to the calculation of proportional-control signals or offset or integral terms.

A second point is that because any system involving proportional corrections is basically an analogue system, then digital measurement or control signals have to be converted to analogue form sooner or later and invariably result in a hybrid solution. As discussed previously, the increased resolution obtainable from a digital transducer, compared with an analogue solution, coupled with the benefit of improved repeatability of digital storage for datum set points, enables the accuracy of a conventional analogue control loop to be considerably enhanced by the addition of digital reference systems.

Digital controllers or controller sub-systems are generally based on three main elements that can be identified regardless of the system application. These are:

1.

A uni-directional or bi-directional (reversible) counter having as many decades as is necessary to accommodate the span of control required.

2.

A time-base reference for gating the counter; for example, in one of the several modes of operation detailed below.

3.

A digital storage reference (such as edge switches, diode pinboard, core store, magnetic or punched tape) for simulating a numerical datum to which the counter may be referred. Either method enables the number corresponding to the required datum to be represented in parallel-coded binary form in whatever code is preferred.

Associated with the above three main elements will be the necessary interfaces between signal input and control output, such as a digital-to-analogue (d-a) converter, and system logic such as coincidence detection, gating and data transfer.

The form of counter used will depend on the application, but consideration of a few examples of the alternative ways of using counters might now be appropriate. The simplest form of counter is an asynchronous or 'ripple through' uni-direc-tional counter that will continue to totalize a train of pulses presented at the input until the train of pulses is discontinued or the counter input is clamped in response to some external control signal such as a timing signal. The counter stages may be preset to some required limiting value and coincidence logic added to enable this value to be detected when reached or exceeded. This arrangement is popular for batch counting and presetting a datum, in certain counting and presetting indicating-and-control systems, such as numerical coincidence, totalizing and off-limit detection in alarm supervisory systems.

As an alternative to the use of external preset data that may be adjustable, the counter may simply be required to totalize and clear to zero when a full-scale reading is obtained. One useful feature of the latter mode of operation is in representing a 'suppressed nominal' datum; the counter is programmed to count a signal frequency input against a time base and arranged so that the output of the counter is ignored until after it has filled and recycled to count for the second time; the second count corresponds to the error from the required nominal represented by full scale of the counter.

A number obtained after recycling corresponds to a positive difference signal, and if the counter fails to total, i.e. it does not recycle, the complement is taken to obtain a negative difference signal. The nominal may be represented by 3 decades, i.e. 999, and a count of 1000 units resets the counter to zero. Thus, in a finite gating time of, say, 1 second, a frequency of 1020 Hz will show as a positive error of 20 bits and a frequency of, say, 980 Hz will show as a negative error of 20 bits.

A bi-directional counter is more expensive than a uni-directional counter but may be preferred for certain applications where a continuous or semi-continuous difference signal is required. There are two principal modes of operation that enable a reversible counter to be used to obtain a difference or error signal, or a signal that may be used as the integral of an error. In the first, the counter is arranged to count up in one direction from one input source for a fixed time interval, then count down in the reverse direction from the second input source in an identical time interval, the resulting number being the difference between the two.

In the second mode of operation the counter is used as an integrator. Both inputs are presented simultaneously so that the counter is adding or subtracting, depending on the respective frequencies of the inputs. Where there is a likelihood of the two pulses arriving simultaneously, as indeed they would if the system eventually reached perfect equilibrium with no frequency difference, a special input circuit eliminates pulses that are coincident to prevent the counter from locking up. The design of such a coincident pulse-canceller circuit is critical however, because the tolerance of pulse widths determines the ultimate resolution of the difference count; whereas the former mode of operation, in which the counter is cycled alternately in one direction and then the other, produces an absolute difference count.

When digital control is added to improve the accuracy of the datum position, the simplest form of control (i.e. integral term) may be obtained by totalizing in a reversible counter; but for applications requiring a position reference to be obtained from a frequency source such as a digital tachometer, the controller will require to be related to absolute time and a crystal timebase or similar time reference will be required, depending on the accuracy required. The accuracy of a crystal may be typically a few parts in a million which may be improved by temperature control. The accuracy of an electromechanical time reference such as tuning fork may be one order smaller but adequate for many applications. The electromechanical solution generally tends to be cheaper because the natural frequency of electromechanical devices is relatively low, and less division is required to produce suitably spaced timing signals (e.g. of the order of 0.1 to 1 s for tachometry).

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Digital Controllers

David M. Auslander , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

III.C Simulation of Digital Control Systems

Digital control systems are usually have a mixture of continuous-time and discrete-time behavior. The control object exists continuously in time (thus the differential equation representation) while the controller is active only for brief periods of time. The simulation software can recognize this situation by embedding a mini-event-manager into the main simulation loop.

The first part of the simulation program sets parameter values, initializes, states variables, etc. The working part of the program is the simulation loop. The main simulation loop has two parts: the event manager which controls execution of the discrete functions (those that the control computer would be doing) and the ODE solver for the continuous part of the system (normally the control object).

The event manager generally handles execution of one or more control (feedback) loops, data logging, external events that might affect operation, and when to terminate the simulation. On each pass through the event manager, code associated with all relevant events is executed and each of those events sets the time at which it will again need attention. The ODE simulation section then solves the system equations for the control object up to the time of the next event. Any ODE algorithm can be used; the sample software uses both fixed step size Euler solvers coded inline with the event loop and MATLAB solvers that use a separate file for the differential equation right-hand sides.

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Mathematical Background and State of the Art

Soliman Abdel-hady Soliman Ph.D. , Ahmad M. Al-Kandari Ph.D. , in Electrical Load Forecasting, 2010

1.12 Difference Equations

In modern control systems, digital processors are used to perform the task of control, so it is important to establish equations that relate digital and discrete time signals. Whereas differential equations are used to represent systems with analog signals, difference equations are used for systems with discrete data. It is easier to use difference equations than differential equations on a digital computer, and these equations are generally easier to solve.

As an example of discrete approximation, we can use a forward difference process to approximate the derivative of a function at a given instant; that is,

(1.57) ${\frac{dy\left(t\right)}{dt}|}_{t=kT}=\frac{y\left[\left(k+1\right)T\right]-y\left(kT\right)}{T}$

where T is chosen to be some small value that will lead to a good approximation. When we use equation (1.57), a first-order differential equation in the form

(1.58) $\frac{dy\left(t\right)}{dt}+ay\left(t\right)=f\left(t\right)$

can be transformed into

(1.59) $\frac{y\left[\right(k+1\left)T\right]-y\left(kT\right)}{T}+ay\left(kT\right)=f\left(kT\right)$

(1.60) $y\left[\right(k+1\left)T\right]=[1+aT]y\left(kT\right)+Tf\left(kT\right)$

The preceding equation is called a first-order difference equation, and it gives the value of y one step ahead as a function of y and f one step back.

In general, a linear nth-order difference equation with constant coefficients can be written as

(1.61) $a_{n+1}y(k+n)+a_{n}y(k+n)+\dots +a_{2}y(k+1)+a_{1}y\left(k\right)=f\left(k\right)$

where y(i), i = k, k + 1,…,k + n denote the values of the discrete dependent variable y at the ith instant if the independent variable is discrete time. In general, the independent variable can represent any real physical quantity.

In a manner similar to analog systems, it is convenient to use a set of first-order difference equations (state equations) to represent a high-order difference equation. For the difference equation in equation (1.61), if we let

(1.62) $\begin{array}{l}{x}_1\left(k\right)=y\left(k\right)\\ x_2\left(k\right)=x_1(k+1)=y(k+1)\\ ⋮\\ x_{n-1}\left(k\right)=x_{n-2}(k+1)=y(k+n-2)\end{array}$

then the equation is written as

(1.63) $x_n(k+1)=-\frac{a_1}{a_{n+1}}{x}_1\left(k\right)-\frac{a_2}{a_{n+1}}{x}_2\left(k\right)-\cdots -\frac{a_n}{a_{n+1}}{x}_n\left(k\right)+\frac{1}{a_{n+1}}f\left(k\right)$

The first n − 1 state equations are taken directly from equation (1.62), and the final one is given by equation (1.63). Writing these n first-order difference state equations in vector-matrix form, we have

(1.64) $x\left[\right(k+1\left)T\right]=Ax\left(kT\right)+Bu\left(kT\right)$

where

(1.65) $x\left(kT\right)=\left[\begin{array}{l}{x}_1\left(kT\right)\\ x_2\left(kT\right)\\ ⋮\\ x_n\left(kT\right)\end{array}\right]$

is the n × 1 state vector, and

(1.66) $A=\left[\begin{array}{cccccc}0& 1& 0& 0& \dots & 0\\ 0& 0& 1& 0& \dots & 0\\ 0& 0& 0& 1& \dots & 0\\ 0& 0& 0& 0& \dots & 1\\ -\frac{a_1}{a_{n+1}}& -\frac{a_2}{a_{n+1}}& -\frac{a_3}{a_{n+1}}& \cdot & \dots & -\frac{a_n}{a_{n+1}}\end{array}\right]$

(1.67) $B=\left[\begin{array}{c}0\\ 0\\ 0\\ ⋮\\ 0\\ \frac{1}{a_{n+1}}\end{array}\right]$

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Digital Control System Design

M. Sami Fadali , Antonio Visioli , in Digital Control Engineering, 2009

Publisher Summary

To design a digital control system, a z-domain transfer function or difference equation model of the controller that meets given design specifications, is seeked. The controller model can be obtained from the model of an analog controller that meets the same design specifications. Alternatively, the digital controller can be designed in the z-domain using procedures that are almost identical to s-domain analog controller design. This chapter elucidates both these approaches. It begins with an explanation of the z-domain root locus and describes the method of sketching the z-domain root locus for a digital control system or obtaining it using MATLAB. This approach is based on the relation between any time function and its s-domain poles and zeros. If the time function is sampled and the resulting sequence is z-transformed, the z-transform contains information about the transformed time sequence and the original time function. The poles of the z-domain function can therefore be used to characterize the sequence, and possibly the sampled continuous time function, without inverse z-transformation. However, this latter characterization is generally more difficult than characterization based on the s-domain functions.

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High switching frequency three-phase current-source converters and their control

Dapeng Lu , Frede Blaabjerg , in Control of Power Electronic Converters and Systems, 2021

12.3.1.4 Time delay effect on virtual impedance

Due to the digital control system, a finite time delay, e.g., $G_{del}=e^{-1.5s{T}_s}$ , is introduced, which affects the performance of the active damping through varying the implemented virtual impedance [21,22]. Firstly, considering pure virtual resistances in Eqs. (12.15) and (12.18), by involving the time delay effect, they change to

(12.23) $Z_{adi}=R_{adi}{e}^{1.5s{T}_s}\Rightarrow Z_{adi}\left(j\omega \right)=R_{adi}\cos \left(1.5\omega T_s\right)+j{R}_{adi}\sin \left(1.5\omega T_s\right),i=1,2$

It can be obtained that real terms are positive in the frequency range of (0, 1/6ω _s ), and turn to be negative in the frequency range of (1/6ω _s , 1/2ω _s ). Regarding to the imaginary terms, they are positive from 0 to 1/3ω _s , while they become negative from 1/3ω _s to 1/2ω _s . Positive real terms provide the damping effect to the LC resonance. However, negative real terms will lead to open-loop right half plane (RHP) poles to the single-loop DC-link current control, which counteract with the fast dynamic response and even result in instability. Therefore, positive real terms should preferable be guaranteed. In the case of imaginary terms, the positive brings inductance and the negative brings capacitance behaviors virtually, which shift the resonance frequency [22].

According to the stability analysis in 12.2.2, active damping is essential when the LC resonance frequency is in the high frequency range, which may be in the range of negative real terms. To solve this, a simple method is to invert the polarity of the feedback gain, which thus could provide positive real terms at the resonance frequency.

Next, as with the cases of virtual impedance in Eqs. (12.13), (12.17), and (12.20), it is still required to have a positive Re{Z _adi (jω)}, i   = 1, 2, 3 in order to avoid a nonminimal-phase property and potential instability. After considering the time delay effect, real parts of the virtual impedance in Eqs. (12.13), (12.17), and (12.20) become Eqs. (12.24), (12.25), and (12.26), respectively.

(12.24) $\mathrm{Re}\left\{Z_{ad1}\left(j\omega \right)\right\}=\frac{\omega L_{dc}{I}_{dc}}{k_{ad1}[{\left(D_d{V}_{cd}\right)}^2+{\left(\omega L_{dc}{I}_{dc}\right)}^2]}\left[\underset{{\tau }_1}{\underbrace{\omega L_{dc}\cos \left(1.5\omega T_s\right)-D_d{V}_{cd}\sin \left(1.5\omega T_s\right)}}\right]$

(12.25) $\mathrm{Re}\left\{Z_{ad2}\left(j\omega \right)\right\}=\frac{L_{dc}{I}_{dc}}{k_{ad2}C[{\left(D_d{V}_{cd}\right)}^2+{\left(\omega L_{dc}{I}_{dc}\right)}^2]}\underset{{\tau }_2}{\underbrace{[D_d{V}_{cd}\cos \left(1.5\omega T_s\right)+\omega \sin \left(1.5\omega T_s\right)]}}$

(12.26) $\mathrm{Re}\left\{Z_{ad3}\left(j\omega \right)\right\}=\frac{{\omega }^{2}L{L}_{dc}^2}{k_{ad3}[{\left(D_d{V}_{cd}\right)}^2+{\left(\omega L_{dc}{I}_{dc}\right)}^2]}\underset{{\tau }_3}{\underbrace{[-D_d{V}_{cd}\cos \left(1.5\omega T_s\right)-\omega L_{dc}{I}_{dc}\sin \left(1.5\omega T_s\right)]}}$

It can be seen that τ _i , i  =   1, 2, 3, determines the polarity of Re{Z _adi (jω)}, i   = 1, 2, 3, which are plotted in Fig. 12.23. Compared to 1/6ω _s in the pure resistance cases, the ranges of positive real terms vary in the proportional feedbacks, which are (0.35ω _s , 1/2ω _s ), (0, 0.33ω _s ), and (0.26ω _s , 1/2ω _s ) in capacitor-voltage, capacitor-current, and inductor-current feedback, respectively. Moreover, the boundaries depend on the steady-state values and DC-link inductor of the CSC, which increases the complexity of the active damping design.

Figure 12.23. Critical frequency of Re{Z _adi (jω)} (V _g   =   173   V, L _dc   =   3   mH, I _dc   =   10A, V _dc   =   100   V).

Thus, based on involving the DC-link dynamics and the simplified model, it is possible to achieve active damping with proportional feedbacks of capacitor-voltage, capacitor-current, and inductor-current in the perspective of controller design, which becomes more flexible in practical applications. Furthermore, by employing a high-pass filter in the capacitor-voltage feedback or a low-pass filter in the capacitor-current feedback, pure virtual resistance can be obtained, which is also simple to be designed including the time delay effect. The solution to maintain a positive real term is to change the polarity of the feedback gain. The design principle of the high-pass filter and the low-pass filter are presented, which depends on the DC-link current dynamics. Lastly, it is noted that by using the inductor-current feedback, which is also an inner-control loop of regulating the inductor-current, performances of the input current can be improved, e.g., implementing harmonic suppression by adding resonant controllers to the feedback loop.

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Digital Controls Specification

The reading text for this course was originally written by, ... Robert McDowall P. Eng. , in Fundamentals of HVAC Control Systems, 2008

12.1 Benefits and Challenges of DDC

DDC systems can be extremely sophisticated and perform control logic that was quite impossible with any previous system. Unfortunately, many owners do not have operating and maintenance staff that are trained to understand and utilize the available potential of DDC. In addition, owner's staff often do not understand even the simplest ways of monitoring system energy performance. As a result, the anticipated control performance may not be met and the energy consumption may be much higher than expected or is necessary.

To deal as effectively as possible with this issue, a designer must start a project by finding out the owner's business motivation, and likely operation and maintenance abilities. The initial owner may be a developer who is going to sell the building and is not particularly concerned about performance or energy costs. At the other extreme are the owners who are building a performance and energy showpiece for their head office. In the first case, a simple system with little sophistication will likely suit the situation. In the second case it will be valuable to work up the project objectives with the owner, the architect, and electrical designer so that there is a common, agreed set of project objectives and strategies.

This cooperative approach will help use a systems approach to design rather than the (oversimplified) situation where the architect designs the building and then mechanical and electrical designers prepare their design. By systems design we mean considering the building as a whole system and designing the architecture, mechanical and electrical, to be efficient rather than considering each component separately. A mechanical example for a cooler climate is putting funds into high performance windows and omitting perimeter heating. This example does not produce a more complex system for operation and maintenance.

A combined mechanical and electrical example is using occupancy sensors to switch off lights and to provide a signal to the HVAC system to reduce outside air and allow temperatures change to the unoccupied limits. Although this approach has the potential to reduce lighting and HVAC energy consumption, it also raises the complexity of the HVAC DDC controls. Is the owner committed to providing the resources to run such a system effectively?

When first discussing the DDC system with an owner it is useful to have a clear understanding of the benefits and challenges of DDC. The following set of nine benefits is taken from the ASHRAE GPC-13-2007 Guideline to Specifying DDC Systems, with additional comments.

1.

DDC systems can reduce energy costs by enabling mechanical systems to operate at peak efficiency. Equipment can be scheduled to run only when required and therefore generate only the required capacity at any time. Additional savings are possible if the DDC system is used for more sophisticated purposes than timeclocks or conventional controls. If the DDC system simply duplicates the function of these devices, there may not be a significant reduction in energy consumption.

"Operate at peak efficiency" what does this mean? How will the client know what the initial efficiency is and what it is in a month, a year, a decade? Efficiency is generally defined as useful output divided by total inputs. In HVAC systems "useful output" is keeping a process or people in the required environmental conditions. Unfortunately, we cannot measure comfort and are limited to measuring cooling output, which is not an entirely satisfactory metric. We will consider the efficiency of a chilled water plant in a later section and how that may be monitored.

When designing a system think about "What has to be provided when?" In addition to time scheduling, there is the issue of "How much must be provided?" Simple examples include adjusting the outside air volume to match actual occupancy needs and adjusting duct static pressures down so as to provide only the required airflow.

Even scheduled running hours can often be significantly reduced if occupants can turn on the system for their area for a limited time (2 h, for example) when they are present earlier, or later, than the core operating hours.

Every watt of lighting in hot weather adds to the cooling load, so cooperating with the lighting designer to minimize lighting loads and lighting on time also reduce cooling loads. People are generally poor at turning off unnecessary lighting, so automating switch-off can produce lighting power savings of 50% or more.

2.

DDC systems have extensive functionality that permit the technology to be used in diverse applications, such as commercial HVAC, surgical suites, and laboratory clean rooms. For example, they can be used to measure the amount of air delivered to each area, compare this to the ventilation needs of the building, and then vary the amount of outdoor air introduced to meet the ventilation requirements of ANSI/ASHRAE Standard 62-2007. This level of control is not practical with pneumatic or electronic controls.

The level of additional functionality available through the use of DDC often requires a more carefully designed HVAC system and better trained staff to effectively operate it. Be very careful not to over design for the staffing capabilities, and to ensure that the control logic is clear to the operating staff.

Note that there is considerable scope for automatic fault detection in DDC systems but it is not for free. Manufacturers and independent organizations are now offering fault detection programs and independent data analysis tools that will become more readily available and easier to implement in the future. Software can be provided on the system or the system can be interrogated over the internet and the analysis and reports generated at the office of the service provider.

3.

A DDC system that controls HVAC systems in commercial, institutional, and multi-family residential buildings will provide tighter control over the building systems. This means that temperatures can be controlled more accurately, and system abnormalities can be identified and corrected before they become serious (e.g., equipment failure or dealing with occupant complaints).

A DDC system cannot overcome system design problems. Under capacity, poor airflow in occupied zones, and an inability to remove moisture without overcooling are design flaws not control flaws. No additional DDC sophistication will resolve these issues. Be very careful to avoid blaming the DDC controls until it is clear that the system has the capacity to perform. This will avoid wasting considerable resources, making the DDC system seem unsatisfactory, and generally frustrating all parties.

In addition to providing tighter control, DDC can also be used to provide occupant control not practical with other systems. For example, providing occupant adjustable thermostats is a real challenge in many buildings as the occupants make extreme changes. With a DDC system, the thermostat limits can be individually set with a narrow band, say 73–78°F in summer. This prevents anyone turning the thermostat down to 65°F or even lower and having the cooling running flat out to produce a lower-than-necessary temperature and excessive energy consumption.

Systems with internet access can be set up so that occupants can adjust, within limits, the temperature in their space using their own PC and standard internet browser. This may be considered as a valuable feature to the owner of a high end office tower with demanding tenants, or as a complete waste of money by another owner.

4.

In addition to commercial HVAC control, DDC can be used to control or monitor elevators and other building systems including fire alarm, security, and lighting. This enhances the ability of maintenance staff/building operators to monitor these systems.

Note that the speed of response and circuit monitoring usually requires specialist DDC equipment to control non-HVAC systems. Thus, the controller running an HVAC plant may well be a different controller than the controller running the security system in that area. Both controllers may use the same data and communication protocols on the same network and appear as the same for the operator at their workstation.

More and more frequently the security, fire, and other systems provide inputs to the DDC system, even if they are not integral with the building HVAC system.

Several HVAC controls manufacturers are now providing lighting and access control. Note that open bidding becomes more difficult with increasing integration of disciplines, and requirements for all areas must be very clearly identified for getting bids. This integration is moving towards a Cybernetic Building System (CBS) which involves intelligent control, operation, and reporting of performance and problems. The capability exists with DDC systems to include self-commissioning. Examples include controllers which retune parameters as conditions change. A useful example is on a large air-handling unit, altering PID loop parameters when changing from controlling the heating coil in winter to cooling coil in summer.

5.

DDC allows the user to perform intricate scheduling and collect alarms and trend data for troubleshooting problems. The alarm and trend features permit the operator to learn the heartbeat of the system. Observing how a mechanical system performs under different load conditions allows the programming to be fine-tuned and permits the operator to anticipate problems. This allows the occupants' comfort concerns to be dealt with promptly before the problems become serious.

The ability to collect trend data is particularly valuable both in setting up the system to operate but also in ongoing checking on performance. For ongoing checking, some standard trends and reports should be established and included in the installation contract requirements.

6.

Many DDC systems can provide programming and graphics that allow the system to serve as the building documentation for the operator. This is valuable since paper copies of the system documentation are often misplaced.

The graphic display of the systems with current operating data is a major advantage of DDC. On larger systems it is very important that the hierarchy of displays and their format makes it easy for the staff to navigate quickly and reliably to what they want displayed.

7.

A DDC system requires significantly less maintenance than pneumatic controls. Pneumatic controls need regular maintenance and must be periodically recalibrated. The use of DDC results in lower preventive maintenance costs due to calibration and also lower repair costs for replacement of pneumatic or electromechanical devices that degrade over time. All systems require some maintenance. Sensors such as relative humidity and pressure sensors require regular calibration regardless of what type of system they serve.

The actuators and mechanical parts of the system, valves and dampers, will still need maintenance and will still fail. Automatic detection of failure is particularly valuable for these components.

8.

DDC systems reduce labor costs through remote monitoring and troubleshooting. Paying a technician to drive to the building to deal with every problem can be minimized with DDC. In many cases, on-site operations can be eliminated or reduced to a single shift. Problems with the building are called out to a central monitoring service or to a technician with a pager, and often may be rectified from remote locations by modem.

External monitoring and maintenance can be contracted to an external organization. The monitoring can be done from anywhere in the world where internet service is available, although the hands-on maintenance will need to be from a local provider.

9.

DDC systems are programmable devices. As the needs of a building change, the system can be reprogrammed to meet the new requirements.

This is both an advantage and, unfortunately, a disadvantage. Unless the operating staff really understand and agree with the control processes they can, and do, modify things. Often the result is very poor building performance.

One of the advantages of having the system formally commissioned (having a separate commissioning agent test and verify the correct installation and performance of the system) is that there is a clear definition of how the system was operating when installed. This knowledge makes later checking on performance much easier as one has verified initial operation.

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Practical Issues

M. Sami Fadali , Antonio Visioli , in Digital Control Engineering, 2009

Publisher Summary

The designer of a digital control system must be mindful of the fact that the control algorithm is implemented as a software program that forms part of the control loop. Successful practical implementation of digital controllers requires careful attention to several hardware and software requirements. During the design phase, designers make several simplifying assumptions that affect the implemented controller. They usually assume uniform sampling with negligible delay due to the computation of the control variable. Thus, they assume no delay between the sampling instant and the instant at which the computed control value is applied to the actuator. This chapter discusses the most important of these requirements and their influence on controller design. It then analyzes the choice of the sampling frequency in more detail in the presence of antialiasing filters and the effects of quantization, rounding, and truncation errors. In particular, it examines the effective implementation of a proportional–integral–derivative (PID) controller. Finally, it examines changing the sampling rate during control operation as well as output sampling at a slower rate than that of the controller.

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Sensors and Auxiliary Devices

The reading text for this course was originally written by, ... Robert McDowall P. Eng. , in Fundamentals of HVAC Control Systems, 2008

Electrical Resistance

Modern analog electronic and digital control systems generally rely on devices that resistance changes with temperature. Listed roughly in the order of commonality and popularity, these include thermistors, resistance temperature detectors (RTDs), and integrated circuit temperature sensors.

Thermistors are semiconductor compounds ( Figure 4-7 ), that exhibit a large change in resistance, with changes in temperature usually decreasing as the temperature increases. The Y-axis of Figure 4-7 is the ratio of resistance compared to the resistance at 77°F. The characteristic resistance-temperature curve is nonlinear. The current, passed through the sensor to establish resistance, heats the sensor offsetting the reading to some extent (called self-heating). In electronic applications, conditioning circuits are provided in the transmitter to create a linear signal from the resistance change. In digital control systems, the variable resistance is often translated to a temperature signal by using a software look-up table that maps the temperature corresponding to the measured resistance, or by solving an exponential equation using exponents and coefficients provided by the thermistor manufacturer. Their main advantages and disadvantages are tabulated in Table 4-2 .

Figure 4-7. Thermistor Characteristic

Table 4-2. Thermistor – Advantages and Disadvantages

Thermistor
Advantages	Disadvantages
High resistance change	Nonlinear
Fast response	Fragile
Two-wire measurement	Current source required
Low cost	Self-heating

Thermistors typically have an accuracy around ±0.5°F, but they can be as accurate as ±0.2°F. They have a high sensitivity, in other words they have a fast and detailed response to a change in temperature. However, they drift over time, and regular calibration is required to maintain this accuracy. At one time, calibration was required about every six months or so, but the quality of thermistors has improved in recent years, reducing the frequency interval to once every five years or more. For instance, commercial grade thermistors are now available with a guaranteed maximum drift of 0.05°F over a five-year period. They now have long-term stability and a fast response at a low cost.

The RTD is another of the most commonly used temperature sensor in analog electronic and digital control systems because it is very stable and accurate, and advances in manufacturing techniques have rapidly brought prices down. As the name implies, the RTD is constructed of a metal that has a resistance variation as a direct acting function of temperature that is linear over the range of application ( Figure 4-8 ). Common materials include platinum, copper-nickel, copper, tungsten, and some nickel-iron alloys. In HVAC applications, RTDs are often in a wound wire configuration, with the RTD metal formed into a fine wire and wrapped around a core. Coil wound RTDs cost more than thermistors but they are more stable, so regular recalibration is not usually required. Standard platinum RTDs have a reference resistance of 100 ohms at 0°C. This low resistance (compared to 10,000–100,000 ohms for thermistors) typically requires that the measurement circuit compensate for or eliminate the resistance of the wiring used to connect the RTD to the detector, because this resistance will be on the same order of magnitude as the RTD. To do this either the detector must be calibrated to compensate for wiring resistance or, more commonly, three-wire or four-wire circuits are used that balance or eliminate wiring resistance. For HVAC applications, platinum RTDs rated at 100 ohms are typically about ±0.5°F at the calibration point to ±1.0°F accuracy over the application range. However, high purity platinum sensors can have an accuracy of ±0.02°F or even better.

Figure 4-8. Thermistor and RTD Resistance Change with Temperature

A recent development is the thin-film platinum RTD, which has a reference resistance of about 1000 ohms. Made by deposition techniques that substantially reduce the cost, these sensors are one of the primary reasons why RTDs began to replace thermistors in electronic and digital control systems. Thin-film RTDs have accuracies on the order of ±0.5°F to ±1.0°F at their calibration point. As the units are dependent on the behavior of platinum metal they have a very, very low drift. The main advantages and disadvantages of RTDs are shown in Table 4-3 .

Table 4-3. RTD – Advantages and Disadvantages

RTDs
Advantages	Disadvantages
Most stable	Expensive
Most accurate	Current source required
Most linear	Coiled type low resistance, 100 ohms, requires good temperature compensation
	Film type has relatively low resistance
	Self-heating

Integrated circuit (IC) temperature sensors (also called solid-state temperature sensors or linear diodes) are based on semiconductor diodes and transistors that exhibit reproducible temperature dependence. They are typically sold as ready-made, packaged integrated circuits (sensor and transmitter) with built-in conditioning to produce a linear resistance to temperature signal. Solid-state sensors have the advantage of requiring no calibration, and their cost and accuracy are on the order of thin-film platinum RTDs. See Table 4-4 for their main advantages and disadvantages.

Table 4-4. Linear Diodes – Advantages and Disadvantages

Linear Diodes
Advantages	Disadvantages
Most linear	Use up to 330°F
Inexpensive	Power supply required
	Slow
	Self-heating
	Limited configurations

The process to select an appropriate sensor type should be concerned with the economics, accuracy, and long-term reliability of the sensor. A summary of sensor characteristics is shown in Table 4-5 . In HVAC systems, for the most part, extremely accurate devices are not usually needed to produce the required actions. All of the above sensor types are within this acceptable window of requirements. Different manufacturers of controls typically carry the capability to use any of these sensors.

Table 4-5. Summary of Sensors

Temperature Sensors Comparison
Type	Primary Use	Advantages	Disadvantages	Response Time
Thermocouple	Portable units and high temperature use < 5000°F	Inexpensive Self-powered for average accuracy	Very low voltage output	Slow to fast depending on wire gauge
Thermistor	High sensitivity General use < 300°F	Very large resistance change	Nonlinear Fragile Self-heating	Fast
RTD	General purpose < 1400°F	Very accurate Interchangeable Very stable	Relatively expensive	Long for coil Medium/fast for foil Short for thin film
Integrated circuit	General purpose < 400°F	Linear output Relatively inexpensive	Not rugged Limited selection	Medium/fast

If extreme accuracy or extreme reliability is required, specify these requirements and highlight them in the design specifications.

The final commissioning process is critical, and necessary for the assurance of proper control system and sensor performance to the process.

The useful accuracy of temperature sensors varies considerably. Early in this text it was mentioned that room temperature sensors need to be reliable but not accurate if the occupant can adjust them. The occupant will adjust the thermostat to their comfort and accuracy of calibration in degrees Fahrenheit is not the issue.

Now consider an air-conditioning plant which includes an air economizer (uses outside air for cooling when appropriate) and a cooling coil. The plant supplies air at a constant temperature of 54°F. There are two temperature sensors we are going to consider: outside air and supply air. The outside air sensor is used for information and controls when the change is made from 100% outside air to minimum outside air.

From the point of view of plant performance, the outside air temperature matters at the changeover point but not elsewhere. Any nonlinearity would be irrelevant as long as it is set correctly at the changeover point. Even at the changeover point, an error only matters for a relatively few hours in the year in most climates.

Now consider the performance of the supply air temperature sensor. It is to maintain 54°F and let us suppose the return air temperature is 75°F. The cooling effect is the air being heated in the spaces from 54°F to 75°F, a temperature rise of 21°F. Now let us suppose the temperature sensor is just 1°F off and the supply temperature is 55°F. The cooling capacity is down from 55°F to 75°F, or 20°F, a drop of 5% in cooling capacity. An error of 2°F produces a reduction of 10%. Accuracy really does matter here. However, the accuracy is only needed at 54°F not at higher or lower temperatures.

This issue of accuracy is particularly important where sensors are replaced without site calibration. A higher supplied accuracy is needed in this change-it-out situation without site calibration.

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Software Technologies for Complex Control Systems

Bonnie S. Heck , in The Electrical Engineering Handbook, 2005

14.6 Real-Time Applications

As mentioned previously, digital control systems are much more sensitive to time delays than are other software applications. There are three types of timed events that occur in typical control system: periodic events (such as updating control signals at a given rate), asynchronous events (such as set point adjustments from a supervisory control), and sporadic events (such as a fault occurrence). A long time delay that is incurred when processing an event could cause catastrophic behavior, such as instability or failure to recover from a fault. Making the problem more difficult to manage and to analyze is the fact that operations in typical networked computer systems occur in a nondeterministic manner, giving rise to varying time delays. The concern over how this nondeterministic behavior affects different software applications has prompted a great deal of research in the area of real-time computing.

Consider the real-time operation of a complex control system where different components are connected via a network. There are three places where the time delays occur: the data transfer over the network, the software applications running on the processors, and the middleware that integrates the components.

Consider first the network communications. Performance of a communication network is commonly measured in terms of time delays (e.g., due to access delay, message delay, transmission delays), reliability or accuracy of data, and throughput (e.g., max data rate divided by the data size). For control systems, the time delay is the most critical measure due to stability and performance considerations. To address the special concerns of control systems, some specialized protocols have been written that provide for a dedicated control area network (such as DeviceNet). The alternative is to use a standard data network such as ethernet. Data networks typically use protocols (such as TCP/IP) that result in a nondeterministic transfer of data, as opposed to the smaller but more frequent data packet transfers conducted by control area networks. Results examining the use of ethernet for real-time applications are given in Schneider et al. (2000), while a comparison of ethernet (using CSMA/CD) to a token ring bus and to control area networks is given in Fend-Li Lian et al. (2001). Networks using ATM technology have good performance with other real-time applications, such as video and voice, and have good potential for future use in control applications.

Next, consider real-time behavior of software applications. While high-level languages do have function calls that seem to imply real-time, the timings are not exact. For example, a sleep(20) command implemented in a Java thread would seem to make the thread pause for 20 msec. However, the thread may actually sleep for 19 msec or 21 msec. This can be explained by examining how an operating system on a processor handles several tasks that are running concurrently (known as multitasking or multithreading: the system must schedule a dedicated processor time to each of these different tasks. As a result, tasks are typically scheduled in a nondeterministic manner, which gives rise to the resulting soft real-time behavior.

While the processor itself does have an absolute clock, a realtime operating system (e.g., LinuxOS, VxWorks, Sun/Chorus ClassiX, and QNX) must be used to get hard real-time performance out of the applications. There are products (rather kernels) that can be installed on a system with a non-real-time operating system, such as the Real-Time Linux modification of the popular Linux Operating System. Here, hard real-time applications coexist with the normal linux kernel and hard real-time tasks are always given priority for execution. The normal linux kernel as a whole is only executed when slack time is available. Most computers use hardware components to perform tasks that need real-time performance, such as video cards on desktop machines or DSP boards in signal processing applications. These tasks can be performed in parallel without the need for the nondeterministic uncertainty introduced by scheduling of processor time. Another alternative is to use separate dedicated processors to perform each concurrent task, such as having one processor perform low-level control loops and a separate processor performing high-level loops such as fault detection and supervisory control. Moreover, the software application must be able to take advantage of realtime operations. While C has been used successfully in control system implementation, Java lacks good real-time applicability. This may change in the near future because there is a large effort to develop real-time Java capability as evidenced by the release of Real-Time Java Specification (Bollella et al., 2000).

Finally, the middleware must run in real-time. For example, the original version of CORBA did not support hard real-time operations because of overhead in the client/server implementation due to the ORB intercepting the remote method calls, redirecting them, and marshalling or demarshalling data to translate data values from their local representations to a common network protocol format. CORBA also lacked realtime scheduling capabilities. Extensions have been made to build a real-time CORBA (also known as TAO) (O'Ryan et al., 2000). As described previously, using local replicas helps speed up the real-time behavior of the middleware. Further, real-time applications require the middleware to schedule events based on quality of service (QoS) measures. Typical QoS parameters include desired rates for periodic events (e.g., updates from sensors), deadlines for when a task must be completed (e.g., when the new command must be sent to the actuator), and priority values indicating the relative importance of different procedures. A real-time middleware must also time stamp transactions and prescribe efficient and fast memory usage.

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Digital Control Systems Chapter 3 Solution Fadali

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