Control Systems Engineering - Science topic

Yes, you can use a Triggered Subsystem in MATLAB Simulink to implement a low-pass filter on a PWM signal. The Triggered Subsystem allows you to execute a block of Simulink model when a specific condition is met, which in this case would be the PWM signal exceeding a certain frequency threshold.

A high-level overview of how you can design this technique in MATLAB Simulink is:

1 - Generate the PWM signal: Start by generating the PWM signal that you want to filter. You can use appropriate blocks or functions in Simulink to generate the PWM waveform based on your requirements.

2 - Measure the frequency: Use a frequency measurement block or algorithm to measure the frequency of the PWM signal. This could be achieved using techniques such as zero-crossing detection or FFT analysis.

3 - Compare with the threshold: Compare the measured frequency with the desired frequency threshold (20 kHz in your case). You can use a Comparison block or an if-else condition to make this comparison.

4 - Triggered Subsystem: Inside the Triggered Subsystem block, design the low-pass filter. You can use standard Simulink blocks, such as a Discrete-Time Integrator and a Discrete-Time Transfer Function block, to implement a simple low-pass filter. Adjust the filter parameters based on your desired cutoff frequency.

5 - Connect the subsystem: Connect the output of the Triggered Subsystem to the appropriate output in your Simulink model, so that the filtered PWM signal can be utilized for further processing or analysis.

6 - Configure the Trigger: Set up the trigger condition for the Triggered Subsystem. In this case, the trigger condition would be the comparison result from step 3.

7 - Simulate and analyze: Simulate the Simulink model and observe the filtered PWM output. Verify that the frequencies above 20 kHz are effectively attenuated while the frequencies below 20 kHz are passed through.

It's important to note that the specific details of your implementation may vary depending on your PWM signal source, sampling rate, desired filter characteristics, and any other additional requirements. Experimentation and fine-tuning may be necessary to achieve the desired filtering behavior.

By utilizing the Triggered Subsystem and appropriate filter blocks, you can implement a low-pass filter on a PWM signal in MATLAB Simulink, removing high frequencies above the specified threshold while allowing frequencies below it to pass through.

With respect to electric motors / drives, we have collected a database of 500+ different parameter sets describing the first principle dynamics of permanent magnet synchronous motors which are publicly available here: https://github.com/upb-lea/meta_RL_PMSM

The parameter set originates from real-world, industry grade motor series. In addition, we offer an upsampling scheme to produce additional ‘artificial’ parameter sets if desired.

For simulation, the parameter sets can be loaded and processed using the following framework: https://github.com/upb-lea/gym-electric-motor

Although this model set only describes one specific kind of ODE structure, the dynamics of the covered motors (from few watts to hundreds of kilowatts) varies strongly.

Sorry, it is jesus.mh@chihuahua.tecnm.mx

Dear Saifur Rahman Khan and Al,

lI wish you Happy New Year: success in your spiritual achievement, good health and prosperity for it!

I found the next on the facbook (https://www.facebook.com/USGSVolcanoes):

'A volcanologist is a person who studies volcanoes, but there are many different specialties within the field of volcanology. Which interests you and what steps should you take to achieve your goal? Find out more in #VolcanoWatch.

Earthquakes are one primary tool used to study volcanoes. A volcano seismologist studies the earthquakes that are generated as magma moves through Earth’s crust.

A volcano geodesist studies the deformation, or change in shape, of a volcano caused by the movement of magma and gases beneath the surface. Many features of volcanoes can be studied from space, as well, using satellite sensors. Tools like these provide clues about the state of the volcano.

Geologists and geochemists study the composition of lavas and gases to understand the source and style of the eruption. Measuring gas emissions is especially important, as the vog (volcanic air pollution) caused by toxic volcanic gases can contribute to breathing problems, acid rain, and agricultural problems downwind, especially during long-lived eruptions.

If you are interested in becoming a volcanologist, you’ll need to work toward a bachelor’s degree, preferably in a STEM field (Science, Technology, Engineering, and Math). Volcanologists frequently pursue degrees in geology, chemistry, physics, and/or mathematics, but that is not always the case. Oceanography, computer science, engineering, environmental science are all potential pathways, and the list goes on. Explore different fields to find what interests you most.

After achieving a bachelor’s degree, consider options for advanced degrees like a Masters or Doctorate. Many advanced degree programs in the sciences are funded, meaning tuition may be waived, and you might get a stipend for doing the work. Basically, you get paid instead of having to pay for school, and you gain valuable work experience at the same time.

You might consider working for the USGS or other agencies and companies. You have seen many photos of HVO scientists working during the eruptions of Mauna Loa and Kīlauea. The National Park Service also offers a variety of positions for people with either bachelor’s or advanced degrees, such as park geologists, archaeologists, botanists, guides, interpretive rangers, and law enforcement rangers. Science writing and journalism are also excellent ways to explore the excitement of volcanology, natural disasters, and cutting-edge science, while encouraging those passions in others. Similarly, eco- and geo-tourism are great ways to get close to the action and work outdoors, while also meeting, educating, and inspiring people from all over the world. Careers in emergency management will have you helping people stay informed during crises.

Check out usajobs.gov for positions within the federal government. There is even a special section for students and recent grads.

Volcano Activity Updates

#MaunaLoa is not erupting. Webcam imagery shows weak, residual incandescence intermittently in the inactive Northeast Rift Zone fissure 3 lava flow at night. Seismicity remains low and ground deformation rates have decreased. Sulfur dioxide (SO2) emission rates are at background levels. For Mauna Loa monitoring data, see: https://www.usgs.gov/volcanoes/mauna-loa/monitoring-data.

#Kilauea is not erupting. Lava supply to the Halemaʻumaʻu lava lake in Hawai‘i Volcanoes National Park ceased on December 9. Sulfur dioxide emission rates have decreased to near pre-eruption background levels and were last measured at approximately 200 tonnes per day (t/d) on December 14. Seismicity is elevated but stable, with few earthquakes. Over the past week, summit tiltmeters recorded several deflation-inflation (DI) events. For Kīlauea monitoring data, see https://www.usgs.gov/.../past-week-monitoring-data-kilauea.

There were three earthquakes with 3 or more felt reports in the Hawaiian Islands during the past week: a M3.3 earthquake 14 km (8 mi) S of Fern Forest at 7 km (4 mi) depth on Dec. 27 at 4:33 a.m. HST, a M3.4 earthquake 7 km (4 mi) WSW of Volcano at 2 km (1 mi) depth on Dec. 24 at 8:31 p.m. HST, and a M2.5 earthquake 1 km (0 mi) S of Mountain View at 11 km (7 mi) depth on Dec. 24 at 9:57 a.m. HST.

In the photo, an HVO technician adjusts a volcanic gas analysis instrument that was specifically designed for this Unoccupied Aircraft System (UAS) unit, which carries three one-liter analysis bags. The instrument transmits gas concentration information in real-time during the flight at Kīlauea summit. USGS has special use permits from the National Park Service to conduct official UAS missions as part of HVO's mission to monitor active volcanoes in Hawaii, assess their hazards, issue warnings, and advance scientific understanding to reduce impacts of volcanic eruptions. Launching, landing, or operating an unoccupied aircraft from or on lands and waters administered by the National Park Service within the boundaries of Hawai‘i Volcanoes National Park is prohibited under 36 CFR § 1.5 - Closures and public use limits.

USGS image taken January 14, 2022 by M. Warren.

#USGS #HVO #HawaiianVolcanoObservatory'

Maybe it can help you!

Regards,

Laszlo

Hi Marco, you have already received good answers, all complementary, check Silvano Meroni and Yew-Chung Chak 's responses.

Clarivate announced that starting with 2023 ESCI-indexed journals will also be assigned an impact factor. See: https://clarivate.com/blog/clarivate-announces-changes-to-the-2023-journal-citation-reports-release/

Use TCP/IP communication.

Dear Italo,

In general, the bounds selection is made empirically because the "suitable range" of a PID controller is problem-dependent. The way I use to select the bounds is: 1. I tune a PID controller that produces a stable response to the closed-loop system. Then, 2. I choose a range around this "nominal" value large enough such that the GA has still some degree of freedom to search in the optimization space. Finally, 3. if the GA converges, I start decreasing/increasing this range till I got a more or less good behavior of the GA, i.e., the GA doesn't stick in a sub-optimal minimum or so.

If you want to use a more rigorous approach, I would suggest computing the set of all stabilizing PID controllers for the particular system. Then, I would establish the bounds for the GA search space to be the this computed set. In that way, you would search for the optimal controller only within those producing a stable closed-loop response.

Best,

Jorge

The ladder

It depends on your sample frequency, which is limited for performance of DAQ. another issue should be concerned is the number analog of a processing control system. the quality of data will be bad or real-time data might be good if you collect as much data in process as possible.

Hi,

file name "sfun_ abcaaaaa.c" is not available in the folder mentioned.

Sounds analogous to an inverted pendulum. Here is one way to construct something with closely similar dynamics. Take a knee (90° angle) connector for sewage pipe, cut it nicely and mount it on a toy car such that the knee is pointing straight upwards. Place a small ball (i.e. from a roll-on bottle) on top of the knee and try to keep it there stably by moving the toy car back, forth, and sideways (hard to do manually without automated feedback control loop). I guess many feedback control schemes would keep the ball at the top of this curve subject to micromotion.

Aside: If the knee was facing down, mounted on a motor, and the ball is placed inside the 90° curve while the motor is rotated at some angular frequency, the resulting dynamics would resemble Mathieu equations whose stability analysis could be checked from Floquet theory. Incidentally, these are the same dynamics exhibited by a single ion trapped in a linear Paul trap.

Disturbance is sometimes helpful while dealing with multicollinearity. Also helpful may be keeping the number of degrees of freedom minimum.

Maybe you can alernatively consider the recursive least squares algorithm (RLS). RLS is the recursive application of the well-known least squares (LS) regression algorithm, so that each new data point is taken in account to modify (correct) a previous estimate of the parameters from some linear (or linearized) correlation thought to model the observed system. The method allows for the dynamical application of LS to time series acquired in real-time. As with LS, there may be several correlation equations with the corresponding set of dependent (observed) variables. For the recursive least squares algorithm with forgetting factor (RLS-FF), adquired data is weighted according to its age, with increased weight given to the most recent data.

Years ago, while investigating adaptive control and energetic optimization of aerobic fermenters, I have applied the RLS-FF algorithm to estimate the parameters from the KLa correlation, used to predict the O2 gas-liquid mass-transfer, hence giving increased weight to most recent data. Estimates were improved by imposing sinusoidal disturbance to air flow and agitation speed (manipulated variables). The power dissipated by agitation was accessed by a torque meter (pilot plant). The proposed (adaptive) control algorithm compared favourably with PID. Simulations assessed the effect of numerically generated white Gaussian noise (2-sigma truncated) and of first order delay. This investigation was reported at (MSc Thesis):

Dear Sarang, did you find any suitable simulation software? We also need it for a small project we are working on

you can follow the link

i would suggest Udemy courses they are pretty advanced

Hello, if possible, please explain more about this. Thank you

It depends upon the size of the accumulator, and the flowrate required. If you calculate the flowrate required for the PAS pump you could work out the accumulator volume required. See link below for typical (but a little old) energy consumption info.

However, many modern vehicles are moving to electric PAS systems, so the traditional hydraulic system may not be required in future.

Mohamed-Mourad Lafifi Zahoor Ahmed

Thanks alot for your time! These links are really helpful!

Your system is basicly a double integrator, so a borderline stable system. If you look at your phase value of the bode plot for the continious system (bode(G)), you will see that it has a phase of -180° over all frequencies. Since any kind of delay causes the phase of a system to drop, if you introduce an arbitrarily small amount of delay on your continious tf G, and use the command "margin(G)", you will find that your system will always have a negative phase margin. And since discretizing will always introduce delay into your system, closing the loop on it will automatically destabalize it. You can use 'tustin' as you discretizing method, which essentialy uses trapezoidal integration, a more stable integration algorithm, and simulink should probably get your discrete system stable this way.

For more details:

1. Historically twisting have been designed to substitute a discontinuous controller with Lipschitz continuous one keeping the finite-time theretically exact convergence to the sliding output for the systems with relative degree one affected by Lipschitz perturbaitons(see Levant A. IJC, 1993 https://www.tau.ac.il/~levant/slorder93.pdf).

The only adavantage of the twisting controller is that if this algorithm is applied for the systems with relative degree two it ensures theoretically exact finite-time convergence of the sliding output and its derivative to zero in finite-time.

But from the viewpoint of chattering the twiating controller is the worst one because his gains are at least twice bigger, than the upper bound of uncertainties.

Moreover, the amplitude of chattering produced by twisting controller is bigger than the amplitude of chattering produced by relay controller with a linear sliding surface see attached file, table 3

Dear Saeb AmirAhmadi Chomachar ,

The transfer function H(s)= N(s)/D(s)

has the form of filter characteristics.

Such general transfer characteristics has zeros and poles.

The filter which contains zeros and poles are elliptic filters.

Such filters can be implemented by using op amps+ discrete RC elements

They can be built completely integrated using switched capacitor filters

They can be also implemented by using gm-C filters.

For such filter design using active filters please follow the reference: https://www.sciencedirect.com/book/9780750675475/analog-and-digital-filter-design

Bestwishes

Dear Professor Sabatier

First, I'd like to thank you so for asking these kind of challenging questions. These kinds of questions are very important for advancing this field. As you said, our life is full of doubts. Now this question arise that: How can we trust to different integer differential models? The only sin of fractional derivative is that it has been introduced after integer derivative. Why should we accept that the first derivative (u') denotes the velocity? Why another fractional derivative (such u^{0999}) is not?

As we can check in many considerable works, fractional derivatives modify many old results in integer derivatives. As you said, you often use fractional models to build and implement very physical and useful systems (eg: battery state observers, hydrogen generator, ...). We need a comparison in study of these different natural phenomena and so a standard basic theory for the comparison which is usually integer derivative models while we can doubt it. I fee that we could improve our initial mental attitude. Again, I appreciate you.

Dear Adolfo Silva ;

Dear David Mark Koenig ;

We have:

Applications of Double Laplace Transform to Boundary Value Problems DOI: 10.9790/5728-0925760

For more details we have:

Best regards

Try this articles:

1- PID Tuning of Plants With Time Delay Using Root Locus

Greg Baker ,San Jose State University

2- A revision of root locus method with applications

September 2015, Journal of Process Control 34:26-34, DOI: 10.1016/j.jprocont.2015.07.007

Some interesting links dealing with that would be:

Best regards

I agree with Aparna Sathya Murthy

The support in MATLAB support may help you. Adaptive Neuro-Fuzzy Inference Systems (ANFIS) Library for Simulink :

Best regards

T. Bauer, J.T. Betts, W. Hallman, W.P. Huffman, K. Zondervan, Solving the optimal control problem using a nonlinear programming technique, Parts I and II, 1984.

A.E. Bryson Jr.Optimal control – 1950 to 1985

IEEE Control Systems (1996),

For more details, we have:

Monte Carlo Simulation of Real Dynamic Systems :

With Matlab: http://www.lasar.polimi.it/?page_id=611

Best regards

Pass x, instead of tau, as rightly pointed out by Saeb AmirAhmadi Chomachar

syms x y psi tau t

c1 = 1;c2 = 1.5;lambda = 0.1;

x_r(tau) = 0.8486*tau - 0.6949;

y_r(tau) = 5.866*sin(0.1257*tau + pi);

psi_r(tau) = 0.7958*sin(0.1257*tau - pi/2);

x_r_dot = 0.8486;

y_r_dot(tau) = 0.7374*cos(0.1257*tau + pi);

psi_r_dot(tau) = 0.1*cos(0.1257*tau - pi/2);

phrase1 = c1/2*(cos(psi)*(x - x_r) + sin(psi)*(y - y_r))*(cos(psi)*x_r_dot + sin(psi)*y_r_dot);

phrase2 = c1/2*(-sin(psi)*(x - x_r) + cos(psi)*(y - y_r))*(-sin(psi)*x_r_dot+cos(psi)*y_r_dot);

phrase3 = 0.5*(psi - psi_r)*psi_r_dot;

eq = -2*(1-lambda)^2*(phrase1 + phrase2 + phrase3) - 2*lambda^2*(t - tau);

eqn = rewrite(eq,'log');

sol = solve(eqn == 0 , x , 'IgnoreAnalyticConstraints',1);

pretty(sol)

HIL or 'hardware-in-the-loop' testing is by its very nature a resource-hungry solution to testing, requiring multi-skilled teams able to set up and configure both the execution platform and the I/Os as well as the modelling environment.

Once your model is verified (i.e., MIL in the previous step is successful), the next stage is Software-in-Loop(SIL), where you generate code only from the Controller model and replace the Controller block with this code.

For Video tutorials , we have:

Introduction | Nonlinear Control Systems: https://www.youtube.com/watch?v=Xgnwn0G9qoo

The effectiveness of tuned mass dampers (TMDs) in reducing the seismic response of civil structures is still a debated issue. The few studies regarding TMDs on inelastic structures indicate that they would perform well under moderate earthquake loading, when the structure remains linear or weakly nonlinear, while tending to fail under severe ground shaking, when the structure experiences strong nonlinearities. TMD seismic efficiency should be therefore rationally assessed by considering to which extent moderate and severe earthquakes respectively contribute to the expected cost of damages and losses over the lifespan of the structure.

Check this chapitre:

Dear Joo ...

Check your the step of discretization and linearisation of your simulation.

Regards

I suggest you follow :

Tableau Desktop is also good one for drawing the scientific illustrations.

Stability Analysis of Discrete Time Systems :https://nptel.ac.in/content/storage2/courses/108103008/PDF/module3/m3_lec1.pdf

Discrete-Time Systems:

Dear Kouba Nour El Yakine

Manual PID tuning is done by setting the reset time to its maximum value and the rate to zero and increasing the gain until the loop oscillates at a constant amplitude. (When the response to an error correction occurs quickly a larger gain can be used.

The best method for tuning PID controller parameters is depend of your system

Regards

You can use control methods in everywhere. Below, there are some areas:

I suggest that you use the 5D BIM technology, which is related to the cost of construction elements. You need to model your project using software as revit, and then link the model's elements to the cost and time schedule using software like Naviswork. This will give you visualization for the construction phases and the project life cycle.

I suggest you to follow https://www.mathworks.com/help/control/ref/ss.html

The conversion from the S-domain to the Z-domain can be accomplished by using the bilinear transformation.

- A transformation for S to Z

S= (Z-1) /(Z+1)

- And frequency prewarping

w analog = tan wd/2,

where wd= 2 pi f/fs

As one sees if one changes fs , one has to change w analog as a consequence of prewarping.

Best wishes

When the energy of a signal is finite, but power is zero, such a signal is called energy signal. for example- a pulse has a finite energy, though power is zero.

On the other hand, if the energy of a signal is infinite, but it has a finite power, it is called as power signal., for ex- sinusoid is a power signal.

The auto-correlation function is simply the convolution of the signal with itself (not flipped), where delay/lag is the parameter. Here we try to find how much a signal overlaps with itself when lag parameter is varied over time. Its Fourier transform is power spectral density, which also does the same thing in frequency domain. So, at t=0, the auto-correlation function provides the energy or power of the signal x(t). (based on whether x(t) is a power or energy signal) which is same as the power in frequency domain obtained by integrating power spectral densities for all frequencies.

so, basically, both tell the same thing, one with respect to time, other with respect to frequency.

I don't understand exactly what you mean? What is your goal? What do you want to do?

Hi, I initially face a same problem in my research. It solved :)

Kindly refer my publication attached.

Significantly very fast. Standing at the control panel of a Vestas 850 kw , I saw the display was showing the instantaneous power production and pitch angle. The display was updating either every second or half second and each reading gave different readings. so a time constant for the pitch of under a second, for 40 m blades is astonishing!

From the IMU we get position and velocity after sensor funsion. The input variable of the plant seems to be the speed of the gimbal motor (since its driven by 3 sine waves), not the voltage as usual in a brushed dc motor setup. So one can not just implement a torque control for holding the position. When I investigated the Storm32BGC, it looks like the PID does influence the position and the speed at the same time: Fast movements leads to fast speed corrections, but also the deviation from the stationary position leads to a given speed.

We have proposed a numerical method (denoted as MEM) for solving the

Nonlinear Dynamic problems. If you would like, you can view our published articles in this field entitled "dynamic analysis of SDOF systems using modified energy method". By the way, the presented idea is also generalized to MDOF systems in another study, which is available on my research-gate account.

if you need more information, please let me know.

Regards,

Jalili

There are 4 issues I have seen in this discussion:

1. Backstepping can compensate theoretically exactly only uncertainties decreasing together with the state variables. No comparison between backstepping and SMC in this sense.

2. Matched and unmatched uncertainties

For perturbed chain of integrators there are no unmatched perturbations. If you will differentiate the output till the system order all the perturbations will be matched! That is why the only reasonable to control the systems with unmatched uncertainties is a combination between backstepping and SMC, where backstepping compensates state dependent part and SMC(see Estrada et al IJRNC,27(4),2017,DOI : 10.1002/rnc.3590,Automatica 2010(11), TAC 2010(11), J. Davila TAC 2013, Ferreira et al Journal of Franklin Institute, 2014, 351(4),doi:/10.1016/j.jfranklin.2013.12.011, Automatica, September 2015, Vol. 59, 10.1016/j.automatica.2015.06.020)

3. SMC can not be smooth. Sliding mode (for definition!) is a motion on the sliding surface!

4. Chattering

4.a. There is no way to keep the sliding mode and eliminate the chattering.

4.b. It is wrong opinion that it continuous HOSM controllers (like super-twisting, Kamal et al DOI:10.1016/j.automatica.2016.02.001, Torres et al doi :10.1016/j.automatica.2017.02.035, Laghrouche et al TAC,2017) everytime produce less chattering that discontinuous ones (see https://www.researchgate.net/publication/317229996_Is_It_Reasonable_to_Substitute_Discontinuous_SMC_by_Continuous_HOSMC).

It is true for the systems with fast actuators only.

You can get it form

If you want non-contact sensor, then proximity displacement sensor such as Bentley Nevada proximity sensors and Omega non-contact sensors can be used, they are cheaper than Laser sensors and have excellent accuracy:

If contact is permissible then you have wider options and LVDT sensors can be used, they have good accuracy and cheaper pices:

or search google or alibaba for LVDT sensor

It is generally difficult to stabilize unstable high-order non-minimum phase plants with the 'ideal' PID controller. Take Lakshmanaprabu's model as an example, which has a real pole and a real zero in the right-half plane:

Gp = (5⋅s − 1)/(16000⋅s⁴ + 41200⋅s³ + 2940⋅s² − 152⋅s − 5).

It is possible to stabilize the plant with a cascading compensator

Gc = k₁ − (k₄⋅s² + k₃⋅s + k₂)/(s³ + N₃⋅s² + N₂⋅s + N₁)

where

k₁ = −4.4690e+04

k₂ = 4.9923e+04

k₃ = −2.6980e+05

k₄ = 1.1096e+05

N₁ = −1.1172

N₂ = 6.0334

N₃ = −2.4079

The closed-loop transfer function (Gc*Gp)/(1 + Gc*Gp) is a 7^th-order system:

Gcl = (−2.235e+05⋅s⁴ + 2.793e+04⋅s³ + 4190⋅s² − 139.7⋅s − 5.586) / (1.6e+04⋅s⁷ + 2674⋅s⁶ + 270.4⋅s⁵ + 17.88⋅s⁴ + 0.8151⋅s³ + 0.02545⋅s² + 0.0004759⋅s + 4.249e-06),

which has two real zeros in the left-half plane and another two real zeros in the right-half plane. Cancelling the two real zeros in the left-half plane with a pre-filter

Gf = 4.249e-06/(s² + 0.125⋅s + 0.0025)

applied to the input command outside of the feedback loop yields Gclf = Gf*Gcl

Gclf = (−5.934e-05⋅s² + 1.483e-05⋅s − 5.934e-07) / (s⁷ + 0.1671⋅s⁶ + 0.0169⋅s⁵ + 0.001118⋅s⁴ + 5.094e-05⋅s³ + 1.59e-06⋅s² + 2.974e-08⋅s + 2.656e-10)

Computing the DC gain of Gclf (−2234.5) and placing a reference scaling factor "1/dcgain(Gclf)" ensures the unity gain when a unit step input is applied.

Dear Faeze Sdi,

Why proximal SVM are you using? You may try other algorithms for fault detection.

You can find a fully developed 14-DOF vehicle model in MATLAB with a time efficient suspension model and a reprogrammed full set MF tire model in https://arxiv.org/abs/1803.09411, in addition, this work formulated and solved the complicated optimal design and control problems of a 4-IWD electric race car on a given track in curvilinear coordinate system.

Dear Jiri Kovar,

I don't know exactly about GETFRAME. Would you please explain more?

I want to create motional time window in order to fault detection in each window.

See for paritial fraction i understand u want the values of a b or the way the roots will be arranged it is simple break roots and put s- a or s- b or whatever it be plus also then in one case put the value of one root in such a way the other root becomes zero and u will obtain value of a and b substitute back then take inverse laplace transform will obtain the transfer function in paritial fraction again the expansion of roots like repeated roots the method needs to differentiae the root once u obtaim the first root ur problem would be solved 100 percent by ogata it has detailed explanation modern control theory by ogata

I totally agree with you in that

Hi Waheed,

Thanks for your reply.

Since you have designed a Lyapunov-based integral backstepping controller for the 4^th-order MISO system, what exactly is your difficulty in checking the stability of the control system?

thankyou all.

The SVD precoding approach comes from Telatar's seminal work: http://web.mit.edu/18.325/www/telatar_capacity.pdf

He shows that it achieved the MIMO capacity. That paper is very nicely written, so I highly recommend it!

Anyway, you can consider derivative filter even in case of fractional PID controller in the design procedure, see e.g. But think carefully whether you really need fractional action at the controller side. It should be usually motivated by contradictory frequency domain requirements that cannot be fulfilled by classical integer order PID.

Dear Boris,

I hope I will be excused for writing a “long message,” (and people who are not interested can just ignore it) yet this to make sure that my messages are not misunderstood as puting “this method against the other.” Actually, I started with the claim that Nyquist theorem and Nyquist plot contain the entire information. The only problem is that, in realistic systems, where the amplitudes may go up quite a lot, even to infinite, when you need to decide what exactly occurs around 1, this could be a tough task in Nyquist plot. Here, Bode, with its logarithmic scale, is best… when it works, and, because of the two separate plots, this usually occurs in not too complex systems. Then, maybe only after some use, you learn that Nichols combines both advantages. Before Matlab (and other tools), plotting Nichols used to be a terrible task and this may explain why people might have remained reluctant to use it.

Please see my message about the Company which started with “No, we don’t use Nichols” and ended using Nichols as the main tool. So, maybe you just try to write (or ask students to write) Nichols along with Bode and just see what Nichols gives in all other than simple cases.

As you know, my main “academic” business has been (simple) adaptive control. However, as I use to tell, although I have been playing Prof. and Dr., all my years I have been mainly an engineer. In this context, first I actually refused to use adaptive control, because control practitioners do not think that, for a given specific case, one cannot find enough information to do a good fixed controller. Even when one has to fit parameters to a given specific plant, one can do it off-line. Actually, PID is so widely used because many systems are not too complicated; they could even be stable or close to stability to start with and then, you can safely fit the right parameters to give you the desired performance. If you don’t need too much, same controller can serve all machines of same type.

The problems start when you really need good performance, which usually means fast response and also very precise tracking. While doing manual tuning on one or two machines may still be not too difficult, when you produce 20, 100, or 400 machines per month, this becomes a very tough job, in particular when your customers, maybe thousands of kilometers away, may replace a motor or any other piece and the performance deteriorates. So, you may also have to keep sending your best guys all over the world.

The same complex systems and/or many not-exactly-identical systems are also the reason for my present use of adaptive control.

Same Company, which started with “we don’t use Nichols,” also responded to my idea of using automatic (adaptive) tuning with “Adaptive? Ha-Ha! You, Professor, please keep writing papers. We have real problems.” However, because the real problem was really too tough, they finally thought that they could not lose too much letting this funny Professor play for a little while.

The result? Now there is a button on each machine and all that they have to do at the end of production, or what the operator has to do when any change leads to deterioration of performance, is to push the button and wait a minute or so for the parameters to be adapted to the specific machine. Their technology, which was “dead” by 1994, is alive and kicking today and performance is something else.

This was only automatic tuning per machine, not what we understand by adaptive control, yet this was also what made me see that, when high performance is needed, on-line (simple) adaptive control is needed to maintain performance even for one given machine under uncertainty and under various and (real-time) changing operational environments. Results justify this and will still be seen.

With best regards, Itzhak

Hi Dawn,

I can see where you're coming from. You are trying to generalize that 

\dot{e_n} = \dot{x_n} − y_d⁽ⁿ⁾ = x₁⁽ⁿ⁾ − y_d⁽ⁿ⁾ .

However, 

\dot{x_n} ≠ x₁⁽ⁿ⁾ because of 

\dot{x_n} = ∂(x) + β(x)u + η.

The two concepts are closely related but not completely identical. We may take a deep look at them from two perspectives. 

1. The first is from the perspective of pure physics. Passivity can be considered as universal in mechanical systems, e.g., robot manipulators and satellites. Specifically,  a satellite in orbit, without active control, is stable in terms of both its position and velocity and in addition the velocity of the satellite would finally decay since the system mechanical energy (kinetic and potential energy) gradually decreases due to the atmosphere drag forces (whose is role is similar to damping control). This phenomenon is due to the fact that a satellite in free motion is (output strictly) passive with respect to its output (i.e., velocity). Another common example is the bicycle and it would finally become still on a horizontal road if we do not insert energy by our legs. Passivity may also be thought to be the will of the system (free) to stabilize itself.

2. The second perspective is based on the passivity formula. Passivity is typically defined in terms of the input (u) and output (y) of a system, namely

int_0^t y^Tu dr >=E(t)-E(0)

where E(t) is the system storage. The interpretation is that the injected energy from the external input is not less than the variation of the system storage. "Not less than" here implies that the injected energy by the input is partitioned into two parts: 1) variation of the system storage and 2) the nonnegative second part. This intuitively means that there is some energy dissipated (by the system) and only part of the energy is converted to the system storage. 

The passivity formula, however, does not naturally implies stability of the system under the external input (u) but it does imply that the free system (i.e., without the external input) is stable. The stability of the free system, yet, has a weak connection with the typical equilibrium-based Lyapunov stability since the system storage does not explicitly specify that it is defined based on certain equilibrium of the system.

If  the external input is allowed to be designed, the connection between passivity and equilibrium-based Lyapunov stability can be explicitly established (in part). For instance, with a negative output feedback for the external input, output regulation (to its equilibrium zero) can typically be achieved.

In summary, we might say that passivity often implies (equilibrium-based Lyapunov) stability, but it also tells many other significant things concerning the physics/dynamics of the system.

I hope the above points would complement the existing answers and be of some help.