I think, one of the simplest non-minimum phase system is the well -known allpass filter, which can be used for smoothing the delay properties of a system.

sir is there any mechanical system that i can visualize ?? combining many subsystems may result in non-minimum phase system..

Earlier when i read about oscillations in system i was surprised but later that seemed obvious. In the same way is there any system that shows non-minimum phase character due to some obvious reason ?

Dear Sarvesh

The Drum-boiler dynamics is one of the practical example of non-minimum phase system. Actually, the complicated shrink and swell dynamics are present which creates a non minimum phase behavior which changes significantly with the operating conditions. I think the attached file (K.J. Astrom,, R.D. Bell, Drum-boiler dynamics, Automatica 36 (2000) 363}378) may solve your problem.

Thanks

I think, in general we can say that each delay (wanted or not) in a system with feedback (electrical, mechanical,...) can be described by a non-minimum phase block.

Reversing a car is the example I use in my course. Imagine parallel parking where the output is the distance from the kerb. When you first start to move backwards while turning the steering wheel, the driver moves away from the kerb before he/she moves closer.. This is the reason why parallel parking is hard and is a part of the UK driving test.

Altitude of an aircraft is another example.

When a pilot wants to gain altitude, he/she first has to rotate the aircraft to increase the angle of attack. To rotate the aircraft, a downward force of the tail is obtained by raising the elevator. That causes an overall downward force on the aircraft that initially lowers the centre of gravity before the increased upward force on the main wings from the increased angle of attack raises the aircraft. High performance fighter aircraft often use canards (e.g. Gripen) instead to remove this non-minimum phase behaviour.

Also remember that the closed-loop sensitivity function of open loop unstable systems become non-minimum phase when the loop is closed. So there are many examples of this that are easy to imagine, like the pole balancing problem. This is why it is hard to maneuver a bicycle away from the kerb if you get forced into the gutter.

Thanks u sir.. These were the perfect examples i were looking for..

The "four tank" system is a classical example of a MIMO system, with a intuitively stable nature, which can be configured to be minimum or non-minimum phase. Other examples may be found when there are unstable dynamics in the feedback path of the diagonal (or direct) dynamics. Nonetheless, this doesn't always results on a non-minimum phase transfer zero (i.e. a MULTIVARIABLE non-minimum phase dynamic).

"Artificial" non-minimum phase dynamics appear when sampling is involved due to the time delays, their nature is not within the "physical" nature of the phenomenon.

Non-minimum behaviours include time-delay and right-half-plane zeros. Time-delay is more understantable than RHP zeros. An RHP zero can be understandood to have opposite response directions in steady-state and transcient. A simple example is an instance hot water system, where hot water is directly heated up from cold water without a hot water tank. In such a system, if you want to increase water temperature by openning the hot water tap, you will firstly get temperature decreased due to increased cold water flowrate although gradually, the water temperature will increase because the increase heat supply will eventually heat water up.

Aircraft or missile with attitude controls situated behind the center of gravity are non-minimum phase when it comes to controlling the flight path angle or the load factor.

In order to increase the lift the pilot seeks to increasing the angle of attack and for that matter he needs to create a positive moment. This positive moment is achieved by reducing the pitch moment of the elevator, that is by reducing the elevators contribution to total aircraft lift. This creates an initial response that is opposed to final response sought.

How about this picturesque examples, as well as an explanations: SEE THE FOLLOWING LINK! Honestly, many of them are available on net! Regards!

http://control.ee.ethz.ch/~ifa_cs2/RS2_lecture5.small.pdf

suppose for instance the dynamics of an aircraft is non-minimum phased. If the input to the aircraft is for turning on the right side then it initially moves towards the left side before proceeding in the correct direction.

Ali Arshad correctly identifies the non-minimum phase behaviour in the lateral dynamics of an aircraft. This is known as "adverse yaw", for which the Wikipedia page gives a good explanation

http://en.wikipedia.org/wiki/Adverse_yaw

Roll on a ship has a non-minimum phase (zero in the right half plane). When making a course alteration, you pull out the rudder, say to starboard (right). The rudder then procides a force towards port (left). The rudder force will make the ship roll towards starboard down since the rudder is significantly below the water surface and the ship's centre of gravity is above the water surface. The turn rate of the ship bulds up more slowly than roll because the ratio between torque and moment of inertia is much larger for yaw (turning of the vessel) than for roll (angular acceleration in roll is higher for a specific rudder force). When turn rate eventually builds up, the centrifugal force on the gentre of gravity will make the ship heel outwards in the turn, i.e. starboard side up. This is a non-minimum phase behaviour. Everybody who have been passengers onboard a ship have experienced that the ship makes a damped oscillation in roll when the heading is altered. The non-minimum phase behaviour described above is the reason for this phenomenon.

Further reading: Perez and Blanke: Ship Roll Motion Control. Annual reviews in control, vol 36 (2012), pp. 129-147. DOI: 10.1016/j.arcontrol.2012.03.010

Imagine heating a house with coal furnace. If the temperature is too low, you add more coal to heat the furnace up. But at first step, you actually achieve the opposite, the temperature is reduced, because added coal dumps the fire. In the second step, fire gets more power and the temperature begins to rise. (This is an example from the control lessons that I had taken during my study).

Nonminimum-phase zeros - much to do about nothing - classical control - revisited part II

Hoagg, J.B. ; Bernstein, D.S.

Control Systems, IEEE

Volume: 27 , Issue: 3

Digital Object Identifier: 10.1109/MCS.2007.365003

is a good read.

Another example could be the feeding of bacteria (their number is evaluated through a mean value accounting for births and deaths within the overall considered population). Roughly speaking when you feed bacteria they start to eat and then "forget" to reproduce themselves. The mean number of bacteria first decreases (they are still dying with the same rate) and then increases (as they are stronger) until a neaw equilibrium. From a mathematical point of view it makes appear an unstable zero in the transfer function when the model is linearized ...

There were lots of good examples and explanations.

For a simple illustration, I take two simple one-pole transfer functions, of the form

G(s)=K/(1+sT), so T gives us the time-constant and K the dc gain.

The step response of

g1(s)=1/(1+2s)

would end at 1, with a time constant of 2 secs

The step response of

g2(s)=-1/(s+2)=-0.5/(1+0.5s)

would only end at -0.5 (in the negative direction), with a faster time constant of 0.5 secs

The sum is

g(s)=g1(s)+g2(s)= 1/(1+s/0.5)- 0.5/(1+s/2)=(1-s)/((s+1)(s+2))

Everything is stable, including, of course, the sum, or in other words "stable pole-dynamics." The negative sign in the nominator of the summation means a non-minimum phase zero, or "unstable zero-dynamics."

The two components together result in a faster yet ultimately lower effect in the negative direction, which is felt first, before the slower yet ultimately dominant efect in the positive direction becomes dominant.

The result is an initial bump in the negative direction that ultimately ends in the intended positive direction.

Hope it helps

(I wanted to add a figure, yet I see that RG does not allow it)

Well,very good examples, and I may add one another; a time-delayed system which is so often:

Imagine a pipe which its valve is just on the one side of it that leads water to the other side and into a Tank; By calculating its time-domain equations,you will see that there is a delay, proportional to the length of the pipe which causes the transfer function to be non-minimum phase.

Why? Because even after closing the valve, the remaining water in the pipe will still flow into the tank; Remember the behavior of a Non-minimum phase system:

when your input is positive, It will response in the negative direction, for a small time period!

I may tip that although non-minimum phase TF is not a rare circumstance, in practical applications, it may be neglected in some applications.

I collect these examples for my system dynamics courses at Penn State, and the mathematical answer is that a non-minimum phase system is one where there are any zeros of a transfer function in the right-half s-plane. More practically, the cause of non-minimum behavior is due to phenomenon where the rate dependence of a system response is in the opposite direction of the steady response. In layman's terms, a non-minimum phase system is one where the initial response of a system works in the opposite directions as the long-term response of a system.

Some great physical examples are listed below that exhibit non-minimum-phase behavior, though I'm not certain that all have been fully analyzed to be CERTAIN that there's non-minimum-phase zeros lurking in the system transfer function. Further, while the examples below try to illustrate the effects of the initial behavior moving against the long-term behavior, the mathematical definition of a non-minimum phase system does not REQUIRE that the non-minimum phase effect necessarily be observable in the response.

* A boat turning right or left - a boat steered to the left (port) will initially move starboard (right) and then swing the boat to the left (port) because the initial forces on the rudder are opposite in direction of the forces on the prow of the boat. This is why you MUST have a tugboat to move large ships away from docks, since the ship must initially swing into the dock to turn away from dock after an initial transient. The same phenomenon happens with rear-steered passenger vehicles (or front-steered vehicles that are backing up). This also occurs in aircraft turning right/left or up/down (noted next).

* An aircraft gaining altitude - an elevator must tilt upward to eventually point the nose of the aircraft up. But in pointing upward, it reduces the net lift on the aircraft and as a result, the aircraft initially decreases altitude before gaining altitude.

* Digestion and most metabolic processes are non-minimum phase in terms of energy - in order to gain calories from food, you body must expend calories to break the food down (chewing, digestion, etc) but in the long term you should gain more energy from the food than you expended. Thus it can be very dangerous to feed complex foods to someone on the verge of starvation - the act of eating can kill them, as will their eating too much initially. Similarly, if one is doing an activity that requires large energy use, it may make sense NOT to eat since eating takes energy away initially after eating - this is why it is a bad idea to eat prior to most sports. This non-minimum phase effect is also why endurance sports where eating is required during the activity (ultra-marathons, full triathlons, etc) are as hard to train for in terms of eating as they are in terms of physical fitness!

* Reproduction is generally non-minimum phase in terms of survival of the individuals - plants that produce seeds, humans that produce babies, etc are more at risk for death than those that save their energies / efforts / time to NOT produce offspring. However, to state the obvious, reproduction provides long-term benefits to the propagation of a species.

* Pheromone bug traps are non-minimum phase - a gardener who puts out traps that attract bugs to their death by using odors will initially attract MORE insects to their garden area. However, after a time, these bugs will be (hopefully) caught in the trap and thereby the long-term presence of bugs around the garden will be reduced.

* Balancing a pole on your hand is non-minimum phase - in order to tilt the pole to the right, you initially have to swing your hand to the left and then to the right. This is a great example of using of non-minimum-phase zeros in the control loop to stabilize an otherwise unstable system. This effect also comes into play with rocket launches, since a rocket engine must balance the rocket above it during take-off, just like a pole on one's hand. If the rocket motor cannot be steered fast enough to account for this behavior, the rocket will "fall over" and thus crash.

* A shower with combined hot/cold flow valves where hot/cold supplies are both very different than ambient temperature will exhibit non-minimum-phase behavior - this effect is very noticeable in cold climates with poorly insulated walls and fixtures where the shower head is far from the flow control faucets. For example, if a shower is too cold, you turn on the hot water valve. This increases the flow rate through the pipes from the valve to the shower head. This results in less time for the cold water already in the pipe to exchange heat, resulting in an initially COLDER shower and then hotter shower. Similarly, if the water is too hot, increasing the cold water flow increases the flow rate of the hot water already in the pipes, resulting in less heat loss to ambient air, resulting in HOTTER shower temperatures for a short time before the water becomes colder. This is a wonderful example of how time-delays in a system can cause behavior that is non-minimum phase. It's also a good motivation for the use on non-minimum phase zeros in the Pade transfer function approximation of a time delay.

* A jet-ski's ride height above the water is non-minimum phase, since the inlet sucks from underneath while the speed lifts the entire ski relative to the water - if you increase the throttle on a jet ski, the engine will initially intake more water underneath and actually sink lower into the water. But once up to speed, the jet ski rides much higher atop the moving water. Similarly, when at high speeds, shutting off the throttle initially causes less suction of water through the intake underneath the ski, resulting in higher ride heights before the jet ski slows down and sinks further into the water.

* Most medical practices are non-minimum phase in terms of the health of the patient - for example, to make someone with a diseased appendix healthier, you initially cut them open (which is decidedly unhealthy) to remove the appendix. Similarly, many cancer cures require the ingestion of toxic poisons that initially make the patient feel much worse. This effect brings irony to the medical motto of "first, do no harm." And exercise is similar to medical practice: it usually makes you feel worse (less energy) immediately after exercising, while making you feel better in the long run.

* Vacuuming a house to reduce dust in the air - initially, running a vacuum cleaner will produce more particulates in the air before (hopefully!) sucking up enough dust and dirt to eventually reduce the net particulates (after cleaning is done). Similarly, shutting down a factory's air-handling system may initially reduce air flow and thus particulates moving about, but in the long term may result in significantly more dust particles since they aren't being filtered anymore by the air system.

* Hiring a new employee (or making an investment, or buying new equipment, etc) - the goal in hiring someone is to increase productivity, but usually productivity in a group with a new hire initially DECREASES due to the training required. Similarly, if companies stop hiring new people, they may see temporary productivity INCREASES due to the time savings of not having to continually train new people. The same trends are seen with savings, large equipment purchases, etc - initially, there are short term losses that may lead to long-term gains. Indeed, many company managers make tons of money by swinging the fundamental trade-off between short and long-term performance, e.g. sacrificing long-term performance of a company for short-term gain (which is ultimately horrible for the company) versus those making short-term sacrifices for long-term benefit. There are plenty of examples of CEOs who use this short- versus long-term trade-off for both good and ill.

* A hot air balloon where the hot air is obtained from a small rocket motor that is pointed downward such that the flame of the rocket feeds into the balloon. The initial firing of the rocket will push the balloon downward, but the heating of the air (assuming the balloon is large enough) will eventually lift the entire balloon upward (even if the rocket is left on!)

The following examples have been submitted to me by students and seem to have non-minimum phase effects, but I've not see enough math on either to confirm:

* The flushing a domestic toilet seems like it is non-minimum-phase - the goal is to empty the tank, but to flush it you initially need to dump water into it which makes the take more full (initially).

* The effect of a volcano (or large scale fires) on climate heating may be non-minimum phase - the initial event produces particles that may block the sun producing colder climates immediately, but a volcano (or fire) also emits large quantities of CO2 that trap heat and thus have a net warming effect over decades and centuries.

There are wonderful papers written by Karl Astrom on the mathematical limits of stabilizing non-minimum phase systems and their similarity to time-delay systems, but I think this is beyond the scope of your question.

Great examples and discussion here.

One of the most well-known non-minimum phase responses (I prefer the more intuitive term "inverse response) is boiler drum level. Steam drums in large industrial and power boilers are a saturated mixture of liquid and vapor. When steam out of the drum increases due to demand, level control must increase feedwater flow, which, initially, cools and collapses bubbles in the saturated mixture in the drum causing level to temporarily drop (shrink) before rising, with also an inverse response going the other direction (decreasing feedwater causes temporary level increase, or swell). The phenomena is sometimes referred to as "shrink/swell".

You can find a good example (aircraft altitude change to elevator deflection, as many researchers exemplied below) for Boeing 747 from Franklin's book (Feedback Control of Dynamic Systems). The attachment in my class material illustrates the transfer function and MATLAB plots/m-file showing the NMP system's response. You can also find the good explanation using inverse Laplace transform on why RHP zeros makes an undershoot from the Franklin book at p. 141 in Ch 3.5. Wish this be helpful.

The best example I have seen was one I used when I was a TA for the MIT "Digital Control of Dynamic Systems" course: an inverted pendulum.

Imagine supporting an inverted object, such as (1) holding the end of a broom balanced on the palm of your hand, or (2) a rigid beam connected by a one degree of freedom hinge to a motorized horizontal drive. To move the object in a given direction requires an initial movement in the opposite direction followed by a movement in the desired direction, hence the "non-minimum" phase zero.

-Dan

I teach System Dynamics, and we have found examples with ideas from the students, most of which are given by people enclosed to this question. However, I think the examples given below would be interesting.

1) One of the most popular examples in economics is that an enterprise (business/production/any economic activity) always needs to increase costs in a short period to gain long term of revenue or cost reduction.

Suppose the dynamics for investment period is: -1/(s+1) with negative sign, because it is the cost.

And for the revenue there will be a slower dynamic with positive magnitude, say: 3/(2s+1). Then the sum is a non-minimum phase system with s=5 as its zero.

2) Ice increases in volume when frozen. It's similar to any stuff arrangement process, that needs more space before than you can optimize your space utilization.

3) To gain success in any part of your life, which means welfare and calmness, you need a period of hard working that may accompany troubles.

"There will be convenience after all troubles".

I would like to know if these are good examples or not, so I appreciate whoever can correct me. I particular I'd like to hear from whom has voted this answer opposing.

On January 9th this thread was opened with a question from our indian memeber Sarvesh Singh. He got in total 27 answers.

For my feeling, it is a bit disappointing not to hear from the questioner if one or more of the answers did satisfy him.

I do not give an example since there are several ones already proposed in the enclosed discussions to this question.

However, i want to point out that a minimum phase system is really one which is BOTH STABLE AND OF STABLE INVERSE ( all poles and zeros being stable) . Remember that in this case , it is very easy to plot an approximate phase Bode plot from its modulus counterpart. That is , from the whole family of frequency responses we can obtain from a given one by changing the signs in zeros and poles, there is just one where the phase plot can be calculated graphically, approximately directly from the modulus plot according to the empirical rules of averaging decibels to calculate approximately phases. The total maximum variation of phase is minimum for this paprticular plant within the whole family.

However, it has been very common in the adaptive control community to call minimum phase system to one just with STABLE ZEROS. This appears in that way in papers and books. Of course, this name used in this way is not in accordance with a minimum phase variation on the whole range of frequencies. The approprite way for this is INVERSELY STABLE or of stable inverse, also used in other many books and papers on the subject.

@Manuel De la Sen

You are right, of course, and in Physics they (correctly) call minimum-phase only systems with all poles and all zeros in the LHP, because they are the only systems that indeed exhibit what we can call “minimum-phase.”

However, in Engineering, even long before any Adaptive Control, because both poles in the RHP and zeros in the RHP have special effects, people tried to give both categories special names.

Now, because the evident effect of poles in the RHP on systems is that they are unstable, while the phase effect is only secondary to instability, “unstable” has become their name.

Even if it is not entirely correct, “non-minimum-phase” remained the name for the special effect of zeros in RHP because even if the sytems are stable, they still still exhibit nonminimum-phase and a diffrent time-response.

Trying to be more rigorous, people may call them systems with STABLE or UNSTABLE poles or zeros and yet, at a second thought, actually this is just funny, because all poles and zeros are always "stable." Only the system may show stable or unstable behavior.

Because similar effects also appear in nonstationary systems, where we cannot talk about poles or zeros, the correct name for systems with only zeros in RHP, which then can also be extended to nonstationary systems, is probably, as you wrote, systems with STABLE or UNSTABLE INVERSE or, maybe even better, systems with STABLE or UNSTABLE INTERNAL DYNAMICS.

One of the most common non-minimum examples are aircraft control of the vertical trajectory. Normal acceleration is mostly created by the augmentation of the angle of attack.

Modification of the angle of attack results from the pitch rate which is controlled by elevator position which creates required pitch moment.

Most aircraft empennage (tail) is situated behind the center of gravity, and as result in order to create a positive pitch moment that will increase the angle of attack, thereby creating a positive (upward) load factor the empennage lift variation must be negative and thus is minimum phase.

From a performance point of view this causes the contribution of the tail to be detrimental to aircraft global lift and from a stability point of view to be minimum phase which may be a problem with control techniques such as sliding mode control.

This is why, aircraft manufacturers, seeking short landing performance, in Europe had aircraft equipped with canard controls such at the SAAB Viggen.

Also small canards are sometimes used concurrently with rear elevons in the Dassault Rafale and BEA Typhoon, here the idea is to mitigate the non-minimum phase response of the elevons.

Christian:

Can you please provide some information or bibliography on the detrimental influence of the tail in the case of sliding mode control?

Kaplan turbine is yet one example. When the blade angle is reduced, the power momentarily increases due to kinetic energy of the water.

You can think of an invesre pendulum, e.g., on a chart. When you want to move the pendulum to the left, first you have to move the chart to the right. The pendulum will tilt to the left. Now you have to move the chart fast to the left and overrun the center of gravity to stabilize the pendulum at the new position.

A nonminimum-phase stable system is one that has a zero in the right half of the plane. Physical phenomena that give rise to nonminimum-phase stable behavior include control of the level of a volume of boiling water and hydroelectric power generation.

A very simple case is constituted by (traditional) aircraft control surfaces.

Suppose you want to climb, for that matter you try to increase your lift by augmenting the angle of attack. This is achieved by pulling the stick which creates a positive pitch moment aimed at increasing the angle of attack by reducing the lift of the tail and thus the contribution that the tail brings to the total lift is reduced.

This means that the initial effect that the pulling of the stick is to reduce the flight path angle and this effect is opposite to the global effect sought which is to increase the flight path angle.

Let me further ask two things:

(a) Is NMP a general property of real systems (like nonlinearity)? I would expect so since every system has some delay.

(b) Could an analogy be drawn between the time response of a NMP system and an optimal control policy? Both are trading off immediate penalties with long-term cost savings, as I understand from reading the many examples you have kindly given.

Best regards!

Hi Yutao (or Alex),

a) There are lots of good answers to this question. I hope you do spend time to read and digest them, slowly-slowly and one-by-one. 

As people showed, there are bodies where the physics is such that, when you give a command to move upwards, the first reaction is downwards, but only before the body starts moving to the desired upwards direction. So, yes, this is a property of some real systems, such as planes and missiles and other processes.

In a previous response, I tried to give some simple mathematical representation.

If your plant is G1(s)=K1/(1+sT1)=3/(1+2s), this is a simple first-order-pole with gain K1=3 and time constant T1=2. If you give it a unit-step command u(t), the plant will move upwards and end at y(t)=3.

If your plant is G2(s)=K2/(1+sT2)=-2/(1+s), this is another simple first-order-pole with gain K2=-2 and time constant T2=1. If you give it the same unit-step command u(t), the plant will move downwards and end at y(t)=-2.

If your plant is the combination

G(s)=G1(s)+G2(s)=3/(1+2s)-2/(1+s), there are two opposite actions. Although the second term gain is smaller, the shorter time constant acts first and the plant moves downwards. However, then the larger positive gain takes over and the plant starts moving upwards. ending at y(t)=3-2=1.

I won't do the computations (which I did in the previous response), yet if you make the computations, you end with a denominator with two stable poles, while the nominator contains a non-minimum phase zero.

b) I am not sure I understand your second question. NMP is the property of the system, while, as you indeed write, optimal control is a policy.

Best regards,

Itzhak

A simple electrical circuit example consists of a Bridge circuit. One leg consists of R in series with C, the other leg consists of R in sieries with L. The two legs are parallel connected and fed by the Input voltage. The Output voltage is taken from the Bridge diagonal. The Transfer function has a right Hand Zero.

As given examples, in non minimum phase system the initial response of a system works in the opposite directions as the long-term response of a system. However, I don't see that behavior in the non minimum phase system with transfer function (s-1)/(s+2). I tried to plot the unit step response and see if there is any overshoot or undershoot (as the behavior of non-minimum phase system). If you look at the plot attached below, there is no overshoot/undershoot. Can anyone explain why?

https://s24.postimg.org/cjxa288yt/Non_minimum_phase_system.png

Hi Nguyen,

As I tried to explain, the initial "opposite" move of a simple nonminimum-phase systems can be explained by the competition of a negative faster step-response with lower final amplitude and a positive slower time rersponse with higher final amplitude. So, the decompositon first mainly gives you the fast response towards the negative direction that in time becomes dominated by the slower yet higher amplitude in the positive direction.

Your transfer function is just too simple to show such things. The direct input-output term (the c0onstant) of your decomposition does not have this dynamic effect.

Add just another pole and see the step response of (s-1)/(s+2)/(s+3).

Becaue of the signs, it witl first go positive and then end negative, so maybe yoy may want to also see (1-s)/(s+2)/(s+3) 

Have Fun!

Itzhak

Hi Itzhak Barkana,

Thanks for the explanation! I've got it now. So not every non minimum phase system will show that behavior as those mentioned examples implied. By reading previous posts, I thought that every non minimum phase system should show that behavior. 

I understood it and just tried to analyse and plot two transfer functions you gave in the attached below. It is interesting to know this. Thanks again!

Hi Nguyen,

You are alright and there is not much to add and yet, I wouldn't want to leave your message without a response.

So, in the denominator, the negative coefficient means a right-half-plane (RHP) pole and instability.

Instead, in the nominator, the negative coefficient of the RHP zero results in a stable result yet in a different direction. How visible the time-response effect is depends on the relative amplitudes of various "fighting" terms. Even in my "clear" example, you can play with the coefficients until you may see no effect or almost no effects.

For eventual future tasks, it is good to take into account that the real problems with NMP systems start when this is only your open-loop and you want to use it in closed loop, in particular with nonlinear or adaptive control. Then, NMP seems to make the mission impossible, unless you can compensate for it.

Pitch control of an autonomous underwater vehicle: if it doesn't have forward dive planes, only rear planes or a pivoting rear thruster, the tail end of the vehicle has to rise before the center of mass starts to descend. 

Continuing this topic...

Let's say I have a non-minimum phase system with zeros in RHP. I am from Power Electronics where this behavior is common for some switching power converters (after their linearization). Usually one would just limit bandwidth of a controller to stabilize a closed-loop system. Is there any controller that can provide faster response despite non-minimum phase nature of a system?

 Hi Andrii,

I am working in what is called Simple Adaptive Control (SAC). Here, basically, non-minimum phase (NMP) is just taboo. However a simple device called parallel-feedforward configuration (PFC) overcomes this issue and ultimately ends in performance much beyond what could be expected otherwise. Look for

I Barkana: "Classical and Simple Adaptive Control Design for a Non-Minimum Phase Autopilot," AIAA Journal of Guidance, Control and Dynamics, Vol. 28, No. 4, pp. 631-638, 2005.

The specific NMP object is a UAV, yet the treatment is for any NMP system.

Simply, the steam engine. The principle is the same as "coal-heated house" as above.

A simple dc-dc boost converter is a NMP system which can reliably be controlled by a numerous techniques such as SMC , Synergetic , Fuzzy....

Hi Mr Barkana

I know that by parallel feedforward compensator and simple adaptive control it is quite straight forward to stabilize non-minimumphase systems. however, in tracking cases the problem becomes difficult because it will result in output tracking errors. The problem becomes even harder when the system has nonlinearities and uncertain terms. It is also known that one way to design PFC is to find a stabilizing controller and use its inverse in parallel with the system to constitute an ASP augmented plant. I think for NMP systems one can rarely find stable PFCs or at least PFCs with small steady state gain. As a result, it seems that in this method, tracking performance for nonminimum phase systems (especially not LTI NMP systems) gets poor. Now the question is, since you are so active on this topic, is there any way to find PFC which result in good tracking for NMP systems (Preferably non-LTI uncertain or nonlinear ones)?

Itzhak Barkana

Hi Surena,

(and I am Itzhak for all, no need for Prof, Dr., Mr., or Sir, except for strictly official occasions)

While linear means the “simple” first-order polynomial, nonlinear means anything else, and I never thought to propose a general solution for the general problem. Any nonlinear system is a world of itself and before being able to even start thinking about controlling a system, one must first see what one can know about the system.

When you complain that tracking with PFC ends in errors, I dare to ask: “compared to what?” and I understand that you might have something better in your mind.

Now, more seriously, if we don’t know anything about the system, we cannot do much (unless you believe that you do know the exact rank of your system, can build a perfect identifier, and use this for your controller).

So, what I think we do have in real-world systems is some way to know that the system is stabilizable (or not) and, even better, to be able to reach some stabilizing controller. If we do, then we can add fictitious controllers (including improper controllers) that would allow us to use high gains. Then, instead of actually implementing and using those fictitious controllers with high gains, we can use their inverse, the PFC, with small gains.

This is the general idea, yet to really get results with any specific system, we have to deal with its specific problems in detail.

I don’t know if you know the papers below.

[A] made the extension of the theory to nonlinear and time-varying systems. This found Steve Ulrich exactly when he needed to end his Ph.D. work on flexible robots and led to a few papers, such as [B] and {C} and others, then I thought to deal with the good tracking problem in [D].

I am not sure if this helps, yet I cannot provide a short answer and an easy solution to a difficult problem.

Please feel free to always come back with any specific questions or comments.

With best regards,

Itzhak

[A] I. Barkana: Output feedback stabilizability and passivity in nonstationary and nonlinear systems. International Journal of Adaptive Control and Signal Processing 24(7), 568–591 (2010)

[B] Ulrich, S., Sasiadek, J., Barkana, I.: Modeling and direct adaptive control of a flexible-joint manipulator. AIAA Journal of Guidance, Control and Dynamics 35(1), 25–38 (2012)

[C] Ulrich, S., Sasiadek, J., Barkana, I.: Nonlinear adaptive output feedback control of flexible joint space manipulators with joint stiffness uncertainties. AIAA Journal of Guidance, Control and Dynamics 37(6), 441–449, 2014. DOI: http://arc.aiaa.org/doi/abs/10.2514/1.G000197

[D] Itzhak Barkana, ”Robustness and perfect tracking with simple adaptive control in nonlinear systems.” Mathematics in Engineering, Science and Aeronautics - MESA, 9(1):21–45, 2018