[Martin Taylor 960723 11:00]

I thought that the following message, which was distributed to the system

dynamics (SD) mailing list, might be of interest, given the occasional

discussions we have had about the veracity of discrete simulations of

continuous control loops.

Martin

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Many many moons ago, there were discussions on discrete/continuous simulation:

discrete in time or integral-valued variables. I was going to offer a control

engineer's perspective: computer-controlled systems, but somehow never got

around to doing it. While thinking about setting up a computer-controlled

system for my experiment, I recalled the unfinished business on my part (Gene,

here it is, I am finally able to write down some insights from computer-

controlled systems). So, I shall finish this one first before moving on to

HyPE.

I think there are many insights to be gained from this perspective; some

include discrete-time measurements and the interval between measurements.

Note that the word "discrete" only applies to time, i.e. time is represented

as a sequence of integers 1, 2, ..., with the duration between two sequential

counts is called the sampling interval (or sampling time, Ts).

Contents:

0. Background.

1. Discrete- v. Continuous-Time.

2. Necessity for Added Complexity due to Hybrid Formulation.

3. Food for Thoughts.

0. Background.

The advent of fast digital computers increases the ease of implementation of

controllers. Fast digital computers also guarantees practical application of

nonconventional control algorithms such as intelligent and adaptive. However,

there is added complexities, such as finite precision arithmetic, resulting

from the use of digital computers in the control of engineering systems.

Finite precision arithmetic is due to the way digital computers represent

a real number in a string of zeros and ones. This phenomenon is called the

quantized effect of finite precision arithmetic.

As an example, a computer equipped with a 10V-bipolar 12-bit analog-to-digital

converter represents -10 V as 0, 0 V as 2048, and 10 V as 4096. In the above

example, a 2.75 V signal is recognized as an integer (2.75 - (-10))/20*4096 =

2611. The representation of any signal ranging from -10V to 10V has integral

value ranging from 0 to 4096 (for a 12-bit ADC, 2^N for an N-bit ADC). Note

the similarity between this representation and human counts; there can only be

1, 2, or any number of persons.

The computer then performs arithmetic operations to calculate the amount of

energy for the actuator(s) to expend in an effort to drive the plant. The

amount of time required to perform this computation depends on its

complication. A PID (proportional-integral-derivative) control algorithm

is one of the fastest to perform. Many times, the computation is performed

in software. Consequently, the control action is only available discretely

in time. Compare this to an analog controller in which control action is

available continuously; examples of analog controller is the electronic

circuit implementation of a PID controller with the use of resistors,

capacitors and transistors.

The control action out of the computer should be continuous in time because

of the continuous operations of actuator(s). There is also an added

complication related to the reconstruction of a continuous-time from

discrete-time signal that I opt not to discuss at this time. In a nutshell,

it is necessary that the control action be available at regular intervals.

Selection of the sampling frequency is also quite involved as it has a lot to

do with the speed of response of the plant to control and the computational

speed of the computer. We also must select it in such a way that aliasing,

fast changing signals appearing as slow changing ones, doesn't occur. So far,

I haven't been able to find such phenomenon in other fields than engineering.

Ideally the sampling frequency should be ten to twenty times the plant

natural or fastest frequency. This also ensures relevancy of the fed-back

signal.

Another direct result of this computation is the control action always lags

the measurement; the lag time depends on the speed of computation. For this

reason, in the early days of computer-controlled systems, engineers also

employed hardware-assisted computing with the use of digital signal

processing hardwares. The point is there exists a delay between measured

behavior and any action intended to affect the system's future behavior.

Note the similarity between this and an inventory control systems in

several manufacturing industries: a production output depends on

inventory and sales reports which are available only periodically. The

similarity becomes apparent as one has to wait for the product to finish

all the operations. Similar phenomenon is also observable in the fastfood

world or other service industries.

It is not very clear, in the socio-economic realm, the impact of less-than-

frequent measurements on the success of any measurement-based policies to

affect the behavior of the system. In the science/engineering realm, however,

the impact is better understood because of the existence of well-formulated

theories and their mathematical representations.

In the world of control engineering, the following terminologies are used:

-Discrete-time signal: signal available periodically at specific intervals.

-Quantized signal: integer-valued signals.

-Digital signal: discrete-time and quantized signals.

1. Discrete- v. Continuous-Time.

The name of the game is "Computer-Control that Process" which means

implementing a control algorithm to influence the behavior of

hardwares. The resulting computer-controlled system must not cause

damage to the hardware(s) or injury to the operator(s). This is the

role of modeling for control--useful for obtaining operating parameters

that yield a desired behavior.

Many physical systems operate continuously in time, yet our computer-

controlled system takes measurement and sends control command at

periodic intervals. Managers will realize that this is also one way

they operate (periodic measurement and policy implementation). They

look at monthly or quarterly reports and decide what steps to take to

go after their goal(s), e.g., bottom line, etc. So do people in

government in making policies involving economy (interest rates, taxes,

commerce, etc.), crime and policing, etc. Of course here the measurements

are also used for prediction that plays a role in selection of policy or

decision.

Having known the purpose of modeling and also operations of the system,

we are now ready to discuss the role of continuous and discrete modeling

in the design and implementation of control algorithm(s) for the intended

purpose.

Regardless of the mode, continuous or discrete time, many controllers

have been designed initially with the use of modeling and simulation

that can be performed on analog computers or using numerical integration

on digital computers. Let's ignore the fact that numerical integrations

on digital computers are also approximate! The main focus is on the

mathematical representation of the system. A continuous-time system is

typically represented as

dx1c/dt = f1[x1c(t),x2c(t),...,xNc(t),u(t),t]

dx2c/dt = f2[x1c(t),x2c(t),...,xNc(t),u(t),t]

...

dxNc/dt = fN[x1c(t),x2c(t),...,xNc(t),u(t),t]

whereas a discrete-time system is given by

x1[(k+1)] = F1[x1(k),x2(k),...,xN(k),u(k),k,T]

x2[(k+1)] = F2[x1(k),x2(k),...,xN(k),u(k),k,T]

...

xN[(k+1)] = FN[x1(k),x2(k),...,xN(k),u(k),k,T]

The relationship between the two representation is t = kT, k = 0,1,...

T is the sampling interval (time).

Physical plants operates continuously in time. Their variables (states)

have non-quantized numerical values. Some plants operates continuously

in time, yet some of their variables have quantized numerical values even

though most of the other variables have non-quantized values. Examples

of these "plants" are in manufacturing and service industries.

The measurement and controller portion of the "closed-loop" system

generally operates discretely in time. This is particularly true in a

computer-controlled system as well as in some of the aforementioned

social, economic and political systems.

In simulating the closed-loop systems, control engineers will do either

(1) transform the continuous-time representation of the "plants" into

its discrete-time form (2) insert a zero-order-hold between the controller

and the plant. A zero-order-hold extrapolates a discrete-time signal such

that it would appear continuous by holding the numerical value of the

signal at one time step constant over an interval until its value changes

at the next time step.

In other words,

for kT <= t < (k+1)T, xc(t) = x(k)

This is actually what happens in a computer-controlled system; the digital

control command from the computer is converted into an analog actuating

signal with the use of some logic circuits that perform the zero-order-hold

operations described above.

Let's look at the question that control engineers asked themselves during

the infancy of digital control theory, "Is the added complexity of hybrid

modeling and simulation necessary?" or "What is the criterion for designing

a computer-controlled system using a continuous-time formulation only?"

Note the question above is the result of my paraphrasing their quest for

better understanding at the time. Note also its similarity to the

questions in this list some time ago; such as "Is SD non-quantized formulation

sufficient to depict the behavior of the rabbit-fox dynamics in which the

rabbit and the fox have clearly integral values?" or "Can SD formulation be

used to describe the borrowing-and-paying-off debt in which interests on the

principal is calculated periodically?", etc.

2. Necessity for Added Complexity due to Hybrid Formulation.

The direct correlation between the continous- and discrete-time domain is

described in details as we explore the conformal mapping from the Laplace-

to Z-transform complex domain. Rather than spending time on Laplace and

Z-transforms, I urge whoever is interested to consult the literature on the

subjects. Instead, I will mention intuitively the basic transformation that

also determines a condition that allows the use of continous-time modeling

and simulation to design the controller of the system. Intuitively, the

determinant factor here is the sampling time T.

Ref --> Controller --> Switch --> Zero Order Hold --> Plant --> Output

> >

> >

><-------------------- Measurement <--------------------|

The switch closes when time is an integer multiple of T; it is open

otherwise. When it is closed, the system operates in a closed-loop mode

--control action is based on the deviation of output from its desired

(reference) value (negative feedback loop). When it is open, the plant

operates in an open loop mode in which the input into the plant is constant,

at the value of the control command at the previous time step. The longer

the system operates open-loop, the less efficient is the feedback action!

Also, as the sampling time depends on the speed of processing the fed-back

signal into the control command, the longer it is the controller becomes

less effective as it tries to affect the output based on obsolete measurement.

The result is a system that is unstable--blowing up or exhibiting "violent"

oscillation with growing amplitude until the system fails. Many times,

though, the outcome is not so dramatic that the output "only" exhibit damped

oscillation.

As discussed earlier, the dependency of the sampling time T on the speed of

computing and the lowest characteristic time in the system complicates the

design of a computer-controlled system. There are two contradicting criteria:

(1) sampling as fast as possible yet (2) allowing sufficient time to process

the measured data. As T gets closer to the lowest charasteristic time, we

can use continuous-time modeling and simulation to design our control

algorithm. Otherwise, control engineers have to use the hybrid modeling and

simulation described above.

3. Food for Thoughts.

In an effort to "control" the behavior of social system, people have employed

the feedback control methodology. They take measurements, compare them to

the desired behavior and, finally, make decision (or implement policy) that

will bring the behavior closer and closer to the target. Many times, we have

witnessed the lead time required to process the raw data into useful

indicators. This lag is analogous to the computing time in a computer-

controlled system. Essentially there is no control imposed on any social

system in between measurement time which can vary from weeks to months to

quarters.

In socioeconomic/sociopolitical systems, the impact of such a delay is less

understood that it is in science/engineering systems. Now, my questions are

(1) Has anyone of the esteemed SD practitioners or policy makers on this list

studied such an impact?

(2) Has anyone used SD to estimate the fastest changing dynamics in the system

of interests?

(3) How does one determine the frequency of measurement and control?

(4) Are there studies on "aliasing" phenomenon in systems other than science/

engineering?

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Subject: Insights from computer-controlled systems. (SD0367)

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Thanks for your attention.

Regards,

-B-

bbens@mit.edu

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