Università degli Studi

di Modena e Reggio Emilia

Automation

Robotics and

System

CONTROL

Modeling and Simulation

Cesare Fantuzzi

University of Modena and Reggio Emilia

02/11/2011

1

Course syllabus

Part 1: Theory of Modeling and Simulation.

Part 2: Numerical simulation.

Part 3: How develop simulation projects using Matlab

and Simulink.

Part 4: Case studies.

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Part 2: Numerical Simulation

The purpose of this part is to discuss the properties of numerical Ordinary Differential Equation (ODE) solvers and their codes, as well as the algorithms behind these solvers.

Numerical simulation. Simulation time and physical time (simulation clock, time step).

Integration using Euler formula. Numerical stability of algorithm. Stiff Differential Equation. Fixed and variable simulation step.

Simulation Error Analysis.

Multi-domain simulation. Co-Simulation (Cooperative simulation).

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Time

When we simulate a continuous-time system on a digital

computer, some quantity will have to be discretized, as

we cannot update the state variables infinitely often

within a finite time period.

Most numerical ODE solvers discretize the time axis, i.e.,

they advance the simulation clock using finite time steps.

The time step, h, may either be fixed or variable.

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Principles of Numerical

Simulation

Let’s consider a generic ODE (ordinary differential

equation) which represents first order ordinary

differential equation (could be non-linear).

where x is the state vector, u is the input vector, and t

denotes the time, the independent variable across which

we wish to simulate.

We also require initial conditions for the state variables:

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)),(),(()( ttutxftx

00)( xtx

Taylor Series Expansion

The model can be simulated using a Taylor series

expansion. If we know the state vector at a certain

instant of time, t, the state vector can be calculated at

some later time instant, t + h by means of a Taylor series

expansion:

The state-space model is used to compute the first

derivative in the Taylor series:

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...!2

)()()()(

2

2

2

h

dt

txdh

dt

tdxtxhtx

...!2

)()),(),(()()(

2

2

2

h

dt

txdhttutxftxhtx

The Truncation Error

Numerical integration approximate a certain number of

terms of the Taylor series.

This number can be either fixed or variable.

Approximation order of the numerical method.

An algorithm that approximates the terms of the Taylor

series up to the third derivative:

is thus an algorithm of third-order.

The truncation error of the method grows proportionally

with the fourth power of the integration step size, h.

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)...(!3

)(

!2

)()()()( 4

3

2

22

hoh

dt

tfdh

dt

tdfhtftxhtx

The Explicit Euler Integration

The most simple numerical ODE solver is based on the

explicit so-called “Forward Euler” (FE) formula

First-order integration method:

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httxftxhtx

htxtxhtx

)),(()()(

)()()(

Iterative integration

When using explicit integration methods, the simulation

doesn’t require any iteration within an integration step,

unless the model contains algebraic loops:

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Time and Again

The step size, h, is not necessarily identical with the time

advance, dt, of model evaluations.

Many integration algorithms, such as the famous Runge-

Kutta algorithms, perform multiple model evaluations

within a single time step.

Thus, each time step, h, contains several micro-steps,

dt, whereby dt is not necessarily a fixed divider of h.

Instead, the simulation clock may jump back and forth

within each individual time step.

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Variable step solvers in

Simulink

The variable-step solvers in the Simulink product

dynamically vary the step size during the simulation.

Each of these solvers increases or reduces the step

size using its local error control to achieve the tolerances

that you specify.

Computing the step size at each time step adds to the

computational overhead but can reduce the total

number of steps, and the simulation time required to

maintain a specified level of accuracy.

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Time and Again

Even if the integration algorithm used is such that dt remains

positive at all times, the simulation clock does not necessarily

advance monotonously with real time.

There are two types of error-controlled integration algorithms

that differ in the way they handle steps that exhibit an error

estimate that is too large.

– Optimistic algorithms simply continue, in spite of the exceeded error

tolerance, while reducing the step size for the subsequent step.

– conservative algorithms reject the step, and repeat it with a smaller

step size.

Thus, whenever a step is rejected, the simulation clock in a

conservative algorithm turns back to repeat the step, while

not committing the same error.

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Time and Again

Even if an optimistic algorithm with positive dt values is being

employed, the simulation clock may still not advance

monotonously with real time.

The reason is that integration algorithms cannot integrate

across discontinuities in the model.

Thus, if a discontinuity is encountered somewhere inside an

integration step, the step size must be reduced and the

step must be repeated, in order to place the discontinuity in

between subsequent steps.

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Example

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1

s+1

Transfer Fcn

Time

To Workspace1

simout

To WorkspaceCoulomb &

Viscous Friction

Clock

Chirp Signal

Variable Step

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Prompt Matlab

>> whos

>> % how long are the vector Time and simout?

>> plot(Time,’x’)

>> % how is spaced the vector Time?

>> plot(Time, simout, ‘x’)

>> % Do you see the variation on space in time are related

with the discontinuities in simout?

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Fixed Step

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Programmatic simulation

Entering simulation commands in the MATLAB

Command Window or from a MATLAB file enables you

to run unattended simulations.

For example, you can perform Monte Carlo analysis by

changing the parameters randomly and executing

simulations in a loop.

You can use the sim command to run a simulation

programmatically.

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Example

Run several simulations to test PID tuning parameters.

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Kg

s +(1/tau)s2

Transfer Fcn

SimTime

To Workspace1

simout

To WorkspaceStep

PID(s)

PID Controller

Clock

Add

Algebraic Loop

An algebraic loop is formed when two or more blocks

with direct feedthrough form a feedback loop.

The basic problem with algebraic loops is that the output,

y, at time, t, is a function of itself

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y(t)Ku(t) - Ky(t) = K 211

Simulink and Algebraic Loops

In general, at each time t, SIMULINK assumes that the initial guess for y at the current integration step, yguess, is equal to the value at the previous integration step, y[t-1]:

The current calculated output of the K1 Gain block is then computed as

If ycalc = yguess: yguess is accepted as the output, y, at time t.

If ycalc != yguess, then an iterative Newton-Raphson technique is used to modify yguess until either the solution converges (within some tolerance) or the maximum number of iterations has been reached.

The maximum number of iterations, 200, is hard-coded into SIMULINK.

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1]-y[t = yguess

yguessKK -u K = ycalc 211

An «unsuspecting» algebraic

loop

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s+1

s+10

Transfer Fcn

SimTime

To Workspace1

simout

To WorkspaceStep

Clock

Backward Euler method

Another numerical integration method of first order is the

“Backward Euler” (BE) method:

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hhthtxftxhtx )),(()()(

The Numerical Stability

Domain

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The Numerical Stability Domain II

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Simulation With the FE Algorithm

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Computation of the Largest Numerically

Stable Integration Step Size for FE

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Simulink Example

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Improve accuracy

Improve solution accuracy with an integration algorithm

consisting in a prediction step using FE (Forward E.)

correction step using BE (Backward E.):

After some math:

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The Heun Integration Algorithm

The method, called Heun integration algorithm,

combines FE e PE algorithms:

To obtain:

Which is a second-order approximation of the ODE

solution.

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The Family of Explicit Runge-

Kutta Methods

If we allow more than one prediction stage, it is possible

to obtain higher-order integration algorithms:

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4th-order accurate Runge-

Kutta (RK4) algorithm

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RK4 algorithm

The next value (xn + 1) is determined by the present value (xn) plus

the product of the size of the interval (h) and an estimated slope

The slope is a weighted average of slopes:

is the slope at the beginning of the interval;

is the slope at the midpoint of the interval, using slope to

determine the value of x at the point tn + h / 2 using Euler's method;

is again the slope at the midpoint, but now using the slope

to determine the x-value;

is the slope at the end of the interval, with its y-value

determined using

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kx

1Px

2Px

3Px

kx

1Px

2Px

History of Explicit Runge-

Kutta Methods

The number of non-linear equations grows rapidly with the

order of the methods. Already for RK methods of order 5, there

no longer exists a solution in 5 stages.

More stages must be added in order to increase the number of

parameters.

Because of the many non-linear equations to be solved, it took

a long timebefore higher-order RK methods were found.

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Concluding Remarks

Explicit Runge-Kutta algorithms of various orders of

approximation accuracy were developed.

All FRK algorithms except FE are multi-stage algorithms that

require internal function evaluations.

FRK algorithms do not preserve any information across multiple

steps, i.e., these algorithms are self-starting and start afresh

with each new integration step.

The class of explicit RK algorithms are among the most widely

used numerical ODE solvers on the market today.

For most engineering problems, 4th-order FRK algorithms offer

a good compromise between the needed accuracy and the

economy of simulating across a single step.

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Stiff Systems

We call a linear system stiff, if it is stable and its eigenvalues vary a lot in terms of their real parts (have parts with very different dynamics).

Non-linear systems are called stiff, if they are stable and exhibit both fast and slow modes in their behavior

– The linearization of such systems leads to stiff linear systems.

These systems cannot be simulated efficiently by means of any explicit RK algorithm, because we would need very small time steps to move those eigenvalues (of either the system itself or of its linearization) that are located most to the left in the complex plane into the numerical stability domain.

This is not only a problem of the explicit RK algorithm, but of all explicit numerical ODE solvers in general.

For the efficient simulation of stiff systems, implicit integration algorithms are required.

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An example

A stiff ODE is an ordinary differential equation that has a

transient region whose behavior is on a different scale from

that outside this transient region.

A physical example of a stiff system involves chemical

reaction rates, where typically the convergence to a final

solution can be quite rapid.

Simulate with Matlab/Simulink the system and display the

length of the output array with the command ‘whos’:

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1

s +1001s+10002

Transfer Fcn

SimTime

To Workspace1

simout

To WorkspaceStep

Clock

Rules of thumb

A nonstiff solver is primarily concerned with accuracy.

When encountering a stiff problem, a nonstiff solver will

reduce its step size accordingly, making it much more

inefficient.

Identifying a stiff system is one of the more important

steps in the process of numerical integration, as a

nonstiff solver is much less efficient than a stiff solver.

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Rules of thumb (cont.)

If the eigenvalues are obtainable, or available, a measure of stiffness can be calculated.

This stiffness ratio is the ratio of the eigenvalue with the largest magnitude to the eigenvalue with the smallest magnitude.

If the region of integration is on a region with no transient, the equation is not stiff. A stiff equation must have a transient.

Understanding what you are modeling is a great advantage when you are choosing a solver, especially in this instance. If you expect behaviors on different scales, you might want to choose a stiff solver.

Finally, and somewhat unfortunately, you might want to choose a stiff solver if you have tried a nonstiff solver and found it to be very inefficient and time-consuming.

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System with jumps and

discontinuities.

All of the simulation methods that we encountered until now

operate, in one form or another, on polynomial extrapolations

using Taylor series expansions.

A fundamental property of polynomials is that they don’t exhibit

discontinuities.

Consequently, a model that contains discontinuities cannot be

simulated across these discontinuities using polynomial

extrapolations.

A large majority of engineering systems exhibit many

discontinuities that need to be included in their models.

The numerical integration methods for dynamic systems that

we discussed until now can thus not be used for the simulation

of hybrid systems.

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Abusing the Step-size Control

What happens if we simply close our eyes and integrate across a discontinuity of the model using any one of the numerical ODE solvers with step-size control introduced earlier?

The algorithm doesn’t know that there exists a discontinuity. What it does notice is a rapid change in the trajectory.

The algorithm concludes that a new eigenvalue appeared far out to the left in the complex plane.

Consequently, the algorithm reduces the step size in order to capture this eigenvalue in its accuracy domain.

However, the new “eigenvalue” is a joker. It doesn’t allow itself to be captured.

Irrespective of how much the step size is being reduced, the eigenvalue remains outside the accuracy domain.

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When does the iteration on the

rejected step end?

The step-size reduction continues until one of two things

happens:

– The step size is reduced to the smallest value specified in the step-size

control algorithm.

– The step-size control algorithm decides that the integration accuracy is

acceptable. This will eventually happen, as by reducing the step size, the

non-linear terms in the Taylor series expansion lose their importance.

With a sufficiently small step size, every algorithm behaves like either

Forward or Backward Euler.

Once the discontinuity lies in the past, the evasive eigenvalue

disappears as miraculously as it had shown up before.

The step-size control algorithm slowly increases the step size

again, until it reaches its optimal value . . . or until it encounters

the next discontinuity, whichever happens first.

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Discontinuity

Abuse of the step-size control algorithm for the localization of

discontinuities often works quite well, and it is for this reason

that many of the more primitive environments for the modeling

and simulation of dynamic systems don’t offer any special

provisions for handling discontinuities in the model.

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Time Event

Our problems arose from the fact that we didn’t instruct the

ODE solver that there were discontinuities. The ODE solver is

incapable of reading and interpreting models. It can only

execute them.

What we need is a syntactical element in the model description

language that enables us to explicitly inform the integrator of

occurrences of discontinuities.

From now on, we shall call discontinuities discrete events. What

we need are explicit mechanisms for the description and

handling of events.

In some cases, the time of occurrence of an event is known in

advance. In this case, we talk about time events.

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Time Event Scheduling

One way to advise the simulation of forthcoming time events is

by offering in the language a schedule statement that can be

used to schedule future time events.

For example, we might include in the initial section of the

program the following statements:

State = Open;

schedule CloseGateEvent at tevent;

and in the code of the CloseGateEvent function:

State = Closed;

schedule CloseGateEvent at time=t + period ;

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Integrate between

discontinuities

It should be mentioned that the discontinuity associated with the event is not part of the continuous-time model. All simulation trajectories in between events are perfectly continuous.

For this reason, the ODE solver won’t encounter any problems. It never attempts to simulate across a discontinuity in the model.

The localization of a time event is trivial. All we need in order to localize time events accurately is an ODE solver that provides dense output.

We proceed in exactly the same manner that we use to localize a communication instant, at which we wish to report the values of the output variables.

If the integration algorithm reduces the step in order to hit the communication instants (common for single-step algorithms), we shall do the same in order to arrive at the time of the next time event.

On the other hand, if the integration algorithm uses interpolation for the purpose of calculating the values of the output variables at communication instants (common for multi-step algorithms), we shall do the same for localizing time events.

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Time events

Once an event has been localized, the actions associated with the event are being executed.

Afterwards, we are dealing with a new continuous-time simulation with new initial conditions.

The time events are stored in an event calendar, a linear linked list, in such a way that the next time event is always stored at the beginning of the list.

The continuous simulation only needs to know the instant of time when the next time event is scheduled to occur.

The continuous simulation proceeds until the time of occurrence of the next event. At that moment, the continuous simulation terminates, the actions associated with the event (i.e., the discontinuity) are being processed, and a new continuous simulation starts with new initial conditions.

21/03/2011 47

Execute in segments

Consequently, the continuous simulations are now

executed in segments. The individual continuous-time

simulation segments are interrupted for handling discrete

events that occur.

A model that specifies explicitly the discrete events is

called hybrid model.

The simulation of a hybrid model is called hybrid

simulation.

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State event

Often the time of occurrence of an event is not known in

advance. What is known is only the condition under which the

event is to take place.

Such events are called state events.

State events cannot be planned.

To this end, we might include in the section where the

simulation equations are being described by Boolean equation

as:

schedule OpenEvent when Condition is True;

and the code of the OpenEvent function consists of:

Gate = open;

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Discrete event solver

State events are specified using event conditions, usually

formulated in the form of zero-crossing functions.

Such a function is called event detection function.

During the continuous simulation, all active event detection

functions must be constantly monitored to check whether any of

the zero crossings has taken place.

Once one of the event detection functions has found a zero

crossing, an event localization algorithm is triggered that

iterates on the exact time of the zero crossing.

In the mathematical literature, the event localization algorithm is

often also referred to as root finding algorithm.

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How Simulink detects

discontinuities?

The Simulink software uses a technique known as zero-crossing detection to accurately locate a discontinuity without resorting to excessively small time steps. Usually this technique improves simulation run time, but it can cause some simulations to halt before the intended completion time.

A block can register a set of zero-crossing variables, each of which is a function of a state variable that can have a discontinuity. The zero-crossing function passes through zero from a positive or negative value when the corresponding discontinuity occurs. The registered zero-crossing variables are updated at the end of each simulation step, and any variable that has changed sign is identified as having had a zero-crossing event.

If any zero crossings are detected, the Simulink software interpolates between the previous and current values of each variable that changed sign to estimate the times of the zero crossings (that is, the discontinuities)

21/03/2011 51

Real Time Simulation

Conceptually, the implementation of real-time simulation

software is straightforward.

The difficulties of real-time simulation are not of a

conceptual nature. They have to do with keeping track of

real time.

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Real Time Clock

The real-time clock is responsible for the synchronization of real time

and simulated time.

The real-time clock is programmed to send a trigger impulse once

every h time units of real time, where h is the current step size of the

integration algorithm, and the simulation program is equipped with a

busy waiting mechanism that is launched as soon as all computations

associated with the current step have been completed, and that

checks for arrival of the next trigger signal. The new step will not

begin until the trigger signal has been received.

A input data channel that feeds input signals at the beginning of each

integration step to update the values of all external driving functions.

The inputs are updated once at the beginning of every integration

step and are then kept constant during the entire step.

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A output data channel, that feeds output signals at the end of

each integration step, i.e., the newest output information is put

out through the D/A converters for inspection by the user, or for

driving real hardware (for so-called hardware-in-the-loop

simulations.

External events are time events that are generated outside the

simulation. External events are used for asynchronous

communication with the simulation program, e.g. for the

modification of parameter values, or for handling asynchronous

readout requests, or for communication between several

asynchronously running computer programs either on the same

or different computers. External events are usually postponed

to the end of the current step and replace a portion of the busy

waiting period.

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how event handling may

proceed under real-time

constraints. We perform a regular integration step, advancing the

simulation from time tn to time tn+1.

At the end of the step, we discover that a zero crossing has taken place.

We interpolate to the next event time, tnext in [tn, tn+1], using a formula of the same order of accuracy as that of the integration method in use.

We repeat the last integration step to advance the entire state vector from time tn to time tnext.

We then perform the actions associated with the event, and compute a new consistent initial state.

Starting from the new initial state, we perform a partial step advancing the state vector from time tnext to time tn+1.

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As no iteration takes place, the amount of work, i.e., the total number of floating-point operations needed, can be estimated accurately.

Assuming that only one state event is allowed to occur within a single integration step, we can thus calculate, how much extra time we need to allot, in order to handle single state events within an integration step adequately.

We perform three integration steps instead of only one, and we have to accommodate the additional computations needed to process the event actions themselves.

Thus, the total effort may grow by about a factor of four.

For this reason, the allowed resource utilization for regular integration steps needs to drop from about 80% to about 20%.

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END OF PART 2

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57