Department of Electronics and Electrical Engineering

1
Department of Electronics and Electrical
Engineering
Inverse Simulation Methods and Applications
TUTORIAL PRESENTATION 6th EUROSIM CONGRESS,
LJUBLJANA, SEPTEMBER 2007
David J. Murray-Smith
E-mail djms_at_elec.gla.ac.uk
2
Contents and Organisation of the Tutorial

• An introduction to inverse simulation ideas and
  the methods of inverse simulation developed
  initially for aeronautical engineering
  applications but now applied more widely
  differentiation based, integration based and
  optimisation-based techniques.
• Some preliminary applications and case studies.
• A brief introduction to some other methods of
  inverse simulation using feedback principles,
  DAE-based methods.
• Links between inverse simulation and model
  inversion methods from control engineering.
• Model-following control system design using
  inverse simulation.
• Applications and case studies.
• Links between nonlinear model predictive control,
  receding horizon concepts and inverse simulation
  methods.
• Further case studies.
• Discussion.

3
What is Inverse Simulation?

• Conventional modelling and simulation a process
  of finding a model output for a given set of
  initial conditions and a prescribed time history
  of inputs.
• Inverse modelling and simulation a process
  through which inputs are found that will
  produce a prescribed model output.

4
The Inverse Simulation Process
5
Model Inversion (the linear case)

• Consider the SISO linear system described by the
  transfer function
• Then (subject to some restrictions in terms of
  realisability) the inverse is

6
Model Inversion (the simple linear case)

• Inverse is realisable only if the order of the
  numerator is equal to the order of the
  denominator
• If the inverse transfer function is not
  realisable we can make it realisable by adding
  sufficient additional poles to the denominator of
  the inverse to make the order of denominator at
  least as high as that of the numerator. Those
  extra poles in E(s) must lie far to the left in
  the s-plane compared with poles of N(s).
• Must then use integration methods suitable for
  stiff systems to generate an inverse simulation.
  

7
Why use Inverse Simulation?

• Puts emphasis on the control action needed to
  achieve a particular output response.
• Specially important in the case of systems
  described by nonlinear dynamic models.
• Allows investigation of characteristics needed in
  a control actuator to achieve a required output.
• Allows investigation of limitations of human
  operators in closed-loop manual control systems.
  Particularly valuable in the case of constrained
  responses.

8

• Much of the original interest in inverse
  simulation came from the helicopter design
  community. Many of the issues relevant to
  helicopters carry over to design problems for
  other types of application such as robotics and
  underwater vehicles.
• Unlike fixed-wing aircraft, helicopters can hover
  and move sideways or fly at low speed close to
  trees, buildings and the ground. Equivalent
  situations arise in the case of underwater
  vehicles and robots.
• Precise control of helicopter flight presents
  particular design challenges that necessitate
  extensive use of computer simulation techniques
  at the design stage. Inverse simulation has been
  found to have a particularly valuable role in
  this.

9
Why Simulation? Why not Inverse Models?

• Important to distinguish between inverse
  simulation methods and inverse models derived by
  analytical methods.
• Analytical methods can be applied to linear
  models and to certain forms of nonlinear model
  described by equations having specific forms.
• Methods of inverse simulation exist that can be
  applied to models of all kinds.

10
Inverse Simulation Concepts

• Initial value problem
• For inverse simulation u(t) has to be found for a
  given y(t). Differentiating gives
• If this equation is invertible we can write

The inverse model has dynamic properties that can
differ significantly from the original model. The
forcing function is dy/dt rather that y(t).
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Inverse Simulation Methods

• There are at least five broad classes of
  method
• Direct techniques involving numerical
  differentiation.
• Iterative techniques involving numerical
  integration.
• Techniques based on optimisation methods.
• Methods based on properties of high-gain feedback
  systems.
• Methods based on DAE solutions.

12
Numerical Differentiation Approach of Thomson
and Bradley

• At time step n
• where yn is known.
• Then we have to find xn and un such that F1 and
  F2 are approximately zero.
• Then, based on a Newton-Raphson type of approach
  at the mth iteration, we have

13
Issues Arising with the Differentiation Approach

• The differentiation approach is model specific.
• Numerical differencing can give rise to problems
  of rounding error and thus the choice of step
  size ?t in numerical differentiation is very
  important.
• Increments used in calculation of Jacobian can
  present problems.
• If measured data are being used (e.g. for model
  validation applications) differentiation can
  amplify high frequency components and adversely
  affect signal to noise ratios.

14
Numerical Integration Approach of Hess, Gao and
Wang

• Involves repeated solution of the initial value
  problem.
• At the mth estimate at the nth time point we
  define an error en as the difference between yn
  and the required output ydn.
• i.e. (en)m (yn)m ydn
• A Newton-Raphson approach can then be used to
  find a new estimate of u as
• ( un-1)m1 (un-1)m- J-1(en)m
• where J is the Jacobian. New estimates of the
  state
• vector, its derivative and the output are
  obtained. The process continues until all
  elements of en fall below a preset threshold
  value.

15
Description of the GENISA Algorithm
Estimates of state and output vectors
Dynamic system
Error function
State and control vectors
Newton-Raphson method
16
Essentials of the Integration-based Iterative
Approach

• The desired trajectory is discretised (as in the
  differentiation-based approach).
• At each time point an estimate is made of the
  amplitude of step change of each input to move
  system to next point.
• The resulting output is found and error between
  actual and desired output calculated.
• An iterative process then used to minimise error.
• By taking inputs over all time periods can find
  input time histories needed to make the system
  follow the desired trajectory.

17
Features of the Integration-based Iterative
Approach

• The integration based approach is generic and can
  be applied directly to any existing simulation
  model.
• The algorithm is slow compared with numerical
  differentiation approach.
• Issues of numerical accuracy have given cause for
  concern.

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Limitations

• There are three categories of inverse
  simulation problem
• Number of inputs gt Number of outputs
• (no solution available).
• Number of inputs Number of outputs
• Number of inputslt Number of outputs
• (so-called redundant case where no inverse
  of the Jacobian exists, requiring use of the
  Moore-Penrose generalised inverse).

19
Numerical Issues

• Algorithms based on numerical differencing can
  give rise to problems of rounding error. This
  presents potential difficulties for the
  differentiation approach.
• Problems can also arise through errors in the
  calculation of the Jacobian. These can affect
  both the differentiation and integration based
  approaches.
• Potential issues of non-uniqueness of solutions

20
Calculation of the elements of the Jacobian

• In most cases of practical importance an
• approximation method must be used to
• obtain the Jacobian
• When the Jacobian matrix is not square
• (number of inputs ? number of outputs)
• may use the Moore-Penrose pseudo-
• inverse approach to obtain a solution.

21
Input Saturation Effects

• Input saturation and limiting present a challenge
  to traditional inverse simulation methods that
  depend on gradient information.
• Traditional methods depend on smooth properties
  of both model and the manoeuvre.
• Complications of the Jacobian can be avoided
  through use of the constrained Nelder-Mead (NM)
  method. This gives a derivative-free approach
  based on optimisation.

22
System with input constraints using
Newton-Raphson approach for Inverse
SimulationSituation at the kth discretized
interval with input saturation levels umax and
umin elements of Jacobian can become zero,
leading to problems of singularity and thus
non-convergence of the algorithm.
23
Options available

• Use the NR algorithm for the model without
  saturation constraints examination of time
  histories of inputs can then provide useful
  physical insight as limiting values are
  approached.
• Use other forms of inverse simulation method than
  can accommodate input limits particularly
  relevant for control system analysis and design.

24
Optimisation-based methods

• In 1999 Celi demonstrated that the Newton-Raphson
  type of iterative scheme normally used in the
  standard integration-based approach could be
  replaced by a numerical optimisation method.
  
• Since 2005 Lu and Murray-Smith have been
  successfully applying the Nelder-Mead simplex
  type of search-based optimisation algorithm,
  which requires no gradient information, to
  problems of inverse simulation.
• Celi, R., Optimization-based inverse simulation
  of a helicopter maneuver, In Proc. 25th Eur.
  Rotorcraft Forum, Rome, Italy, H12.1-H12.12,
  1999.
• Lu, L., PhD Thesis, University of Glasgow, 2007.
  

25
The Nelder-Mead Approach Applied to Inverse
Simulation

• Cost function typically takes form
• This must be minimised, subject to
• Process involves forward simulation from tk to
  tk1 and then finding u(tk) that miminizes L
  using the value of x(tk1) found from the forward
  simulation.

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Essentials of the Constrained Nelder-Mead Approach

• The constrained Nelder-Mead algorithm is a
  well-known method for minimizing a scalar-valued
  nonlinear function involving only function
  values.
• It can handle discontinuities satisfactorily,
  particularly if they do not occur near the
  optimum solution.
• For inverse simulation it combines optimization
  with the integration method and, like the
  Newton-Raphson approach, is applied over the
  interval tk , ,tk1 .
• Does not involve any analytic or numerical
  gradient information.
• Essentially a direct-search downhill simplex
  method for function minimization slow but
  robust.

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Outline of the NM Algorithm

• Algorithm first characterises a simplex in
  q-dimensional space by q1 vertices.
• Applying four rules involving reflection,
  expansion, contraction and shrinkage a new point
  in or near to the current simplex is found.
• A new simplex is then formed by replacing a
  vertex in the current simplex by this new point.
• Cost function values at vertices of the old
  simplex are compared with those for the new
  simplex.
• The process is continued until the diameter of
  the simplex is less than a specified tolerance.

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Discretisation Processes in Inverse Simulation

• Analysis in the nonlinear case is difficult so
  more appropriate to first consider a linear
  system
• Procedures based on integration divide into two
  steps a) discretisation and b) solution.
• Stability and performance of the second stage
  depends on numerical properties of the algorithm.
  Questions of dynamical stability are important
  only for the first stage.
• After discretisation the linear model above
  becomes
• where
• and

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Application to a Simple Surface Ship Model with
Rudder Deflection and Rate Limits

• Norrbin model

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ROV Zeefakkel(Photograph United Dutch Navy and
Maritime Forum (http//www.dutchfleet.net))
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Assessment Scheme for Inverse Simulations
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Inverse Simulation Resultsfor ROV Zeefakkel -
Relatively Small Manoeuvre
No saturation limits. Newton-Raphson method,
(?t0.2s, 20 degrees heading change)
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Inverse Simulation Results for ROV Zeefakkel
Larger Manoeuvre
No saturation limits. Newton-Raphson method,
(?t0.2s, 50 degrees heading change)
34
Inverse Simulation Results for ROV Zeefakkel
Saturation limits included. Nelder-Mead method,
?t0.2s
35
Assessment of Results from ROV Zeefakkel Study

• The results from inverse simulation and forward
  simulation agree well for cases where there is no
  saturation limit.
• With a rudder limit of 35 deg. and rudder rate
  limit of 7 deg./s. NR method fails to converge if
  U lt 9 m./s. for heading change of 20 deg. and for
  U lt 15 m./s. for heading change of 50 deg..
  However, the NM method does provide useful
  results in such cases.
• As expected, the required inputs become larger as
  speed falls and the larger the demanded heading
  change.
• The more the input involves nonlinear effects the
  more difficult it is to follow the ideal (linear)
  trajectory.

36
Inverse Simulation - Revisiting the Fundamentals
Initial value problem
control input
output
model
Conventional simulation
desired output
control input
model
Inverse simulation
37
Comparisons of Model Inversion and Inverse
Simulation Approaches

• Model Inversion
• Highly mathematical basis for nonlinear case.
• Quite extensively used for aerospace
  applications.
• Tends to be rather complex for application to
  full-scale nonlinear models such as helicopter
  and ship models.
• Most approaches are noncausal for
  nonminimum-phase systems.
• Inverse Simulation
• Easy and feasible for implementation for
  minimum-phase systems.
• Involves causal inversion for some linear
  nonminimum-phase systems.
• Can be applied to nonlinear models without
  difficulty.

38
Inversion by Inverse Simulation

• Suggestion
• For discontinuous complex models such as
  helicopter or ship models, it may be convenient
  to use inverse simulation instead of model
  inversion for FF control system design, provided
  a suitable sampling interval ?t can be chosen.
• Rules for selection of the interval ?t
• Need to guarantee the convergence of inverse
  simulation process
• Must attempt to keep the zeros within Unit Circle
  in the discretisation process.

39
Example 1- A Simple Linear NMP System
This model has two RHP zeros 0.5000 7.0534i
Investigated Model
Adopted Manoeuvre

Z
The hurdle-hop manoeuvre
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Example 1- A Simple Linear NMP System (Cont.)
Distribution of zeros in the z-plane (left) and
variation of zero magnitude with ?t (right). In
the z-plane plot, the lighter dashed line (red)
represents the unit circle.
41
Example 1- A Simple Linear Non-minimum Phase
System (Cont.)
The calculated inputs from inverse simulation
Comparisons of the calculated outputs with the
ideal manoeuvres ?????? Ideal manoeuvre
--- Forward simulation
42
A Feedback Systems Approach to Inverse Simulation.

• Based on principles of analog dividers and
  inverse function generators and also ideas
  published in the 1990s by Gray and von Grünhagen
  at DLR in Germany.
• Uses high gain feedback principles to generate
  inputs required to produce model output that
  matches the required output.

43
Block Diagram for Feedback Approach to Inverse
Simulation
Model input needed to produce required model
output
Required output of model
Model output
Plant model
Gain factor

-
44
Case Study A Two-Tank Liquid Level System

• Laboratory system (Tequipment Ltd) intended
  for control system experiments.

45
Classical Model of the System
A1 dH1/dt Qvi Cd1 a1 2g (H1 H2) ½ A2
dH2/dt Cd1 a1 2g (H1 H2) ½ - Cd2 a2 2g
(H2 H3) ½
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Experimental Data Large Transient(Set 6 ----
liquid depth (mm) versus time (seconds), H1-
blue, H2- red plugs in two largest holes)
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Comparison for data set 6(Simulation results on
left for Cd1 0.43,Cd2 0.48. Experimental
results on right)
48
Input from Feedback Approach

• Plot of input flow (m3/s) as a function of
  time (sec) calculated by inverse simulation using
  feedback approach

49
Theoretical input (cm3/min) versus time (sec)
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Comments on Feedback System Approach

• Obvious issues of stability (both model stability
  and numerical stability).
• Obvious issues of non-uniqueness of solutions
  (depend on design method used).
• Feedback analysis and design more straightforward
  in this case than for closed-loop control systems
  (no issues of robustness).
• Computation time very much less than by other
  approaches to inverse simulation.

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Differential Algebraic Equations (DAE) Approach

• Allows an inverse simulation model to be derived
  directly from the structure of the conventional
  system simulation model.
• Readily available for users of simulation and
  modelling tools such as Modelica/Dymola or
  Scilab/Scicos that incorporate DAE solvers.
• An inverse model of a DAE is constructed simply
  by changing the meaning of variables. The result
  is still a DAE which can be dealt with using
  standard DAE solution methods.
• See Thümmel, M., Looye, G., et al. Nonlinear
  inverse models for control, Proceedings 4th
  Intl. Modelica Conference, Hamburg, March 7-8,
  2005, pp267-279.

52
Inverse Simulation for Control Systems
Applications

• Many applications of model inversion are
  associated with control system design.
• Can inverse simulation techniques replace methods
  of model inversion for control design
  applications? For example, in combined
  feed-forward/ feedback model following control
  systems.
• Conversely, can techniques and concepts developed
  in areas such as nonlinear model-based predictive
  control be used with benefit in inverse
  simulation methods?

53
Traditional Model-Following Structure
The Feedforward (FF) Feedback (FB) Structure
54
Strengths of the FFFB Structure

• Feedforward Channel (FFC)
• Designed to compensate for the dynamics of the
  plant.
• May assist in providing precision tracking.
• Feedback Channel (FBC)
• Provides robust stability against uncertainties
  caused by external disturbances and reduces
  sensitivity to sensor noise.
• Reduces the risks of long-term drifts in the
  overall system response by minimizing the
  feedforward inaccuracies.

55
Model Following Control Applications

• Case One
• An 8th-order linearised Lynx-like helicopter
  model
• Four outputs heave velocity, roll rate (p),
  pitch rate (q), and heading rate
• Four inputs the traditional four control
  channels of the helicopter
• Case Two
• A full nonlinear container ship model (Son
  and Nomoto) with
• constraints on rudder angle, propeller speed
  and rudder rate.
• One output the heading angle
• One input the rudder angle

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Control Structure
FFC inverse simulation FBC the K/KS designed
by H8 approach with disturbance and measurement
noise
57
Results from Case One (Helicopter)
Ideal values results without FFC stars results
with FFC solid line (?t 0.01s)
58
Results from Case Two (Container ship with
heading and roll control)
Linear FFC solid Nonlinear FFC dot solid No
FFC dashed (Left, ?n0.1 rad/s right, ?n0.015
rad/s)
The feedback signals for channels p and F
(rudder-roll subsystem) are switched on from 300
s to 500 s and are switched off at all other
times
59
Results from Case Two (Container Ship)
60
Links with Model Predictive Control and Receding
Horizon ConceptsApplication Helicopter in
pop-up manoeuvre
61
Restrictions of Conventional Inverse Simulation
Lynx flying a pop-up manoeuvre (flight velocity
120 knots, height 40 m, manoeuvre time 5 sec).

• Possible constraints
• Mechanical limitations of control surfaces
• Control rates (actuator saturation
  characteristics)
• Limitations of main rotor and tail rotor torque
• Human pilot limitations
• Handling Qualities requirements
• Structural limits of critical components
• Limits of state values

62
Idea of Predictive Inverse Simulation
Control Input
Receding Horizon
Time
Predictive Inverse Simulation
Inverse Simulation
Predictive Control Receding Horizon


Calculates feasible solution based on heuristic
approach.
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Predictive Inverse Simulation Algorithm
Manoeuvre Modelling
Inverse Simulation GENISA Algorithm
Predictive Part Receding Horizon
States andControl Inputs
Trim Algorithm
Constraints Exceeded?
NO
Helicopter Model
YES
Decision Tree Algorithm
A Priori KnowledgeDatabase
64
Mathematical Representation of Pop-Up Manoeuvre
Boundary conditions
Exit
Boundary conditions
Start
65
Mathematical Representation of Pop-Up Manoeuvre
Boundary conditions
Exit
Boundary conditions
Start
Decision point
66
Helicopter Model
67
Decision Tree Approach
flight velocity profile
heading/sideslip constraint
heading/sideslip profile
I
Decision point
Decision Point
II
I
II
III
m
III
1
2
n
In general there might be m different modifiers.
Decision point
A Priori Knowledge database allows to reduce
dramatically a number of possible changes n.
68
Simulation Results Avoiding the Collective Pitch
Limit
Lynx flying a pop-up manoeuvre (flight velocity
120 knots, height 40 m, manoeuvre time 5 sec).
receding horizon
decision point
Results of predictive inverse simulation (blue
line) using a 3 sec prediction horizon (red
line).
69
Avoiding the Longitudinal Cyclic Pitch Limit
Lynx flying a pop-up manoeuvre (flight velocity
60 knots, height 34 m, manoeuvre time 5 sec).
decision point
receding horizon
Results of predictive inverse simulation (blue
line) using a 2 sec prediction horizon (red line).
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Avoiding the Longitudinal Cyclic Pitch Limit
Lynx flying a pop-up manoeuvre (flight velocity
80 knots, height 37 m, manoeuvre time 5 sec).
decision point
receding horizon
Results of predictive inverse simulation (blue
line) using a 4 sec prediction horizon (red line).
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Avoiding the Lateral Cyclic Pitch Limit
Lynx flying a lateral realignment manoeuvre
(flight velocity 100 knots, manoeuvre time 5 sec).
receding horizon
decision point
Results of predictive inverse simulation (blue
line) using a 2 sec prediction horizon (red line).
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Conclusions relating to predictive inverse
simulation

• Method developed achieves the aim of improving
  the realism of conventional inverse simulation
  results by implementing the receding horizon
  approach.
• A process of constraint handling is incorporated
  into the inverse simulation algorithm.
• Predictive inverse simulation can reduce a number
  of iterative calculations in application to the
  conceptual design of helicopters.
• Proposed methodology can form the basis for a
  trajectory generation algorithm for use in
  tunnel in the sky systems.

Future Work
1. Investigation of techniques that can provide
higher speed of solution for inverse simulation
problems. 2. Implementation of other helicopter
manoeuvres. 3. Incorporation of new constraints,
such as control rates limits, limitations of
rotor and tail rotor torque, limits of state
values(for example, pitch attitude).
73
Areas of Current Research at the University of
Glasgow

• Further investigation of numerical issues for
  iterative methods of inverse simulation.
• Further research on the application of ideas from
  the Model Predictive Control field to development
  of improved inverse simulation algorithms and
  piloting strategies (with D. Thomson, D.
  Anderson, M.Bagiev and in collaboration with M.
  Grimble and colleagues (University of
  Strathclyde) with financial support from UK
  EPSRC). Application to helicopter-ship landing
  problems.
• Development of inverse simulation methods based
  on a feedback approach. Emphasis is on providing
  inverse solutions for real-time applications such
  as those arising in the receding horizon work
  involving constrained helicopter flight.
• Application of inverse simulation methods to
  simulation model validation.
• Investigation of the potential of DAE-based
  methods applied to inverse simulation for
  aeronautical and marine applications.

74
Conclusions

• Inverse simulation can reduce need for repeated
  conventional simulation runs in the investigation
  of many engineering problems involving nonlinear
  dynamic systems.
• Inverse simulation can replace analytical
  methods of model inversion for minimum-phase
  problems and some kinds of linear non-minimum
  phase systems.
• Use of inverse simulation coupled with receding
  horizon concepts provides potential for new forms
  of decision support system for a range of
  applications involving piloted vehicles.
• However, challenging problems remain in order to
  convert inverse simulation techniques into design
  tools for routine use by engineers in industry.

75
Acknowledgements

• I wish to thank my research student Linghai Lu
  who has contributed the applications relating to
  ships and has undertaken much of the work
  involving feedforward and feedback control system
  design.
• I also wish to thank my colleague Dr Douglas
  Thomson of the Dept. of Aerospace Engineering who
  was responsible for much of the fundamental
  research on inverse simulation methods for
  helicopter applications carried out at the
  University of Glasgow.
• I also thank Dr David Anderson and Dr Marat
  Bagiev (Dept. of Aerospace Eng.) for their
  contributions relating to the recent research on
  predictive methods for helicopter piloting and to
  Professor Mike Grimble and his colleagues at the
  University of Strathclyde with whom we
  collaborate.
• I acknowledge the support of the UK Engineering
  and Physical Sciences Research Council through
  grant GR/S91024/01. This tutorial has been
  developed with the support of that grant.

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