looking at problems the other way round: engineering applications of inverse simulation based on...

Looking at Problems the Other Way Round:

Engineering Applications of Inverse Simulation Based on Continuous System Simulation Methods

David J. Murray-Smith,

Emeritus Professor and Honorary Senior Research Fellow,

School of Engineering, Rankine Building

University of Glasgow,

Glasgow G12 8QQ, Scotland, U.K.

E-mail: [email protected]

Keynote Lecture: The inverse simulation approach


ORGANISATION

Part 1: Why use an inverse simulation approach? Areas where it has proved useful. E.g. helicopter flight dynamics; land vehicles; surface ships and underwater vehicles

• Part 2: A brief introduction to inverse simulation methods. Iterative methods; • methods based on continuous system simulation tools.• Part 3: Inverse simulation based on continuous system simulation methods. Methods based on feedback principles; other approaches.• Part 4: Experience. Multi-input multi-output systems; applications involving actuator dynamics as in fixed-wing aircraft, helicopters and underwater vehicles. • Part 5: Discussion and conclusions

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Keynote Lecture : The inverse simulation approach

Part 1: Why use an inverse simulation approach?

What do we mean by “inverse simulation” and how can it be used?

.


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”.

I

• Inputs: rudder, stern planes, top and bottom bow-planes, port and starboard bow-planes, propeller.

Limits: 20 degrees for control surfaces; 1500 rpm for propeller.

(From Healey , A. J. and Lienard, D., Multivariable sliding mode control for autonomous diving and steering of unmanned underwater vehicles, IEEE J. Ocean Engineering, Vol.18, No. 3, pp. 327-339, 1993)

Example: An Unmanned Underwater Vehicle (UUV) control surfaces shown


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Essentials of the inverse approach

Use of inverse models or inverse simulations changes how we look at a problem and can provide insight that is not so readily available from forward simulation. Emphasises the control action to achieve a given output. Has been used especially in systems with a human operator to investigate levels of difficulty for specific tasks and to establish operator limits.

Useful also for external validation, especially where measured responses involve a drift component due to inherent integral action within the system under test. Again, provides different kinds of physical insight from forward simulation.

• Initial value problem:

Conventional simulation:

Inverse simulation:

modelmodel

output

Inverse modelInverse model

desired output Input needed


• Some of first applications involved aircraft and helicopter manoeuvrability and handling qualities investigations.

• For example:

• Approach is also appropriate for many other application areas where actuators may reach limits in terms of amplitudes or rates. E.g, the underwater vehicle case.


. Take-off procedures from an offshore platform to ensure that a helicopter can recover following a failure of one engine in the second phase of climb.


Recovery procedures and pilot work-load studies

Both diagrams from Thomson, D. and Bradley, R., Inverse simulation as a tool for flight dynamics research-Principles and applications, Progress in Aerospace Sciences, 42, 174-210, 2006.

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Another example: Investigation of the potential performance of a helicopter in the early stages of design

First must choose series of manoeuvres appropriate to the proposed role of the helicopter.

For a given helicopter model, inverse simulation can then help answer questions about performance limits, performance sensitivities and possible design changes.

Applications in conceptual design


• Can the helicopter with known power and control limits fly the manoeuvre without exceeding vehicle limits?

If the answer is NO then can consider• What changes of design can allow requirements to be met?

If the answer is YES then can consider issues such as:• What is the “margin” of control that the pilot will have?

• What are the mass and centre of gravity limitations for each manoeuvre?


From: Cameron, N.,Thomson, D.G. and Murray-Smith, D.J., ‘Pilot modelling and inverse simulation for initial handling qualities assessment’, Aeronautical J., 107, 511-520, (2003).

Slalom manoeuvre


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Left: Collective input; Right: Lateral cyclic input From: Thomson and Bradley, Proc. ERF, (1990)

Control inputs for a slalom manoeuvre

From: Thomson, D.G. and Bradley, R. ‘The use of inverse simulation for conceptual design’ Proc. European Rotorcraft Forum, Glasgow, UK, 16-18 September 1990, Royal Aeronautical Society, London, UK (1990).


From: Thomson, D.G. and Bradley, R. ‘The use of inverse simulation for conceptual design’ Proc. European Rotorcraft Forum, Glasgow, UK, 16-18 September 1990, Royal Aeronautical Society, London, UK (1990).

Lateral cyclic input for a slalom manoeuvre: second and third attempts

Inverse Simulation for Model Validation

• Involves using measured response data from experiments on real system as input data for an inverse simulation based on the available model.

• The difference between known experimental input and input obtained from the inverse simulation algorithm may provide insight not readily available from comparisons of system and model outputs from forward simulation runs (e.g when drift in experimental response data is an issue).

• Particularly useful in pointing to structural inadequacies in a model. • Work relates mainly to helicopter flight mechanics modelling.

External validation of models through inverse simulation


Inverse Simulation in Integrated System Design

Particularly appropriate where measured responses can be affected by offsets and resultant drift effects.

Inverse simulation within the external validation process

1/(s+1) 1/sInverse Model

offset

+

+

+

-

Inverse ModelSystem


Inverse Simulation in Integrated System DesignInverse simulation within the external validation process

1/(s+1) 1/sInverse Model

offset

+

+

+

-


From: Thomson and Bradley, Aeronautical J., 94 (1990).


Quick-hop manoeuvre

• Results are for Lynx helicopter in 300ft quick-hop manoeuvre.

• Results show the four helicopter control inputs (collective, longitudinal cyclic, lateral cyclic and tail rotor) from flight data and from inverse simulation model.

• Inverse simulation correctly predicts overall control strategy used by the pilot. But there are differences: are these due to parametric discrepancies in the model or are there issues with the model structure, or both?

• Could apply qualitative approaches based on parameter sensitivity analysis to gain more insight.


Control inputs for quick-hop manoeuvre

Figure from: Bradley, Padfield, Murray-Smith and Thomson, Trans. Inst. Measurement and Control, 12(4), 1990.

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?

Tutorial (4): Control and model validation applications

Control systems applications


Traditional model-following structure with FFC and FBC

The Feedforward (FF) +Feedback (FB) Structure

-

+Controller

YPlant

U

Uff

+

+

Model Inversion

UfbYdTrajectory

Generation


Features of the FF+FB 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.


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Part 2:A brief review of inverse simulation methods

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Inverse modelling/simulation concepts

Initial value problem:

For inverse solution u(t) has to be found for a given y(t). Differentiating gives:

If this equation is solvable for u we can write:

)(

)0( )( 0

xgy

xx x,ufx

),( uxfdx

dg

dt

dx

dx

dyy

),( yxhu

),()),(,( yxFyxhxfx The inverse model properties may be significantly different from the original model. Also the forcing function is now dy/dt rather that y(t).

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 model such as helicopter and ship models. Most approaches only applicable with minimum-phase models.

• Inverse Simulation

Easy and feasible for implementation for minimum-phase systems (more difficult for non-minimum phase systems).

Can be applied to many forms of complex nonlinear model without difficulty


Iterative methods with discretised models

• The “differentiation approach” (e.g. Kato and Sugiura1, Thomson2). This involves replacing derivatives with a discrete equivalent and then solving the resulting nonlinear algebraic equations iteratively to fins the necessary inputs.

• The “integration approach” (e.g. Hess, Gao and Wang3; Rutherford and Thomson4). This involves repeated solution of the initial value problem and gradient or search-based methods to find the inputs needed to achieve a specified set of outputs.

• “Optimisation-based approaches”: e.g. Celi’s work on helicopter applications5.

1. Kato, O. and Sugiura, I, AIAA J. Guidance, Control and Dynamics, 9(2), (1986),2. Thomson, D.G., In Proc. 12th European Rotorcraft Forum, 1986, Paper 45.3.Hess, R.A., Gao, C. and Wang, S.H., AIAA J. Guidance, Control and Dynamics, 14(5), (1991), 733-7.4. Rutherford, S. and Thomson, D.G., Aeronautical J. 100(993), 1996. ,5. Celi, R., Optimization-based inverse simulation of a helicopter maneuver, In: Proc. 25th Eur. Rotorcraft Forum, Paper H12,1999.


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 for gradient-based optimisation. These can affect both the differentiation and integration based approaches.

• Potential issues of non-uniqueness of solutions.


Numerical issues in the iterative approach

Results illustrating numerical problems)


From: Rutherford, S. and Thomson, D.G., ‘ Improved methodologies for inverse simulation’, J. of Aircraft, 100, 79-86,(1996)

Numerical instability of type observed by Gao and Hess (1993) and by Rutherford and Thomson (1995) in the application of the iterative integration-based type of approach. Thomson and Bradley (2006) suggest modifying the error function (e.g. using acceleration rather than velocity) to overcome this difficulty when it is encountered. Convergence problems associated with calculation of Jacobian elements are also found in many nonlinear problems.

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Part 3: Inverse simulation based on continuous system simulation methods


Differential Algebraic Equations (DAE) Approach

• Allows an inverse simulation model to be derived directly from the structure of the conventional continuous system simulation model.

• METHOD 1: An inverse model of a Differential Algebraic Equation (DAE) may be constructed simply by changing the meaning of variables. Tools such as Modelica/Dymola or Scilab/Scicos incorporate DAE solver algorithms.

• METHOD 2: An inverse model can be derived using an approximate continuous differentiator - and an inverse solution can then be found using a continuous equivalent of the discrete “differentiation” approach (Kato and Saguira/Thomson and Bradley) outlined previously.

• METHOD 3: A continuous systems type description for an inverse model can be constructed through the properties of high gain feedback systems. This is the approach considered in detail in this presentation.

Methods based on use of Continuous System Simulation Tools


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.

• Published results to date are for cases involving relatively simple simulation models.

• See (or example)Thümmel, M., Looye, G., et al. “Nonlinear inverse models for control”, Proceedings 4th Intl. Modelica Conference, Hamburg, March 7-8, 2005, pp267-279.

The Differential Algebraic Equations (DAE) approach


• This is a continuous simulation equivalent of the discrete “differentiation method”.

• The basic idea is to re-arrange the model equations so that the inputs of interest appear on the left hand side. Derivatives of state variables on the right hand side can then be approximated using a simple continuous representation based on the use of an integrator block and feedback (provided T is small in terms of the other dynamics of the system being modelled).

An approximate differentiation approach.


1/T 1/s w(t)v(t)

dw/dt


Example: a linear state-space model

Poles at s = -1, s = -2, s = -3; zeros at s=-0.5 ± j7.0534Range of frequencies of interest 0 to 30 rad/s.


Re-arrangement of equations +

For time constant T small compared with the dynamics of the model this new state variable representation closely approximates the original and we get a satisfactory inverse solution.

for the first state equation and a new output equationWe now have

Let the desired output be defined by

The derivative Is then approximated by

+

+


The equations for the inverse simulation +

+

Zeros at s = -1, s = -2, s = -3. Poles at s=-0.5 ± j7.0534 and at s= -1/T.Original (forward) model had poles at s = -1, s = -2, s = -3. Zeros at s=-0.5 ± j7.0534.

The linear state-space model results

Model output generated from input found by inverse simulation

Input time history found from inverse simulation


• Based on principles of analogue dividers and inverse function generators and also ideas applied and published by engineers at DLR in Germany in the 1990s (e.g., Hamel (1994) andGray and von Grünhagen (1998)). Quite separately similar ideas ( “inverse dynamics compensation via simulation of feedback control systems” (IDCS))appeared in Japan in the 1990s through the work of Tagawa and Fukui.

• These methods use high gain feedback principles to generate inputs required to produce a model output that matches a given required output.

A feedback systems approach to inverse simulation.


• •

Model

Feedback gain factor

Referenceinput: required output of model

Model input needed to produce require output from the model

+

_

The feedback approach: the basic idea


• G(s)

• K

• v

• w

• +

• - )(1)(

)(

sKG

K

sV

sW

)(1

1

sGK

For the case where K is very large this gives:

)(

1

)(

)(

sGsV

sW

The feedback method for the linear case


Brief History of the Feedback System Approach

• Can be traced to feedback methods in analogue computing.

• The extension to inverse simulation is hard to pin down exactly but much early work was undertaken at the DLR aerospace laboratories at Braunschweig in Germany (Hamel, von Grűnhagen and colleagues). Mainly concerned with aircraft and helicopter flight control.

• There was independent work in Japan on an approach termed

“inverse dynamics compensation via simulation of feedback control systems”. Initially concerned with robotics applications, mainly. (Tagawa and colleagues).

• More recent work has attempted to generalise the approach and consider a broader range of applications (Murray-Smith)


Points to be noted:i• G

(s)

• K

• v

• w

• +

• -

1) The variable v in this representation is the form of output required from the model while w is input that must be applied to the model (under open-loop conditions) to produce this output.

2) Number of poles of closed-loop system always same as number of zeros and there is no issue of realisability for the inverse.

3) Feedback system design for inverse simulation is distinctly different from control system design – no external disturbances, no issues of robustness. The model G is known exactly - hence high gain solutions and other relatively simple methods of feedback system design (e.g. eigenstructure assignment) can be applied successfully.


The linear state-space model again

Poles at s = -1, s = -2, s = -3; zeros at s=-0.5 ± j7.0534Range of frequencies of interest 0 to 30 rad/s.


The linear state-space example (continued)

Proportional feedback control applied to modelusing gain factor of 1000 gives closed-loop poles ats = -1005, s = -0.5 ± j7.0 and transmission zeros ats = -3.000, s = -2.000 and s = -1.000

From Murray-Smith, D. J. Feedback methods for inverse simulation of dynamic models for engineering systems applications, Math. And Computer Modelling of Dynamical Systems,, 17(5), 515-541, 2011



Input time history found from inverse simulation showing oscillations that are associatedwith complex poles in inverse model.

Model output generated from the input time history found by inverse simulation.




Bode diagram for combined system involving model and the inverse simulation in cascade



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Part 4: Experience with applications


Case study: Model of a 2-tank system

Cdt

dHA

viQ 1

A1 dH1/dt = Qi1 – Cd1 a1 [2g (H1 – H2)] ½

A2 dH2/dt = Qi2 + Cd1 a1 [2g (H1 – H2)] ½ - Cd2 a2 [2g (H2 – H3)] ½

Tutorial (3): The feedback approach

Analysis of the linearised version of this model shows that it has two poles and no zeros.

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


A two-input two-output version of thesystem


Test of inv. sim. by feedback method: input, simulation, inverse and then comparison


Inverse simulation involving a required pattern of output levels


Inverse simulation by the approximate differentiation method


Inverse simulation by feedback with required output levels and input limiting


Inverse simulation by differentiation withrequired output levels and input limiting

As compared with the previous case using the feedback approach:


• Both the approximate differentiation and feedback methods give very similar results for cases where there are no limits. Both are approximate inversion techniques.

• The two approaches give different results when limits are encountered. Which method you apply must depend on the objectives of the investigation.

• With the feedback method there are obvious issues of non-uniqueness of solutions (depend on feedback design method used).

• Similarly, with the differentiation approach the result depends on the magnitude of the time constant in the differentiation loop.

Comments on this case study


Case Study: Fixed-wing aircraft model using the feedback approach


Fixed-wing aircraft example (continued)

Analysis of linearised model:

Root locus for the feedback system using pitch rate with proportional control and gain factor of 1000



Fixed-wing aircraft example (continued)From Murray-Smith, D. J. Feedback methods for inverse simulation of dynamic models for engineering systems applications, Math. And Computer Modelling of Dynamical Systems,, 17(5), 515-541, 2011



Root locus plot involving pitch angle feedback pathway and proportionalcontrol

Results from inverse simulation with pitch attitude feedback

From Murray-Smith, D. J. Feedback methods for inverse simulation of dynamic models for engineering systems applications, Math. and Computer Modelling of Dynamical Systems,, 17(5), 515-541, 2011



Results for state-variable feedback from pitch angle, pitch rate, vertical velocity and forward velocity. Poles of closed-loop system lie at or very close to zeros of model. All other poles of closed-loop system lie far from these points.

Effect of an elevator deflection lIimit of ±0.02 rad. Required pitch shown by dashed line. Achievablepitch with this limit shown by continuous line.

From Murray-Smith, D. J. Feedback methods for inverse simulation of dynamic models for engineering systems applications, Math. and Computer Modelling of Dynamical Systems,, 17(5), 515-541, 2011


• Issues of stability can arise with the feedback approach (both model stability and numerical stability).

• Obvious issues of non-uniqueness of solutions (depend on feedback design method used or form of approximate differentiator).

• Feedback analysis and design for inverse simulation is more straightforward than for closed-loop control systems (no issues of measurement noise, robustness or disturbance rejection performance).

• Computation time with continuous system simulation methods can be less than with other approaches.

Comments on these case studies


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Part 5: Discussion


Ideal Manoeuvre or Expected

Control Trajectory

Inverse SimulationAlgorithms

Required Inputs

Iterative discretemethods

A summary of inverse simulation methods

Feedback methodOptimisation

based methods

Approximate differentiation

method

DAE basedmethods


.

Difficulties and limitations

• Some inherent difficulties and limitations apply to all the approaches in terms of numbers of inputs and outputs etc. These are well covered in the literature.

• There are fundamental problems with non-minimum-phase systems (and nonlinear equivalents). Inversion of a non-minimum-phase model leads to an unstable inverse model.

• Differentiation method has difficulties since inversion process involves model manipulation each time the model structure is changed. The feedback approach avoids that but feedback gains may need adjustment.

• The feedback approach has potential problems of instability and limit cycle phenomena. These problems are most evident when actuator rate limits are present – can be understood using describing function methods.


Areas of current research on inverse simulation at the University of Glasgow

• Further investigation and comparisons in terms of numerical issues for the various different methods of inverse simulation.

• Further development of inverse simulation methods based on the continuous system simulation approaches. Emphasis is on providing inverse solutions for real-time applications such as those arising in control applications.

• Application of inverse simulation methods to simulation model validation. Extending previous work involving two-tank process system as well as work involving helicopter flight test data.

• Use of inverse simulation to investigate actuator rate and amplitude limits for aircraft and underwater vehicle applications.


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Conclusions

• Although it involves approximations inverse simulation provides a potentially useful approach to model inversion that avoids mathematical complexities.

• Continuous system simulation methods offer alternatives to iterative methods based on discrete models. They avoid known problems of these iterative approaches (e.g. difficulties in determining elements of the Jacobian and associated issues of non-convergence)

• The approximate differentiation and feedback approaches have both been shown here to be useful for a number of relatively simple engineering applications. Their use with more complex applications (such as a high-order nonlinear UUV model and a nonlinear AUV model) have also proved successful.

• The two approaches considered have their own strengths and weaknesses. With care, they can offer important physical insight for complex nonlinear analysis and design problems, especially for vehicle systems and control.


e

• I wish to thank Dr Douglas Thomson of the University of Glasgow who was responsible for much of the fundamental research on inverse simulation methods for helicopter applications and has been an active collaborator in a number of applications projects, including some reported in this lecture.

• I wish also to thank my colleague Dr Euan McGookin of the University of Glasgow who has stimulated my interest in the application of inverse simulation methods to actuator problems associated with the control of ships and underwater vehicles.

• I wish to acknowledge the support of the US Office of Naval Research in terms of funding to Dr Thomson, Dr McGookin and myself which (among other things) supported my research on the continuous systems simulation approaches to inverse simulation over the period 2009-11. That funding involved awards to California State University, Chico (CSUC) and associated sub-contracts placed at the University of Glasgow by CSUC.

AAcknowledgements


looking at problems the other way round: engineering applications of inverse simulation based on...

Documents

conventional simulation

forward simulation

inverse modelling

inverse simulations

keynote lecture

iterative methods methods

unmanned underwater

prescribed model output