Optimal and Robust Control of Active Suspension for Tracked Vehicles

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Optimal and Robust Control of Active Suspension
for Tracked Vehicles

• Kunal
• Nov 1st 2004
• University of Illinois at Urbana-Champaign

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Motivation

• Conventional Passive suspension employing
  Springs, Dampers and Torsion bars are incapable
  of effective vibration isolation.
• Ineffective vibration isolation limits the
  vehicle mobility, causes discomfort and seriously
  affects the drivers response.
• Active suspension has been employed in
  automobiles has shown a remarkable improvement in
  suspension performance.
• Tracked vehicles like armored Tanks and military
  vehicles operate in a much more stringent
  environment and carry sensitive equipment which
  must be isolated from sudden jerks and excessive
  vibrations.
• Passive suspension limits the vehicles
  maneuverability in a battle environment which is
  a crucial criterion.

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Issues in design (Measures of Performance)

• The wheel-terrain force should be limited. This
  is to ensure that the wheel always maintains
  contact with the ground.
• The suspension workspace is restricted. This is
  to ensure that the suspension doesnt experience
  unduly large deflections.
• Hull vibrations are reduced to ensure the driver
  comfort.
• The control force is limited. The energy or force
  available to operate the suspension is limited by
  the capacity of the engine and has to be
  minimized.
• The suspension should be adaptable. This is to
  ensure uniform performance under varying
  conditions.
• The performance should be maintained under
  uncertainty and changing parameters

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Challenges in suspension design for tracked
vehicles

• Large number of degrees of freedom compared to
  four wheel automobiles.
• The presence of tracks adds considerable
  complexity. Modeling of tracks difficult.
• Still no widely acceptable way to model track
  terrain interaction.
• The performance measures are conflicting in
  nature and its impossible to improve all the
  indices.
• Choice of weights to emphasize relative
  importance of indices is tricky.
• Changing operating conditions and complex vehicle
  dynamics leads to improper under modeled plants.

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Overview

• Development of the Full Car Dynamic Model.
• Controller design using full car model of the
  tracked vehicle for Full State feedback, PID and
  sky hook.
• Development of the 3-D ride dynamic non-linear
  model for the tracked vehicle.
• Incorporation of the Driver seat modeling in the
  non linear vehicle model.
• Performance evaluation of the suspension on three
  standard test tracks.
• Post processing of the results to realistically
  emulate the operating conditions.
• Comparison of different controllers performance
  on the three tracks over a range of operating
  conditions.
• The Robust control problem and half car
  formulation.
• References.

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The Full Car Model

• The Full car model is the most complex model
  which is able to incorporate all the modes of the
  vehicle motion.
• Includes the pitch and roll motion of the hull
  sprung mass along with the vertical heave motion.
• The Full Car model is important for containing
  the pitch as well as the roll acceleration of the
  vehicle while traveling on uneven terrains.
• Present model uses a series of 12 un-sprung
  masses to model the 12 wheels of the tracked
  vehicle.

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Full Car Model for a tracked vehicle.
The schematic diagram of the full car model for
the tracked vehicle
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The State Space Formulation


terrain input
Ref Hac A. ,1970Journal of Sound and Vibration
terrain roughness factor
variance of terrain irregularities
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Full Car Model Full State Feed Back

• The full state feed back solution for the Full
  Car model is based on the Linear Quadratic
  Regulator (LQR) concept in the Optimal control
  theory.
• Aim is to decide a control law u(t) - K.x(t)
  which minimizes the defined performance index
  (Cost Function).
• Full State Feed Back assumes the availability of
  all the state variables for measurement.

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The Cost Function
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Cost Function contd..
Cannot be solved directly as numerical
instabilities occur
Partition the matrices and solve separately
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Cost Function contd..
The Final Solution
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State Covariance Matrix
The Average system performance can be evaluated
by calculating the System Covariance matrix
Covariance matrix for the input
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Sub-optimal Output Feed Back Design

• The output feed back problem is posed and solved
  as an optimal regulator problem after
  incorporating additional states.
• K Solution of this reformulated problem
• The feedback matrix K is obtained by Kosuts
  method by minimizing the norm given by.
• where ? is the set of all admissible
  controllers and C is the controller constraint
  matrix.

Solution for full-state feedback
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Full Car Model PID Control

• Suspension deflection measurement is the measured
  output.
• The general form of the equations is
• The state matrix is modified to include the
  additional integral state.
• 
  

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Full Car Model Design
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Full Car Model Sky Hook Damping

• A Damper is placed between the sprung mass and
  the sky.
• This amounts to the negative feed back of the
  sprung mass velocity.
• There are 2 cases of the sky hook damping
• Ideal sky hook No control force on the un
  sprung mass
• Practical sky hook Actuator applies force on
  both the sprung as well as the un-sprung

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Full Car Model Ideal Sky Hook Damping
Suspension deflection transmissibility
Sprung Mass acceleration
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Full Car Model Practical Sky Hook Damping
Suspension Deflection transmissibility
Sprung Mass acceleration frequency response
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Full Car Model Sky Hook Conclusions

• The Ideal sky hook results indicate better sprung
  mass vibration response than the other out put
  feed back methods though its worse than that of
  the full state feedback case.
• The un-sprung mass response hasnt deteriorated
  showing excellent terrain holding ability.
• In the practical sky hook case the sprung mass
  transmissibility deteriorates a bit from the
  ideal case.
• The un-sprung mass response is worse with
  oscillatory motion of the un sprung mass.
• The road holding deteriorates considerably at the
  high velocity values in the practical sky hook
  case.

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Full Car Model Dynamic Simulation

• The Full Car model based controller design has
  been tested using a 3 D Ride Dynamic Model
  of the tracked vehicle.
• The equations describing the hull accelerations
  of the vehicle are as given below.

Hull vertical acceleration
Hull Pitch acceleration
Hull Roll acceleration
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Full Car Model Dynamic Simulation

• The full car active control laws mentioned have
  been tested by incorporating the active
  suspension in the 3 D Non Linear Ride Dynamic
  model.
• Power and Acceleration calculation is done based
  on the RMS estimate of the simulated values.
• where x (i) is the instantaneous acceleration
  value and N is the number of data points.

RMS value of the actuator force
RMS value of the relative velocity
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Driver Seat Modeling

• Usually the driver seat has some cushioning
  effect which acts as a low pass filter reducing
  the level of the acceleration actually
  experienced by the driver.
• This effect can be modeled by assuming a passive
  spring mass system as shown below.

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Driver Seat Modeling

• Assuming Ms to be the mass of the driver with the
  seat, Ks to be the seat spring stiffness and Bs
  to be the seat spring damping coefficient the
  driver acceleration ad is given by
• where ls, ld, vs, vd, are the positions and
  velocities of the hull at the seat and the driver
  respectively.

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Track-Terrain Interaction
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Post-Processing of Simulation results

• The post processing of the obtained vehicle
  dynamic variables is necessary to make the
  results realistic. The most important need is to
  filter out the high frequency components from the
  obtained results. This is important as the huge
  vehicle hull structure has a rapid attenuating
  effect beyond the 2nd natural frequency or the
  wheel-hop frequency on any dynamic parameter.
• Further the insensitivity of the human body to
  high frequency vibrations suggests the removal of
  these components for obtaining a practical
  estimate of the vehicle dynamic performance from
  the simulation results.

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Full Car Model Dynamic Simulation

• The designed controllers are tested by
  incorporating the active suspension system for
  the Full Car in the Ride Dynamic Model for the 3
  standard terrains

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Full Car Dynamic Simulation Terrain 1 at 72
km/hTime plot comparisons for different
controllers
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Robust Controller design for a Half-Car Model
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The Robust control problem

• The real problem in robust multivariable feedback
  control system design is to synthesize a control
  law which maintains system response and error
  signals to within pre-specified tolerances
  despite the effects of uncertainty on the system.
• Uncertainty may take many forms but among the
  most significant are noise/disturbance signals
  and transfer function modeling errors. Another
  source of uncertainty is un-modeled nonlinear
  behavior.
• The benefit of using a frequency domain approach
  is that we can specify frequency dependent
  weights and thus have a better frequency loop
  shaping.
• For the current problem a weighted mixed
  sensitivity formulation is used as it is a direct
  and effective way of achieving Multivariable Loop
  Shaping

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General framework
Sensitivity function d ? y
r ? u
Complementary Sensitivity function r ? y
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Uncertainty Description
Robustness Theorem 1 If the plant is nominally
stable then the size of the smallest
for which the system becomes unstable is given by
Robustness Theorem 2 If the plant is nominally
stable then the size of the smallest
for which the system becomes unstable is given by
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The Weighted Mixed Sensitivity formulation
Design Objective
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The Weighted Mixed Sensitivity formulation
The singular values of determine the
disturbance attenuation since is in fact the
closed-loop transfer function from disturbance u1
to plant output y1a. Thus a disturbance
attenuation performance specification may be
written as
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Singular value specification on S and T
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The System Interconnection structure
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The Augmented System Structure
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The Actuator
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Performance and Robustness
Wp Performance Criterion
Wr Robustness Criterion
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Perturbed Plant
Heave acceleration plot for the perturbed plant
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Perturbed Plant
Pitch acceleration plot for the perturbed plant
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Perturbed Plant
Suspension Deflection plot for the perturbed
plant
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Controller Design

• Two Controllers were designed

Fh is a suboptimal controller designed using
iteration
Ref Glover and Doyle Systems and control
letters vol11, 1988
Fmu is designed using D-K iteration approach to
mu-synthesis
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Design using D-K iteration

• The Objective is to find a controller F(s) and a
  Diagonal scaling matrix D(s) such that

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Nominal and Robust Performance of the controllers
Nominal Performance
Robust Performance
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Controller order reduction
Plot comparing the full order and the reduced
order controller
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Digital Implementation
Plot comparing the continuous time and the
discrete time controller
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Simulation Results (Sinusoidal terrain)
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Simulation Results (Trapezoidal terrain)
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Conclusion

• The robust control solution for active suspension
  design achieves the desired criterion of
  consistent performance under perturbations.
• Two controllers were designed and their mu-plots
  were compared to show their performance under
  nominal and perturbed circumstances.
• As D-K iteration gives controllers with huge
  sizes, controller order reduction was carried on
  and the performance of the reduced order
  controller was compared with the full order
  controller.
• Simulation results carried on the test tracks
  provided by CVRDE underlines the successful
  design of the controller.

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References

• Srinivasa Y G and Ravi Teja S, "Investigations on
  the Stochastically Optimal PID Controller for a
  Linear Quarter Car Road Vehicle Model, Vehicle
  System Dynamics, 26(1996), pp. 103 116.
• A.Dhir and S.Sankar, Ride dynamics of high-speed
  track vehicles Simulation with field validation,
  Vehicle system dynamics PP 379 409.
• A. Dhir and S Sankar ,Assessment of tracked
  vehicle suspension system using a validated
  computer simulation model, Journal of
  Terramechanics, Volume 32, Issue 3, May 1995,
  Pages 127-149,
• Hac A, 1985, Suspension optimization of a 2 DOF
  vehicle model using stochastic optimal control
  technique, Journal of sound and vibration,
  100-(3). , 343 357.
• Kosut, R.L. (1970) Sub- Optimal Control of linear
  time-invariant systems subject to control
  structure constraints. IEEE Transactions on
  Automatic Control. AC-15,5,557-563

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Referencescontd.

• J. Doyle, K. Glover, P. Khargonekar, and B.
  Francis, "State-space solutions to standard H2
  and H-infinity control problems," IEEE Trans.
  Automat. Contr., AC-34, no. 8, pp. 831-847, Aug.
  1989.
• Antonio Moran, Masao Nagai, Analysis and Design
  of Active Suspensions by H-infinity Robust
  Control Theory, JSME International Journal
  Series III, Vol.35, No. 3, 1992
• Yamashita etal, Application of H-infinity
  control to Active Suspension Systems,
  Automatica, Vol 30, No.11,1994

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• Thank You