adaptive dynamic inversion control of a linear scalar plant ......adaptive dynamic inversion control...

Adaptive Dynamic Inversion Control of a LinearScalar Plant with Constrained Control Inputs

Monish D. Tandale & John Valasek

Aerospace Engineering

Paper # ThA10.224th American Control Conference

Portland, Oregon June 8 to 10, 2005

American Control Conference: 9th June 2005Monish Tandale & John Valasek

Overview

Problem Definition

Adaptive Dynamic Inversion ControlNo Control Saturation.

Control Saturation. (Modified Update Law)

Trackable TrajectoriesStable, Neutrally stable and Unstable plants

Instability Protection: Switching Control Law

Numerical Simulations

Conclusions and Future Work


Problem Definition

Linear Scalar Plant:where and

Control:

Objective: 1. Track reference feasible wrt control limits.2. Non feasible trajectories : track as close as possible,

maintain stability and ensure that all signals remain bounded.

ubxax ** +=&1, ℜ∈ux

)0*(],,[* and ],[* maxminmaxmin ≠∈∈ bbbbaaa

)0( , ],[ >+−∈ mmm uuuu

rr xx &,

Let us ignore problem of feasible trajectories (for the time being)

and consider trajectory tracking problem


Adaptive Dynamic Inversion ControlNo Control Saturation - 1

Objective : Trajectory Tracking for Dynamic Systems.

Control is calculated by : Dynamic Inversion and Sliding Mode Control.

Dynamic Inversion requires system parameters which are uncertain.

Adaptive Learning Parameters are used for the Dynamic Inversion which are updated in real time.

The Adaptation Mechanism is driven by the error between the actual plant trajectory and the reference trajectory


Adaptive Dynamic Inversion ControlNo Control Saturation - 2

Defining trajectory tracking error

The control law

along with the update law

When (no control saturation)

ensures converges to asymptotically.

To ensure is bounded zero crossing of must be avoided by parameter projection.

rxxe −=1

)ˆ)(ˆ/1( 1exxabu rc λ−+= &

ˆ ,ˆ 1211 uebxea aγγ =−= &&

ac uu =rxx

cu b̂


Problem: Control SaturationParameter drift in adaptive systems because of errors arising from control saturation .

Tracking error has contributions due toInitial Condition Error.

Parametric Uncertainties in the model.

Control Saturation.

We are adapting only to parametric uncertainties in the system model.

Including the error component due to saturation will cause parameter drift.

Objective : Isolate the component of the tracking error arising due to Control Saturation


Modified Reference and Modified Tracking Error

This estimated lack of acceleration is subtracted from the original reference.

The modified reference can be tracked within saturation limits.

The tracking error between the plant trajectory and the modified reference does not have the error component due to saturation.

This modified tracking error is used to update the adaptive parameters.

Thus error due to control saturation is isolated.

Calculated Control – Applied Control Lack of Acceleration


Adaptive Dynamic Inversion ControlControl Saturation - 1

Defining modified reference trajectory

Tracking errors

The control law

along with the update law

ensures converges to asymptotically.

Modified Reference

mx)ee(exxexxe rmm 32132 and +=−=−=

ˆ ,ˆ 2221 uebxea aγγ =−= &&

mxx

)ˆ)(ˆ/1( 1exxabu rc λ−+= &

acrmrm uuxxbxx −=−−−= δλδ ),(ˆ&&


Adaptive Dynamic Inversion ControlControl Saturation - 2

Modified reference converges to the original desired reference when control is out of saturation

We need to ensure that modified reference is bounded. Show plant state is bounded.

δλ bee ˆ33 −−=&

t

x

xrxm

x

x

Can we identify the feasible / non feasible trajectories?


Trackable Trajectory: Stable Plant

State: Stabilizing tendency

Control: Limited

destablilizing tendency

Conclusion:

For

open loop stable plants

with bounded control

State is Bounded


Trackable Trajectory: Neutrally Stable Plant

is not a function of

the state

Direction of is not limited

due to bounded control.

Only magnitude of

is limited

Conclusion:

Entire State-Space is

accessible

x&

x&

x&


Trackable Trajectory: Unstable Plant

State: Destabilizing tendencyControl: Limited stablilizing tendency‘Boundary of No Return’

ConclusionEntire state space is accessible.Loss of control of Direction ofLimit state withinBoundary of No Return

x&


Instability Protection

Switch Control from as state approaches boundary of no return

st uu to

Numerical Simulations(Unstable Plant)

Case # 1

Instability Protection Turned Off


Phase PortraitCase 1: Instability Protection is Switched Off


State and Tracking Error Case 1: Instability Protection is Switched Off


Control and Adaptive ParametersCase 1: Instability Protection is Switched Off

Numerical SimulationsCase # 2

Instability Protection Turned On

Reference Modification Turned Off


Phase PortraitCase 2: Instability Protection On, Reference Modification Off


State and Tracking Error Case 2: Instability Protection On, Reference Modification Off


Control and Adaptive ParametersCase 2: Instability Protection On, Reference Modification Off

Numerical SimulationsCase # 3

Instability Protection Turned On

Reference Modification Turned On


Phase PortraitCase 3: Instability Protection and Reference Modification On


State and Tracking Error Case 3: Instability Protection and Reference Modification On


Control and Adaptive ParametersCase 3: Instability Protection and Reference Modification On


Conclusions

If the control is unsaturated, the tracking errorasymptotically goes to zero and all signals in the closed-loop are bounded.If the control is saturated, all signals are bounded and the state asymptotically approaches the modified reference.The switching control strategy successfully restricts the state within the boundary of no return and the control does not show any chattering.If the reference trajectory is persistently exciting, the adaptive parameters converge to the true parameters, even in the presence of control saturation.


Future Work

Multi-input-Multi-Output Systems.Extension is not straightforward – Increase in Complexity.

Direction Consistent Control Constraint Mechanism.

Parameter Projection to prevent singularity in the inversion of the estimated control effectiveness matrix.

Estimate the Boundary of No Return for multi-state-multi-input open loop unstable systems: Difficult.

Nonlinear Systems.Extension to open loop stable plants: Not Difficult.

Rate Saturation and Position Saturation


Thank You

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adaptive dynamic inversion control of a linear scalar plant ......adaptive dynamic inversion control...

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