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PO
LIPO
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di
di M
IM
Itecn
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tecn
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lano
lano
ADAPTIVE AUGMENTED CONTROL OF UNMANNED ROTORCRAFT VEHICLES
C.L. Bottasso, R. Nicastro, L. Riviello, B. Savini
Politecnico di Milano
AHS International Specialists' Meeting onUnmanned Rotorcraft
Chandler, AZ, January 23-25, 2007
ADAPTIVE AUGMENTED CONTROL OF UNMANNED ROTORCRAFT VEHICLES
C.L. Bottasso, R. Nicastro, L. Riviello, B. Savini
Politecnico di Milano
AHS International Specialists' Meeting onUnmanned Rotorcraft
Chandler, AZ, January 23-25, 2007
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POLITECNICO di MILANO DIA
Rotorcraft UAVs at PoliMIRotorcraft UAVs at PoliMI
• Low-cost platform for development and testing of navigationnavigation and controlcontrol strategies (including vision, flight envelope protection, etc.);
• Vehicles: off-the-shelf hobby helicopters;
• On-board control hardware based on PC-104 standard;
• Bottom-up approach: everything is in-house developedeverything is in-house developed (Inertial Navigation System, Guidance and Control algorithms, Linux-based real-time OS, flight simulators, etc. etc.)
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POLITECNICO di MILANO DIA
OutlineOutline
• Non-linear model predictive control;
• Reference Augmented Predictive Control (RAPC): motivations;
• Reference Augmented Model Identification;
• Reference Augmented Neural Control;
• Results;
• Conclusions and outlook.
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POLITECNICO di MILANO DIA
UAV Control ArchitectureUAV Control Architecture
Target
Obstacles
Hierarchical three-layer control architectureHierarchical three-layer control architecture (Gat 1998):
Vision/sensor range
• Strategic layer: assign mission objectives (typically relegated to a human operator);
•Tactical layer: generate vehicle guidance information, based on input from strategic layer and sensor information;• Reflexive layer: track trajectory generated by tactical layer, control, stabilize and regulate vehicle. In this paper: Adaptive Non-linear Model Predictive Adaptive Non-linear Model Predictive ControlControl.
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POLITECNICO di MILANO DIA
Non-Linear Model Predictive Control
Non-Linear Model Predictive Control
Non-linear Model Predictive ControlNon-linear Model Predictive Control (NMPC):
Find the control action which minimizes an index of performance, by predicting the future behavior of the plant using a non-linear reduced modelnon-linear reduced model.
- Reduced model:
- Initial conditions:
- Output definition:
Cost:
with desired goal outputs and controls.
Stability resultsStability results: Findeisen et al. 2003, Grimm et al. 2005.
L(y;u) = (y ¡ y¤)T Q(y ¡ y¤) + (u ¡ u¤)T R (u ¡ u¤)
minu ;x ;y
J =
Z t0+Tp
t0
L(y;u)dt
s.t.: f ( _x;x;u) = 0 t 2 [t0;t0 +Tp]
x(t0) = x0
y = g(x) t 2 [t0;t0 + Tp]
(¢)¤
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POLITECNICO di MILANO DIA
Model-Adaptive Predictive ControlModel-Adaptive Predictive Control
1. Tracking problem
Plant response
3. Reduced model update
Predictive solutions
2. Steering problem
Prediction window
Steering window
Tracking cost
Prediction error
Prediction window
Tracking cost
Steering window
Prediction error
Tracking costPrediction window
Steering window
Prediction error
Goal trajectory
Receding horizon controlReceding horizon control:
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MotivationMotivation
• For any given problem: wealth of knowledgeknowledge and legacylegacy methods which perform reasonably well;
• Quest for better performance/improved capabilities: undesirableundesirable and wastefulwasteful to neglect valuable existing knowledge;
Reference Augmented Predictive ControlReference Augmented Predictive Control (RAPCRAPC): exploit available legacy methods, embedding them in a non-linear model predictive control framework.
Specifically:
• ModelModel: augment flight mechanics rotorcraft models (BEM+inflow theories) to account for unresolved or unmodeled physics;
• ControlControl: design a non-linear controller augmenting linear ones (LQR) which are known to provide a minimum level of performance about certain linearized operating conditions.
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OutlineOutline
• Non-linear model predictive control;
• Reference Augmented Predictive Control (RAPC): motivations;
• Reference Augmented Model Identification;
• Reference Augmented Neural Control;
• Results;
• Conclusions and outlook.
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POLITECNICO di MILANO DIA
GoalGoal:
• Develop reduced modelreduced model capable of predicting the behavior predicting the behavior of the plantof the plant with minimum error (same outputs when subjected to same inputs);
• Reduced model must be self-adaptiveself-adaptive (capable of learning) to adjust to varying operating conditions.
Predictive solutions
Prediction (tracking) window
Steering window
Prediction Prediction error to be error to be minimizedminimized
Reference Augmented Model Identification
Reference Augmented Model Identification
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POLITECNICO di MILANO DIA
Neural augmented reference modelNeural augmented reference model:
reference (problem dependent) analytical model,
RemarkRemark: reference model will notnot, in general, ensure adequateadequate predictions, i.e.
when = system states/controls,
= model states/controls.
Augmented reference model:Augmented reference model:
where is the unknownunknown reference model defectdefect that ensures
when
Hence, if we knewif we knew , we would have perfect predictionperfect prediction capabilities.
d
d
eu = u.
eu = u;
f ref( _x;x;u) = 0:
ex 6= x
x;u
ex; eu
f ref( _x;x;u) = d(x;u);
ex = x
Reference Augmented Model Identification
Reference Augmented Model Identification
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POLITECNICO di MILANO DIA
Approximate with single-hidden-layer neural networkssingle-hidden-layer neural networks:
where
and
= functional reconstruction error;
= matrices of synaptic weights and biases;
= sigmoid activation functions;
= network input.
The reduced model parametersreduced model parameters
are identified on-line using Kalman filteringKalman filtering.
d
¾(Á) = (¾(Á1); : : : ;¾(ÁN n))T
d(y;u) = dp(x;u;pm) +";
dp(x;u;pm) = WmT ¾(Vm
T i +am) + bm;
"
Wm;Vm;am;bm
i = (xT ;uT )T
pm = (::: ;Wmi k;Vmi k
;ami;bmi
; : : :)T
Reference Augmented Model Identification
Reference Augmented Model Identification
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Model Augmentation ResultsModel Augmentation ResultsPitch rate for plant, reference, and neural-augmented reference with same prescribed inputs.
Short Short transient = transient =
fast adaptionfast adaption
Black: plant
Red: reference model
Blue: reference model +neural network
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OutlineOutline
• Non-linear model predictive control;
• Reference Augmented Predictive Control (RAPC): motivations;
• Reference Augmented Model Identification;
• Reference Augmented Neural Control;
• Results;
• Conclusions and outlook.
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Prediction problem:
Enforcing optimalityEnforcing optimality, we get:
Non-Linear Model Predictive Control
Non-Linear Model Predictive Controlminu ;x ;y
J =
Z t0+Tp
t0
L(y;u)dt
s.t.: f ( _x;x;u) = 0 t 2 [t0;t0 +Tp]
x(t0) = x0
y = g(x) t 2 [t0;t0 + Tp]
f ( _x;x;u;pm) = 0; t 2 [t0;t0 +Tp];
x(t0) = x0;
¡d(f T
; _x ¸ )
dt+ f T
;x ¸ + yT;x L ;y = 0; t 2 [t0;t0 +Tp];
¸ (t0 +Tp) = 0;
L ;u +f T;u ¸ = 0; t 2 [t0;t0 +Tp]:
• Model equations:
• Adjoint equations:
• Transversality conditions:
• State initial conditions:
• Co-state final conditions:
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FuturePast
Prediction window
x0
t0
t0 t0 +Tp
Goal response x¤(t)
Goal control u¤(t)
Non-Linear Model Predictive Control
Non-Linear Model Predictive Control
u(t)
Optimal control u(t)
x(t); t < t0
u(t); t < t0
Â(¢;¢;¢;¢)
It can be shown that minimizing minimizing controlcontrol is See paper for details.
u(t) = ¡x0;y¤(t);u¤(t);t
¢
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Reference augmented form:Reference augmented form:
where is the unknown control defect.
RemarkRemark: if one knew , the optimal control would be available without having to solve the open-loop optimal control problem.
IdeaIdea:
- ApproximateApproximate using an adaptive parametric element:
- IdentifyIdentify on-line, i.e. find the parameters which minimize the reconstruction error .
pc"
Reference Augmented Predictive Control
Reference Augmented Predictive Control
u(t) = uref(t) +À¡x0;y¤(t);u¤(t);t
¢
À(¢;¢;¢;¢)
À(¢;¢;¢;¢)
À(¢;¢;¢;¢)À
¡x0;y¤(t);u¤(t);t
¢= Àp
¡x0;y¤(t);u¤(t);t;pc
¢+"c
Àp(¢;¢;¢;¢)
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POLITECNICO di MILANO DIA
Iterative procedureIterative procedure to solve the problem in real-time:
• Integrate reduced model equations forwardforward in time over the prediction window, using and the latest available parameters (state prediction):(state prediction):
• Integrate adjoint equations backward backward in time (co-state (co-state prediction):prediction):
• CorrectCorrect control law parameters , e.g. using steepest descent:
pc
uref pc
_pc = ¡ ´ J ;pc ! pnewc = pold
c ¡ ´ J ;pc
¡d(f T
; _x ¸ )
dt+ (f ;x +uT
;x f ;u )T ¸ + yT;x L ;y + uT
;x L ;u = 0 t 2 [t0;t0 + Tp]
¸ (t0 + Tp) = 0
f ( _x;x;u;pm) = 0 t 2 [t0;t0 + Tp]
x(t0) = x0
On-line Identification of Control Parameters
On-line Identification of Control Parameters
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POLITECNICO di MILANO DIA
RemarkRemark: the parameter correction step
seeks to enforce the transversality conditiontransversality condition
Once this is satisfied, the control is optimaloptimal, since the state and co-state equations and the boundary conditions are satisfiedsatisfied.
_pc = ¡ ´ J ;pc
Z t0+Tp
t0
ÀT;pc
(L ;u + f T;u ¸ ) dt = 0
On-line Identification of Control Parameters
On-line Identification of Control Parameters
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Tracking cost
Future Target
• PredictPredict state forward• PredictPredict co-state backwards
• PredictPredict control action
• UpdateUpdate estimate of control action, based on transversality violation
_pc = ¡ ´ J ;pc
• AdvanceAdvance plant• UpdateUpdate model, based on prediction error
Past
Optimal control
Prediction error
• RepeatRepeat
FuturePast
Prediction horizonSteering window
State
Control
On-line Identification of Control Parameters
On-line Identification of Control Parameters
x(t)
¸ (t)
u(t)
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Neural-Network-Based Implementation
Neural-Network-Based Implementation
- Drop dependence on time history of goal quantities:
- Approximate temporal dependence using shape functions:
- Associate each nodal value with the output of a single-single-hidden-layerhidden-layer feed-forward neural networkneural network, one for each component:
where
Output:
Input:
Control parameters:
Àp¡x0;y¤(t);u¤(t);t;pc
¢¼Àp
¡x0;y¤(t0);u¤(t0);t;pc
¢
Àp¡x0;y¤(t0);u¤(t0);¿;pc
¢¼
(1¡ ») Àpk
¡x0;y¤(t0);u¤(t0);pc
¢+»Àpk + 1
¡x0;y¤(t0);u¤(t0);pc
¢
oc = W Tc ¾(V T
c i c +ac) +bc
oc = (ÀTp0
;ÀTp1
; : : : ;ÀTpM ¡ 1
)T
i c =¡xT
0 ;x¤T (t0);u¤T (t0)¢T
pc = (::: ;Wci j; : : : ;Vci j
; : : : ;aci; : : : ;bci
; : : :)T
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FuturePast
Prediction window
x0
t0
t0 t0 +Tp
x(t); t < t0
u(t); t < t0
u¤(t0)
x¤(t0)
NNÀpk
Neural-Network-Based Implementation
Neural-Network-Based Implementation
x¤(t)
u¤(t)
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OutlineOutline
• Non-linear model predictive control;
• Reference Augmented Predictive Control (RAPC): motivations;
• Reference Augmented Model Identification;
• Reference Augmented Neural Control;
• Results;
• Conclusions and outlook.
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Vehicle Model and Simulation Environment
Vehicle Model and Simulation Environment
Vehicle modelVehicle model:
• Blade element and inflow theory (Prouty, Peters);
• Quasi-steady flapping dynamics, aerodynamic damping correction;
• Look-up tables for aerodynamic coefficients of lifting surfaces;
• Effects of compressibility and downwash at the tail due to main rotor;
• Process and measurement noise, delays.
Reflexive controllerReflexive controller:
• State reconstruction by Extended Kalman Filtering;
• Reference controller: output-feedback LQR at 50 Hz;
• Goal trajectory planned as in Bottasso et al. 2007.
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ResultsResults
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ResultsResults
Integral tracking error vs. length of prediction window:
Significant Significant improvement over improvement over LQRLQR
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ResultsResults
Turn rate vs. time:
RAPC
LQR
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ResultsResults
Integral tracking error vs. model mismatch parameter:
RAPC without model adaption
RAPC with model adaption
Significant Significant improvement over improvement over LQRLQR
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ResultsResults
Main rotor collective & norm of control network parameters:
Initial Initial transienttransient
AdaptedAdapted
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ConclusionsConclusions
• Non-linear reduced model identificationreduced model identification for capturing unmodeled or unresolved physics;
• Linear controller promotedpromoted to non-linear;
• Hard real-timereal-time capable (fixed number of ops, no iterations);
• Adaption of control action can be performed independentlyindependently from adaption of reduced model;
• Reference model and reference control ensure good predictions even before adaptionbefore adaption, avoid need for pre-training, simplify adaptionsimplify adaption since defect is small;
• Conceptually possible (but not investigated here) to do adaption diagnosticsadaption diagnostics by monitoring defects;
• Theoretically non-linearly stablenon-linearly stable (if identification of , successful);
• Basic concept demonstrated in a high-fidelity virtual environment.
pcpm
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OutlookOutlook
• Real-time implementation and integration in a rotorcraft UAV (in progress) at the Autonomous Flight Lab at PoliMI;
• Testing and extensive experimentation;
• Integration with vision for fully autonomous navigation in complex environments.