21-07-2012, 10:49 AM
Asymptotic Tracking for Aircraft via Robust and Adaptive Dynamic Inversion Methods
[attachment=28265]
INTRODUCTION
FEEDBACK linearization is a general control method
where the nonlinear dynamics of a system are canceled
by state feedback yielding a residual linear system. Dynamic
inversion (DI) is a similar concept as feedback linearization
that is commonly used within the aerospace community
to replace linear aircraft dynamics with a reference model
[1]–[8]. For example, in [3] a general DI approach is presented
for a reference tracking problem for a minimum-phase and
left-invertible linear system. A DI controller is designed for a
nonminimum-phase hypersonic aircraft system in [1], which
utilizes an additional controller to stabilize the zero dynamics.
A finite-time stabilization design is proposed in [2], which
utilizes DI given a full rank input matrix. Typically, DI methods
assume the corresponding plant models are exactly known.
ROBUST CONTROL DEVELOPMENT
A contribution of the development in this section is a robust
technique that yields asymptotic tracking for an aircraft in the
presence of parametric uncertainty in a non-square input matrix
and an unknown nonlinear disturbance. To this end, the control
law is developed based on the output dynamics, which enables
us to transform the uncertain input matrix into a square
matrix. By utilizing a feedforward (best guess) estimate of the
input uncertainty in the control law in conjunction with a robust
control term, we are able to compensate for the input uncertainty.
Specifically, asymptotic tracking is proven based on
the assumption that an estimate of the uncertain input matrix can
be selected such that a diagonal dominance property is satisfied
in the closed-loop error system.
ADAPTIVE CONTROL EXTENSION
The robust DI control technique presented in the previous sections
can be extended to an ADI-based control method. Robust
control methods can be utilized to compensate for both structured
and unstructured bounded uncertainty; however, since robust
control methods are based on worst-case uncertainty and
disturbances, high gain or high frequency feedback is often required
to achieve stability. The subsequent analysis illustrates
how asymptotic output tracking can be achieved for the aircraft
system in (1) by utilizing a Lyapunov-based adaptive law to
compensate for the parametric uncertainty in the input matrix.
Assumption 5: The matrix product is assumed to be invertible.
The invertibility condition means that the control development
and stability analysis presented in this paper is applicable
to relative degree 1 systems.
SIMULATION RESULTS
A. Simulation Model
Numerical simulations were created to test the efficacy of the
developed controllers. While numerical simulation results are
being presented as opposed to experimental results, the sensor
noise, actuator saturation and rate limits, and disturbances used
in the numerical simulation are based on detailed analyses and
specifications from typical sensors and actuators for an Osprey
UAV. Table I of [9], [43] shows the simulation parameters used
that correspond to these hardware constraints.
CONCLUSION
A robust aircraft controller is presented, which achieves
asymptotic tracking control of a model reference system where
the plant dynamics contain input uncertainty and a bounded
non-LP disturbance. The developed controller exhibits the
desirable characteristic of tracking the specified decoupled
reference model. An example of such a decoupling is demonstrated
by examining the aircraft response to tracking a roll
rate command while simultaneously tracking a completely
unrelated yaw rate command. This result represents application
of the RISE control strategy in a DI and MRAC framework to a
nonlinear system with additive, non-LP disturbances, where the
control input is multiplied by a non-square matrix containing
parametric uncertainty.
[attachment=28265]
INTRODUCTION
FEEDBACK linearization is a general control method
where the nonlinear dynamics of a system are canceled
by state feedback yielding a residual linear system. Dynamic
inversion (DI) is a similar concept as feedback linearization
that is commonly used within the aerospace community
to replace linear aircraft dynamics with a reference model
[1]–[8]. For example, in [3] a general DI approach is presented
for a reference tracking problem for a minimum-phase and
left-invertible linear system. A DI controller is designed for a
nonminimum-phase hypersonic aircraft system in [1], which
utilizes an additional controller to stabilize the zero dynamics.
A finite-time stabilization design is proposed in [2], which
utilizes DI given a full rank input matrix. Typically, DI methods
assume the corresponding plant models are exactly known.
ROBUST CONTROL DEVELOPMENT
A contribution of the development in this section is a robust
technique that yields asymptotic tracking for an aircraft in the
presence of parametric uncertainty in a non-square input matrix
and an unknown nonlinear disturbance. To this end, the control
law is developed based on the output dynamics, which enables
us to transform the uncertain input matrix into a square
matrix. By utilizing a feedforward (best guess) estimate of the
input uncertainty in the control law in conjunction with a robust
control term, we are able to compensate for the input uncertainty.
Specifically, asymptotic tracking is proven based on
the assumption that an estimate of the uncertain input matrix can
be selected such that a diagonal dominance property is satisfied
in the closed-loop error system.
ADAPTIVE CONTROL EXTENSION
The robust DI control technique presented in the previous sections
can be extended to an ADI-based control method. Robust
control methods can be utilized to compensate for both structured
and unstructured bounded uncertainty; however, since robust
control methods are based on worst-case uncertainty and
disturbances, high gain or high frequency feedback is often required
to achieve stability. The subsequent analysis illustrates
how asymptotic output tracking can be achieved for the aircraft
system in (1) by utilizing a Lyapunov-based adaptive law to
compensate for the parametric uncertainty in the input matrix.
Assumption 5: The matrix product is assumed to be invertible.
The invertibility condition means that the control development
and stability analysis presented in this paper is applicable
to relative degree 1 systems.
SIMULATION RESULTS
A. Simulation Model
Numerical simulations were created to test the efficacy of the
developed controllers. While numerical simulation results are
being presented as opposed to experimental results, the sensor
noise, actuator saturation and rate limits, and disturbances used
in the numerical simulation are based on detailed analyses and
specifications from typical sensors and actuators for an Osprey
UAV. Table I of [9], [43] shows the simulation parameters used
that correspond to these hardware constraints.
CONCLUSION
A robust aircraft controller is presented, which achieves
asymptotic tracking control of a model reference system where
the plant dynamics contain input uncertainty and a bounded
non-LP disturbance. The developed controller exhibits the
desirable characteristic of tracking the specified decoupled
reference model. An example of such a decoupling is demonstrated
by examining the aircraft response to tracking a roll
rate command while simultaneously tracking a completely
unrelated yaw rate command. This result represents application
of the RISE control strategy in a DI and MRAC framework to a
nonlinear system with additive, non-LP disturbances, where the
control input is multiplied by a non-square matrix containing
parametric uncertainty.