DERIVATIVE FREE OUTPUT FEEDBACK ADAPTIVE CONTROL

DERIVATIVE FREE OUTPUT FEEDBACK ADAPTIVE CONTROL Tansel YUCELEN, * Kilsoo KIM, and Anthony J. CALISE Georgia Institute of Technology, Yucelen Atlanta, * GA 30332, USA * tansel@gatech.edu AIAA Guidance, Navigation, and Control Conference 8 11 August 2011, Portland, Oregon

Outline Motivation Adaptive Control Derivative-Free Adaptive Control Illustrative Scalar Example Output Feedback Adaptive Control Problem Formulation Control System Description Adaptive Control Architecture Visualization Wing Rock Dynamics Example Nonlinear Uncertainty External Disturbance Measurement Noise Concluding Remarks

Outline Motivation Adaptive Control Derivative-Free Adaptive Control Illustrative Scalar Example Output Feedback Adaptive Control Problem Formulation Control System Description Adaptive Control Architecture Visualization Wing Rock Dynamics Example Nonlinear Uncertainty External Disturbance Measurement Noise Concluding Remarks

Motivation Models that do not adequately capture the physical system Idealized assumptions and model simplifications Actual dynamics can be nonlinear and uncertain Many loops can be coupled (MIMO) Unknown disturbances (such as turbulance)

Motivation Models that do not adequately capture the physical system Idealized assumptions and model simplifications Actual dynamics can be nonlinear and uncertain Many loops can be coupled (MIMO) Unknown disturbances (such as turbulance) Sudden change in dynamics Reconfiguration Deployment of a payload Structural damage

Motivation Models that do not adequately capture the physical system Idealized assumptions and model simplifications Actual dynamics can be nonlinear and uncertain Many loops can be coupled (MIMO) Unknown disturbances (such as turbulance) Sudden change in dynamics Reconfiguration Deployment of a payload Structural damage Robust controllers? May fail to achieve a given performance criteria Under high levels of uncertainty Require more system modeling information

Motivation Models that do not adequately capture the physical system Idealized assumptions and model simplifications Actual dynamics can be nonlinear and uncertain Many loops can be coupled (MIMO) Unknown disturbances (such as turbulance) Sudden change in dynamics Reconfiguration Deployment of a payload Structural damage Robust controllers? May fail to achieve a given performance criteria Under high levels of uncertainty Require more system modeling information Adaptive controllers?

Adaptive Control Adaptive control is an attractive approach Address system uncertainties and nonlinearities Preserve stability w/o excessively reliance on models

Adaptive Control Adaptive control is an attractive approach Address system uncertainties and nonlinearities Preserve stability w/o excessively reliance on models Indirect/direct adaptive control architectures Indirect architecture: Prm estimation and adapting gains Direct architecture: Adapting gains in resp to sys variations

Adaptive Control Adaptive control is an attractive approach Address system uncertainties and nonlinearities Preserve stability w/o excessively reliance on models Indirect/direct adaptive control architectures Indirect architecture: Prm estimation and adapting gains Direct architecture: Adapting gains in resp to sys variations SYSTEM NOMINAL CONTROL COMMAND

Adaptive Control Adaptive control is an attractive approach Address system uncertainties and nonlinearities Preserve stability w/o excessively reliance on models Indirect/direct adaptive control architectures Indirect architecture: Prm estimation and adapting gains Direct architecture: Adapting gains in resp to sys variations UNCERTAIN SYSTEM NOMINAL CONTROL REFERENCE MODEL COMMAND

Adaptive Control Adaptive control is an attractive approach Address system uncertainties and nonlinearities Preserve stability w/o excessively reliance on models Indirect/direct adaptive control architectures Indirect architecture: Prm estimation and adapting gains Direct architecture: Adapting gains in resp to sys variations UNCERTAIN SYSTEM NOMINAL CONTROL REFERENCE MODEL COMMAND ADAPTIVE CONTROL

Derivative-Free Adaptive Control Derivative-based (standard) adaptive control Based on standard Lyapunov theory Existence of constant unknown ideal set of weights May require unrealistically high adaptation gain May fail to achieve a good perf under failure recovery Require mods in order to prevent from bursting

Derivative-Free Adaptive Control Derivative-based (standard) adaptive control Based on standard Lyapunov theory Existence of constant unknown ideal set of weights May require unrealistically high adaptation gain May fail to achieve a good perf under failure recovery Require mods in order to prevent from bursting Derivative-free adaptive control Based on Lyapunov-Krasovskii theory Guaranteed transient and steady state perf bounds Preserves stability and achieves desired performance Time-varying ideal weights (fast variation is allowed) Adv for sys with sudden change in dynamics Does not need mods in order to prevent from bursting

Derivative-Free Adaptive Control Derivative-free adaptive control Based on Lyapunov-Krasovskii theory Guaranteed transient and steady state perf bounds Preserves stability and achieves desired performance Time-varying ideal weights (fast variation is allowed) Adv for sys with sudden change in dynamics Does not need mods in order to prevent from bursting

Illustration: Constant Ideal Weights

Illustration: Constant Ideal Weights

Illustration: Constant Ideal Weights Low gain ( γ = 25 )

Illustration: Constant Ideal Weights Low gain ( γ = 25 ) Moderate gain ( γ = 125 )

Illustration: Time-Varying Ideal Weights

Illustration: Time-Varying Ideal Weights

Illustration: Time-Varying Ideal Weights Low gain ( γ = 25 )

Illustration: Time-Varying Ideal Weights Low gain ( γ = 25 ) Moderate gain ( γ = 90 )

Outline Motivation Adaptive Control Derivative-Free Adaptive Control Illustrative Scalar Example Output Feedback Adaptive Control Problem Formulation Control System Description Adaptive Control Architecture Visualization Wing Rock Dynamics Example Nonlinear Uncertainty External Disturbance Measurement Noise Concluding Remarks

Output Feedback Adaptive Control Extension of derivative-free adapt ctrl to output fdbk

Output Feedback Adaptive Control Extension of derivative-free adapt ctrl to output fdbk Augmentation of a fixed gain, observer based output fdbk ctrl

Output Feedback Adaptive Control Extension of derivative-free adapt ctrl to output fdbk Augmentation of a fixed gain, observer based output fdbk ctrl Realization of adapt ctrl does not require reference model Observer acts like a reference model

Output Feedback Adaptive Control Extension of derivative-free adapt ctrl to output fdbk Augmentation of a fixed gain, observer based output fdbk ctrl Realization of adapt ctrl does not require reference model Observer acts like a reference model Parameter dependent Riccati equation (PDRE) is used Rather than a Lyapunov equation

Output Feedback Adaptive Control Extension of derivative-free adapt ctrl to output fdbk Augmentation of a fixed gain, observer based output fdbk ctrl Realization of adapt ctrl does not require reference model Observer acts like a reference model Parameter dependent Riccati equation (PDRE) is used Rather than a Lyapunov equation Stability analysis uses a Lyapunov-Krasovskii functional That entails the solution of PDRE

Output Feedback Adaptive Control Extension of derivative-free adapt ctrl to output fdbk Augmentation of a fixed gain, observer based output fdbk ctrl Realization of adapt ctrl does not require reference model Observer acts like a reference model Parameter dependent Riccati equation (PDRE) is used Rather than a Lyapunov equation Stability analysis uses a Lyapunov-Krasovskii functional That entails the solution of PDRE Cost of implementation is far less than that of other methods

Output Feedback Adaptive Control Extension of derivative-free adapt ctrl to output fdbk Augmentation of a fixed gain, observer based output fdbk ctrl Realization of adapt ctrl does not require reference model Observer acts like a reference model Parameter dependent Riccati equation (PDRE) is used Rather than a Lyapunov equation Stability analysis uses a Lyapunov-Krasovskii functional That entails the solution of PDRE Cost of implementation is far less than that of other methods Advantageous for applications to systems with Sudden change in dynamics

Problem Formulation

Problem Formulation

Problem Formulation

Problem Formulation

Remarks

Remarks

Remarks

Control System Description

Control System Description

Control System Description

Adaptive Control Description

Adaptive Control Architecture (PDRE)

Remarks on PDRE

Remarks on PDRE v = v β 2

Remarks on PDRE 0 = A e T P + PA e + v C T PB C T PB T + Q 0

Remarks on PDRE 0 = A e T P + PA e + v C T PB C T PB T + Q 0 If PB = C T (positive-real), then PDRE reduces to Lyapunov eqn and v =

Remarks on PDRE 0 = A e T P + PA e + v C T PB C T PB T + Q 0 If PB = C T (positive-real), then PDRE reduces to Lyapunov eqn and v = This suggests that for the purposes of adaptive control design, when m > 1, it is advantageous to define a new meas by taking a linear combination of existing measurements y o t = My t = MCx t = C o x(t)

Remarks on PDRE 0 = A e T P + PA e + v C T PB C T PB T + Q 0 If PB = C T (positive-real), then PDRE reduces to Lyapunov eqn and v = This suggests that for the purposes of adaptive control design, when m > 1, it is advantageous to define a new meas by taking a linear combination of existing measurements y o t = My t = MCx t = C o x(t) M is a norm preserving transformation that minimizes a norm measure of N o = C o T P o B, 0 = A e T P o + P o A e + Q o

Remarks on PDRE 0 = A e T P + PA e + v C T PB C T PB T + Q 0 If PB = C T (positive-real), then PDRE reduces to Lyapunov eqn and v = This suggests that for the purposes of adaptive control design, when m > 1, it is advantageous to define a new meas by taking a linear combination of existing measurements y o t = My t = MCx t = C o x(t) M is a norm preserving transformation that minimizes a norm measure of N o = C o T P o B, 0 = A e T P o + P o A e + Q o Taking the Frobenius norm as a measure, it can be shown that the solution for M that min N o F subj to the constraint MC F = C F is given by M = C F B T P o C T CC T 1 B T P C o C T CCT 1 F

Visualization

Main Result

Main Result

Main Result

Main Result

Outline Motivation Adaptive Control Derivative-Free Adaptive Control Illustrative Scalar Example Output Feedback Adaptive Control Problem Formulation Control System Description Adaptive Control Architecture Visualization Wing Rock Dynamics Example Nonlinear Uncertainty External Disturbance Measurement Noise Concluding Remarks

Wing Rock Dynamics Wing rock is a nonlinear phenomenon in which an aircraft exhibits an oscillation in roll at high angles of attack

Wing Rock Dynamics Wing rock is a nonlinear phenomenon in which an aircraft exhibits an oscillation in roll at high angles of attack A two state model for wing rock dynamics can be given by where, f 1 t being a square wave having an amplitude of 0.5 and a period of 15 //...seconds, f 2 t = 0.5 sin(1.5t), and d(t) is an external disturbance

Wing Rock Dynamics Wing rock is a nonlinear phenomenon in which an aircraft exhibits an oscillation in roll at high angles of attack A two state model for wing rock dynamics can be given by where, f 1 t being a square wave having an amplitude of 0.5 and a period of 15 //...seconds, f 2 t = 0.5 sin(1.5t), and d(t) is an external disturbance x 1 (t) represents the roll angle and x 2 t represents the roll rate

Nominal and Adaptive Control Designs The reference model is selected to be second order with a natural frequency of 1.6 rad/sec and a damping ratio of 0.8, and to have a unity gain from r(t) to y m (t) at low frequency K 1 = 2.56, 2.56 K 2 = 2.56

Nominal and Adaptive Control Designs The reference model is selected to be second order with a natural frequency of 1.6 rad/sec and a damping ratio of 0.8, and to have a unity gain from r(t) to y m (t) at low frequency K 1 = 2.56, 2.56 K 2 = 2.56 We chose L = 12.8, 64.0 T State observer poles are 5 times larger than reference model poles

Nominal and Adaptive Control Designs The reference model is selected to be second order with a natural frequency of 1.6 rad/sec and a damping ratio of 0.8, and to have a unity gain from r(t) to y m (t) at low frequency K 1 = 2.56, 2.56 K 2 = 2.56 We chose L = 12.8, 64.0 T State observer poles are 5 times larger than reference model poles For adaptive control design 1 e Basis function β x = [0.5, x 1, 1 e x2 1+e x 1 1+e x 2 ]T β = 1.5 For μ = 0.05 and Q o = 0.25 I 2, it was determined that v = 124.6 κ 2 < 35.4 We set Ω 1 = 0.95I 3, κ 2 = 35, and τ = 0.01 seconds

Nominal and Adaptive Control Designs The reference model is selected to be second order with a natural frequency of 1.6 rad/sec and a damping ratio of 0.8, and to have a unity gain from r(t) to y m (t) at low frequency K 1 = 2.56, 2.56 K 2 = 2.56 We chose L = 12.8, 64.0 T State observer poles are 5 times larger than reference model poles For adaptive control design 1 e Basis function β x = [0.5, x 1, 1 e x2 1+e x 1 1+e x 2 ]T β = 1.5 For μ = 0.05 and Q o = 0.25 I 2, it was determined that v = 124.6 κ 2 < 35.4 We set Ω 1 = 0.95I 3, κ 2 = 35, and τ = 0.01 seconds Goal: Tracking a reference command

Constant Ideal Weights Nominal and adaptive control responses for the case of constant ideal weights

Time-Varying Ideal Weights Nominal and adaptive control responses for the case of time-varying ideal weights

Time-Varying Ideal Weights and Disturbances Depiction of d(t) and w(t)

Time-Varying Ideal Weights and Disturbances Nominal and adaptive control responses with disturbances for the case of time-varying ideal weights

Outline Motivation Adaptive Control Derivative-Free Adaptive Control Illustrative Scalar Example Output Feedback Adaptive Control Problem Formulation Control System Description Adaptive Control Architecture Visualization Wing Rock Dynamics Example Nonlinear Uncertainty External Disturbance Measurement Noise Concluding Remarks

Concluding Remarks Extension of state feedback, derivative-free adaptive controller to an output feedback form

Concluding Remarks Extension of state feedback, derivative-free adaptive controller to an output feedback form Particularly useful for situations in which Nature of sys uncertainty cannot be adequately represented by a set of basis functions with constant ideal weights

Concluding Remarks Extension of state feedback, derivative-free adaptive controller to an output feedback form Particularly useful for situations in which Nature of sys uncertainty cannot be adequately represented by a set of basis functions with constant ideal weights Level of complexity is far less than many other methods

Concluding Remarks Extension of state feedback, derivative-free adaptive controller to an output feedback form Particularly useful for situations in which Nature of sys uncertainty cannot be adequately represented by a set of basis functions with constant ideal weights Level of complexity is far less than many other methods Can be implemented in a form that augments an observer based linear controller architecture

Concluding Remarks Extension of state feedback, derivative-free adaptive controller to an output feedback form Particularly useful for situations in which Nature of sys uncertainty cannot be adequately represented by a set of basis functions with constant ideal weights Level of complexity is far less than many other methods Can be implemented in a form that augments an observer based linear controller architecture Illustrative example shows that the presented theory and the simulation results are compatible

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