Nested Plant/Controller Codesign Using G-Optimal Design and Continuous Time Adaptation Laws: Theoretical Framework and Application to an Airborne Wind Energy System

Deese, Joe; Vermillion, Chris

doi:10.1115/1.4040759

This paper presents a nested codesign (combined plant and controller design) formulation that uses optimal design of experiments (DoE) techniques at the upper level to globally explore the plant design space, with continuous-time control parameter adaptation laws used at the lower level. The global design space exploration made possible through optimal DoE techniques makes the proposed methodology appealing for complex, nonconvex optimization problems for which legacy approaches are not effective. Furthermore, the use of continuous-time adaptation laws for control parameter optimization allows for the extension of the proposed optimization framework to the experimental realm, where control parameters can be optimized during experiments. At each full iteration, optimal DoE are used to generate a batch of plant designs within a prescribed design space. Each plant design is tested in either a simulation or experiment, during which an adaptation law is used for control parameter optimization. Two techniques are proposed for control parameter optimization at each iteration: extremum seeking (ES) and continuous-time DoE. The design space is reduced at the end of each full iteration, based on a response surface characterization and quality of fit estimate. The effectiveness of the approach is demonstrated for an airborne wind energy (AWE) system, where the plant parameters are the center of mass location and stabilizer area, and the control parameter is the trim pitch angle.

Introduction

Optimal design of active systems presents a number of challenges when the plant and controller exhibit bidirectional coupling. If a system design exhibits bidirectional coupling, the optimal plant design depends on the controller and vice versa. Systems, such as automotive suspension systems [1–3], elevator systems in Ref. [4], advanced powertrain systems in Ref. [5] and airborne wind energy (AWE) systems in Refs. [6] and [7], have been shown to exhibit such coupling. Traditionally, combined plant and controller optimization (termed codesign) has been used to address the challenges associated with the codesign of coupled systems. In general, codesign methodologies can be broken down into four categories: sequential, iterative, nested, and simultaneous (see Ref. [8]). A sequential strategy completes the plant and controller optimization problems in successive order. A variant of the sequential approach, proposed in Ref. [9], makes use of control proxy functions to augment the objective function of the plant, thereby separating the plant and controller optimization problems into two subproblems. The iterative approach fully optimizes the plant design for a given controller, then optimizes the controller design for a fixed plant, then repeats the cycle (see Refs. [10] and [11]). A nested optimization approach contains two loops: an inner loop that completes a full optimization of the controller while keeping the plant fixed and an outer loop that completes an iteration of the plant optimization (see Refs. [1,2], and [4]). This approach has been proven to converge to an optimal design under limited circumstances in Ref. [8]. In a simultaneous optimization strategy, both the plant and controller optimization problems are carried out at the same time (see Refs. [12] and [13]). An efficient decomposition-based variant of simultaneous optimization is proposed in Ref. [14].

Two key gaps exist in the codesign literature:

(1)
The complexity of systems that can be considered by existing codesign tools is limited—often substantially—by restrictions on the underlying optimization tools. For example, LQR-based optimal control design techniques assume a linear system. Plant optimization techniques that use sequential quadratic programming require an accurate estimate of the gradient and Hessian of the objective function or Lagrangian for constrained problems. Practical implementation of Pontryagin's minimum principle (PMP)-based techniques requires a relatively simple, closed-form expression for system dynamics.
(2)
Existing codesign techniques do not leverage the ability to adjust control parameters in real time. In a nested codesign framework, in particular, the ability to adjust controller parameters during the simulations/experiments can dramatically reduce the time and cost of the optimization process.

This paper presents a novel nested codesign methodology that addresses the above issues. The framework presented here fits into the nested methodology proposed by Fathy et al. [8], and the techniques used at both the plant and control design levels are specifically tailored to meet address the gaps in the state of the art discussed above. Since the nested methodology involves a complete controller optimization for every iteration of the plant design, it represents an ideal framework for adapting control parameters during simulations or experiments (where the plant design remains fixed over the course of a simulation or experiment). Table 1 provides a brief comparison of existing nested codesign strategies, including the strategy proposed herein. The current proposed framework presents the following advantages:

Table 1

Comparison of nested codesign techniques

Author	Outer loop optimization: single design or batch	Outer loop technique	Inner loop optimization: offline or real-time	Inner loop technique
Fathy (see Refs. [1,2], and [4])	Single	Interior-point method	Offline	LQR/LQG
Deese (see Ref. [25])	Batch	Optimal DoE	Real-time, continuous	Extremum seeking
Baheri (see Ref. [27])	Single	Bayesian optimization	Real-time, discontinuous	Bayesian optimization
Deese (present work)	Batch	Optimal DoE	Real-time, continuous	Continuous-time DoE

Author	Outer loop optimization: single design or batch	Outer loop technique	Inner loop optimization: offline or real-time	Inner loop technique
Fathy (see Refs. [1,2], and [4])	Single	Interior-point method	Offline	LQR/LQG
Deese (see Ref. [25])	Batch	Optimal DoE	Real-time, continuous	Extremum seeking
Baheri (see Ref. [27])	Single	Bayesian optimization	Real-time, discontinuous	Bayesian optimization
Deese (present work)	Batch	Optimal DoE	Real-time, continuous	Continuous-time DoE

(1)
The use of online adaptation mechanisms to adjust control parameters enables a natural extension of the codesign strategy to the experimental realm, where plant designs can only (typically) be adjusted at discrete iterations, but control parameters can be adapted continuously.
(2)
The proposed framework relies on optimal design of experiments (DoE) techniques that explore a global design space and are not restricted to linear systems or those for which the system can be represented through a small number of closed-form equations.

In the proposed framework, G-optimal DoE is used to select a set of candidate plant designs that cover the design space while maximizing a statistical information metric. By selecting a set of candidate plant designs that cover the design space, this approach considers the possibility of multiple local optima. In the inner loop, a continuous-time adaptation law is used to find the optimal control design during the course of simulating or running an experiment for each candidate plant design. In this work, two continuous-time adaptation laws are proposed for this purpose: extremum seeking and entropy-based design of experiments. Extremum seeking (ES) is a commonly used, though local, adaptation technique for finding the maximizer or minimizer of an system whose extremum is unknown. Entropy-based continuous-time DoE, a novel contribution of this paper, attempts to overcome the limitations of ES (namely locality of the solution), while leveraging the same principles that are used in the outer loop plant DoE. Because information-based design techniques are typically performed in discrete batches (i.e., a finite number of design candidates are selected through each design of experiments), the tailoring of information-based techniques for use in continuous-time control parameter adaptation represents a unique contribution on its own. After each iteration of the full system optimization (G-optimal DoE for the plant and continuous-time control parameter optimization), the plant design space is reduced using a statistical quality of fit metric and hypothesis testing.

In this work, the proposed codesign strategy is validated on an AWE system. AWE systems replace the traditional tower of wind turbines with tethers and a lifting body to harness high altitude winds. In general, the wind at these high altitudes is more consistent and greater in magnitude than ground level winds (see Ref. [15]). The tethers significantly reduce the material costs associated with the tower and foundation of traditional wind turbines and allow for adjustment of the height of the turbine. A substantial quantity of literature has focused on control design for AWE systems, including the use of optimal control techniques for altitude optimization in Refs. [16] and [17], as well as the control of power-augmenting crosswind flight in Refs. [18–22]. However, there has been far less research focused on the combined plant and control design of an AWE system. Deodhar et al. [7] and NikpoorParizi and Vermillion [23] have both shown that a coupling exists between the plant and controller design (i.e., the optimal controller depends on the plant design and vice versa). Consequently, we focus on an AWE system—specifically, the Altaeros' Buoyant airborne turbine (BAT), shown in Fig. 1—as a case study. In particular, we take the longitudinal center of mass location and stabilizer scale factor (which dictates the surface area of the horizontal and vertical stabilizers) as plant parameters to be optimized, whereas we take the trim pitch angle as the controller parameter to be optimized. While these design parameters are the same as those considered in Refs. [7] and [24], the present paper differentiates itself from the aforementioned work through the unique continuous-time controller parameter adaptation that is used in the inner loop of a nested framework. Furthermore, the present paper represents a greatly extended version of our earlier conference paper, [25], which only considers extremum seeking as the inner loop control parameter adaptation strategy.

Fig. 1

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Buoyant airborne turbine of Altaeros Energies [26]

Nested Optimization Framework

The goal of our codesign framework is to determine the plant and controller parameter vectors, denoted by

p_{p}

and

p_{c}

⁠, respectively, that minimize an integral performance index, subject to a dynamic model and bounds on the parameters. Mathematically, the optimization problem can be expressed as follows:

\begin{matrix} \underset{p_{p}, p_{c}}{minimize} J (p_{c}, p_{p}) = \int_{0}^{t_{f}} g (x (t), u (t); p_{p}, p_{c}) d t \\ subject to : p_{p} \in P, p_{c} \in C \\ given : \dot{x} = f (x, u; p_{p}, p_{c}) \end{matrix}

(1)

A general block diagram of the codesign framework considered in this work is given in Fig. 2. This optimization strategy generates a batch of candidate plant designs using G-optimal DoE, which are split into training and validation points. Once those experiments are designed, they are tested via either simulations or experiments, during which an adaptive control law is used to find the optimal controller design. After finding the integral cost associated with each plant design operating at the optimal controller design (⁠ $J (p_{c}^{*} (p_{p}), p_{p})$ ⁠), a response surface characterization is carried out using only the training points. Using the response surface characterization, a quality of fit calculation based on the validation points, and hypothesis testing, the design space is reduced. At the next iteration, a batch of candidate plant designs is generated in the reduced design space. The process is repeated until the original design space is reduced to a sufficiently small design space. In the remainder of this section, we describe each of the elements of the codesign framework in mathematical detail.

Fig. 2

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General process diagram for nested plant and controller optimization

Selecting Candidate Plant Design Parameters: G-Optimal Design of Experiments.

The goal of G-optimal design of experiments is to populate the plant design space with a batch of candidate design points that will be tested in either simulations or experiments in the inner loop. The performance index under optimal control parameters, denoted by

J (p_{c}^{*} (p_{p}), p_{p})

⁠, is expressed as

J (p_{c}^{*} (p_{p}), p_{p}) = \int_{0}^{t_{f}} g (x (t), u (t); p_{c}^{*} (p_{p}), p_{p}) d t

(2)

To begin the process of G-optimal DoE, the performance index given in Eq. is approximated as the inner product of an M element regressor vector,

z (p_{p})

⁠, and M element coefficient vector, β, as follows:

\hat{J} (p_{c}^{*} (p_{p}), p_{p}) = z^{T} (p_{p}) β

(3)

The regressor vectors corresponding to each batch of N experiments can be represented compactly through a regressor matrix, Z

Z = {[\begin{matrix} z_{1} & \dots & z_{N} \end{matrix}]}^{T}

(4)

where each column of Z corresponds to a single candidate design point (i.e., z_i represents the regressor associated with ith candidate design point). The structure of the regressor vector is chosen to account for anticipated dependencies in the plant parameters; e.g., if it is anticipated that the dependency of J on $p_{p}$ will be quadratic, then quadratic terms will be included in the regressor vector. G-optimal DoE populates the design space based on an information metric called the prediction variance, which is equal to the maximum entry in the diagonal of $z_{i}^{T} {(Z^{T} Z)}^{- 1} z_{i}$ ⁠.

The generated set of points are split into training and validation points. Training points are used to characterize the response surface (i.e., the actual dependency of J on $p_{p}$ after experiments have been run), while validation points are used to assess the quality of fit of the generated response surface characterization.

Continuous-Time Control Parameter Optimization: Option 1—Extremum Seeking-Based Adaptation Law.

The nested codesign approach involves a full optimization of the controller for every iteration on the plant design. In complex systems where it is necessary to run simulations and/or experiments to evaluate the performance index under a given design, it becomes advantageous to consider real-time control optimization strategies, which allow the control parameter to be optimized during these simulations/experiments. Extremum seeking is a common adaptive control algorithm used in continuous-time optimization, where there is limited or no knowledge of how the performance index depends upon the control variable(s) (i.e., no first- or second-order information is available). A brief summary of ES is provided here. The general block diagram of multivariable ES from Ref. [28], with added context to the codesign problem at hand, is shown in Fig. 3. In general, ES uses a sinusoidal or random perturbation applied to the approximated optimal controller parameter and feedback of the instantaneous performance index value to converge to the true optimal controller parameter. The control parameter update shown in Fig. 3 corresponds to a single control parameter contained within the control design vector (i.e., $p_{c} = {[\begin{matrix} p_{c, 1} & p_{c, 2} & \dots & p_{c, L} \end{matrix}]}^{T}$ ⁠). In the multiparameter ES case, L perturbation frequencies $ω_{1} < ω_{2} < \dots < ω_{L}$ are used for the identification of L parameters (see Ref. [29]). The perturbation frequencies are selected to be sufficiently small with respect to system dynamics and not equal to the frequencies associated with noise. Each perturbation frequency corresponds to a single element within the control design vector. The filter oscillation frequencies (⁠ $ω_{l, i}$ and $ω_{h, i}$ ⁠) are selected to be slower than the perturbation oscillation frequency. It has been shown that with proper selection of ω_i, k_i, $ω_{h, i}, ω_{l, i}$ ⁠, and a_i (all of the parameters associated with ES are listed in Table 2), ES will converge to a local minimizer or maximizer [28]. ES in this work is used for minimization, which can be handled by ES by changing the integrator gain to $- k_{i}$ ⁠.

Fig. 3

Multiparameter ES block diagram displaying the update for a single element of the control vector. In general, i=1,2,3,…,L and L is the number of elements in the control design vector. For odd i, ωi+1=ωi, βi=0, and βi+1=0.

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Multiparameter ES block diagram displaying the update for a single element of the control vector. In general, $i = 1, 2, 3, \dots, L$ and L is the number of elements in the control design vector. For odd i, $ω_{i + 1} = ω_{i}, β_{i} = 0$ ⁠, and $β_{i + 1} = 0$ ⁠.

Table 2

Description of ES signals and parameters for single control parameter update

Variable	Description
J_inst	Instantaneous performance index value
${\dot{J}}_{inst}$	Filtered derivative of instantaneous performance index value
$p_{c}$	Commanded control parameter vector that has the form: $p_{c} = {[\begin{matrix} p_{c, 1} & p_{c, 2} & \dots & p_{c, L} \end{matrix}]}^{T}$
η	Low-level noise
$p_{c, i}$	Commanded control parameter
${\hat{p}}_{c, i}$	Estimated optimal control parameter
ζ	Low-pass filter output
u	Control command
y	Feedback vector
k_i	Integrator gain vector
a_i	Perturbation amplitude
ω_i	Perturbation frequency
$ω_{l, i}$	Low-pass filter frequency
$ω_{h, i}$	High-pass filter frequency

Variable	Description
J_inst	Instantaneous performance index value
${\dot{J}}_{inst}$	Filtered derivative of instantaneous performance index value
$p_{c}$	Commanded control parameter vector that has the form: $p_{c} = {[\begin{matrix} p_{c, 1} & p_{c, 2} & \dots & p_{c, L} \end{matrix}]}^{T}$
η	Low-level noise
$p_{c, i}$	Commanded control parameter
${\hat{p}}_{c, i}$	Estimated optimal control parameter
ζ	Low-pass filter output
u	Control command
y	Feedback vector
k_i	Integrator gain vector
a_i	Perturbation amplitude
ω_i	Perturbation frequency
$ω_{l, i}$	Low-pass filter frequency
$ω_{h, i}$	High-pass filter frequency

Once standard ES has converged to an optimal solution, the algorithm will continue to perturb the system about the perceived optimal solution. In this framework, the performance index at the optimal controller design (⁠

J (p_{c}^{*} (p_{p}), p_{p}))

is needed for the response surface characterization at the outer loop of the codesign strategy. Therefore, at the inner loop,

p_{c}

must converge to a sufficiently small space around

p_{c}^{*}

⁠. Convergence of

p_{c}

to

p_{c}^{*}

is detected by comparing a filtered derivative of

p_{c}

to a threshold. Denoting this filtered derivative by

{\dot{p}}_{c}^{f i l t}

⁠, convergence is detected at time t_c if

‖ {\dot{p}}_{c}^{filt} ‖ \leq ε, \forall t \in [t_{c}, t_{c} - Δ t]

(5)

Proper tuning of ε and $Δ t$ ensures that ${\dot{p}}_{c}^{filt}$ must remain sufficiently small for a sufficiently long time period before convergence is detected.

For

t \geq t_{c}

⁠, the extremum seeking perturbation is suspended, allowing for calculation of the integral performance index under the optimized control parameter vector,

p_{c}^{*}

⁠. The portion of data acquired after convergence is detected is used for the purpose of calculating the performance index, i.e., the performance index is calculated as

J (p_{c}^{*} (p_{p}), p_{p}) = \int_{t_{c}}^{t_{c} + t_{f}} g (x (t), u (t); p_{c}, p_{p}) d t

(6)

Continuous-Time Control Parameter Optimization: Option 2—Entropy-Based Design of Experiments Adaptation Law.

One clear limitation of the extremum seeking-based adaptation law is that it is local in nature, thereby resulting in a codesign formulation where the outer loop explores the global plant design space but the inner loop is limited to local control parameter optimization. To remedy this limitation, we seek a continuous-time control parameter optimization technique that mirrors the global design space exploration performed by the outer loop plant DoE.

Ultimately, the optimal DoE techniques used at the outer loop are based on selecting designs that maximize a statistical information metric (for G-optimal design, this is prediction variance). At the end of each iteration, the design space is reduced based on a response surface characterization and quality of fit estimate. To mirror these techniques at the inner loop, we adjust the control parameters, $p_{c}$ ⁠, based on a continuous-time information metric. Furthermore, we reject parameters from the design space in continuous time, based on a continuous quality of fit update. A fundamental tenet in performing this continuous-time information-based control parameter adjustment and design space reduction is the continuous-time estimate of normalized information entropy, $H (p_{c})$ ⁠, which characterizes how uncertain we are about the performance index for a given value of $p_{c}$ ⁠. While this process is conceptually similar to existing literature on adaptive DoE ([30,31]), a fundamental distinction lies in the fact that the adaptive DoE results of Refs. [30] and [31] are iteration-based, not continuous time-based. The novel contribution of the entropy-based DoE is the ability to optimize a parameter vector of interest based on an information metric in continuous-time. The entropy-based control parameter optimization framework is shown graphically in Fig. 4.

Fig. 4

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General block diagram of entropy-based DoE

Quantifying Normalized Information Entropy.

In general, information entropy describes the uncertainty associated with a particular variable [32]. Within this framework, information entropy is used to characterize how much we have left to learn about a specific design point. To begin the process, normalized information entropy, denoted by

H (p_{c})

⁠, is initialized to a maximum value of one for all points in the design space. In order to implement an update law for information entropy in continuous time, we begin with an iterative entropy update law from Eq. , where the estimated entropy at each point in the design space (where an arbitrary point in the design space is denoted by

{\bar{p}}_{c}

⁠) is updated based on the proximity of

{\bar{p}}_{c}

to the present design point,

p_{c}

⁠. We then “continualize” the iterative entropy update law of Eq. as follows:

\begin{matrix} H_{k + 1} ({\bar{p}}_{c}) = H_{k} ({\bar{p}}_{c}) exp (\frac{- K_{ent}}{d (p_{c}, {\bar{p}}_{c}) + ε}) \\ \Rightarrow H_{k + 1} ({\bar{p}}_{c}) - H_{k} ({\bar{p}}_{c}) = H_{k} ({\bar{p}}_{c}) (exp (\frac{- K_{ent}}{d (p_{c}, {\bar{p}}_{c}) + ε}) - 1) \end{matrix}

(7)

Because

H_{k + 1} ({\bar{p}}_{c}) - H_{k} ({\bar{p}}_{c}) \approx d H / d t ({\bar{p}}_{c}) Δ T

it follows that:

\Rightarrow \frac{d H}{d t} ({\bar{p}}_{c}) Δ T \approx H ({\bar{p}}_{c}) (exp (\frac{- K_{ent}}{d (p_{c}, {\bar{p}}_{c}) + ε}) - 1)

(8)

Here,

H ({\bar{p}}_{c})

is the normalized information entropy for a candidate design point,

Δ T

is the amount of time between parameter adjustments in the iterative framework, K_ent is a gain that reflects the “value” of a single point,

d (p_{c}, {\bar{p}}_{c})

is the distance between the current control design and candidate control design

{\bar{p}}_{c}

⁠, and ε is a small constant that prevents division by zero. K_ent is tunable by the designer based on the system of interest. Ultimately, the above derivation leads to the following continuous time-normalized entropy update law

\frac{d H}{d t} ({\bar{p}}_{c}) = K H ({\bar{p}}_{c}) (exp (\frac{- K_{ent}}{d (p_{c}, {\bar{p}}_{c}) + ε}) - 1)

(9)

for a tunable gain K. To implement this entropy update law, the control parameter design space is quantized into a finite grid of

{\bar{p}}_{c}

values, and the update law is applied for each value in the grid.

Figure 5 illustrates how information entropy evolves during a ten second sample simulation. The upper half of the figure shows the values of p_c (a scalar in this case) as a function of time. In the lower portion, the entropy at the beginning (left) and end (right) of the simulation can be seen. Because values of p_c greater than 0.2 were not visited during the 10 s period, no reduction in normalized entropy is observed over this range of design parameters.

Fig. 5

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Visualization of normalized entropy evolution over 10 s. The top plot shows the design points that have been visited as a function of time. The bottom plots show the values of entropy at the over the design space.

Continuous-Time Control Parameter Update Law.

Unlike extremum seeking, where the control parameter is driven toward the (local) estimated optimal value, the entropy-based control parameter update law selects the next control parameter based on which regions of the design space are likely to yield the most information about behavior of J with respect to

p_{c}

⁠. Thus, the controller moves toward locations of high entropy, with the understanding that

p_{c}

cannot undergo enormous jumps over very short time intervals, given the time constants of the system. To capture this requirement quantitatively, the following update law is used to determine the best subsequent control design to evaluate, denoted by

p_{c}^{'}

⁠:

p_{c}^{'} = \arg \max_{{\bar{p}}_{c} \in C} H ({\bar{p}}_{c}) exp (- K_{sel} {(d (p_{c}, {\bar{p}}_{c}) d (p_{c}, p_{c, prev}))}^{2})

(10)

where $p_{c, prev} (t) = p_{c} (t - T_{d})$ for a small delay time, T_d. Here, K_sel is a constant gain and $d (p_{c}, p_{c, prev}))$ is the distance between the controller design tested at time $t - T_{d}$ and the controller design presently tested. Inclusion of the term $d (p_{c}, p_{c, prev})$ in Eq. (10) ensures that $p_{c}^{'}$ does not continue to get adjusted after it was recently changed by a large amount. Inclusion of the term $d (p_{c}, {\bar{p}}_{c})$ helps to restrict near-term subsequent candidate designs not to veer too far from the present value of $p_{c}$ ⁠. To ensure that the lower level controller can track the commanded control design, the output of the entropy-based DoE is passed through a first-order filter, as illustrated in Fig. 4.

Continuous-Time Controller Response Surface Characterization and Design Space Reduction.

As discussed in the previous subsection, the entropy-based control parameter adaptation law adjusts the control parameter in the direction of maximum entropy, not necessarily in the direction of the optimal value. To estimate the optimal control parameter, $p_{c}^{*}$ ⁠, another step is needed. This step is the continuous-time response surface characterization and design space reduction, wherein (i) a continuous estimate of $J_{inst} (p_{c} (p_{p}))$ is maintained and (ii) control parameters that are nearly certain to be sub-optimal based on this response surface are excluded from C. It is important to note that the response surface characterization at this stage is performed with respect to the control parameters alone, for a given plant design. Once the inner loop control parameter optimization is complete, a separate response surface characterization, with respect to the plant parameters, is performed.

To perform a response surface characterization, we rely on recursive least squares estimation. To do this, the instantaneous performance index is parameterized as

{\hat{J}}_{inst} (p_{c} (p_{p})) = X^{T} (p_{c}) β

(11)

where X is a regressor vector and

β

is a vector of undetermined coefficients. These undetermined coefficients are estimated through the following discrete-time recursive least-squares update law, detailed in Ref. [33], where k represents the time-step:

V (k + 1) = V (k) - \frac{V (k) X^{T} (k + 1) X (k + 1) V (k)}{1 + X (k + 1) V (k) X^{T} (k + 1)}

(12)

γ (k + 1) = V (k + 1) X^{T} (k + 1)

(13)

e = J_{inst} (p_{c} (p_{p})) - X (k + 1) \hat{β} (k)

(14)

\hat{β} (k + 1) = \hat{β} (k) + γ (k + 1) e

(15)

Once

{\hat{J}}_{inst} (p_{c} (p_{p}))

has been obtained, the corresponding minimizer,

p_{c}^{*} (p_{p})

_, can be found. To avoid future exploration of regions of the design space that cannot possibly be optimal, a mechanism for reducing the size of the design space must be introduced. The goal of the design space reduction is to reject portions of the design space that are determined to be suboptimal, with some amount of certainty. Here, we use normalized entropy as an estimate of uncertainty. In particular, we reject candidate designs for which the following inequality holds:

{\hat{J}}_{inst} ({\hat{p}}_{c}^{*} (p_{p})) < {\hat{J}}_{inst} (p_{c} (p_{p})) - K_{rej} H (p_{c})

(16)

where K_rej is tuned to capture the relationship between normalized entropy and uncertainty. It can be seen that a candidate control design will be rejected if its associated performance index exceeds the estimated optimal performance index by an amount that depends on uncertainty (the larger the uncertainty, the larger this amount will be).

Plant Response Surface Characterization and Design Space Reduction.

After completing a control parameter optimization for each candidate plant design, the plant design space must be reduced prior to the next full iteration of the codesign process. To do this, the training points are used to characterize a response surface. This response surface characterization is performed with respect to the plant parameters, having just optimized the controller parameters. The response surface characterization is performed based on the integral performance index value of each training plant design point. To fit a response surface over the entire plant design space with this data, the estimated performance index at any plant design,

p_{p}

⁠, is computed through the following distance-weighted average:

\hat{J} (p_{c}^{*} (p_{p}), p_{p}) = \frac{\sum_{i = 1}^{N_{t}} w_{i} J (p_{c}^{* i} (p_{p}^{i}), p_{p}^{i})}{\sum_{i = 1}^{N_{t}} w_{i}}

(17)

The weight that point p_i has on p is given by

w_{i} = exp (- K_{r} d {(p_{p}, p_{p}^{i})}^{2})

(18)

where K_r is a tunable gain and

d (p_{p}, p_{p}^{i})

is the Euclidean distance between

p_{p}

and

p_{p}^{i}

⁠. Therefore, as the distance between

p_{p}

and

p_{p}^{i}

increases, the weight of

J (p_{c}^{* i} (p_{p}^{i}), p_{p}^{i})

on

\hat{J} (p_{c}^{*} (p_{p}), p_{p})

decreases. In this work, a golden-section search algorithm is used to select the value of K_r that minimizes the mean-squared error between the validation data and estimated response surface, denoted by S and given mathematically by

S = \sqrt{\frac{1}{N_{val}} \sum_{k = 1}^{N_{val}} (J (p_{c}^{*} (p_{p}^{N_{t} + k}), p_{p}^{N_{t} + k}) - \hat{J} (p_{c}^{k} (p_{p}^{N_{t} + k}), p_{p}^{N_{t} + k}))^{2}}

(19)

Since the overall goal of the codesign framework is to converge to an optimal system design, it is important to reduce the size of the design space at each iteration so as to focus only on plant designs that could possibly be optimal. The main idea of the design space reduction is to reject, with confidence, portions of the design space that produce inferior performance. The response surface computed in the outer loop yields two important quantities that can used to shrink the design space. The first is the approximated performance index for a given plant design operating at the optimal controller design (⁠ $\hat{J} (p_{c}^{*} (p_{p}^{*}), p_{p}^{*})$ ⁠). The second is the quality of fit metric, S, which characterizes how well the response surface approximates the true system performance index at validation points.

By comparing the approximated optimal performance with the approximated performance of all other points in the design space, some plant designs can be rejected. This comparison is carried out by hypothesis testing, which computes a Z-score by

z = \frac{\hat{J} (p_{c}^{*} (p_{p}), p_{p}) - \hat{J} (p_{c}^{*} (p_{p} *), p_{p} *)}{S \sqrt{2}}

(20)

Design points are rejected whenever $z > z_{0}$ ⁠. For this work, a 95% confidence interval, which corresponds to $z_{0} = 1.96$ ⁠, was chosen. The reduced plant design space, which excludes all points rejected by the Z-test, is denoted by P.

Codesign Case Study on an Airborne Wind Energy System

The general optimization strategy detailed in Sec. 2 has been applied to the Altaeros Energies BAT, using the numerical model from Ref. [34] for this case study. The BAT is a good candidate system for this optimization framework because it gives rise to complex dynamics with many nonlinearities, as well as coupling between plant and controller parameters. In fact, the works from Refs. [7] and [24] demonstrate bidirectional coupling for the BAT.

In this case study, we focus on two plant parameters and one controller parameter. It is important to note that this framework can be extended to many parameters; however, the restriction to two plant parameters aids in the visualization of the optimal DoE and design space reduction processes in two dimensions. The specific plant parameters of interest are the stabilizer reference area scale factor (K_A) and the longitudinal location of the center of mass (x_cm). Therefore, $p_{p} = {[\begin{matrix} K_{A} & x_{cm} \end{matrix}]}^{T}$ ⁠. The controller parameter of interest is the pitch angle setpoint (also called the trim pitch angle), denoted by $p_{c} = θ_{s p}$ ⁠.

Table 3 provides a summary of tuning guidelines for both adaption laws presented in Sec. 2. The tuning guidelines for ES are based on ES tuning guidelines from Ref. [28]. For the entropy-based DoE adaption, general tuning guidelines are provided in the table. However, when tuning these parameters, it is useful to consider the units in the given application.

Table 3

Description of tuning parameters for each adaptive control strategy

Adaptive law	Symbol	Description	Tuning description
ES	$ω_{l, i}$	Cutoff frequency of low-pass filter	Tuned to be sufficiently slower than perturbation frequency $O (ω_{l, i}) < O (ω_{i})$
	$ω_{h, i}$	Cutoff frequency of high-pass filter	Tuned to be sufficiently slower than perturbation frequency $O (ω_{h, i}) < O (ω_{i})$
	ω_i	Perturbation oscillation frequency	Tuned to be slower than the system dynamics, including noise, i.e., $O (ω_{i}) < O (ω_{system})$
	k_i	Integrator gain	k_i controls the convergence rate of ES. Larger values of k_i will result in faster convergence so long as k_i is chosen sufficiently small to ensure stability and avoid excessive overshoot of the optimal control parameter value(s).
	a_i	Perturbation amplitude	Large values provide greater levels of excitation, at the expense of requiring more control effort to implement and lower oscillation frequencies in the presence of rate limitations.
Adaptive-DoE	K_ent	Entropy reduction length scale	Reflects the tion length scale vels of excitation, at the expense of requiring more control effort to implement anlonger reduces entropy at another.
	K	Entropy gain	Reflects the significance of a collecting the results at a single design point
	K_r	Exponential regression length scale	Reflection of covariance between design points; tuned using Golden Section Search
	K_sel	Selection gain	Incentives selecting design points at time step k + 1 that are around previous design point at time step k

Adaptive law	Symbol	Description	Tuning description
ES	$ω_{l, i}$	Cutoff frequency of low-pass filter	Tuned to be sufficiently slower than perturbation frequency $O (ω_{l, i}) < O (ω_{i})$
	$ω_{h, i}$	Cutoff frequency of high-pass filter	Tuned to be sufficiently slower than perturbation frequency $O (ω_{h, i}) < O (ω_{i})$
	ω_i	Perturbation oscillation frequency	Tuned to be slower than the system dynamics, including noise, i.e., $O (ω_{i}) < O (ω_{system})$
	k_i	Integrator gain	k_i controls the convergence rate of ES. Larger values of k_i will result in faster convergence so long as k_i is chosen sufficiently small to ensure stability and avoid excessive overshoot of the optimal control parameter value(s).
	a_i	Perturbation amplitude	Large values provide greater levels of excitation, at the expense of requiring more control effort to implement and lower oscillation frequencies in the presence of rate limitations.
Adaptive-DoE	K_ent	Entropy reduction length scale	Reflects the tion length scale vels of excitation, at the expense of requiring more control effort to implement anlonger reduces entropy at another.
	K	Entropy gain	Reflects the significance of a collecting the results at a single design point
	K_r	Exponential regression length scale	Reflection of covariance between design points; tuned using Golden Section Search
	K_sel	Selection gain	Incentives selecting design points at time step k + 1 that are around previous design point at time step k

Note: The tuning of ES is detailed in Ref. [28].

Moving forward, the aforementioned codesign framework will be extended to experimental codesign using the lab scale, water channel-based platform detailed in Ref. [35] at UNC-Charlotte. Thus, the results that are presented here provide a foundation for the extension to the experimental setup at UNC-Charlotte.

Plant Dynamics.

The numerical model of the BAT used for the case study in this work is detailed in Ref. [34] and summarized here (see Fig. 6). The dynamic model of the BAT was derived using an Euler–Lagrange framework, from which the position and orientation of the BAT are described by six generalized coordinates: Θ,

Φ, Ψ

⁠, L_t,

θ'

⁠, and

ϕ'

⁠. The first three of these are uncontrolled angles that describe the orientation of the tether with respect to the ground-fixed coordinate system. Specifically, the azimuth angle, Θ, is angle of the tether projection on the horizontal plane, the zenith angle,

Φ

⁠, is the angle of tether with respect to vertical, and the twist angle,

Ψ

⁠, represents the rotation of the BAT about the tether axis. In an effort to simplify the 3 tether model of the BAT, the 3 tethers are approximated by single tether (of length L_t) and a bridle joint. This allows the dynamics to be described fully through ordinary differential equations, rather than differential algebraic equations. At the bridle joint, two control angles are introduced, denoted by

θ'

and

ϕ'

⁠. These controlled coordinates can be described in terms of the individual lengths of each tether in the three tether model by the following:

ϕ' = {tan}^{- 1} (\frac{l_{3} - l_{2}}{l_{sep}^{lat}})

(21)

θ' = {tan}^{- 1} (\frac{l_{1} - 0.5 (l_{2} + l_{3})}{l_{sep}^{long}})

(22)

L_{t} = \frac{1}{3} (l_{1} + l_{2} + l_{3})

(23)

where

l_{sep}^{long}

and

l_{sep}^{lat}

are longitudinal and lateral tether attachment separation distances, respectively. The control inputs are the tether release speeds,

{\bar{u}}_{i},

⁠, given by:

{\bar{u}}_{i} = \frac{d}{d t} l_{i}, i = 1, 2, 3

(24)

Fig. 6

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Axis system for the BAT dynamic model. This diagram illustrates generalized coordinates (left and bottom middle), Euler angles (top middle), and tether attachment points (right).

The system dynamics, derived using an Euler–Lagrange formulation, are given by

D (Q) \ddot{Q} + C (Q, \dot{Q}) \dot{Q} + g (Q) = τ (Q, \dot{Q}, V, ψ_{wind})

(25)

X = f (Q, \dot{Q})

(26)

Ω = g (Q, \dot{Q})

(27)

where:

Q = [Θ Φ Ψ L_{t} θ' ϕ']

(28)

X = [x y z u v w]

(29)

Ω = [ϕ θ ψ p q r]

(30)

Here, τ represents a vector of generalized forces, V is the wind speed, and ψ_wind is the wind direction. The angle of attack (α) and side-slip angle (β) describe the orientation of the apparent wind vector with the body-fixed coordinates of the BAT. Aerodynamic forces and moments are functions of both α and β.

The surface areas of the BAT's horizontal and vertical stabilizers represent important design parameters. However, prior to this work, the numerical model was a lumped aerodynamic model, meaning that aerodynamic coefficients were characterized for the combined shroud (fuselage) and stabilizers. In order to consider the impact of changes to stabilizers on performance of the system, the aerodynamics of the system were partitioned between the main body, horizontal stabilizer, and vertical stabilizer. In this work, the vertical and horizontal stabilizer areas (A_V and A_H, respectively) are scaled uniformly by the plant design parameter K_A, i.e.:

A_{V} = K_{A} A_{V 0}

(31)

A_{H} = K_{A} A_{H 0}

(32)

It is assumed that the stabilizer reference areas are scaled in a manner such that their aspect ratios are preserved in all instances. Figure 7 displays the plant parameters of interest pictorially. Total aerodynamic coefficients are given by

\begin{matrix} C_{D, L, S}^{total} (α, β) = C_{D, L, S}^{body} (α, β) + C_{D, L, S}^{V} (α, β) \frac{A_{V}}{A_{body}} + C_{D, L, S}^{H} (α, β) \frac{A_{H}}{A_{b o d y}} \end{matrix}

(33)

\begin{matrix} C_{M x, M y, M z}^{total} (α, β) = C_{M x, M y, M z}^{body} (α, β) + C_{M x, M y, M z}^{V} (α, β) \frac{A_{V} l_{V}}{A_{body} l_{body}} + C_{M x, M y, M z}^{H} (α, β) \frac{A_{H} l_{H}}{A_{body} l_{body}} \end{matrix}

(34)

Fig. 7

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Plant design parameters for the Altaeros BAT

Here, C_D, C_L, C_S, C_Mx, C_My, and C_Mz represent the drag, lift, side force, roll moment, pitching moment, and yaw moment coefficients of the BAT, respectively.

Controller Design.

In general, an AWE system contains a flight controller that is used to maintain the attitude of the system to desired setpoints. The flight control architecture for the BAT is shown in Fig. 8. The flight controller of the BAT is augmented with an adaptation law that determines the trim pitch angle at which the system should operate at. Two potential adaptation laws have been presented in Sec. 2. The altitude setpoint (z_sp) and the roll angle setpoint (⁠ $ϕ_{sp}$ ⁠) for this work are taken to be constant.

Fig. 8

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Block diagram of closed-loop flight controller for the BAT. Here, z_sp is a constant altitude setpoint. The control parameter to be optimized (θ_sp) is highlighted.

Based on the tracking error, the flight controller uses three lead filters to compute preliminary control inputs that describe the average tether speed, the forward/aft tether speed difference, and the port/starboard speed difference, denoted by

{\bar{u}}_{z}, {\bar{u}}_{θ}

⁠, and

{\bar{u}}_{ϕ}

⁠, respectively

{\bar{u}}_{z} (s) = \frac{k_{d, z} s + k_{p, z}}{τ_{z} s + 1} (z_{s p} (s) - z (s))

(35)

{\bar{u}}_{θ} (s) = \frac{k_{d, θ} s + k_{p, θ}}{τ_{θ} s + 1} (θ_{s p} (s) - θ (s))

(36)

{\bar{u}}_{ϕ} (s) = \frac{k_{d, ϕ} s + k_{p, ϕ}}{τ_{ϕ} s + 1} (ϕ_{s p} (s) - ϕ (s))

(37)

The command tether release speeds (⁠

{\bar{u}}_{center}, {\bar{u}}_{stbd}

⁠, and

{\bar{u}}_{port}

⁠) are calculated as a linear combination of the preliminary control inputs shown in Eqs. –. This linear combination is illustrated by

{\bar{u}}_{center} = {\bar{u}}_{z} - {\bar{u}}_{θ}

(38)

{\bar{u}}_{stbd} = {\bar{u}}_{z} + {\bar{u}}_{θ} + {\bar{u}}_{ϕ}

(39)

{\bar{u}}_{port} = {\bar{u}}_{z} + {\bar{u}}_{θ} - {\bar{u}}_{ϕ}

(40)

Performance Index.

The Altaeros BAT is unique among AWE concepts in that it (i) is designed to remain substantially stationary (as opposed to numerous AWE concepts that utilize crosswind flight to augment power production) and (ii) is intended to be used for secondary purposes such as telecommunications. These qualitative requirements necessitate a control system, whereby the BAT remains sufficiently motionless in variable wind conditions. Furthermore, if multiple BATs are implemented in a farm, it is beneficial that the “ground footprint” of each (defined as the projected area over which the turbines can float) be as small as possible, so as to pack as many BATs as possible into a tight farm. The ground footprint can be captured by considering the zenith angle (shown as

Φ

in Fig. 6) as a term within the performance index. The quality of flight (i.e., the extent to which the BAT remains stationary) is characterized by penalizing roll angle tracking error (⁠

ϕ_{e} = ϕ_{des} - ϕ

⁠) and yaw angle offset (⁠

ψ_{e} = ψ_{flow} - ψ

⁠). In this specific application,

ϕ_{des}

and ψ_flow are both equal to zero. The instantaneous performance index, which is used to drive the continuous-time control parameter adaptation, is given by

J_{inst} (p_{c}, p_{p}) = k_{1} Φ^{2} + k_{2} {(ψ_{e})}^{2} + k_{3} {(ϕ_{e})}^{2}

(41)

The integral cost, which is used in the plant response surface characterization and is computed after convergence to optimal control parameters has been detected, is given by

J (p_{c}^{*} (p_{p}), p_{p}) = \int_{t_{c}}^{t_{c} + t_{f}} (k_{1} Φ^{2} + k_{2} ψ_{e}^{2} + k_{3} ϕ_{e}^{2}) d t

(42)

where t_f = 240 s.

External Perturbation.

In order to evaluate the performance of the AWE system, the system must be perturbed, either through a nonequilibrium initial condition or external perturbation. Because the continuous-time controller adaptation laws are in effect throughout the entire simulation, up to time t_c, it is essential that the perturbation be consistent throughout the simulation such that the instantaneous performance index depends on the control parameters, not the mildness of the perturbation at the particular time that those control parameters are active. Second, since the goal moving forward is an extension of this framework to an experimental codesign platform, the excitation should be implementable in the lab-scale environment described in Ref. [35]. In light of these requirements, a frequency approximation of vortex shedding off of a cylinder was used as the perturbation. The corresponding flow velocity profiles were based on Strouhal number (St), which is a dimensionless parameter that characterizes the frequency of vortex shedding in the wake of a cylinder. The Strouhal number is given by

St = \frac{f L}{U} = 0.198 (1 - \frac{19.7}{{Re}_{D}})

(43)

where f is the vortex shedding frequency, L is the characteristic length, U is the flow velocity, and Re_D is the Reynolds number associated with L and U. The empirical formula shown in Eq. can be used to solve for the vortex shedding frequency, f, at a given Reynolds number. It is important to note that this empirical formula is only valid for Reynolds numbers in the range

250 < {Re}_{D} < 2 \cdot 10^{5}

⁠, which is applicable to the lab-scale platform to be used in future experiments. In simulation, the velocity components in each direction are functions of the oscillation frequency f, solved for using Eq. . Each of the velocity components is given by

v_{x} = v_{x}^{base} + v_{x 0} sin (ω_{dist} t + \frac{π}{2})

(44)

v_{y} = v_{y 0} sin (ω_{dist} t)

(45)

v_{z} = v_{z 0} sin (ω_{dist} t + \frac{π}{2})

(46)

where $v_{x}^{base} = 0.606 m / s, v_{x 0} = 0.0866 m / s, v_{y 0} = 0.0650 m / s, v_{z 0} = 0.00866 m / s$ ⁠, and $ω_{dist} = 2 π rad / s$ ⁠. These velocities correspond to flow velocities in the lab-scale platform described in Ref. [35]. Through the dimensional analysis discussed in Ref. [35], these lab-scale flow velocities are equivalent to wind speeds of $v_{x}^{base} = 6.06 m / s, v_{x 0} = 0.866 m / s, v_{y 0} = 0.650 m / s, v_{z 0} = 0.0866 m / s$ (i.e., a scale factor of 10) and an oscillation frequency of $ω_{dist} = 0.2 π rad / s$ (i.e., a scale factor of 1/10) on the full-scale BAT, which has 100 times the characteristic length of the models used in the lab-scale setup.

Airborne Wind Energy System Codesign Results

The optimization described in Sec. 2 was applied to the system described in Sec. 3. The summary of results is broken into three subsections: The first section shows the results generated using ES for the inner loop optimization, while the second shows the results when using continuous-time, entropy-based adaptation. The last section provides a comparative assessment of these results.

Optimization Results Using Extremum Seeking.

The results presented in this section were generated using extremum seeking for the control parameter optimization. Figure 9 shows the convergence of the control parameter (trim pitch angle) for a sample plant design. The combined plant/controller optimal designs at each iteration are shown in Table 4 in the first three columns. The final column of Table 4 also lists the total reduction in the size of the plant design space (P) after each iteration, as a percentage of the area of the original plant design space. From these results, it can be seen that the design space has been reduced by 99% by the end of the fourth iteration. In Fig. 10, the candidate design points from the G-optimal DoE at the first four iterations are displayed, along with a contour plot of the response surface characterization. From this series of plots, the shrinking of the space can be visualized.

Fig. 9

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Convergence of the ES algorithm for a sample plant design (⁠ $x_{cm} = 45.4 %$ and K_A = 1)

Fig. 10

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Response surface characterization at first (top left), second (top right), third (bottom left), and fourth (bottom right) iterations with candidate design points overlaid, when ES for the adaptation law in the nested codesign strategy

Table 4

Optimal system design at each iteration when using ES for the adaptation law

Iter.	$K_{A}^{*}$	$x_{cm}^{*}$ (% chord)	$θ_{sp}^{*} (deg)$	% Reduction
1	1.455	52.40	12.99	95.36
2	1.391	52.40	13.10	96.49
3	1.406	52.40	13.08	96.49
4	1.435	52.40	13.03	98.38

Optimization Results Using Continuous-Time Entropy-Based Adaptation.

The results presented in this section were generated using continuous-time, entropy-based adaptation for the control parameter optimization. Figure 11 shows the convergence of the control parameter (trim pitch angle) for a sample plant design (the same sample plant design that was used in Sec. 4.1). The combined plant/controller optimal designs at each iteration are shown in Table 5, in the first three columns. The final column of Table 5 also lists the total reduction in the size of the plant design space (P) after each iteration, as a percentage of the area of the original plant design space. From these results, it can be seen that the design space has been reduced by 99% by the end of the fourth iteration. In Fig. 12, the candidate design points from the G-optimal DoE at the first four iterations are displayed, along with a contour plot of the response surface characterization. From this series of plots, the shrinking of the space can be visualized.

Fig. 11

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Convergence for entropy-based DoE algorithm for a sample plant design (⁠ $x_{cm} = 45.4 %$ and K_A = 1)

Fig. 12

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Response surface characterization at first (top left), second (top right), third (bottom left), and fourth (bottom right) iterations with candidate design points overlaid when entropy-based DoE for the adaptation law in the nested codesign strategy

Table 5

Optimal system design at each iteration when using Entropy-based DoE for the adaptation law

Iter.	$K_{A}^{*}$	$x_{cm}^{*}$ (% chord)	$θ_{sp}^{*} (deg)$	% Reduction
1	1.160	52.4	13.7	75.3
2	1.212	52.4	13.3	81.7
3	1.750	52.4	14.3	99.0
4	1.750	52.4	14.3	99.8

Comparison of Adaptation Strategies.

Table 6 displays a comparison of the system response for multiple configurations, using both continuous-time control parameter optimization strategies. For each optimization strategy, four configurations are compared. The first design is a suboptimal design configuration in terms of both the plant and controller. The plant design was selected as the worst tested plant design throughout each iteration of the nested strategy. The control design was chosen by gridding the controller design space and selecting the worst performing trim pitch angle. The second is the same plant design as the first configuration, while operating at the optimal control design, which is discovered using the corresponding controller optimization strategy. The third is a random plant and controller configuration. The fourth configuration is the optimal design configuration resulting from the nested optimization strategy.

From Table 6, it can be seen this nested codesign framework results in substantial performance improvement and design space reduction, regardless of the chosen optimization technique. Specifically, the nested approach with ES for the inner loop optimization resulted in $43.8 %$ performance improvement when the optimal and worst tested configurations are compared. Similarly, the entropy-based DoE approach resulted in a $44.9 %$ performance improvement. Visualization of the performance improvement when using ES for the inner loop optimization can be seen in Fig. 13. Similarly, the components of the performance index for the entropy-based approach can be seen in Fig. 14.

Fig. 13

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Comparison of performance index components in a sample time interval between the optimal design and a suboptimal design from the design space, while using ES as the adaption law (see Table 6 for specific design configurations)

Fig. 14

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Comparison of performance index components in a sample time interval between the optimal design and a suboptimal design from the design space, while using entropy-based DoE as the adaption law (see Table 6 for specific design configurations)

Table 6

Comparison of system design, integral performance value, and convergence time between optimal (determined by ES and entropy-based DoE) and suboptimal design configurations

Adaptive law	Configuration	K_A	x_cm (% chord)	θ_sp (deg)	$J (p_{c} (p_{p}), p_{p})$	Convergence time (s)
ES	Suboptimal with p_c = 0	0.8673	47.05	0	27.28	N/A
	Suboptimal with $p_{c} = p_{c}^{*}$	0.8673	47.05	8.81	20.4	208.52
	Random system design	1.527	50.91	3.74	18.70	N/A
	Optimal	1.435	52.4	13.04	15.33	750.45
Adaptive-DoE	Suboptimal with p_c = 0	0.8305	47.45	0	27.35	N/A
	Suboptimal, with $p_{c} = p_{c}^{*}$	0.8305	47.45	10.04	20.38	188.53
	Random system design	1.2653	48.4	12.93	18.78	N/A
	Optimal	1.75	52.4	14.26	15.06	201.33

Adaptive law	Configuration	K_A	x_cm (% chord)	θ_sp (deg)	$J (p_{c} (p_{p}), p_{p})$	Convergence time (s)
ES	Suboptimal with p_c = 0	0.8673	47.05	0	27.28	N/A
	Suboptimal with $p_{c} = p_{c}^{*}$	0.8673	47.05	8.81	20.4	208.52
	Random system design	1.527	50.91	3.74	18.70	N/A
	Optimal	1.435	52.4	13.04	15.33	750.45
Adaptive-DoE	Suboptimal with p_c = 0	0.8305	47.45	0	27.35	N/A
	Suboptimal, with $p_{c} = p_{c}^{*}$	0.8305	47.45	10.04	20.38	188.53
	Random system design	1.2653	48.4	12.93	18.78	N/A
	Optimal	1.75	52.4	14.26	15.06	201.33

When comparing the results from both controller optimization approaches, one can see that the ES-based result leads to a slightly higher (worse) performance index value than the entropy-based approach, suggesting that ES might be leading to a local, rather than global, optimum. The last column of Table 6 displays the convergence time associated with the controller optimization on the inner loop. The convergence time associated with entropy-based DoE is faster than that of extremum seeking. One key advantage of extremum seeking over entropy-based DoE is the existence of established and well-defined convergence criteria. However, these convergence criteria only guarantee local convergence, and convergence times can be lengthy. Entropy-based DoE attempts to overcome the local nature of extremum seeking by employing an information-based design metric for design point selection, but rigorous convergence guarantees remain a topic of future research.

Assessment of Net Energy Production.

The focus of the aforementioned results has been on flight quality, not energy generation. An interesting tradeoff has been found to exist between flight quality and net energy generation. To properly consider net energy generation, we examine both the control energy expended and the energy that can be generated by the turbine. For the AWE application, winches are articulated to regulate. The BAT ground station is equipped with three winches. The power consumed by a single winch (⁠

P_{winch, i}

⁠) to maintain the system at a prescribed attitude is given by

P_{winch, i} = {\begin{matrix} - η_{reg, i} F_{t, i} ω_{winch, i} r_{winch, i} & ω_{winch, i} > 0 \\ \frac{- 1}{η_{motor, i}} F_{t, i} ω_{winch, i} r_{winch, i} & ω_{winch, i} \leq 0 \end{matrix}

(47)

Here,

F_{t, i}

is the tension of the tether corresponding to the ith winch,

ω_{winch, i}

is the rotational speed of the ith winch,

r_{winch, i}

is radius of the winch,

η_{reg, i}

is the regenerative efficiency of the winch, and

η_{motor, i}

is the motor efficiency of the winch. The tether tension distribution is approximated by a quasi-static force and moment balance. The energy produced by the AWE system is calculated by

P_{gen} = \frac{1}{2} ρ C_{p} A_{ref} min (v_{rated}, v_{x, app})^{3}

(48)

Here, ρ is the density of the medium, C_p is the power coefficient for the turbine, A_ref is the reference area of the turbine, v_rated is the rated speed of the turbine, and

v_{x, app}

is the apparent wind velocity presented to the turbine. An effective power coefficient was calculated to compare the power generated by the different system configurations. The effective power coefficient is calculated as

C_{p, eff} = \frac{P_{gen}}{\frac{1}{2} ρ A_{ref} min {(v_{rated}, v_{x, free})}^{3}}

(49)

where

v_{x, free}

is the freestream flow velocity. The denominator of Eq. represents the available power in the flow. The ratio of control energy expenditure to total energy produced, listed in column three of Table 7, was calculated as follows:

E_{ratio} = \frac{\int_{t_{c}}^{t_{c} + t_{f}} (\sum_{i = 1}^{3} (P_{winch, i})) d t}{\int_{t_{c}}^{t_{c} + t_{f}} (P_{gen}) d t}

(50)

Table 7

Comparison of energy expenditure ratio and equivalent power coefficient (⁠ $C_{p, eff}$ ⁠) between optimal (determined by ES and entropy-based DoE) and suboptimal design configurations

Adaptive law	System configuration	Control energy consumption ratio (E_ratio)	Effective power coefficient (⁠ $C_{p, eff}$ ⁠)
ES	Suboptimal with p_c = 0	0.004	0.505
	Suboptimal with $p_{c} = p_{c}^{*}$	0.0062	0.490
	Optimal plant design with p_c = 0	0.0045	0.504
	Optimal system design	0.0092	0.468
Adaptive-DoE	Suboptimal with p_c = 0	0.004	0.505
	Suboptimal, with $p_{c} = p_{c}^{*}$	0.007	0.485
	Optimal plant design with p_c = 0	0.005	0.503
	Optimal system design	0.014	0.463

Adaptive law	System configuration	Control energy consumption ratio (E_ratio)	Effective power coefficient (⁠ $C_{p, eff}$ ⁠)
ES	Suboptimal with p_c = 0	0.004	0.505
	Suboptimal with $p_{c} = p_{c}^{*}$	0.0062	0.490
	Optimal plant design with p_c = 0	0.0045	0.504
	Optimal system design	0.0092	0.468
Adaptive-DoE	Suboptimal with p_c = 0	0.004	0.505
	Suboptimal, with $p_{c} = p_{c}^{*}$	0.007	0.485
	Optimal plant design with p_c = 0	0.005	0.503
	Optimal system design	0.014	0.463

From the results presented in Table 7, there is a tradeoff between robust flight control and energy harvesting. This is due to the nature of the control parameter of interest in this application. In particular, pitching up reduces the component of wind presented to the turbine, causing a reduction in energy production. On the other hand, adjustment of the plant parameters alone has minimal impact on net energy production.

Conclusions and Future Work

In this work, a novel nested codesign framework that leverages optimal DoE techniques to adjust plant parameters in an outer loop and continuous-time adaptation to perform control parameter adaptation in an inner loop. The use of DoE techniques enables global exploration of complex design spaces, whereas the continuous-time adaptation enables direct extension of the proposed strategy to the experimental realm, where controller parameters can be adjusted during experiments. The results presented here display the capabilities of the nested codesign framework on an AWE system. From the results presented based on the numerical model of the Altaeros BAT, the plant design space was reduced to less than 1% of the original design after four iterations.

In future work, development of convergence criteria of the entropy-based DoE adaptation law and optimizing the system over a variety of environmental conditions will be explored.

Acknowledgment

This work was supported by eentitled “CAREER: Efficient Experimental Optimization for High-Performance Airborne Wind Energy Systems.”

Funding Data

National Science Foundation award number (1453912).

Nomenclature

$C$ =: control design space
$H$ =: normalized information entropy
$K_{a}$ =: reference area scale factor
$J (p_{c}^{*} (p_{p}), p_{p})$ =: performance index value while operating at the optimal controller design (⁠ $p_{c}$ ⁠) and a candidate plant design (⁠ $p_{p}$ ⁠)
$J_{inst} (p_{c}, p_{p})$ =: instantaneous performance index value
$\hat{J} (p_{c}^{*} (p_{p}), p_{p})$ =: response surface characterization
$N_{val}$ =: number of validation points
$N_{t}$ =: number of training points
$P$ =: plant design space
$p_{p}$ =: candidate plant design
$p_{c}$ =: candidate control design
$S$ =: quality of fit metric for response surface characterization
$x_{cm}$ =: x-location of center of mass
$z$ =: test statistic used for hypothesis testing
$θ_{sp}$ =: trim pitch angle

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Nested Plant/Controller Codesign Using G-Optimal Design and Continuous Time Adaptation Laws: Theoretical Framework and Application to an Airborne Wind Energy System

Introduction

Nested Optimization Framework

Selecting Candidate Plant Design Parameters: G-Optimal Design of Experiments.

Continuous-Time Control Parameter Optimization: Option 1—Extremum Seeking-Based Adaptation Law.

Continuous-Time Control Parameter Optimization: Option 2—Entropy-Based Design of Experiments Adaptation Law.

Quantifying Normalized Information Entropy.

Continuous-Time Control Parameter Update Law.

Continuous-Time Controller Response Surface Characterization and Design Space Reduction.

Plant Response Surface Characterization and Design Space Reduction.

Codesign Case Study on an Airborne Wind Energy System

Plant Dynamics.

Controller Design.

Performance Index.

External Perturbation.

Airborne Wind Energy System Codesign Results

Optimization Results Using Extremum Seeking.

Optimization Results Using Continuous-Time Entropy-Based Adaptation.

Comparison of Adaptation Strategies.

Assessment of Net Energy Production.

Conclusions and Future Work

Acknowledgment

Funding Data

Nomenclature

References

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ASME Journals

ASME Conference Proceedings

ASME eBooks

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Nested Plant/Controller Codesign Using G-Optimal Design and Continuous Time Adaptation Laws: Theoretical Framework and Application to an Airborne Wind Energy System

Introduction

Nested Optimization Framework

Selecting Candidate Plant Design Parameters: G-Optimal Design of Experiments.

Continuous-Time Control Parameter Optimization: Option 1—Extremum Seeking-Based Adaptation Law.

Continuous-Time Control Parameter Optimization: Option 2—Entropy-Based Design of Experiments Adaptation Law.

Quantifying Normalized Information Entropy.

Continuous-Time Control Parameter Update Law.

Continuous-Time Controller Response Surface Characterization and Design Space Reduction.

Plant Response Surface Characterization and Design Space Reduction.

Codesign Case Study on an Airborne Wind Energy System

Plant Dynamics.

Controller Design.

Performance Index.

External Perturbation.

Airborne Wind Energy System Codesign Results

Optimization Results Using Extremum Seeking.

Optimization Results Using Continuous-Time Entropy-Based Adaptation.

Comparison of Adaptation Strategies.

Assessment of Net Energy Production.

Conclusions and Future Work

Acknowledgment

Funding Data

Nomenclature

References

Get Email Alerts

Cited By

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