Abstract
Novel conceptual aircraft designs have been enabled by more electrified aircraft components providing enhanced capability and versatility. Through the advancement of multidisciplinary design optimization, control co-design methods have become a popular approach for system design conceptualization wherein the plant and control actions are designed simultaneously to account for the coupling between vehicle subsystems and power management systems. Many prior efforts have focused on open-loop control co-design that can later be adapted for a more realistic operating case. This work focuses on the development and scalability of closed-loop control co-design that would result in a physically realizable plant and closed-loop control law. The theoretical approach is demonstrated practically through the design of a hybrid electric unmanned air vehicle and two feedback controllers that operate the hybrid power split and propulsion system. The system is designed to complete a dynamic seven phase mission consisting of multiple cruise, dash, engage, dive, and climb segments as quickly as possible. Given the scale of the dynamic design problem, a convergence study is introduced that facilitates accurate and computationally tractable design optimization studies. The study is conducted for independent, sequential, and simultaneous design approaches. The results indicate high-speed motors, high voltage batteries, and responsive control gains result in a fast vehicle with high thrust-to-weight ratio. The simultaneous design solution had the best closed-loop performance, outclassing a baseline system design by over 30%.
1 Introduction
Unmanned aerial vehicles (UAVs) are gaining popularity as cost-effective flight platforms adaptable to a variety of applications such as surveying, firefighting, or communication [1–4]. Recent research in the electrification and hybridization of traditionally fuel powered UAV's aims to enable support of more advanced electrical systems and payloads. Electrified systems also offer increased design flexibility through the development of novel system architectures with electrified power and propulsion systems [5]. However, accounting for the coupling amongst mission dependent loads, power sources, and energy management strategies when conceptualizing electrified system designs can be challenging. Therefore, designing new platforms with conventional design approaches may result in oversized and underperforming systems [6].
With the rise of multidisciplinary design optimization, combined plant and control design (co-design) techniques have been developed where both the plant and control behavior is considered in the design process. A conventional approach would be sequential co-design wherein a plant is designed first, followed by the control development. However, it has been established that this approach is suboptimal because the design process does not account for the coupling between plant and control designs [7]. To address this issue, integrated design optimization approaches have been developed and investigated.
In the plant and controller co-design literature, researchers investigate either open-loop or closed-loop co-design (OLCD and CLCD, respectively). While both approaches identify an optimal plant configuration, OLCD controls the system with an optimal open-loop control action whereas an optimal feedback controller is designed in CLCD. The key difference between the two approaches is in how the control system is optimized. OLCD is useful to generate design insights but has limited application to physical systems where uncertainties are present because open-loop control does not inherently compensate for uncertainty. CLCD directly addresses this limitation because the feedback controller is designed during the co-design process.
While much of the existing co-design literature has focused on various OLCD studies and techniques [8–14], less has work has investigated CLCD. One of the most common approaches to CLCD in the literature is nested co-design techniques wherein a feedback control law is synthesized within an inner loop of a plant design optimization. These methods typically rely on control systems with efficient synthesis techniques, like linear quadratic regulation [15–18] or [19], to develop the control law. With the exception of work by Nash and Jain [19], the controller tuning parameters are fixed over the optimization. Alternatively, some design problems have considered the direct design of proportional-integral-derivative (PID) class controllers within a co-design framework. In Ref. [20], a PID controller for a DC electric motor was designed; work that was later extended to robust design through a min-max co-design optimization [21]. Gradient-free methods were used to design proportional-integral (PI) controllers for a hybrid energy storage system in Ref. [22]. Notably, the authors of Ref. [23] used feedback linearization to improve the co-design of a continuously variable transmission and its PI controller.
Some effort has focused on integrating more advance control methodologies, such as model predictive control (MPC), into co-design. In Ref. [24], the authors defined an explicit MPC parametrically so that the optimal control behavior can be transcribed into a plant design optimization routine. A similar effort designed a robust MPC for a fluid thermal management system in Ref. [25]. Stochastic dynamic programing was used to design a power management strategy for a hybrid electric vehicle in Ref. [26]. Similarly, the authors of Ref. [27] used a shooting method approach within a genetic algorithm optimization to design a hybrid electric vehicle powertrain, cooling system, and its MPC. Notably, these efforts have long computation times. Additionally, most of these studies focus on a fixed controller using a single set of control gains to control a range of candidate plant designs; not accounting for optimizing feedback controller tuning may likely result in suboptimal overall performance.
While some efforts have accounted for feedback control behavior in the co-design process, few studies have designed the controllers by selecting optimal control parameters. Of the studies that tailor the controller tuning to unique plant designs, the scope of the design methods limits their application to more complex system-level design necessary for next-generation aircraft. Conditional logic, which is commonplace in a variety of control systems, has also been neglected from the co-design literature thus far. Additionally, developing a suitable mission trajectory while designing the plant and controller has been neglected from the CLCD literature.
Therefore, this work addresses this research gap by contributing a generalizable approach to closed-loop control co-design with the following unique features:
methods for time-convergence analysis that evaluate the tradeoff between optimization accuracy and time to improve scalability of the co-design formulation to more complex systems,
smoothing functions that approximate conditional logic, like switching, for gradient-based control design, and
mission design with plant and controller design.
The methods are applied to the design of a hybrid electric UAV power and propulsion system, its respective feedback controllers, and the mission, for different co-design methodologies. Note that experimental demonstrations fall outside the scope of this effort due to the expense of creating multiple UAVs. Some co-design validation studies have appeared in the literature [28,29].
The remainder of this paper is organized as follows. Section 2 details the general open-loop and closed-loop co-design optimization problems, the approximated logic functions, and the proposed convergence analysis. Section 3 introduces a candidate hybrid UAV system and its feedback controllers and the system modeling methods. The hybrid UAV case study design problem and convergence analysis results are provided in Sec. 4. Section 5 presents the design optimization results. A summary and proposed future work are described in Sec. 6.
2 Optimization Methods
Numerical optimization methods are a common tool to design size, shape, control, and topology variables for a primary design objective and various predefined constraints representing physical system limitations. These techniques are particularly useful in co-design problems where they can evaluate a range of plant and control design options for either open or closed-loop systems. Sections 2.1 and 2.2 introduce a generalizable form of the OLCD and CLCD problem formulations. Because solving system-level co-design problems can be computationally expensive, a convergence analysis is presented to algorithmically evaluate the tradeoff between computation time and optimization accuracy.
2.1 Open-Loop Co-Design Problem.
where is the optimization objective to be minimized and can be a function of any of the optimization variables, is time, is the plant parameters, is the dynamic system state, is the algebraic/static system state, is the control trajectory, are disturbances, and and are inequality and equality constraints, respectively. Similar to Refs. [11], [13], and [30], time is explicitly included as a decision variable in this work to facilitate mission design wherein the duration of the mission can change depending on system behavior. The other design variables include the plant parameters and the open-loop control trajectory. The objective function is minimized with respect to the plant dynamics and , which are presented in nonlinear differential algebraic equation (DAE) form. Note that the functional time dependencies are implied for dynamic variables (e.g., ).
2.2 Closed-Loop Co-Design Problem.
In this formulation, the control design decisions are represented by the control parameters and are selected subject to the feedback controller equations , which represents controller dynamics common in integral control or filter design, and , which represents the feedback control law. For problems that consider a combination of both open-loop and closed-loop control decisions, may still represent an open-loop control trajectory. As introduced, this formulation is applicable to structured control laws where the controller and its dynamics can be explicitly expressed as a function of system parameters. It may be challenging to design nonstructured or implicit control policies, like model predictive control (MPC), within the proposed framework. Specifically, MPC is challenging to design within a co-design problem because obtaining an MPC solution is computationally expensive, and it may be difficult to derive second-order model information for complex systems.
The key difference between the OLCD and CLCD formulations, and the merit of the proposed CLCD approach, is in how the control behavior is optimized. The CLCD formulation imposes structural constraints on the feasible control action via the controller equations and , while the OLCD formulation does not, thus addressing a practical limitation of OLCD. Because the CLCD and OLCD problems differ, they will generally return unique plant and control design solutions. The addition of control constraints will degrade the CLCD solution's performance, albeit at the benefit of identifying a physically designable plant and implementable optimal feedback control strategy.
The dynamic complexity and large number of design variables makes solving these problems difficult and/or time consuming. Therefore, gradient-based optimization methods are preferred. However, feedback controllers often use piecewise or conditional functions, such as saturation, to manipulate the applied control action. Such nonsmooth functions may negatively impact the convergence of a gradient-based optimization routine, so smooth alternatives are introduced in Sec. 2.2.1.
2.2.1 Smoothing Functions.
where is the scaled input variable, , and .
where is the scaled input variable, , and .

Comparison of the exact and approximate saturation (top), conditional logic (middle), and clamping functions (bottom). The smoothing factors are used for the clamping function.
2.3 Methods for Time-Integration Convergence.
When solving dynamic optimizations, a time-integration method is used to integrate the state (and possibly control) dynamics to generate a representative system response trajectory. Two time-integration methodologies are proposed in the literature: explicit and implicit [31]. While both approaches have their merits, they effectively accomplish the same task through the discretization of the state dynamics. In explicit methods, like Runge–Kutta 4 or Forward Euler, the state equations are discretized directly and then encoded into the optimization routine. In contrast, implicit methods represent the timeseries information as a sequence of continuous polynomials that are constrained to exactly represent the continuous-time system dynamics at discrete points in time. For both approaches, the discretization rate/interval greatly impacts both the optimization time and accuracy. Smaller discretization intervals yield a more accurate representation of the state dynamics at the expense of more design variables and/or constraints that make the optimization process take longer. The converse is also true, which illustrates the tradeoff between optimization accuracy and time. Therefore, a convergence analysis is proposed to assess the compromise between optimization accuracy and time and provide a systematic approach to select an adequate discretization interval.
A conventional approach to selecting a discretization interval is to choose an interval, solve the problem, evaluate the accuracy, and repeat until the desired accuracy specification is met. While effective, this approach does not scale well to larger system-level design problems or multiphase optimizations where multiple discretization intervals could be selected. Instead of solving the full optimization problem repeatedly until a desired optimization accuracy is met, the proposed method reduces the design problem into a simpler form that can be solved many times quickly. The simple problem is used to select an appropriate discretization interval that is used when solving the full problem. The proposed method is presented in the context of implicit time-integration methods because implicit methods can often capture the same system behavior as explicit methods but with a larger discretization interval [31]. The approach is as follows:
Remove all plant and control design variables from the optimization program to convert the optimization into a constraint satisfaction problem. For multi-phase optimizations, remove the constraints that couple the terminal condition of one phase to the initial condition of the following phase. By decoupling the mission phases, the number of discretization steps can be identified for each phase independently. Note that the general optimization formulation has not changed and that only specific design variables and constraints are removed.
Solve the resulting constraint satisfaction problem and record the state trajectory information for a specified discretization interval.
- Evaluate the optimization errorwhere is the polynomial approximation for each state trajectory as illustrated in Fig. 2. The optimization error is only computed at the error nodes, which are located at the midpoint between adjacent discretization nodes, because the implicit integration method guarantees at the discretization nodes (Fig. 2).(11)
If the computed error exceeds the accuracy criteria, decrease the discretization interval, and repeat steps 2 and 3. Otherwise, stop the convergence analysis and apply the identified discretization interval to the full problem. If desired, steps 2 and 3 could also be repeated for a larger discretization interval to reduce computation time of the full optimization.
3 Case Study System Description
Enabled by advancements in electrical power systems, there is increasing interest in the design and manufacture of electrified aircraft as a more sustainable and capable mode of transportation. In particular, electrified unmanned aircraft have received attention for potential applications, including but not limited to, surveying, disaster relief, film and imaging, agriculture, and package delivery [32]. Importantly, certain applications utilize high power electrical sensors or payloads, necessitating a high-power electrical system.
However, the performance and efficiency of these systems are strongly coupled to their design and operation. Therefore, the proposed case study introduced in Sec. 4 will investigate the plant and control design of a hybrid UAV powertrain. The system under study is illustrated in Fig. 3 and is a series hybrid electric UAV powertrain comprised of four subsystems: energy storage, genset, propulsion, and avionics. The energy storage subsystem consists of a battery pack that stores electrical energy and charges/discharges depending on the electrical loading requirements. The genset (internal combustion engine, starter/generator, and an inverter/rectifier) subsystem consumes fuel to generate electrical power. Together, the energy storage and genset subsystems provide electrical power to the propulsion and avionics subsystems. The propulsion system (inverter, propulsion motor, propeller) requires electrical power to generate thrust that propels the aircraft. The avionic subsystem, which is an electrical load, represents the aircraft's electrical payload. Note, this aircraft case study does not represent any specific current or future platform.
The aircraft has two onboard controllers that compute the propulsion and genset subsystem input commands. The genset command is generated by an equivalent consumption minimization strategy (ECMS) controller that controls the hybrid power split between the battery and genset [33]. To track a desired vehicle velocity, a PI controller generates an inverter duty cycle command. Both feedback control approaches are well utilized in the literature and are readily implemented in a co-design problem. To track a desired flight profile defined by velocity and altitude waypoints, the aircraft is sent velocity reference and flight path angle commands from a navigation system. Figure 4 illustrates the exchange of state and control information in the closed-loop control system. Sections 3.1 and 3.2 introduce the mathematical models that can be integrated into an optimization routine. All model variables introduced in Sec. 3 are summarized in Tables 6–9.

Aircraft closed-loop signal routing between the autonomous navigation system, the controllers, and the UAV
3.1 Plant Modeling.
3.1.1 Graph-Based Powertrain Model.
External vertices are treated as disturbances from external states to the system. Similarly, external power flows (dashed edges), represent external power disturbances on the system.
The graph-based model of the UAV powertrain is illustrated in Fig. 6 with Tables 6–9 that define all vertex, edge, input, and parameter information. All dynamic (solid circle), algebraic (double circle), and external (dashed circle) vertices are represented by variables , , and , respectively (Table 6), where is the vertex index as illustrated in Fig. 6.
where is the battery state of charge (SOC) and are the approximation coefficients.
3.1.2 Genset Model.
where is the genset time constant, is the genset input, is the input gain, and is the powertrain bus voltage. Derived from Ref. [33], the fuel consumption dynamics are approximated by a linear relationship of the genset power with constant coefficients . Model parameters are presented in Tables 6–9.
3.1.3 Vehicle Dynamics Model.
In this model, is the vehicle velocity, is the altitude, is the angle of attack, , , , and are the thrust, drag, lift, and gravitational forces, respectively, is the air density as a function of altitude, , , and are the thrust, drag, and lift coefficients, is the wing area, is gravitational acceleration, and are the zero-drag and zero-lift coefficients, respectively, is the maximum angle of attack, is the maximum lift coefficient, and is the drag-due-to-lift factor.
where the subscripts , , , , , , and correspond to the mass of the aircraft's propulsion motor, battery, fuel, genset, avionic load, inverter, and airframe, respectively.
3.2 Controller Modeling.
Proportional-integral and equivalent consumption minimization strategy (ECMS) control laws are commonplace in the literature, and this work introduces a formulation of each that is suited for gradient-based optimization in a CLCD problem. The PI control design expands upon a formulation the authors presented in Ref. [36] and the ECMS formulation is adapted from Ref. [33].
3.2.1 Velocity Proportional-Integral Controller Model.
Proportional-integral control is a suitable feedback controller to track the reference velocity profile commanded by the aircraft's autonomous navigation system. Figure 8 illustrates the PI controller block diagram that incorporates input saturation and clamping antiwindup.
The PI controller reduces the tracking error between the commanded reference velocity and feedback velocity state by leveraging proportional and integral control effort and , respectively. The proportional and integral control gains are and , respectively, and the controller commands the inverter duty cycle .
3.2.2 Equivalent Consumption Minimization Strategy Controller Model.
where is the virtual battery fuel consumption, is the lower heating value of the fuel, is the genset current, is the battery current, is the propulsion load current, is the avionic load current, is the genset power, is the battery power, and is the bus voltage. Note that the total electrical load is and the genset fuel consumption is given in (15b). This optimization problem identifies an optimal genset current that minimizes the total equivalent fuel consumption while current is supplied to the loads and battery current and genset power limitations are respected.
3.3 System Model Summary.
Figure 9 is included to illustrate the coupling and signal passing between each plant and controller model. To summarize, the powertrain exchanges the motor speed and vehicle velocity states with the vehicle dynamics model, and bus voltage and genset current states with the genset model. Additionally, the altitude state of the airframe model calculates the air density that is required by the powertrain model to evaluate the propulsive power. The vehicle dynamics model passes the velocity state to the PI controller to calculate the tracking error, and the PI controller returns the inverter duty cycle command to the powertrain model. The ECMS controller receives instantaneous state of charge , inverter current input , avionic load current , and bus voltage states from the powertrain model and returns the genset command to control the genset. The autonomous navigation system passes a desired velocity reference to the PI controller and a flight path angle command to the airframe model. In total, the plant model has 18 states (7 dynamic, 11 algebraic) and the controller model has 1 state.
4 Design Study
While electrified aircraft are a promising approach to revolutionize transportation, there are still technical barriers such as the limited power and energy density of electronic power systems that constrain the range and performance of modern electrified aircraft. To address this need, it is important to develop more energy dense electronic components and determine how to integrate them most effectively into a vehicle. Through co-design, insights about component design and system operation can be generated.
Often, a key design decision is the choice of battery and motor. While larger batteries and motors have more power and energy capabilities, the resulting increase in system mass is likely to degrade performance. Additionally, control design strongly governs the dynamic system performance, which influences how energy is generated, stored, and delivered to the subsystems. To better understand how to design and operate an electrified aircraft, the following study will consider the design of a hybrid UAV's battery, propulsion motor, and feedback controllers. To demonstrate the effectiveness of the CLCD approach, a hybrid UAV is designed using five different co-design approaches:
Case 1: Simultaneous open-loop co-design (SOL)—Identify an optimal plant and open-loop control decisions by solving (1). While the open-loop control is not physically applicable, the optimal system will describe a best-case design.
Case 2: Sequential co-design (SQ)—The sequential design study represents a conventional design approach wherein the plant is designed first, and the controller designed second. The optimal closed-loop control parameters are identified through (2) using the optimal set of plant parameters from SOL.
Case 3: Simultaneous closed-loop co-design (SCL)—The simultaneous closed-loop problem identifies a set of optimal plant and closed-loop control parameters at the same time by solving (2).
Case 4: Siloed plant design (PCL)—This study identifies an optimal plant design by solving the CLCD optimization problem (2) where the closed-loop control parameters are fixed.
Case 5: Siloed control design (CCL)—This study identifies an optimal closed-loop control parameterization by solving the CLCD optimization problem (2) where the plant parameters are fixed.
Note that siloed design cases 4 and 5 are included for comparison to emphasize the relevance of co-design approaches and illustrate the relative impact of plant and control decisions for the design study. Each case is compared to a baseline (BASE) closed-loop system design. The baseline plant design was developed based on the experimentally validated system parameters from Ref. [34]. The baseline PI control design was developed using pole placement methods, and the ECMS controller was designed to be charge sustaining.
The design problem will determine the number of parallel battery cells , number of series battery cells , the motor constant , and motor coil resistance to size the battery and motor. Additionally, the controllers are tuned by selecting the PI controller's and gains, and the ECMS controller's equivalence factor . Although there is significant flexibility when selecting control gains, the feasible design space is comparatively restricted. Battery packs are comprised of an integer number of cells. While mixed integer or hybrid optimization methods [28,37] could be used to design the battery pack, this effort will allow a noninteger number of cells to facilitate a continuous search of the feasible design space. To generate a physically meaningful design, the motor parameter design space is determined empirically from the commercially sold NeuMotors 8000 Series brushless DC motors. This class of motors has a 4.5–21.4 kW power range, which is well-suited for the proposed application. The motor parameter design space is generated by downloading the motor resistance, motor constant, motor mass, and current limit data from NeuMotors' online catalog. The convex hull of all motor resistance and motor constant value pairs is the feasible motor parameter design space (Fig. 10).

The motor mass and current limit as a function of the convex hull of motor constant and resistance values
where is the battery current and is the prop. current. For these batteries, and .
The aircraft is designed to complete a mission characterized by velocity, altitude, and avionic load power profiles as shown in Fig. 11. The aircraft begins in a cruise condition before diving into the sensing segment where the aircraft activates its avionic load, which represents a high-power sensing device [38–40]. Then the aircraft climbs to a dash condition before climbing a second time to return to cruise. The exact mission conditions are described in Table 1. Note that the climb and dive segments have variable duration and only end when the aircraft reaches the prescribed flight conditions. For example, the sensing segment will not start until the vehicle is flying at 40 m/s at 4 km altitude.
Mission profile conditions (Fig. 11) for the UAV case study
Mission segment | Description | Discretization intervals | Duration (min) | Velocity (m/s) | Altitude (km) | Load power (kW) |
---|---|---|---|---|---|---|
a | Cruise | 10 | 6 | 30 | 5.5 | 0 |
b | Dive | 20 | 30–40 | 5.5–4.0 | 0 | |
c | Sensing | 10 | 2 | 40 | 4.0 | 1.0 |
d | Climb | 22 | 40–45 | 4.0–4.5 | 0 | |
e | Dash | 10 | 4 | 45 | 4.5 | 0 |
f | Climb | 20 | 45–30 | 4.5–5.5 | 0 | |
g | Cruise | 10 | 6 | 30 | 5.5 | 0 |
Mission segment | Description | Discretization intervals | Duration (min) | Velocity (m/s) | Altitude (km) | Load power (kW) |
---|---|---|---|---|---|---|
a | Cruise | 10 | 6 | 30 | 5.5 | 0 |
b | Dive | 20 | 30–40 | 5.5–4.0 | 0 | |
c | Sensing | 10 | 2 | 40 | 4.0 | 1.0 |
d | Climb | 22 | 40–45 | 4.0–4.5 | 0 | |
e | Dash | 10 | 4 | 45 | 4.5 | 0 |
f | Climb | 20 | 45–30 | 4.5–5.5 | 0 | |
g | Cruise | 10 | 6 | 30 | 5.5 | 0 |
The notation represents the time span of the mission segment and is the final time of mission segment .
4.1 Co-Design Optimization Problems.
Note that the key difference between the two problems is the inclusion of the closed-loop control behavior in the CLCD formulations. Variations of these problems that consider different sets of design variables are solved for each of the 5 design cases. The design variables for each of the five design cases are summarized in Table 2.
The free design variables for the hybrid UAV co-design studies
Design study | Optimization problem | Plant decisions | Control decisions | Duration decisions |
---|---|---|---|---|
BASE | Equation (2) | — | — | |
SOL | Equation (1) | |||
SQ | Equation (2) | — | ||
SCL | Equation (2) | |||
PCL | Equation (2) | — | ||
CCL | Equation (2) | — |
Design study | Optimization problem | Plant decisions | Control decisions | Duration decisions |
---|---|---|---|---|
BASE | Equation (2) | — | — | |
SOL | Equation (1) | |||
SQ | Equation (2) | — | ||
SCL | Equation (2) | |||
PCL | Equation (2) | — | ||
CCL | Equation (2) | — |
The SQ problem is initialized with the optimal plant parameters from the SOL design problem.
The dynamic optimizations are built using python-based toolboxes OpenMDAO and Dymos [41,42], and solved using the sequential least squares programing (SLSQP) gradient-based optimizer from SciPyOptimize [43]. They are solved on an Intel i7 CPU workstation with 32GB of RAM. Each design case is warm-started from the baseline optimization solution to guarantee the initial design is feasible. Before solving the optimization problem for each of the design cases, a convergence study is conducted to select an appropriate discretization interval.
4.2 Convergence Study.
The convergence analysis introduced in Sec. 2 is used to determine the number of discretization nodes for each phase of the mission. To simplify the optimization for the convergence study, the 7-segment mission is decomposed into three separate but representative phases: constant speed and altitude (case 1), dive (case 2), and climb (case 3). Additionally, all design variables are fixed except for the state variables, pitch control, and velocity reference, thus reducing the optimization to a constraint satisfaction problem. Each optimization case is optimized for 10 to 50 discretization intervals in two interval increments, and the error statistics for the vehicle velocity and fuel mass are provided in Fig. 12. The velocity and mass statistics are presented because they are the critical vehicle states that demonstrated the largest discretization error amongst all other states. The target accuracy specification is 0.1%.
The convergence analysis demonstrates that few discretization intervals (<10) is required to meet the accuracy specification for case 1 because the aircraft dynamics change slowly in time while the aircraft flies at constant speed and altitude. However, for the more dynamic dive and climb mission segments, the convergence analysis indicates that 20 and 22 discretization nodes are necessary for case 2 (dive) and case 3 (climb), respectively. The number of discretization nodes for each mission segment are reported in Table 1.
5 Design Results
The optimized plant and controller parameters and the mission time for each optimization study is summarized in Table 2. Intuitively, the open loop co-design (SOL) yielded the greatest performance (minimum time) because the optimal control actions were chosen directly. However, the SOL design is only provided for comparison because it does not yield a practical control law. The best performing design, which is physically implementable, is the simultaneous closed-loop (SCL) case because the design process accounts for the coupling between plant and closed-loop controller. As described previously, it is expected that all closed-loop design solutions perform worse than the open-loop solution because the control response is constrained by the control law. The sequential open-loop design (SOL) does not account for plant and closed-loop controller coupling, which results in a slightly suboptimal closed-loop system design. The siloed plant and controller co-design cases perform poorly because they also do not account for coupling between plant and controller. However, it is shown that the plant optimization improves the system performance more than the control optimization for this problem.
Relevant state, control, and output trajectories are illustrated in Fig. 13. Figures 13(a) and 13(b) show that each vehicle completed the full mission profile by following the prescribed velocity and altitude profile. To minimize the total mission time, each vehicle was designed to complete the dive and climb segments as quickly as possible. While Fig. 13(b) shows that each optimized design completes the dive segment within 300-400 s, there is significant disparity in climb performance. The SOL, SCL, and SQ designs each complete the climb segments significantly faster than the BASE, CCL, or PCL designs, and are thus able to complete the mission quicker. Figures 13(c) and 13(d) indicate that faster climb segments are a result of a greater flight path angle achieved through a larger thrust-to-weight ratio (TWR) that enables steeper and faster climbs. Note that improvements in thrust may also result in an increase in vehicle mass. Therefore, leveraging optimization techniques is important to systematically evaluating this tradeoff. The remainder of this section describes how the optimization routine selects design parameters to improve vehicle thrust while minimizing additional weight.

Selected optimal variable trajectories for the aircraft (a) velocity, (b) altitude, (c) flight path angle, (d)thrust-to-weight ratio, (e) battery state of charge, (f) fuel mass, (g) inverter duty cycle, and (h) genset input
As shown by (16c), vehicle thrust is proportional to the prop. motor speed while the vehicle weight is dependent on the battery and motor design parameters and the fuel consumption (15b). To increase the motor speed, the optimizer decreases the motor constant and resistance (Table 3 SCL, SQ, and PCL designs), which improves the motor's electrical and electromechanical conversion efficiency. Therefore, the motor can more efficiently convert electrical energy into mechanical energy and increase thrust.
A comparison of the optimal UAV and control design parameters and performance for each design study
BASE | SOL | SQ | SCL | PCL | CCL | |
---|---|---|---|---|---|---|
Motor constant (V-s/rad) | 0.1238 | 0.087 | 0.087 | 0.0926 | 0.0988 | 0.1238 |
Motor resistance (Ω) | 0.0292 | 0.0158 | 0.0158 | 0.0171 | 0.0186 | 0.0292 |
Series cells | 16 | 20 | 20 | 20 | 20 | 16 |
Parallel cells | 7 | 9.3 | 9.3 | 9.8 | 19.3 | 7 |
Equivalence factor | 18 | — | 16.6 | 16.9 | 18 | 19.3 |
P Gain | 0.0371 | — | 0.001 | 0.001 | 0.0371 | 0.0335 |
I Gain | 0.002 | — | 0.0065 | 0.01 | 0.002 | 0.0042 |
GTOW (kg) | 50.5 | 53.7 | 53.7 | 54.2 | 50.5 | 50.5 |
Mission time (s) | 2434 | 1570 | 1605 | 1590 | 2434 | 2277 |
Improvement (%) | — | 35.5 | 34.1 | 34.7 | 22.6 | 6.45 |
BASE | SOL | SQ | SCL | PCL | CCL | |
---|---|---|---|---|---|---|
Motor constant (V-s/rad) | 0.1238 | 0.087 | 0.087 | 0.0926 | 0.0988 | 0.1238 |
Motor resistance (Ω) | 0.0292 | 0.0158 | 0.0158 | 0.0171 | 0.0186 | 0.0292 |
Series cells | 16 | 20 | 20 | 20 | 20 | 16 |
Parallel cells | 7 | 9.3 | 9.3 | 9.8 | 19.3 | 7 |
Equivalence factor | 18 | — | 16.6 | 16.9 | 18 | 19.3 |
P Gain | 0.0371 | — | 0.001 | 0.001 | 0.0371 | 0.0335 |
I Gain | 0.002 | — | 0.0065 | 0.01 | 0.002 | 0.0042 |
GTOW (kg) | 50.5 | 53.7 | 53.7 | 54.2 | 50.5 | 50.5 |
Mission time (s) | 2434 | 1570 | 1605 | 1590 | 2434 | 2277 |
Improvement (%) | — | 35.5 | 34.1 | 34.7 | 22.6 | 6.45 |
Similarly, the battery can increase motor speed by delivering more electrical power to the motor for a longer duration. To improve capabilities, either the number of series or parallel cells can be increased to increase battery voltage or capacity, respectively. However, increasing the number of cells increases the system mass, which may negatively impact TWR. For all plant designs, the battery voltage is maximized by increasing the pack size to 20 series cells (for SCL, SQ, and PCL designs). When only the plant is designed (PCL), the number of parallel cells is increased to 19.3 in order to better sustain the high operating voltage for longer at the expense of a significant weight penalty. When the plant and control are designed, the number of parallels cells is 9–10 and the ECMS controller equivalence factor is tuned to sustain the battery charge at a higher voltage. This is also observed for the CCL design where the equivalence factor is increased to yield a charge sustaining approach to battery management. While this comparison is fair between the baseline (BASE) and CCL designs because they have the same battery size, it is less fair for the SCL and SQ designs that have dramatically different battery sizes with different power and energy characteristics. The PI gains, which do not impact thrust, do impact the system's transient response. By increasing the I gain and decreasing the P gain, the system becomes more responsive and can more quickly track the desired velocity profile. While more responsive, the optimal gains do yield oscillatory control behavior.
Note the similarity in the optimal parameterization and state trajectories between the SOL and the SCL and SQ vehicle designs. Notably the closed-loop controllers are making similar decisions to the open-loop controller (Figs. 13(g) and 13(h)). This is expected because it has been demonstrated that a properly tuned ECMS controller can perform similarly to the open-loop optimal controller [33]. Additionally, recall that in the simultaneous and sequential design cases, an optimal navigation system generates a velocity reference passed to the PI controller. If the PI controller tracks references quickly, which the optimizer tunes them to do, the closed-loop response will be similar to the open-loop response.
In summary, the simultaneous closed-loop (SCL) vehicle design approach provided the highest performing closed-loop system design. In comparison to the other closed-loop design methodologies, the sequential design has similar performance with similar design characteristics for the motor, battery, and control parameters. For the minimum time mission, design decisions were made to increase the transient response rate and thrust-to-weight ratio.
To verify the results of the convergence study, the velocity and fuel mass discretization error statistics for each closed-loop problem is presented in Table 4. In general, the average error is within ∼0.1% for each case which meets the accuracy criteria defined previously. This implies less than 0.1% design error. Additionally, the computational metrics of each optimization study is presented in Table 5. It should be clear that a convergence study is a valuable preprocessing step because identifying the discretization interval through trial and error would be computationally infeasible for problems of this scale.
The optimization accuracy error statistics for aircraft velocity and fuel mass states
Velocity error | Fuel mass error | |||
---|---|---|---|---|
Mean (%) | Std. Dev. (%) | Mean (%) | Std. Dev. (%) | |
BASE | 0.01 | 0.03 | 0.01 | 0.04 |
SQ | 0.08 | 0.32 | 0.09 | 0.29 |
SCL | 0.02 | 0.06 | 0.02 | 0.11 |
PCL | 0.05 | 0.47 | 0.27 | 1.33 |
CCL | 0.11 | 0.53 | 0.12 | 0.41 |
Velocity error | Fuel mass error | |||
---|---|---|---|---|
Mean (%) | Std. Dev. (%) | Mean (%) | Std. Dev. (%) | |
BASE | 0.01 | 0.03 | 0.01 | 0.04 |
SQ | 0.08 | 0.32 | 0.09 | 0.29 |
SCL | 0.02 | 0.06 | 0.02 | 0.11 |
PCL | 0.05 | 0.47 | 0.27 | 1.33 |
CCL | 0.11 | 0.53 | 0.12 | 0.41 |
The computation metrics for the closed-loop optimization problems
SQ | SCL | PCL | CCL | |
---|---|---|---|---|
Design variables | 2155 | 2159 | 2156 | 2155 |
Constraints | 3204 | 3204 | 3204 | 3204 |
Iterations | 79 | 55 | 46 | 98 |
Function evaluations | 98 | 62 | 52 | 196 |
Gradient evaluations | 79 | 55 | 46 | 98 |
Solution time (hr) | 5.5 | 3.8 | 3.2 | 6.6 |
SQ | SCL | PCL | CCL | |
---|---|---|---|---|
Design variables | 2155 | 2159 | 2156 | 2155 |
Constraints | 3204 | 3204 | 3204 | 3204 |
Iterations | 79 | 55 | 46 | 98 |
Function evaluations | 98 | 62 | 52 | 196 |
Gradient evaluations | 79 | 55 | 46 | 98 |
Solution time (hr) | 5.5 | 3.8 | 3.2 | 6.6 |
Note that the optimizer treats the dynamic states as optimization variables.
6 Conclusion
While electrified aircraft offer a novel suite of capabilities with increased efficiency, it is challenging to account for the inherent system complexity and coupling between subsystems and their controllers. Using traditional sequential design strategies yields feasible but relatively less functional vehicle designs. Therefore, this work proposed an integrated design approach for simultaneous plant sizing and closed-loop control design and demonstrated its effectiveness in application to a hybrid electric UAV.
The traditional open-loop co-design problem formulation and the proposed closed-loop formulation were presented wherein the closed-loop problem accounts for practical constraints on the control architecture. To account for saturation and conditional logic commonplace in feedback control design, various smoothing functions were introduced. Because system-level co-design problems are computationally expensive, a novel convergence analysis was proposed to systematically evaluate the tradeoff between optimization time and accuracy. The methods were applied to design a hybrid electric UAV powertrain and its feedback controller to complete a candidate mission in minimum time. Intuitively, the analysis showed that the simultaneous closed-loop co-design optimization yielded the greatest practical system performance. To complete the mission faster, the optimizer identified a set of plant parameters and control gains that increased the vehicle's thrust-to-weight ratio and transient response speed.
While this work demonstrates the feasibility of plant and controller co-design, future work in coupled aircraft and subsystem design and design under uncertainty are necessary. A natural extension is to consider additional aspects of the hybrid aircraft such as airframe and cooling system design and explore how different mission objectives and constraints impact the optimal system design. Furthermore, to address uncertainty in physical systems, integrating formal robustness metrics and constraints into the optimization is necessary to develop a robust plant and closed-loop control design. Finally, these methods should be validated with a physical test platform to evaluate the feasibility and accuracy of this design approach.
Acknowledgment
Distribution Statement A; PA# AFRL-2022-3883.
Funding Data
The National Science Foundation (NSF) Engineering Research Center (ERC) for Power Optimization of Electro-Thermal Systems (POETS) (Contract No. EEC-1449548 Funder ID: 10.13039/100000001).
Appendix
Hybrid UAV graph model vertex and state information
Vertex | Variable | Description | Units | Capacitance | Lower bound | Upper bound |
---|---|---|---|---|---|---|
1 | Battery state of charge | — | 0.3 | 0.9 | ||
2 | RC pair voltage 1 | V | −10 | 10 | ||
3 | RC pair voltage 2 | V | −10 | 10 | ||
4 | Battery current | A | 0 | |||
5 | Bus voltage | V | 0 | — | — | |
6 | Inverter DC current | A | 0 | — | — | |
7 | Inverter DC link voltage | V | 0 | — | — | |
8 | Inverter q-axis current | A | 0 | — | — | |
9 | Inverter q-axis voltage | V | 0 | — | — | |
10 | Prop. current | A | 0 | 0 | Equation (24) | |
11 | Prop speed | rad/s | 0 | — | — | |
12 | Avionic load current | A | 0 | — | — | |
13 | Avionic load voltage | V | 0 | — | — | |
14 | Genset current | A | — | 0 | 50 | |
15 | Vehicle velocity | m/s | — | 25 | 50 | |
16 | Thermal sink | — | — | — | — | |
17 | Thermal sink | — | — | — | — | |
18 | Thermal sink | — | — | — | — | |
19 | Thermal sink | — | — | — | — |
Vertex | Variable | Description | Units | Capacitance | Lower bound | Upper bound |
---|---|---|---|---|---|---|
1 | Battery state of charge | — | 0.3 | 0.9 | ||
2 | RC pair voltage 1 | V | −10 | 10 | ||
3 | RC pair voltage 2 | V | −10 | 10 | ||
4 | Battery current | A | 0 | |||
5 | Bus voltage | V | 0 | — | — | |
6 | Inverter DC current | A | 0 | — | — | |
7 | Inverter DC link voltage | V | 0 | — | — | |
8 | Inverter q-axis current | A | 0 | — | — | |
9 | Inverter q-axis voltage | V | 0 | — | — | |
10 | Prop. current | A | 0 | 0 | Equation (24) | |
11 | Prop speed | rad/s | 0 | — | — | |
12 | Avionic load current | A | 0 | — | — | |
13 | Avionic load voltage | V | 0 | — | — | |
14 | Genset current | A | — | 0 | 50 | |
15 | Vehicle velocity | m/s | — | 25 | 50 | |
16 | Thermal sink | — | — | — | — | |
17 | Thermal sink | — | — | — | — | |
18 | Thermal sink | — | — | — | — | |
19 | Thermal sink | — | — | — | — |
Hybrid UAV graph model edge and power flow information
Edge | Description | Power flow |
---|---|---|
1 | Battery power | |
2 | Electrical power | |
3 | Electrical power | |
4 | Resistive loss/heat load | |
5 | Resistive loss/heat load | |
6 | Resistive loss/heat load | |
7 | Resistive loss/heat load | |
8 | Electrical power | |
9 | Electrical power | |
10 | Electrical power | |
11 | Controlled electrical power | |
12 | Electrical power | |
13 | Switching and conduction loss | |
14 | Electrical power | |
15 | Electromagnetics | |
16 | Resistive loss/heat load | |
17 | Friction loss | |
18 | Propulsion power | |
19 | Electrical power | |
20 | Electrical power | |
21 | Electrical power | |
22 | Avionic load power |
Edge | Description | Power flow |
---|---|---|
1 | Battery power | |
2 | Electrical power | |
3 | Electrical power | |
4 | Resistive loss/heat load | |
5 | Resistive loss/heat load | |
6 | Resistive loss/heat load | |
7 | Resistive loss/heat load | |
8 | Electrical power | |
9 | Electrical power | |
10 | Electrical power | |
11 | Controlled electrical power | |
12 | Electrical power | |
13 | Switching and conduction loss | |
14 | Electrical power | |
15 | Electromagnetics | |
16 | Resistive loss/heat load | |
17 | Friction loss | |
18 | Propulsion power | |
19 | Electrical power | |
20 | Electrical power | |
21 | Electrical power | |
22 | Avionic load power |
Hybrid UAV model input information
Input | Description | Units | Lower bound | Upper bound |
---|---|---|---|---|
Inverter duty cycle | — | 0.01 | 0.99 | |
Genset command | — | 0.01 | 0.99 | |
Flight path angle | rad |
Input | Description | Units | Lower bound | Upper bound |
---|---|---|---|---|
Inverter duty cycle | — | 0.01 | 0.99 | |
Genset command | — | 0.01 | 0.99 | |
Flight path angle | rad |
Hybrid UAV model parameter information
Parameter | Description | Units | Default value | Lower bound | Upper bound |
---|---|---|---|---|---|
Airframe mass | kg | 18 | N/A | N/A | |
Bus resistance | mΩ | 0.5 | N/A | N/A | |
Cell RC capacitance 1 | kF | 1.97 | N/A | N/A | |
Cell RC capacitance 2 | kF | 52.9 | N/A | N/A | |
Cell RC resistance 1 | mΩ | 27.9 | N/A | N/A | |
Cell RC resistance 2 | mΩ | 34.2 | N/A | N/A | |
Cell capacity | Ah | 3.05 | N/A | N/A | |
Cell mass | g | 45.9 | N/A | N/A | |
Cell series resistance | mΩ | 20.8 | N/A | N/A | |
Current limit coefficient | A | 160 | N/A | N/A | |
Current limit coefficient | A-rad/V-s | 601 | N/A | N/A | |
Current limit coefficient | A/Ω | -3440 | N/A | N/A | |
Drag-due-to-lift factor | — | 0.05 | N/A | N/A | |
Engine time constant | s | 20 | N/A | N/A | |
Equivalence factor | — | 18.0 | 9 | 30 | |
Fuel consumption coefficients | kg | N/A | N/A | ||
Fuel consumption coefficients | kg/W | N/A | N/A | ||
Fuel lower heating value | MJ/kg | 43.4 | N/A | N/A | |
Genset input gain | A | 63.3 | N/A | N/A | |
Genset mass | kg | 12 | N/A | N/A | |
Genset power | kW | - | N/A | N/A | |
Gravitational constant | m/s2 | 9.81 | N/A | N/A | |
Integral gain | — | 0.0020 | 0 | 0.01 | |
Inverter mass | kg | 1.0 | N/A | N/A | |
Inverter resistance | mΩ | 80.8 | N/A | N/A | |
Load mass | kg | 7.1 | N/A | N/A | |
Max AoA | ° | 8 | N/A | N/A | |
Max lift | — | 1.23 | N/A | N/A | |
Motor friction | mNm s/rad | 0.36 | N/A | N/A | |
Motor mass coefficients | kg | 0.81 | N/A | N/A | |
Motor mass coefficients | kg rad/V s | 21.3 | N/A | N/A | |
Motor mass coefficients | Kg/Ω | −39.5 | N/A | N/A | |
Number of parallel cells | — | 7 | 1 | 20 | |
Number of series cells | — | 16 | 1 | 20 | |
OCV coefficients | V | 13.75 | N/A | N/A | |
OCV coefficients | V | −40.12 | N/A | N/A | |
OCV coefficients | V | 44.14 | N/A | N/A | |
OCV coefficients | V | −22.98 | N/A | N/A | |
OCV coefficients | V | 6.73 | N/A | N/A | |
OCV coefficients | V | 2.66 | N/A | N/A | |
Prop. coil resistance | mΩ | 29.2 | Figure 10 | Figure 10 | |
Prop. motor constant | V s/rad | 0.1238 | Figure 10 | Figure 10 | |
Propeller diameter | m | 0.7 | N/A | N/A | |
Propeller efficiency | — | 0.8 | N/A | N/A | |
Proportional gain | — | 0.0371 | 0.001 | 0.1 | |
Thrust coefficient | — | N/A | N/A | ||
Velocity reference | m/s | — | 25 | 50 | |
Viscous friction | mNm s/rad | 0.36 | N/A | N/A | |
Wing area | MJ/kg | 3.4 | N/A | N/A | |
Zero lift coefficient | — | 0.49 | N/A | N/A. |
Parameter | Description | Units | Default value | Lower bound | Upper bound |
---|---|---|---|---|---|
Airframe mass | kg | 18 | N/A | N/A | |
Bus resistance | mΩ | 0.5 | N/A | N/A | |
Cell RC capacitance 1 | kF | 1.97 | N/A | N/A | |
Cell RC capacitance 2 | kF | 52.9 | N/A | N/A | |
Cell RC resistance 1 | mΩ | 27.9 | N/A | N/A | |
Cell RC resistance 2 | mΩ | 34.2 | N/A | N/A | |
Cell capacity | Ah | 3.05 | N/A | N/A | |
Cell mass | g | 45.9 | N/A | N/A | |
Cell series resistance | mΩ | 20.8 | N/A | N/A | |
Current limit coefficient | A | 160 | N/A | N/A | |
Current limit coefficient | A-rad/V-s | 601 | N/A | N/A | |
Current limit coefficient | A/Ω | -3440 | N/A | N/A | |
Drag-due-to-lift factor | — | 0.05 | N/A | N/A | |
Engine time constant | s | 20 | N/A | N/A | |
Equivalence factor | — | 18.0 | 9 | 30 | |
Fuel consumption coefficients | kg | N/A | N/A | ||
Fuel consumption coefficients | kg/W | N/A | N/A | ||
Fuel lower heating value | MJ/kg | 43.4 | N/A | N/A | |
Genset input gain | A | 63.3 | N/A | N/A | |
Genset mass | kg | 12 | N/A | N/A | |
Genset power | kW | - | N/A | N/A | |
Gravitational constant | m/s2 | 9.81 | N/A | N/A | |
Integral gain | — | 0.0020 | 0 | 0.01 | |
Inverter mass | kg | 1.0 | N/A | N/A | |
Inverter resistance | mΩ | 80.8 | N/A | N/A | |
Load mass | kg | 7.1 | N/A | N/A | |
Max AoA | ° | 8 | N/A | N/A | |
Max lift | — | 1.23 | N/A | N/A | |
Motor friction | mNm s/rad | 0.36 | N/A | N/A | |
Motor mass coefficients | kg | 0.81 | N/A | N/A | |
Motor mass coefficients | kg rad/V s | 21.3 | N/A | N/A | |
Motor mass coefficients | Kg/Ω | −39.5 | N/A | N/A | |
Number of parallel cells | — | 7 | 1 | 20 | |
Number of series cells | — | 16 | 1 | 20 | |
OCV coefficients | V | 13.75 | N/A | N/A | |
OCV coefficients | V | −40.12 | N/A | N/A | |
OCV coefficients | V | 44.14 | N/A | N/A | |
OCV coefficients | V | −22.98 | N/A | N/A | |
OCV coefficients | V | 6.73 | N/A | N/A | |
OCV coefficients | V | 2.66 | N/A | N/A | |
Prop. coil resistance | mΩ | 29.2 | Figure 10 | Figure 10 | |
Prop. motor constant | V s/rad | 0.1238 | Figure 10 | Figure 10 | |
Propeller diameter | m | 0.7 | N/A | N/A | |
Propeller efficiency | — | 0.8 | N/A | N/A | |
Proportional gain | — | 0.0371 | 0.001 | 0.1 | |
Thrust coefficient | — | N/A | N/A | ||
Velocity reference | m/s | — | 25 | 50 | |
Viscous friction | mNm s/rad | 0.36 | N/A | N/A | |
Wing area | MJ/kg | 3.4 | N/A | N/A | |
Zero lift coefficient | — | 0.49 | N/A | N/A. |