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This lecture presents a Hamiltonian control method and a distributed optimal control method for distributed Lagrangian systems. The distributed optimal control theory is formulated using a semi-group abstraction resulting in an integro-differential Riccati equation.
The optimal control of structures which consist of composite materials or of perforated materials is discussed. Asymptotic formula, derived from the so-called homogenization theory, are presented which allow the replacement of very complicated problems by much simpler ones.
Optimal control of distributed parameter systems using multilevel techniques
An approximate method is presented for the synthesis of an optimal control of distributed parameter systems, based on a combination of the ideas of Lyapunov's direct method and those of Bellman's successive approximation in policy space. The method is such that the approximate equation is improved after each iteration in the sense of the performance index being used.
A Chandrasekhar-type factorization method is applied to the linear-quadratic optimal control problem for distributed parameter systems. An aeroelastic control problem is used as a model example to demonstrate that if computationally efficient algorithms, such as those of Chandrasekhar-type, are combined with the special structure often available to a particular problem, then an abstract approximation theory developed for distributed parameter control theory becomes a viable method of solution. A numerical scheme based on averaging approximations is applied to hereditary control problems. Numerical examples are given.
The efficient numerical treatment of optimal control problems governed by elliptic partial differential equations (PDEs) and systems of elliptic PDEs, where the control is finite dimensional is discussed. Distributed control as well as boundary control cases are discussed. The main characteristic of the new methods is that they are designed to solve the full optimization problem directly, rather than accelerating a descent method by an efficient multigrid solver for the equations involved. The methods use the adjoint state in order to achieve efficient smoother and a robust coarsening strategy. The main idea is the treatment of the control variables on appropriate scales, i.e., control variables that correspond to smooth functions are solved for on coarse grids depending on the smoothness of these functions. Solution of the control problems is achieved with the cost of solving the constraint equations about two to three times (by a multigrid solver). Numerical examples demonstrate the effectiveness of the method proposed in distributed control case, pointwise control and boundary control problems.
An extension of classical methods of optimal control theory for systems described by ordinary differential equations to distributed-parameter systems described by partial differential equations is presented. An application is given involving the minimum-mass design of a simply-supported shear plate with a fixed fundamental frequency of vibration. An optimal plate thickness distribution in analytical form is found. The case of a minimum-mass design of an elastic sandwich plate whose fundamental frequency of free vibration is fixed. Under the most general conditions, the optimization problem reduces to the solution of two simultaneous partial differential equations involving the optimal thickness distribution and the modal displacement. One equation is the uniform energy distribution expression which was found by Ashley and McIntosh for the optimal design of one-dimensional structures with frequency constraints, and by Prager and Taylor for various design criteria in one and two dimensions. The second equation requires dynamic equilibrium at the preassigned vibration frequency.
This paper presents an analytical solution to the Riccati equation for self-adjoint systems such as beams, plates, strings and membranes moving in space, and shows how the optimal control law can be implemented using the given solution. It is then shown that there always exists a self-adjoint operator describing the distribution of potential energy if the state space is appropriately augmented. A beam-like gravity-stabilized satellite moving in a circular orbit around the earth is used to illustrate the main results in this paper.
The problem of characterizing optimal controls for a class of distributed-parameter systems is considered. The system dynamics are characterized mathematically by a finite number of coupled partial differential equations involving first-order time and space derivatives of the state variables, which are constrained at the boundary by a finite number of algebraic relations. Multiple control inputs, extending over the entire spatial region occupied by the system ("distributed controls') are to be designed so that the response of the system is optimal. A major example involving boundary control of an unstable low-density plasma is developed from physical laws.
The vibration controller design for a thin flexible surface with a given boundary is formulated as an unconstrained optimization problem. It is shown that this formulation is equivalent to a system of uncoupled optimization problems each of which is a linear regulator design problem for a second-order modal system. The regulator solutions give the gains on the modal amplitude and velocity associated with each modal system. For the constrained control case corresponding to a finite number of actuators, an approximation to the optimal distributed control is given. For the case of a limited number of physical sensors, a dual approach yields a system of second-order observers or filters. Simulations indicate that the constrained controller has a satisfactory performance as long as the number of actuators is more than half the number of controlled modes.
The SUbsonic Single Aft eNgine (SUSAN) Electrofan is a NASA concept transport aircraft representative of technology anticipated for a 2040 entry-into-service date. The powertrain consists of a single thrust-producing geared turbofan engine with generators driving a series/parallel partial hybrid power/propulsion system. The architecture includes 16 underwing contrarotating fans, eight on each side. The distributed fans can be used by the flight control system to augment or replace the rudder function. This paper sets up the optimal control problem of setpoint determination for individual wingfans in the distributed propulsion system, accounting for electrical string efficiencies, saturations, and failures. The solution minimizes power consumption while maintaining thrust and torque on the airframe for maneuvering. Additionally, thrust that would have been lost due to temporary fan speed or power saturation is optimally redistributed to maintain overall desired thrust and torque on the aircraft. A simulation of a coordinated turn utilizing the distributed electric propulsion for yaw rate control in a multiple wingfan failure scenario demonstrates the robustness of the powertrain design to failures and helps define its limitations.
The SUbsonic Single Aft eNgine (SUSAN) Electrofan is a NASA concept transport aircraft representative of technology anticipated for a 2040 entry-into-service date. The powertrain consists of a single thrust-producing geared turbofan engine with generators driving a series/parallel partial hybrid power/propulsion system. The architecture includes 16 underwing contrarotating fans, eight on each side. The distributed fans can be used by the flight control system to augment or replace the rudder function. This paper sets up the optimal control problem of setpoint determination for individual wingfans in the distributed propulsion system, accounting for electrical string efficiencies, saturations, and failures. The solution minimizes power consumption while maintaining thrust and torque on the airframe for maneuvering. Additionally, thrust that would have been lost due to temporary fan speed or power saturation is optimally redistributed to maintain overall desired thrust and torque on the aircraft. A simulation of a coordinated turn utilizing the distributed electric propulsion for yaw rate control in a multiple wingfan failure scenario demonstrates the robustness of the powertrain design to failures and helps define its limitations.
Parabolic partial differential equations for optimal pointwise feedback control of distributed parameter systems
The SUbsonic Single Aft eNgine (SUSAN) Electrofan is a NASA concept jet transport aircraft with a 2040 entry-into-service date. It utilizes electrified aircraft propulsion (EAP) to enable propulsive and aerodynamic benefits to reduce fuel usage, emissions, and cost. The powertrain consists of a single thrust producing, boundary layer-ingesting (BLI) turbofan gas turbine engine (GTE) with generators driving a series/parallel partial hybrid EAP system. The architecture includes 16 underwing contrarotating BLI fans, eight on each side, in a mailslot configuration. The 16 fans run on power extracted from the GTE through four 5 MW motor/generators connected to the Low-Pressure Spool, and a single 1 MW motor/generator on the High-Pressure Spool. The distributed fans can be used by the flight control to augment or replace the rudder function. At top of climb, the power extracted from the GTE for the fans is boosted by batteries. The design provides redundancy, and the capacity for boost means that the fans are designed to be able to provide additional thrust when necessary. These features can be leveraged in case of a fan or generator failure. This paper sets up the optimal control problem of setpoint determination for individual fans in the distributed propulsion system, accounting for electrical string efficiencies, saturations, and failures. The solution minimizes power consumption while maintaining thrust and torque on the airframe for maneuvering. Additionally, thrust that would have been lost due to temporary fan speed or power saturation is optimally redistributed to maintain overall desired thrust and torque on the aircraft. The power extraction range constraints derive from the gas turbine engine design and the small amount of variation allowed for the engine to maintain operability. The problem formulation allows the number and location of fan failures for which the thrust and torque can be maintained to be investigated, which has implications for certification. Simulations of a coordinated turn utilizing the distributed electric propulsion for yaw rate control under different failure scenarios demonstrate the robustness of the powertrain design to failures and help define its limitations.
The behavior of hyperbolic and parabolic partial differential equations is contrasted by studying the point-to-point time-optimal control problem for the equation of heat conduction and the equation of motion of a vibrating string. A maximal principle is obtained for the time-optimal control of the one-dimensional heat equation, and it is proven that time optimal controls are weakly bang-bang. The bang-bang principle is proven to be invalid for hyperbolic equations because of the finite speed of wave propagation. In the case of boundary value control of the vibrating spring, the latter is demonstrated by deriving an explicit formula for the time optimal control.
Control moment gyro /CMG/ for spacecraft attitude control, determining optimal gimbal angle rate for desired torque
This paper describes a thrust allocation scheme that minimizes power consumption in an electrified powertrain with Distributed Electric Propulsion. It takes advantage of an observation about component efficiency maps, that efficiency is often highest at high torque/high speed conditions. This optimal approach is demonstrated to satisfy total thrust and net yaw axis torque requirements, making it suitable for utilizing differential thrust for maneuvering.
This paper describes a thrust allocation scheme that minimizes power consumption in an electrified powertrain with Distributed Electric Propulsion. It takes advantage of an observation about component efficiency maps, that efficiency is often highest at high torque/high speed conditions. This optimal approach is demonstrated to satisfy total thrust and net yaw axis torque requirements, making it suitable for utilizing differential thrust for maneuvering.