The matrix minimum principle.
Pontryagin maximum principle restated to apply to systems described by matrix, optimizing performance of linear system
Athans, M.↗
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Analysis of Control Behavior in Eco-Driving Speed Optimization Using Pontryagin’s Minimum Principle
The energy efficiency of autonomous vehicles can be improved by selecting an optimized speed profile. Energy savings can be maximized by performing control optimization with knowledge of the powertrain characteristics and future driving conditions. Previous studies have shown that Pontryagin’s minimum principle (PMP) performs well in vehicle speed optimization problems. Building on the methods proposed in previous studies, the contribution of this study is to derive meaningful observations from the concepts and results of PMP to enhance the understanding of the control problem. In particular, the switching behavior of the control mode is analyzed with supportive variables, such as ξ and mv, which dictates the changes in the control modes. Additionally, the existence of the singular control is analyzed, which helps in understanding the cruise driving in the control problem. Finally, we obtain several solutions that satisfy various boundary conditions along with a map of the reachable states, and discuss the impact of cruise driving. This is helpful for designing practical control concepts for real-world applications based on this map. Previous studies have contributed significantly to this control problem; however, this study provides a better understanding of the issue and offers guidance and inspiration for future real-world applications based on these meaningful observations.
33 ADVANCED PROPULSION SYSTEMS↗
Energy-Optimal Vehicle Longitudinal Motion Control via Pontryagin’s Minimum Principle and Ultra-Local Model
Longitudinal vehicle motion control is essential for enhancing performance and optimizing a vehicle’s energy usage. However, it remains a challenging task due to the nonlinear and uncertain nature of vehicle dynamics, along with varying driving conditions. This paper presents a novel ultra-local optimal control approach based on Pontryagin’s Minimum Principle (PMP) that circumvents the need for detailed system identification by employing an ultra-local model. The control objective is to minimize the total energy consumption under boundary conditions while ensuring smooth traction force generation. The proposed approach is evaluated using a high-fidelity vehicle model in three representative scenarios: (i) nominal driving, (ii) a change in tire road friction coefficient (TRFC) from 0.5 to 0.65 and road slope from 0% to 5% during the maneuver, with target velocity unchanged, and (iii) a change in target velocity from 20 m/s to 0 m/s during the maneuver, while maintaining nominal TRFC and slope conditions. The simulation results demonstrate that the proposed method delivers robust performance, effectively balancing consumption and tracking accuracy in all tested scenarios.
Waleed khan, Muhammad [The University of Texas at ↗
Computation of fuel-optimal control via Newton's method - Theory.
Fixed time fuel optimal control of linear time invariant plant, using Pontryagin minimum principle and Newton method
Athans, M.↗
The matrix minimum principle
Extension of Pontryagin maximum principle and application of matrix minimum principle in solving simple optimal linear regulator problem
MATRIX ANALYSIS↗
A Component-Sizing Methodology for a Hybrid Electric Vehicle Using an Optimization Algorithm
Many leading companies in the automotive industry have been putting tremendous effort into developing new powertrains and technologies to make their products more energy efficient. Evaluating the fuel economy benefit of a new technology in specific powertrain systems is straightforward; and, in an early concept phase, obtaining a projection of energy efficiency benefits from new technologies is extremely useful. However, when carmakers consider new technology or powertrain configurations, they must deal with a trade-off problem involving factors such as energy efficiency and performance, because of the complexities of sizing a vehicle’s powertrain components, which directly affect its energy efficiency and dynamic performance. As powertrains of modern vehicles become more complicated, even more effort is required to design the size of each component. This study presents a component-sizing process based on the forward-looking vehicle simulator “Autonomie” and the optimization algorithm “POUNDERS”; the supervisory control strategy based on Pontryagin’s Minimum Principle (PMP) assures sufficient computational system efficiency. We tested the process by applying it to a single power-split hybrid electric vehicle to determine optimal values of gear ratios and each component size, where we defined the optimization problem as minimizing energy consumption when the vehicle’s dynamic performance is given as a performance constraint. The suggested sizing process will be helpful in determining optimal component sizes for vehicle powertrain to maximize fuel efficiency while dynamic performance is satisfied. Indeed, this process does not require the engineer’s intuition or rules based on heuristics required in the rule-based process.
33 ADVANCED PROPULSION SYSTEMS↗
Energy-Efficient Driving in Connected Corridors via Minimum Principle Control: Vehicle-in-the-Loop Experimental Verification in Mixed Fleets
Connected and automated vehicles (CAVs) can plan and actuate control that explicitly considers performance, system safety, and actuation constraints in a manner more efficient than their human-driven counterparts. In particular, eco-driving is enabled through connected exchange of information from signalized corridors that share their upcoming signal phase and timing (SPaT). This is accomplished in the proposed control approach, which follows first principles to plan a free-flow acceleration-optimal trajectory through green traffic light intervals by Pontryagin's Minimum Principle in a feedback manner. Urban conditions are then imposed from exogeneous traffic comprised of a mixture of human-driven vehicles (HVs) - as well as other CAVs. As such, safe disturbance compensation is achieved by implementing a model predictive controller (MPC) to anticipate and avoid collisions by issuing braking commands as necessary. The control strategy is experimentally vetted through vehicle-in-the-loop (VIL) of a prototype CAV that is embedded into a virtual traffic corridor realized through microsimulation. Up to 36% fuel savings are measured with the proposed control approach over a human-modelled driver, and it was found connectivity in the automation approach improved fuel economy by up to 26% over automation without. Additionally, the passive energy benefits realizable for human drivers when driving behind downstream CAVs are measured, showing up to 22% fuel savings in a HV when driving behind a small penetration of connectivity-enabled automated vehicles.
33 ADVANCED PROPULSION SYSTEMS↗
On the application of deterministic optimization methods to stochastic control problems.
A technique is presented by which one can apply the Minimum Principle of Pontryagin to stochastic optimal control problems formulated around linear systems with Gaussian noises and general cost criteria. Using this technique, the stochastic nature of the problem is suppressed but for two expectation operations, the optimization being essentially deterministic. The technique is applied to systems with quadratic and non-quadratic costs to illustrate its use.
Kramer, L. C.↗
Aircraft Range Optimization Using Singular Perturbations
An approximate analytic solution is developed for the problem of maximizing the range of an aircraft for a fixed end state. The problem is formulated as a singular perturbation and solved by matched inner and outer asymptotic expansions and the minimum principle of Pontryagin. Cruise in the stratosphere, and on transition to and from cruise at constant Mach number are discussed. The state vector includes altitude, flight path angle, and mass. Specific fuel consumption becomes a linear function of power approximating that of the cruise values. Cruise represents the outer solution; altitude and flight path angle are constants, and only mass changes. Transitions between cruise and the specified initial and final conditions correspond to the inner solutions. The mass is constant and altitude and velocity vary. A solution is developed which is valid for cruise but which is not for the initial and final conditions. Transforming of the independent variable near the initial and final conditions result in solutions which are valid for the two inner solutions but not for cruise. The inner solutions can not be obtained without simplifying the state equations. The singular perturbation approach overcomes this difficulty. A quadratic approximation of the state equations is made. The resulting problem is solved analytically, and the two inner solutions are matched to the outer solution.
Oconnor, Joseph Taffe↗
Range optimization for a supersonic aircraft
Range optimal trajectories for an aircraft flying in the vertical plane are obtained from Pontryagin's Minimum Principle. Control variables are load factor n which appears nonlinearly in the equations of motion and throttle setting eta, which appears only linearly. Both controls are subject to fixed bounds, namely eta between values of 0 and 1 and absolute value of n not greater than n(max). Additionally, a dynamic pressure limit is imposed, which represents a first-order state-inequality constraint. For fixed flight time, fixed initial coordinates, and partially fixed final coordinates, the effect of the load factor limit absolute value of n not greater than n(max) is studied. Upon varying n(max), six different switching structures are obtained. All trajectories involve singular control along arcs with active dynamic pressure limit.
Seywald, Hans↗
Investigation of optimal control with a minimum-fuel consumption criterion for a fourth-order plant with two control inputs - synthesis of an efficient suboptimal control.
Pontryagin optimal principle applied to minimum fuel satellite attitude control with two feedback controls
OPTIMAL CONTROL↗
Thrust vectoring effect on time-optimal 90 degrees angle of attack pitch up maneuvers of a high alpha fighter aircraft
Using thrust vectoring, the problem of pitching up a high alpha fighter aircraft to 90 degrees angle of attack in minimum time is considered. Pontryagin's maximum principle is used together with a two-point boundary value numerical algorithm to derive open-loop controls for various parameterized limits on thrust vectoring angles. Without thrust vectoring, a high alpha fighter with initial conditions 0.6 Mach number and 15,000 feet requires 7.0 seconds to pitch-up to 90 degrees. However, with 20-degree angle thrust vectoring, the pitch-up takes only 1.5 seconds with a final Mach number of 0.47. Also, the gain in altitude is less than 100 feet. More important, additional thrust vectoring yields litle benefit beyond the 20-degree angles. This paper describes control solutions in detail for thrust vectoring angles between 0 and 90 degrees.
Stalford, Harold↗
THE OPTIMAL CONTROL OF SOME ATTITUDE CONTROL SYSTEMS FOR DIFFERENT PERFORMANCE CRITERIA
Pontryagin maximum principle to optimize attitude control systems on the basis of minimum fuel or energy consumption
OPTIMAL CONTROL↗
A combined algorithm for minimum time slewing of flexible spacecraft
The use of Pontryagin's Maximum Principle for the large-angle slewing of large flexible structures usually results in the so-called two-point boundary-value problem (TPBVP), in which many requirements (e.g., minimum time, small flexible amplitude, and limited control powers, etc.) must be satisfied simultaneously. The successful solution of this problem depends largely on the use of an efficient numerical computational algorithm. There are many candidate algorithms available for his problem (e.g., quasilinearization, gradient, and shooting, etc.). In this paper, a proposed algorithm, which combines the quasilinearization method with a time shortening technique and a shooting method, is applied to the minimum-time, three-dimensional, and large-angle maneuver of flexible spacecraft, particularly the orbiting Spacecraft Control Laboratory Experiment (SCOLE) configuration. Theoretically, the nonlinear TPBVP can be solved only through the shooting method to find the 'exact' switching times for the bang-bang controls. However, computationally, a suitable guess for the missing initial costates is crucial because the convergence range of the unknown initial costates is usually narrow, especially for systems with high dimensions and when a multi-bang-bang control strategy is needed. On the other hand, the problems of near minimum time attitude maneuver of general rigid spacecraft and fast slewing of flexible spacecraft have been examined by the authors through a numerical approach based on the quasilinearization algorithm with a time shortening technique. Computational results have demonstrated its broad convergence range and insensitivity to initial costate choices. Consequently, a combined approach is naturally suggested here to solve the minimum time slewing problem. That is, in the computational process, the quasilinearization method is used first to obtain a near minimum time solution. Then, the acquired converged initial costates from the quasilinearization approach are transformed (tailored) to and used as the initial costate guess for starting the shooting method. Finally, the shooting method takes over the remaining calculations until the minimum-time solution converges. The nonlinear equations of motion of the SCOLE are formulated by using Lagrange's equations, with the mast modeled as a continuous beam subject to three-dimensional deformations. The numerical results will be presented and some related computational issues will also be discussed.
Bainum, P. M.↗
Minimum-fuel attitude control of a spacecraft by an extended method of steepest-descent.
Minimum fuel attitude control of spacecraft by Pontryagin principle and extended steepest descent method
Fluegge-Lotz, I.↗
The dynamics and control of the in-orbit SCOLE configuration
The study of the dynamics of the Spacecraft Control Laboratory Experiment (SCOLE) is extended to emphasize the synthesis of control laws for both the linearized system as well as the large amplitude slewing maneuvers required to rapidly reorient the antenna line of sight. For control of the system through small amplitude displacements from the nominal equilibrium position LQR techniques are used to develop the control laws. Pontryagin's maximum principle is applied to minimize the time required for the slewing of a general rigid spacecraft system. The minimum slewing time is calculated based on a quasi-linearization algorithm for the resulting two point boundary value problem. The effect of delay in the control input on the stability of a continuously acting controller (designed without considering the delay) is studied analytically for a second order plant. System instability can result even for delays which are only a small fraction of the natural period of motion.
Bainum, Peter M.↗
An improved shooting method for solving minimum-time maneuver problems
The minimum-time attitude maneuvers of a rigid spacecraft are considered. Generally, the associated nonsingular, nonlinear two-point boundary-value problem, derived by using Pontryagin's Maximum Principle, can be solved through shooting methods to find the switching times for the bang-bang control. However, a good initial guess for the missing initial costates is important because the convergence range of the initial costates is very narrow, especially when a multi-bang-bang control strategy is needed. This paper proposes a numerical approach to generate the initial costates so that they are very close to the converged values. The initial costates are first generated by using a recently developed quasi-linearization technique for the near-minimum-time problem and then reduced in magnitude, through some properly designed scales, before the shooting method is used. Numerical examples indicate the closeness of the results from these two techniques (quasi-linearization and shooting).
Li, Feiyue↗
The dynamics and control of large flexible space structures - 12, supplement 11
The rapid 2-D slewing and vibrational control of the unsymmetrical flexible SCOLE (Spacecraft Control Laboratory Experiment) with multi-bounded controls is considered. Pontryagin's Maximum Principle is applied to the nonlinear equations of the system to derive the necessary conditions for the optimal control. The resulting two point boundary value problem is then solved by using the quasilinearization technique, and the near minimum time is obtained by sequentially shortening the slewing time until the controls are near the bang-bang type. The tradeoff between the minimum time and the minimum flexible amplitude requirements is discussed. The numerical results show that the responses of the nonlinear system are significantly different from those of the linearized system for rapid slewing. The SCOLE station-keeping closed loop dynamics are re-examined by employing a slightly different method for developing the equations of motion in which higher order terms in the expressions for the mast modal shape functions are now included. A preliminary study on the effect of actuator mass on the closed loop dynamics of large space systems is conducted. A numerical example based on a coupled two-mass two-spring system illustrates the effect of changes caused in the mass and stiffness matrices on the closed loop system eigenvalues. In certain cases the need for redesigning control laws previously synthesized, but not accounting for actuator masses, is indicated.
Bainum, Peter M.↗