Nonlinear optimal control for triangular tethered multi-satellite formations

Transportation of heavy loads is often performed by multi-axle multi-steered heavy duty vehicles In this article a novel nonlinear optimal control method is applied to the kinematic model of the five-axle and three-steering coupled vehicle system. First, it is proven that the dynamic model of this articulated multi-vehicle system is differentially flat. Next. the state-space model of the five-axle and three-steering vehicle system undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization is based on Taylor series expansion and on the associated Jacobian matrices. For the linearized state-space model of the five-axle and three-steering vehicle system a stabilizing optimal (H-infinity) feedback controller is designed. This controller stands for the solution of the nonlinear optimal control problem under model uncertainty and external perturbations. To compute the controller’s feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The stability properties of the control method are proven through Lyapunov analysis. The proposed nonlinear optimal control approach achieves fast and accurate tracking of setpoints under moderate variations of the control inputs and minimal dispersion of energy by the propulsion and steering system of the five-axle and three-steering vehicle system.

Nonlinear optimal control for a gas compressor driven by an induction motor

Nonlinear optimal control of the acute inflammatory response

Control and stabilization of parallel robotic manipulators is a non-trivial problem because of nonlinearities and the multi-variable structure. In this article, a nonlinear optimal control approach is proposed for the dynamic model of such robotic systems, using as a case-study the model of a five-link parallel robot. The dynamic model of the parallel robotic manipulator undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization relies on Taylor series expansion and on the associated Jacobian matrices. For the linearized state-space model of the system, a stabilizing optimal (H-infinity) feedback controller is designed. This controller stands as a solution to the nonlinear optimal control problem under model uncertainty and external perturbations. To compute the controller’s feedback gains, an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The stability properties of the control method are proven through Lyapunov analysis. Finally, to implement state estimation-based control without the need to measure the entire state vector of the parallel robotic manipulator, the H-infinity Kalman filter is used as a robust state estimator. Among the advantages of this control method, one can note that (i) unlike the popular computed-torque method for robotic manipulators, the new control approach is characterized by optimality and is also applicable when the number of control inputs is not equal to the robot’s number of DOFs and (ii) it achieves fast and accurate tracking of reference set points under minimal energy consumption by the robot’s actuators.

Nonlinear optimal and multi-loop flatness-based control of induction motor-driven desalination units

A new nonlinear optimal control method is proposed for solving the problem of control and stabilization of a two-link robotic manipulator where the first link is rigid and the second link is flexible. The dynamic model of this robot describes both the rotational motion of the two links and the vibration modes which are excited at the flexible link. The control and stabilization problem of this 2-DOF rigid-flexible robot is nontrivial due to complex nonlinear dynamics and underactuation. To apply the proposed nonlinear optimal control method, the dynamic model of the two-link rigid-flexible robotic manipulator undergoes first approximate linearization around a temporary operating point that is updated at each iteration of the control algorithm. The linearization takes place through first-order Taylor series expansion and through the computation of the Jacobian matrices of the flexible manipulator’s state-space description. For the approximately linearized model of the two-link rigid-flexible robot, an H-infinity feedback controller is designed. Actually, the H-infinity controller stands for the solution of the optimal control problem for the rigid-flexible robot under uncertainty and external perturbations. For the computation of the feedback gains of the H-infinity controller, an algebraic Riccati equation is solved at each time-step of the control method. The stability properties of the control algorithm are proven through Lyapunov analysis. First, it is shown that the control scheme achieves H-infinity tracking performance which signifies robustness for the control loop of the two-link rigid-flexible robot under uncertainties and external perturbations. Next, it is also shown that the control loop of the 2-DOF rigid-flexible robot is globally asymptotically stable. The proposed control method achieves fast and accurate tracking of setpoints under moderate variations of the control inputs. The concept of the proposed nonlinear optimal control method has been to approximate locally and at each sampling interval the nonlinear dynamics of the rigid-flexible robotic manipulator by an equivalent linear state-space description and to solve the H-infinity control problem for the linearized model. Finally, the control inputs are applied to the initial nonlinear dynamics and stability proof is given for the initial nonlinear state-space model.

Tower cranes find wide use in construction works, in ports and in several loading and unloading procedures met in industry. A nonlinear optimal control approach is proposed for the dynamic model of the 4-DOF underactuated tower crane. The dynamic model of the robotic crane undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization relies on Taylor series expansion and on the associated Jacobian matrices. For the linearized state-space model of the system a stabilizing optimal (H-infinity) feedback controller is designed. To compute the controller’s feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The stability properties of the control method are proven through Lyapunov analysis. The proposed control approach is advantageous because: (i) unlike the popular computed torque method for robotic manipulators, the new control approach is characterized by optimality and is also applicable when the number of control inputs is not equal to the robot’s number of DOFs, (ii) it achieves fast and accurate tracking of reference setpoints under minimal energy consumption by the robot’s actuators, (iii) unlike the popular Nonlinear Model Predictive Control method, the article’s nonlinear optimal control scheme is of proven global stability and convergence to the optimum.

This paper proposes a nonlinear optimal control approach for mulitple degrees of freedom (DOF) brachiation robots, which are often used in inspection and maintenance tasks of the electric power grid. Because of the nonlinear and multivariable structure of the related state-space model, as well as because of underactuation, the control problem of these robots is nontrivial. The dynamic model of the brachiation robots undergoes first approximate linearization with the use of Taylor series expansion around a temporary operating point which is recomputed at each iteration of the control method. For the approximately linearized model, an H-infinity feedback controller is designed. The linearization procedure relies on the Jacobian matrices of the brachiation robots’ state-space model. The proposed control method stands for the solution of the optimal control problem for the nonlinear and multivariable dynamics of the brachiation robots, under model uncertainties and external perturbations. For the computation of the controller’s feedback gains an algebraic Riccati equation is solved at each time-step of the control method. The global stability properties of the control scheme are proven through Lyapunov analysis. The new nonlinear optimal control approach achieves fast and accurate tracking for all state variables of the brachiation robots, under moderate variations of the control inputs.

In this article, the control problem for omnidirectional 3-wheel autonomous mobile robots is solved with the use of (i) a nonlinear optimal control method (ii) a flatness-based control approach which is implemented in successive loops. To apply method (i) that is nonlinear optimal control, the dynamic model of the omnidirectional 3-wheel autonomous mobile robots undergoes approximate linearization at each sampling instant with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrix. The linearization point is defined by the present value of the system's state vector and by the last sampled value of the control inputs vector. To compute the feedback gains of the optimal controller an algebraic Riccati equation is repetitively solved at each time-step of the control algorithm. The global stability properties of the nonlinear optimal control method are proven through Lyapunov analysis. To implement control method (ii), that is flatness-based control in successive loops, the state-space model of the omnidirectional 3-wheel autonomous mobile robot is separated into chained subsystems, which are connected in cascading loops. Each one of these subsystems can be viewed independently as a differentially flat system and control about it can be performed with inversion of its dynamics as in the case of input-output linearized flat systems. The state variables of the preceding (i-th) subsystem become virtual control inputs for the subsequent (i+1-th) subsystem. In turn, exogenous control inputs are applied to the last subsystem. The whole control method is implemented in successive loops and its global stability properties are also proven through Lyapunov stability analysis. The proposed method achieves trajectory tracking and autonomous navigation for the omnidirectional 3-wheel autonomous mobile robots without the need of diffeomorphisms and complicated state-space model transformations.

The safety of driving two-wheel vehicles, such as motorcycles, can be significantly improved through electronic control of the their stability properties. This will also allow for precise path following and capability for dexterous maneuvering. In this article, a nonlinear optimal control method is developed for solving the stabilization and path following problem of autonomous motorcycles. The joint kinematic and dynamic model of the motorcycle undergoes approximate linearization around a temporary operating point which is recomputed at each iteration of the control algorithm. The linearization takes place using Taylor series series expansion and the computation of the Jacobian matrices of the system's states-space model. For the approximately linearized model of the motorcycle an H-infinity feedback controller is designed. The computation of the feedback gain of the controller requires the repetitive solution of an algebraic Riccati equation, taking again place at each time-step of the control method. The concept of the control method is that at each time instant the system's state vector is made to converge to the temporary equilibrium, while this equilibrium is shifted towards the reference trajectory. Thus, asymptotically the state vector of the motorcycle converges to the reference setpoints. Through Lyapunov stability analysis the global asymptotic stability properties of the control method are proven.

A nonlinear optimal control method is developed for autonomous truck and trailer systems. Actually, two cases are distinguished: (a) a truck and trailer system that is steered by the front wheels of its truck, (b) an autonomous fire-truck robot that is steered by both the front wheels of its truck and by the rear wheels of its trailer. The kinematic model of the autonomous vehicles undergoes linearization through Taylor series expansion. The linearization is computed at a temporary operating point that is defined at each time instant by the present value of the state vector and the last value of the control inputs vector. The linearization is based on the computation of Jacobian matrices. The modeling error due to approximate linearization is considered to be a perturbation that is compensated by the robustness of the control scheme. For the approximately linearized model of the autonomous vehicles an H-infinity feedback controller is designed. This requires the solution of an algebraic Riccati equation at each iteration of the control algorithm. The stability of the control loop is confirmed through Lyapunov analysis. It is shown that the control loop exhibits the H-infinity tracking performance which implies elevated robustness against modeling errors and external disturbances. Moreover, under moderate conditions the global asymptotic stability of the control loop is proven. Finally, to implement state estimation-based control for the autonomous vehicles, through the processing of a small number of sensor measurements, the H-infinity Kalman Filter is proposed.

In this paper, the nonlinear optimal control problem is investigated for spacecraft formation flying with collision avoidance. Based on a nonlinear model of formation flying, two optimal tracking control laws are proposed to ensure the formation pattern converges to the predetermined configuration. The first optimal control law which combines Lyapunov optimizing control with a trajectory-following optimization technique is developed to solve the finite-time nonlinear optimal control problem. For the second controller, an extended [Formula: see text]-[Formula: see text] method is applied to design a closed-form feedback control scheme for nonlinear control problem of spacecraft formation flying with non-standard cost functions. By taking into account the flying safety, a repulsive control scheme is incorporated in the optimal controller to ensure collision avoidance and minimize the performance index. Finally, a numerical example is performed to demonstrate the effectiveness of the proposed approaches.

A EROSPACE engineering applications greatly stimulated the development of optimal control theory during the 1950s and 1960s, where the objective was to drive the system states in such a way that some defined cost was minimized. This turned out to have very useful applications in the design of regulators (where some steady state is to be maintained) and in tracking control strategies (where some predetermined state trajectory is to be followed). Among such applications was the problem of optimal flight trajectories for aircraft and space vehicles. Linear optimal control theory in particular has been very well documented and widely applied, where the plant that is controlled is assumed linear and the feedback controller is constrained to be linear with respect to its input. However, the availability of powerful low-cost microprocessors has spurred great advantages in the theory and applications of nonlinear control. The competitive era of rapid technological change, particularly in aerospace exploration, now demands stringent accuracy and cost requirements in nonlinear control systems. This has motivated the rapid development of nonlinear control theory for application to challenging, complex, dynamical real-world problems, particularly those that bear major practical significance in aerospace, marine, and defense industries. Infinite-time horizon nonlinear optimal control (ITHNOC) presents a viable option for synthesizing stabilizing controllers for nonlinear systems by making a state-input tradeoff, where the objective is to minimize the cost given by a performance index. The original theory of nonlinear optimal control dates from the 1960s. Various theoretical and practical aspects of the problem have been addressed in the literature over the decades since. In particular, the continuous-time nonlinear deterministic optimal control problem associated with autonomous (time-invariant) nonlinear regulator systems that are affine (linear) in the controls has been studied by many authors. The long-established theory of optimal control offers quite mature and well-documented techniques for solving this control-affine nonlinear optimization problem, based on dynamic programming or calculus of variations, but their application is generally a very tedious task. Bellman’s dynamic programming approach reduces to solving a nonlinear first-order partial differential equation (PDE), expressed by the Hamilton–Jacobi–Bellman (HJB) equation. The solution to the HJB equation gives the optimal performance/cost value (or storage) function and determines an optimal control in feedback form under some smoothness assumptions. Alternatively, in the classical calculus of variations, optimal control problems can be characterized locally in terms of the Hamiltonian dynamics arising from Pontryagin’s minimum principle. These are the characteristic equations of the HJB PDE, which result in a nonlinear, constrained two-point boundary value problem (TPBVP) that, in general, can only be solved by successive approximation of the optimal control input using iterative numerical techniques for each set of initial conditions. Numerically, even though the nonlinear TPBVP is somewhat easier to solve than the HJBPDE, control signals can only be determined offline and are thus best suited for feedforward control of plants for which the state trajectories are known a priori. Therefore, contrary to the dynamic programming approach, the resultant control law is not generally in feedback form. Open-loop control, however, is sensitive to random disturbances and requires that the initial state be on the optimal trajectory. In contrast, nonlinear optimal feedback has inherent robustness properties (inherent in the sense that it is obtained by ignoring uncertainty and disturbances). The potential difficulty with the HJB approach is that no efficient algorithm is available to solve the PDE when it is nonlinear and the problem dimension is high, making it impossible to derive exact expressions for optimal controls for most nontrivial problems of interest. The optimal can only be computed in special cases, such as linear dynamics and quadratic cost, or very low-dimensional systems. In particular, if the plant is linear time invariant (LTI) and the (infinite-time) performance index is quadratic, then the corresponding HJB equation for this infamous linear-quadratic regulator (LQR) problem reduces to an algebraic Riccati equation (ARE). Contrary to the well-developed and widely applied theory and computational tools for theRiccati equation (for example, see [1]), theHJB equation is difficult, if not impossible, to solve for most practical applications. The exact solution for the optimal control policies is very complex

In this article, we propose an adaptive robust nonlinear optimal sliding mode control (SMC) using the optimal homotopy asymptotic method (RNOSC‐OHAM) for maximizing wind power capture. Because of the unstable nature of the wind and the presence of uncertainties and disturbances in the structure of the wind turbine, the optimal controller cannot provide robustness against uncertainties. Hence, we integrated this approach with a proportional–integral–derivative SMC (PID‐SMC) scheme to ensure the robustness of the system. The presented hybrid control input signal involves two components: a nonlinear optimal control law that provides the optimal performance of the nominal system and an adaptive PID‐SMC law. The adaptive PID‐SMC law employs a PID sliding surface along with adaptive mechanisms to create the robustness of the nonlinear wind turbine system. The nonlinear optimal control policy is designed by addressing the partial differential Hamilton–Jacobi–Bellman (HJB) equation. This equation is approximated using the OHAM. This combination strategy reduces the final time of the optimal control problems, provides desired responses and improves performance. Additionally, the RNOSC‐OHAM controller facilitates the safe performance of the wind turbine under uncertainties and maximizes the wind power captured. To evaluate the performance of RNOSC‐OHAM, the results of the presented algorithm are compared with some existing control schemes. The results indicate that the designed RNOSC‐OHAM controller is very rapid and needs few iterations and computational costs. Indeed, the present control scheme exhibits the best characteristics and has the fastest transient response. The RNOSC‐OHAM controller offers a proper balance between enhancing aerodynamic power capture and minimizing low‐speed shaft fluctuations using a small control input.

Nonlinear optimal control problems are challenging to solve due to the prevalence of local minima that prevent convergence and/or optimality. This paper describes nearest-neighbors optimal control (NNOC), a data-driven framework for nonlinear optimal control using indirect methods. It determines initial guesses for new problems with the help of precomputed solutions to similar problems, retrieved using k-nearest neighbors. A sensitivity analysis technique is introduced to linearly approximate the variation of solutions between new and precomputed problems based on their variation of parameters. Experiments show that NNOC can obtain the global optimal solution orders of magnitude faster than standard random restart methods, and sensitivity analysis can further reduce the solving time almost by half. Examples are shown on two optimal control problems in vehicle control.