Nonlinear Quadratic Optimal Control Research Articles

State‐dependent indirect hp‐pseudospectral method for rigid spacecraft attitude optimal control problem

AbstractThe state‐dependent indirect ‐pseudospectral (SDIHP) method for rigid spacecraft attitude control is presented in this paper. The optimal controller is obtained by the SDIHP method through classifying the attitude optimal control problem as a nonlinear quadratic optimal control problem. In the proposed control scheme, successive state‐dependent coefficient (SDC) parameterization is initially applied to the spacecraft dynamics to derive an available linear system to decrease the calculation cost and speed. Then, the indirect ‐adaptive spectral discretization is developed to convert the resultant optimal control problem into a sequence of linear equations, which can improve the global computational efficiency and guarantee the solving precision. Hence, the optimal control can be efficiently solved using basic matrix operations under the necessary collocation points based on the ‐adaptive features. Considering the inertia uncertainties and external disturbances, a state feedback controller is generated by the receding horizon control (RHC) framework to improve the robustness. Simulation results are included to show the effectiveness and performance of the suggested optimal method.

Constrained nonlinear high-efficiency model predictive technics for test mass capture

This paper presents a nonlinear high-efficiency model predictive control (NHMPC) with constraints designed for the test mass (TM) capture phase of the drag-free satellite about gravitational wave observatory. To avoid collisions between test mass and satellite cavity, TM capture is the essential technology for drag-free satellite. Test mass is located inside an electrostatic suspension and locked by a clamp mechanism initially. The test masses are released with high initial offsets and velocities when the mechanism is retracted. The purpose of this phase is to guarantee the TM to be positioned at the cage center and attitude aligned with the local cage frame. Due to the low actuation authority of electrostatic suspension along with critical initial offsets and velocities, it is a challenging task to design the attitude and translation control schemes with simultaneous consideration of system performance and energy consumption. For the problem given above, the TM capture can be reformulated into a nonlinear quadratic optimal control problem with the state and input constraints. A nonlinear model predictive control (NMPC) structure is also designed to handle the noises by forming a closed loop. This control framework can realize optimality and robustness in a compromise. To improve the speed of solving high-dimensional nonlinear optimal control online for MPC, the indirect Chebyshev pseudospectral method with constraints is employed. The convergence and stability of the control loop are analyzed. Simulation examples show that the feasibility and performance of the designed control loop are verified compared with the existing methods and computation times in the millisecond range can be achieved.

Novel closed-loop controllers for fractional nonlinear quadratic systems

<abstract><p>A novel closed-loop optimal controller for fractional nonlinear quadratic optimal control problems is introduced. By using a new idea, the optimality conditions for the fractional nonlinear problems are derived. The linearized Riccati fractional order differential equation is derived and a new solution method is given for the first time, which can be applied to integer order nonlinear optimal control problems. The proposed closed-loop controller is applied to illustrative examples. Novel unprecedented processes of designing a variable linear controller and of finding the optimal performance index for integer order nonlinear systems are presented.</p></abstract>

Discrete LQR and ILQR methods based on high order Runge-Kutta discretizations

In this paper, discrete linear quadratic regulator (DLQR) and iterative linear quadratic regulator (ILQR) methods based on high-order Runge-Kutta (RK) discretizations are proposed for solving linear and nonlinear quadratic optimal control problems respectively. As discovered in Hager (2000) [11], direct approaches with RK discretization are equivalent with indirect approaches based on symplectic partitioned Runge-Kutta (SPRK) integration. In this paper, we will reconstruct this equivalence by the analogue of continuous and discrete dynamic programming. Then, based on the equivalence, we discuss the issue that the internal-stage controls produced by direct approaches may have lower order accuracy than the RK method used. We propose order conditions for internal-stage controls and then demonstrate that third or fourth order explicit RK methods cannot avoid the order reduction phenomenon. To overcome this obstacle, we calculate node control instead of internal-stage controls in DLQR and ILQR methods. Another advantage of our methods is high computational efficiency which comes from the usage of feedback technique. A simple numerical example will illustrate the validity of ILQR in solving nonlinear optimal control problems with high-order RK discretizations. In this paper, we also demonstrate that ILQR is essentially a quasi-Newton method with linear convergence rate.

Approximate Solution of Quadratic Time Varying Optimal Control Problems via Differential Transform Method

Because there is no general analytical method for solving the quadratic time varying optimal control problems (QTVOCPs), many authors adopt the numerical methods for this purpose. This motivates some authors to use the differential transform method (DTM) for finding an approximate solution for the linear quadratic optimal control problems (LQOCPs) and a class of nonlinear quadratic optimal control. In this paper, we propose an algorithm depend on applying DTM about the two end points of the time horizon to find an approximate solution for the linear or nonlinear QTVOCPs. To show the reliability, accuracy and efficiency of suggested method, some illustrated examples have been provided.

Indirect robust suboptimal control of two-satellite electromagnetic formation reconfiguration with geomagnetic effect

As a novel approach to control the relative motion of a satellite formation, electromagnetic formation flight (EMFF) has some prominent advantages, such as no propellant consumption and no plume contamination, and has a broad prospect of application in such fields as on-orbit detection and optical interferometry. The current paper investigates the optimal control for the reconfiguration of a two-satellite electromagnetic formation using the nonlinear quadratic optimal control technique. Specifically, the effects of the Earth’s magnetic field on the EMFF satellites are analyzed, and then the nonlinear translational dynamic model of a two-satellite electromagnetic formation is derived by utilizing the analytical mechanics theory. Considering the high nonlinearity and coupling in the dynamic model and the actuator saturation, a closed-loop robust suboptimal control strategy based on the indirect robust control scheme and the θ-D technique is proposed with robust stability and optimality. To ensure a further reduction of control input, the designed suboptimal controller is modified by applying the Tracking-Differentiator. The feasibility of the derived translational dynamics and proposed control strategy for the robust reconfiguration mission is validated through theoretical analysis and numerical simulations.

On optimal periodic solution of differential equations

this paper dedicated to the construction of solution of a three time scale periodic singular perturbed non-linear quadratic optimal control problem by using the direct method. The algorithm of the method is the direct substitution of the postulated asymptotic expansion of the solution of the problem and then by the conditions of the problem we constructed of a series of problem and find terms of the asymptotic. We find the solution by using the Hamilton's function and maximum principle

Theoretical study of the Nonlinear Quadratic Optimal Control implementation

Abstract In this brief, I study the finite and infinite nonlinear discrete time optimal control. The quadratic control problem for nonlinear case can be solved with different methods such as: linearization of the system model around each operation point or some different methods, where should be used an on-line parameter identification algorithm. In this paper, I study some properties of these algorithms in order to improve the control efficiency of the nonlinear process control. In this paper, I supposed the all states are accessible, so there is not necessary any state estimation algorithm for the implementation of the proposed optimal control (LQR - Linear Quadratic Regulators) methods.

Nonlinear Constrained Optimal Control Problems and Cardinal Hermite Interpolant Multiscaling Functions

AbstractIn this paper, a numerical method for solving nonlinear quadratic optimal control problems with inequality constraints is presented. The method is based upon cardinal Hermite interpolant multiscaling function approximation. The properties of these multiscaling functions are presented first. These properties are then utilized to reduce the solution of the nonlinear constrained optimal control to a nonlinear programming one, to which existing algorithms may be applied. Illustrative examples are included to demonstrate the efficiency and applicability of the technique.

Indirect robust control of agile missile via Theta-D technique

Abstract An agile missile with tail fins and pulse thrusters has continuous and discontinuous control inputs. This brings certain difficulty to the autopilot design and stability analysis. Indirect robust control via Theta- D technique is employed to handle this problem. An acceleration tracking system is formulated based on the nonlinear dynamics of agile missile. Considering the dynamics of actuators, there is an error between actual input and computed input. A robust control problem is formed by treating the error as input uncertainty. The robust control is equivalent to a nonlinear quadratic optimal control of the nominal system with a modified performance index including uncertainty bound. Theta- D technique is applied to solve the nonlinear optimal control problem to obtain the final control law. Numerical results show the effectiveness and robustness of the proposed strategy.

An Iterative Technique for Solving a Class of Nonlinear Quadratic Optimal Control Problems Using Chebyshev Polynomials

In this paper, a method for solving a class of nonlinear optimal control problems is presented. The method is based on replacing the dynamic nonlinear optimal control problem by a sequence of quadratic programming problems. To this end, the iterative technique developed by Banks is used to replace the original nonlinear dynamic system by a sequence of linear time-varying dynamic systems, then each of the new problems is converted to quadratic programming problem by parameterizing the state variables by a finite length Chebyshev series with unknown parameters. To show the effectiveness of the proposed method, simulation results of a nonlinear optimal control problem are presented.

Closed Form Solution of Nonlinear-Quadratic Optimal Control Problem by State-control Parameterization using Chebyshev Polynomials

In this paper the quasilinearization technique along with the Chebyshev polynomials of the first type are used to solve the nonlinear-quadratic optimal control problem with terminal state constraints. The quasilinearization is used to convert the nonlinear quadratic optimal control problem into sequence of linear quadratic optimal control problems. Then by approximating the state and control variables using Chebyshev polynomials, the optimal control problem can be approximated by a sequence of quadratic programming problems. The paper presents a closed form solution that relates the parameters of each of the quadratic programming problems to the original problem parameters. To illustrate the numerical behavior of the proposed method, the solution of the Van der Pol oscillator problem with and without terminal state constraints is presented.

ANALYTIC-APPROXIMATE SOLUTION FOR A CLASS OF NONLINEAR OPTIMAL CONTROL PROBLEMS BY HOMOTOPY ANALYSIS METHOD

In this paper, we apply the homotopy analysis method (HAM) for nonlinear quadratic optimal control problems (OCP's). This method is employed to solve extreme conditions obtained from the Pontryagin's maximum principle (PMP). Applying the HAM, we in essence transfer a nonlinear two-point boundary value problem (TPBVP), into an infinite number of linear sub-problems, and then use the sum of the solutions of its first several sub-problems to approximate the exact solution. Note that such a kind of transformation does not need the existence of any small or large parameters in governing equation and initial/boundary conditions. The comparison of the HAM results with the Measure theory method, Modal series and collocation method solutions are made. Some illustrative examples are given to demonstrate the simplicity and efficiency of the proposed method.

Adaptive fully-discrete finite element methods for nonlinear quadratic parabolic boundary optimal control

The aim of this work is to study adaptive fully-discrete finite element methods for quadratic boundary optimal control problems governed by nonlinear parabolic equations. We derive a posteriori error estimates for the state and control approximation. Such estimates can be used to construct reliable adaptive finite element approximation for nonlinear quadratic parabolic boundary optimal control problems. Finally, we present a numerical example to show the theoretical results.

Solving A Class Of Nonlinear Optimal Control Problems By Differential Transformation Method

Based on the Differential Transformation Method (DTM), a solution procedure for solving a class of nonlinear quadratic optimal control problems is presented in this paper. The reason for selecting this solution procedure is the less computational cost in comparison with the ordinary solution methods of original problem. First, the problem is converted to a two-point boundary value problem then the new problem is transferred into a set of algebraic equations by applying the differential transformation properties. By presenting the algorithmic solution procedure, two numerical examples are given to demonstrate the simplicity and efficiency of the new method.

Solving a class of nonlinear optimal control problems via he’s variational iteration method

This paper presents an analytical approximate solution for a class of nonlinear quadratic optimal control problems. The proposed method consists of a Variational Iteration Method (VIM) together with a shooting method like procedure, for solving the extreme conditions obtained from the Pontryagin’s Maximum Principle (PMP). This method is applicable for a large class of nonlinear quadratic optimal control problems. In order to use the proposed method, a control design algorithm with low computational complexity is presented. Through the finite iterations of algorithm, a suboptimal control law is obtained for the nonlinear optimal control problem. Two illustrative examples are given to demonstrate the simplicity and efficiency of the proposed method.

A priori error analysis of mixed methods for nonlinear quadratic optimal control problems

In this paper, we study an a priori error analysis for the quadratic optimal control problems governed by nonlinear elliptic partial differential equations using mixed finite element methods. The state and the costate are approximated by the lowest order Raviart-Thomas mixed finite element spaces, and the control is approximated by piecewise constant functions. A priori error estimates for the mixed finite element approximation of nonlinear optimal control problems are obtained. Some numerical examples are presented to confirm our theoretical results.

On infinite-time nonlinear quadratic optimal control

This work presents an approximate solution method for the infinite-time nonlinear quadratic optimal control problem. The method is applicable to a large class of nonlinear systems and involves solving a Riccati equation and a series of algebraic equations. The conditions for uniqueness and stability of the resulting feedback policy are established. It is shown that the proposed approximation method is useful in determining the region in which the constrained and unconstrained optimal control problems are identical. A reactor control problem is used to illustrate the method.

Discrete-time optimal fuzzy controller design: global concept approach

Proposes a systematic and theoretically sound way to design a global optimal discrete-time fuzzy controller to control and stabilize a nonlinear discrete-time fuzzy system with finite or infinite horizon (time). A linear-like global system representation of a discrete-time fuzzy system is first proposed by viewing such a system in a global concept and unifying the individual matrices into synthetic matrices. Then, based on this kind of system representation, a discrete-time optimal fuzzy control law which can achieve a global minimum effect is developed theoretically. A nonlinear two-point boundary-value-problem (TPBVP) is derived as a necessary and sufficient condition for the nonlinear quadratic optimal control problem. To simplify the computation, a multi-stage decomposition of the optimization scheme is proposed, and then a segmental recursive Riccati-like equation is derived. Moreover, in the case of time-invariant fuzzy systems, we show that the optimal controller can be obtained by just solving discrete-time algebraic Riccati-like equations. Based on this, several fascinating characteristics of the resultant closed-loop fuzzy system can easily be elicited. The stability of the closed-loop fuzzy system can be ensured by the designed optimal fuzzy controller. The optimal closed-loop fuzzy system can not only be guaranteed to be exponentially stable, but also stabilized to any desired degree. Also, the total energy of system output is absolutely finite. Moreover, the resultant closed-loop fuzzy system possesses an infinite gain margin, i.e. its stability is guaranteed no matter how large the feedback gain becomes. An example is given to illustrate the proposed optimal fuzzy controller design approach and to demonstrate the proven stability properties.

A descriptor system approach to nonlinear singularly perturbed optimal control problem

This paper considers the infinite horizon nonlinear quadratic optimal control problem for a singularly perturbed system which is nonlinear in both, the slow and the fast variables. The relationship between this problem and the analogous one for a descriptor system is investigated. Parameter-independent controllers are constructed that solve the problem for the descriptor system and lead the full-order system to the near-optimal performance. Estimates on the closeness of the cost under near-optimal controllers to the optimal one are obtained.