High precision control in PTP trajectory planning for nonlinear systems using on high‐degree polynomial and cuckoo search

In this paper, in order to fulfill requirements for high precision control, high‐degree polynomial and cuckoo search algorithms are used in point‐to‐point (PTP) trajectory planning for nonlinear systems. The high‐degree polynomial describes the PTP trajectory and the cuckoo search is applied to obtain the coefficients of the high‐degree polynomial with constraints. This method is verified in the hydraulic toggle mechanism. Hamilton's principle and a Lagrange multiplier help establish the dynamic and kinematic equations in the hydraulic toggle mechanism. Position accuracy and the convergence effect of different degrees of polynomials are then compared through experiment. Specifically, this paper discusses the error of the 8‐degree polynomial trajectory between the optimization and practical values and the repeated accuracy of the optimization trajectory, which verify the method's high precision control in regard to the PTP trajectory planning for nonlinear system proposed in this paper. This method could be used in any type of PTP trajectory planning for nonlinear system.     

Optimal control of earth pressure balance of shield tunneling machine based on dual-heuristic dynamic programming

Earth pressure balance (EPB) shield tunneling machine has been widely used in underground construction. To avoid the catastrophic accidents caused by earth pressure imbalance, the earth pressure on excavation face must be controlled balance to that in chamber. To solve this problem better, a multi-variable data-driven optimal control method for shield machine based on dual-heuristic programming (DHP) is proposed. The DHP controller is constructed with action network, model network, and critic network based on back-propagation neural networks (BPNNs). Following Bellman's principle of optimality, a cost function of DHP controller for the chamber's earth pressure is presented, which simplifies a multi-level optimization to a single-level optimization. To minimize the cost function, the action network utilizes the critic network's error to achieve multi-variable optimization, and the optimal control parameters for the tunneling process are obtained at last. The simulation results show that the method can effectively control the earth pressure balance. Even in case of disturbance, the system has strong anti-interference ability and the control process is also quicker and steadier.     

A BMI approach to guaranteed cost control of discrete‐time uncertain system with both state and input delays

In this study, the guaranteed cost control of discrete time uncertain system with both state and input delays is considered. Sufficient conditions for the existence of a memoryless state feedback guaranteed cost control law are given in the bilinear matrix inequality form, which needs much less auxiliary matrix variables and storage space. Furthermore, the design of guaranteed cost controller is reformulated as an optimization problem with a linear objective function, bilinear, and linear matrix inequalities constraints. A nonlinear semi‐definite optimization solver—PENLAB is used as a solution technique. A numerical example is given to demonstrate the effectiveness of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.     

Iterative model and trajectory refinement for orbital trajectory optimization

An iterative model and trajectory refinement (IMTR) strategy is proposed for trajectory optimization of nonlinear systems. A high‐ and a low‐fidelity models are used. The high‐fidelity model accurately represents the system but is not easily amenable to trajectory optimization, because of degree of nonlinearity, computational cost, or to being of “black‐box” type. The low‐fidelity model is suitable for numerical optimization but approximates the system dynamics with an error. The IMTR method is proposed to systematically iterate between the 2 models and efficiently converge on a control solution. Examples are drawn from orbital mechanics. The IMTR approach is compared to optimal nonlinear quadratic control using Pontryagin maximum principle. A convergence criterion for the IMTR iterations is established.     

A direct search approach to optimization for nonlinear model predictive control

Nonlinear model predictive control (NMPC) depends on performing a constrained nonlinear optimization, based on predictions of future system behavior, during a sampling interval to determine the control action to be applied to the system during the next time step. The difficulty in designing an optimization procedure to solve a constrained NMPC problem is due to the finite time horizon to which the predictive model is evaluated, the state and control actuator constraints, and sampling interval length. The resulting objective function, which is to be optimized is typically not differentiable. Although there are many commercial, shareware, and open‐source optimization packages available that can perform a nonlinear constrained optimization for most cases, there are NMPC implementations requiring embedded code or that must meet stringent timing requirements that preclude the use of off‐the‐shelf packages. In cases where the predictive model is known, such as aerodynamic or hydrodynamic systems, a direct‐search optimization algorithm can perform well enough in a real‐time environment. Direct search algorithms are simple to implement and can be made more efficient by applying differential geometric techniques to the search methodology. The typical smoothness of the equations of motion for vehicular systems allows the objective function's stationarities to be handled in a straight‐forward way. Copyright © 2013 John Wiley & Sons, Ltd.     

Quadratic and linear controls developing an optimal treatment for the use of BCG immunotherapy in superficial bladder cancer

Our prime objective of the study is to exhibit the advantage to introduce a quadratic control in place of linear control in a cost function to be minimized, and that is associated to an optimal control problem that we formulate for a pre‐validated model of bacillus Calmette‐Guérin (BCG) immunotherapy in superficial bladder cancer. The compartmental model of interest is in the form of a nonlinear system of four ordinary differential equations that describe interactions between the used BCG strain, tumor cells, and immune responses. Previous studies reported that the optimal dose of BCG for treating bladder cancer is yet unknown. Hence, we aim to establish the optimization approach that can be applied for determining the values of the optimal BCG concentration along the therapy period to stimulate immune‐system cells and reduce cancer cells growth during BCG intravesical therapy. Pontryagin's maximum principle and the generalized Legendre–Clebsch condition are employed to provide the explicit formulations of the sought optimal controls. The optimality system is resolved numerically based on a fourth‐order iterative Runge–Kutta progressive‐regressive scheme, which is used to solve a two‐point boundary value problem. Copyright © 2015 John Wiley & Sons, Ltd.     

Nonlinear optimal control of washing machine based on approximate solution of HJB equation

Performance of a washing machine depends on different parameters, namely water temperature, degree of soiling, water volume, and motor speed. In this paper, firstly, a nonlinear model is derived based on the empirical data. In addition, validity of the proposed model is also verified. Secondly, to optimize the performance, a quadratic energy function is considered, which is minimized subject to the nonlinear model of the Washing Machine. Finding the corresponding control law requires solving a partial differential equation called Hamiltonian–Jacobi–Bellman (HJB) equation. An approximate solution for HJB equation using the second‐, third‐, and fourth‐order terms of Taylor's series expansion is utilized. Simulation results reveal that the higher‐order controller leads not only to a lower cost function but also to a better performance. Copyright © 2007 John Wiley & Sons, Ltd.     

Stability analysis and observer design for discrete‐time systems with interval time‐varying delay

The problem of observer design for a class of nonlinear discrete‐time systems with time‐varying delay is considered. A new approach of nonlinear observer design is proposed for the class of systems. By using Lyapunov's stability theory and Schur complement lemma, the novel sufficient conditions that guarantee the observer error converge asymptotically to zero are obtained in terms of linear matrix inequalities. The computing method for observer gain matrix is given. Finally, simulation examples are provided to demonstrate the effectiveness of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

H∞ model reduction for T‐S fuzzy systems over finite frequency ranges

This paper investigates the H_∞ model reduction problem over finite frequency ranges for continuous time Takagi‐Sugeno (T‐S) fuzzy systems. Given an asymptotically stable system, our aim is to find a stable reduced‐order system in such a way that the error of the transfer function between the original system and the reduced‐order one is bounded over a finite frequency range. By using the Finsler's lemma, new sufficient conditions in terms of linear matrix inequalities are derived in different frequency ranges. These conditions provide extra degrees of freedom in the optimization of the H_∞ performance when the frequency range of noises is known beforehand. Finally, we demonstrate via numerical examples that our method can achieve much smaller approximation error than existing approaches.     

Optimal design of trajectory parameters and position tracking with balance for riderless bicycle

Riderless bicycles are typically nonholonomic, underactuated, and nonminimum‐phase systems. The instability and complex dynamic coupling make the trajectory generation and tracking of the bicycles more challenging. In this paper, we consider both the trajectory generation and position tracking of a riderless bicycle. To achieve smooth motion performances, the desired planar trajectory of the contact point of the bicycle's rear wheel is constructed using a parameterized polynomial curve that can connect two given endpoints with associated tangent angles. The optimal parameters of the polynomial curve are obtained by minimizing the maximum of the roll angle's quasistatic trajectory of the bicycle, and this problem is solved by the particle swarm optimization algorithm. Then, position tracking of the desired planar trajectory with balance is converted into an optimization problem subject to the dynamic constraints. The cost function is designed as the combination of the position errors and the roll acceleration of the bicycle, in order to achieve an accurate tracking performance and to prevent the bicycle from falling down. This optimization problem is solved by the Gauss pseudospectral method. Simulation results are presented to demonstrate the effectiveness of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.     

Globalized and bounded Nelder‐Mead algorithm with deterministic restarts for tuning controller parameters: Method and application

This paper develops and examines an optimization algorithm for simulation‐based tuning of controller parameters. The proposed algorithm globalizes the Guin augmented variant of Nelder–Mead's nonlinear downhill simplex by deterministic restarts, linearly growing memory vector, and moving initial simplex. First, the effectiveness of the algorithm is tested using 10 complex and multimodal optimization benchmarks. The algorithm achieves global minima of all benchmarks and compares favorably against the evolutionary, swarm, and other globalized local‐search multimodal optimization algorithms in probability of finding global minimum and numerical cost. Next, the proposed algorithm is applied for tuning sliding mode controller parameters for a servo pneumatic position control application. The experimental results reveal that the system with sliding mode controller parameters tuned using the proposed algorithm targeting smooth position control with maximum possible accuracy, performs as desired and eliminates the need of manual online tuning for desired performance. The results are also compared with the performance of the same servo pneumatic system with parameters tuned using manual online tuning in an earlier published work. The system with controller parameters tuned using the proposed algorithm shows improvement in accuracy by 28.9% in sinusoidal and 42.2% in multiple step polynomials tracking.     

A discrete-time optimal adaptive control law for a robot arm

The equations of motion describing a robot's dynamics are coupled and non-linear, making the design of an optimum controller difficult using classical techniques. In this work an explicit adaptive control law is proposed based on a discrete linear model for each link and on the minimization of a quadratic performance criterion for position error and total control effort. The system parameters are recursively estimated at each control step by use of least squares, with a typical sample time of 0.02 s. A computer simulation of the resulting scheme is performed to evaluate the controller. The simulation model, based on the first three links of an existing robot, includes detailed motor dynamics and treats the wrist assembly as a load mass. Simulated test paths requiring movement of the outer two links indicate that the controller adapts and that its behaviour is stable and convergent.     

Optimizing autocatalysis with uncertainty by derivative‐free estimators

A derivative‐free estimator was introduced to alleviate the drawbacks of the conventional Kalman filter when performing nonlinear analyses under different circumstances. In this work, the scaled Unscented Kalman Filter, Divided Difference Kalman filter, and Cubature Kalman filter (CKF) were selected to investigate the effectiveness of these filters in predicting the states of a complex semi‐batch reaction between propionic anhydride and 2‐butanol. The estimator's performance was evaluated under four different case studies, that is, under normal condition, under poor estimator initialization, under disturbances, and under parameter uncertainty. Results from the study show that CKF was the best option for an online dynamic optimization because of its highest degree of accuracy and stability under the normal and noisy conditions. Under normal condition, CKF yielded the lowest root mean square error of 0.61 × 10^?2. Under uncertain initial condition, disturbance and parameter uncertainty, the lowest error of 1.83 × 10^?2, 1.04 × 10^?2, and 0.81 × 10^?2 were obtained, respectively.     

Inverse optimal controller based on extended Kalman filter for discrete‐time nonlinear systems

In this study, we present an inverse optimal control approach based on extended Kalman filter (EKF) algorithm to solve the optimal control problem of discrete‐time affine nonlinear systems. The main aim of inverse optimal control is to circumvent the tedious task of solving the Hamilton‐Jacobi‐Bellman equation that results from the classical solution of a nonlinear optimal control problem. Here, the inverse optimal controller is based on defining an appropriate quadratic control Lyapunov function (CLF) where the parameters of this candidate CLF were estimated by adopting the EKF equations. The root mean square error of the system states is used as the observed error in the case of classical EKF algorithm application, whereas, here, the EKF tries to eliminate the same root mean square error defined over the parameters by generating a CLF matrix with appropriate elements. The performance and the applicability of the proposed scheme is illustrated through both simulations performed on a nonlinear system model and a real‐time laboratory experiment. Simulation study demonstrate the effectiveness of the proposed method in comparison with 2 other inverse control approaches. Finally, the proposed controller is implemented on a professional control board to stabilize a DC‐DC boost converter and minimize a meaningful cost function. The experimental results show the applicability and effectiveness of the proposed EKF‐based inverse optimal control even in real‐time control systems with a very short time constant.     

Hamiltonian two‐degrees‐of‐freedom control of chemical reactors

A novel unified approach to two‐degrees‐of‐freedom control is devised and applied to a classical chemical reactor model. The scheme is constructed from the optimal control point of view and along the lines of the Hamiltonian formalism for nonlinear processes. The proposed scheme optimizes both the feedforward and the feedback components of the control variable with respect to the same cost objective. The original Hamiltonian function governs the feedforward dynamics, and its derivatives are part of the gain for the feedback component. The optimal state trajectory is generated online, and is tracked by a combination of deterministic and stochastic optimal tools. The relevant numerical data to manipulate all stages come from a unique off‐line calculation, which provides design information for a whole family of related control problems. This is possible because a new set of PDEs (the variational equations) allow to recover the initial value of the costate variable, and the Hamilton equations can then be solved as an initial‐value problem. Perturbations from the optimal trajectory are abated through an optimal state estimator and a deterministic regulator with a generalized Riccati gain. Both gains are updated online, starting with initial values extracted from the solution to the variational equations. The control strategy is particularly useful in driving nonlinear processes from an equilibrium point to an arbitrary target in a finite‐horizon optimization context. Copyright © 2010 John Wiley & Sons, Ltd.     

Symmetries and analytical solutions of the Hamilton–Jacobi–Bellman equation for a class of optimal control problems

The main contribution of this paper is to identify explicit classes of locally controllable second‐order systems and optimization functionals for which optimal control problems can be solved analytically, assuming that a differentiable optimal cost‐to‐go function exists for such control problems. An additional contribution of the paper is to obtain a Lyapunov function for the same classes of systems. The paper addresses the Lie point symmetries of the Hamilton–Jacobi–Bellman (HJB) equation for optimal control of second‐order nonlinear control systems that are affine in a single input and for which the cost is quadratic in the input. It is shown that if there exists a dilation symmetry of the HJB equation for optimal control problems in this class, this symmetry can be used to obtain a solution. It is concluded that when the cost on the state preserves the dilation symmetry, solving the optimal control problem is reduced to finding the solution to a first‐order ordinary differential equation. For some cases where the cost on the state breaks the dilation symmetry, the paper also presents an alternative method to find analytical solutions of the HJB equation corresponding to additive control inputs. The relevance of the proposed methodologies is illustrated in several examples for which analytical solutions are found, including the Van der Pol oscillator and mass–spring systems. Furthermore, it is proved that in the well‐known case of a linear quadratic regulator, the quadratic cost is precisely the cost that preserves the dilation symmetry of the equation. Copyright © 2015 John Wiley & Sons, Ltd.     

Neural net control of nonlinear plants through state feedback

A heuristic design method for state feedback fixed (non‐adaptive) neural net controller in nonlinear plants is presented. The design method evolves as a natural extension of the optimal control strategies employed in linear systems. A multi‐layered feed‐forward neural network is used as the feedback controller. The controller is trained to directly minimize a suitable cost function comprised of the plant output, states and the input. The optimization is carried out using a gradient scheme that employs the recently developed concept of block partial derivatives. The applicability of the proposed design method is demonstrated through simulated examples. Simulation studies include a variety of optimal control problems in nonlinear plants such as: minimum energy and minimum fuel problems, state tracking, output servo with integrator, and unconstrained and constrained regulation. Copyright © 2000 John Wiley & Sons, Ltd.     

An approximate method for solving a class of nonlinear optimal control problems

This paper presents a novel computational approach to generate the suboptimal solutions for a class of nonlinear optimal control problems (OCP's) with a quadratic performance index. Our method is based on the one‐dimensional differential transform method (DTM) and new polynomials that are called DT's polynomials. This method simplifies the difficulties and massive computational work for calculating the differential transform of nonlinear function. The convergence of proposed method are discussed in detail. This method consists of a new modified version of the DTM together with a shooting method such as procedure, for solving the extreme conditions obtained from the Pontryagin's maximum principle. The results reveal that the proposed methods are very effective and simple. Comparisons are made between new DTM generated results, results from literature, and MATLAB bvp4c generated results, and good agreement is observed. Copyright © 2013 John Wiley & Sons, Ltd.     

Intensity‐based 2D‐3D registration for an ACL reconstruction navigation system

To improve the positioning accuracy of tunnels for anterior cruciate ligament (ACL) reconstruction, we proposed an intensity‐based 2D‐3D registration method for an ACL reconstruction navigation system. Methods for digitally reconstructed radiograph (DRR) generation, similarity measurement, and optimization are crucial to 2D‐3D registration. We evaluated the accuracy, success rate, and processing time of different methods: (a) ray‐casting and splating were compared for DRR generation; (b) normalized mutual information (NMI), Mattes mutual information (MMI), and Spearman's rank correlation coefficient (SRC) were assessed for similarity between registrations; and (c) gradient descent (GD) and downhill simplex (DS) were compared for optimization. The combination of splating, SRC, and GD provided the best composite performance and was applied in an augmented reality (AR) ACL reconstruction navigation system. The accuracy of the navigation system could fulfill the clinical needs of ACL reconstruction, with an end pose error of 2.50 mm and an angle error of 2.74°.