On the optimal control of nonlinear systems

The Lie algebra of tensors on a Hilbert space is used to obtain optimal controls for a class of nonlinear systems.     

An evolutionary approach to cancer chemotherapy scheduling

In this paper, we investigate the employment of evolutionary algorithms as a search mechanism in a decision support system for designing chemotherapy schedules. Chemotherapy involves using powerful anti-cancer drugs to help eliminate cancerous cells and cure the condition. It is given in cycles of treatment alternating with rest periods to allow the body to recover from toxic side-effects. The number and duration of these cycles would depend on many factors, and the oncologist would schedule a treatment for each patient’s condition. The design of a chemotherapy schedule can be formulated as an optimal control problem; using an underlying mathematical model of tumour growth (that considers interactions with the immune system and multiple applications of a cycle-phase-specific drug), the objective is to find effective drug schedules that help eradicate the tumour while maintaining the patient health’s above an acceptable level. A detailed study on the effects of different objective functions, in the quality and diversity of the solutions, was performed. A term that keeps at a minimum the tumour levels throughout the course of treatment was found to produce more regular treatments, at the expense of imposing a higher strain on the patient’s health, and reducing the diversity of the solutions. Moreover, when the number of cycles was incorporated in the problem encoding, and a parsimony pressure added to the objective function, shorter treatments were obtained than those initially found by trial and error.

Optimal robust control of drug delivery in cancer chemotherapy: A comparison between three control approaches

During the drug delivery process in chemotherapy, both of the cancer cells and normal healthy cells may be killed. In this paper, three mathematical cell-kill models including log-kill hypothesis, Norton–Simon hypothesis and E_max hypothesis are considered. Three control approaches including optimal linear regulation, nonlinear optimal control based on variation of extremals and H_∞-robust control based on μ-synthesis are developed. An appropriate cost function is defined such that the amount of required drug is minimized while the tumor volume is reduced. For the first time, performance of the system is investigated and compared for three control strategies; applied on three nonlinear models of the process. In additions, their efficiency is compared in the presence of model parametric uncertainties. It is observed that in the presence of model uncertainties, controller designed based on variation of extremals is more efficient than the linear regulation controller. However, H_∞-robust control is more efficient in improving robust performance of the uncertain models with faster tumor reduction and minimum drug usage.     

Resolving actuator redundancy—optimal control vs. control allocation

This paper considers actuator redundancy management for a class of overactuated nonlinear systems. Two tools for distributing the control effort among a redundant set of actuators are optimal control design and control allocation. In this paper, we investigate the relationship between these two design tools when the performance indexes are quadratic in the control input. We show that for a particular class of nonlinear systems, they give exactly the same design freedom in distributing the control effort among the actuators. Linear quadratic optimal control is contained as a special case. A benefit of using a separate control allocator is that actuator constraints can be considered, which is illustrated with a flight control example.     

Locally optimal controllers and globally inverse optimal controllers

In this paper we consider the problem of global asymptotic stabilization with prescribed local behavior. We show that this problem can be formulated in terms of control Lyapunov functions. Moreover, we show that if the local control law has been synthesized employing an LQ approach, then the associated Lyapunov function can be seen as the value function of an optimal problem with some specific local properties. We illustrate these results on two specific classes of systems: backstepping and feedforward systems. Finally, we show how this framework can be employed when considering the orbital transfer problem.     

A new approach to nonlinear optimal feedback law

For a fixed-time free endpoint optimal control problem it is shown that the optimal feedback control satisfies a system of ordinary differential equations. They are obtained using an elimination procedure of the adjoint vector which appears linearly in a set of differential equations. These equations, involving Lie brackets of vector fields, are derived from the Maximum Principle. An application of this approach to robotics is given.     

Stochastic optimal control of networked control systems with control packet dropouts

This paper considers the stochastic optimal control problem for networked control systems(NCSs)with control packet dropouts.The proportional plus up to the third-order derivative(PD3)compensation strategy is adopted to compensate for control packet dropouts at the actuator by using the past control packets stored in the buffer.Based on the strategy,a new NCS structure model with packet dropouts is provided,where the packet dropout is assumed to obey the Bernoulli random binary distribution.In terms of the given model,the stochastic optimal control law is proposed. Numerical examples illustrate the effectiveness of the results.     

Hybrid extended Fourier series for optimal control of nonlinear algebraic dynamical systems

The paper introduces a new method for finding optimal control of algebraic dynamic systems. The structure of algebraic dynamical systems is nonlinear with quadratic and bilinear terms. A new hybrid extended Fourier series is introduced, and state and control variables of the system are expanded by this series. Moreover, properties of new series are presented, and integration and product operational matrices are obtained. Using operational matrices, optimal control of the systems is converted to a set of simultaneous nonlinear algebraic relations. An illustrative example is included to compare our results with those in the literature.     

Nonlinear optimal tracking control with application to super-tankers for autopilot design

A new method is introduced to design optimal tracking controllers for a general class of nonlinear systems. A recently developed recursive approximation theory is applied to solve the nonlinear optimal tracking control problem explicitly by classical means. This reduces the nonlinear problem to a sequence of linear-quadratic and time-varying approximating problems which, under very mild conditions, globally converge in the limit to the nonlinear systems considered. The converged control input from the approximating sequence is then applied to the nonlinear system. The method is used to design an autopilot for the ESSO 190,000-dwt oil tanker. This multi-input-multi-output nonlinear super-tanker model is well established in the literature and represents a challenging problem for control design, where the design requirement is to follow a commanded maneuver at a desired speed. The performance index is selected so as to minimize: (a) the tracking error for a desired course heading, and (b) the rudder deflection angle to ensure that actuators operate within their operating limits. This will present a trade-off between accurate tracking and reduced actuator usage (fuel consumption) as they are both mutually dependent on each other. Simulations of the nonlinear super-tanker control model are conducted to illustrate the effectiveness of the nonlinear tracking controller.     

Minimizing control variation in nonlinear optimal control

In any real system, changing the control signal from one value to another will usually cause wear and tear on the system’s actuators. Thus, when designing a control law, it is important to consider not just predicted system performance, but also the cost associated with changing the control action. This latter cost is almost always ignored in the optimal control literature. In this paper, we consider a class of optimal control problems in which the variation of the control signal is explicitly penalized in the cost function. We develop an effective computational method, based on the control parameterization approach and a novel transformation procedure, for solving this class of optimal control problems. We then apply our method to three example problems in fisheries, train control, and chemical engineering.     

A bio-inspired pursuit strategy for optimal control with partially constrained final state

We discuss a biologically inspired cooperative control strategy which allows a group of autonomous systems to solve optimal control problems with free final time and partially constrained final state. The proposed strategy, termed “generalized sampled local pursuit” (GSLP), mimics the way in which ants optimize their foraging trails, and guides the group toward an optimal solution, starting from an initial feasible trajectory. Under GSLP, an optimal control problem is solved in many “short” segments, which are constructed by group members interacting locally with lower information, communication and storage requirements compared to when the problem is solved all at once. We include a series of simulations that illustrate our approach.     

Pseudospectral methods for solving infinite-horizon optimal control problems

An important aspect of numerically approximating the solution of an infinite-horizon optimal control problem is the manner in which the horizon is treated. Generally, an infinite-horizon optimal control problem is approximated with a finite-horizon problem. In such cases, regardless of the finite duration of the approximation, the final time lies an infinite duration from the actual horizon at t=+∞. In this paper we describe two new direct pseudospectral methods using Legendre–Gauss (LG) and Legendre–Gauss–Radau (LGR) collocation for solving infinite-horizon optimal control problems numerically. A smooth, strictly monotonic transformation is used to map the infinite time domain t∈[0,∞) onto a half-open interval τ∈[−1,1). The resulting problem on the finite interval is transcribed to a nonlinear programming problem using collocation. The proposed methods yield approximations to the state and the costate on the entire horizon, including approximations at t=+∞. These pseudospectral methods can be written equivalently in either a differential or an implicit integral form. In numerical experiments, the discrete solution exhibits exponential convergence as a function of the number of collocation points. It is shown that the map ?:[−1,+1)→[0,+∞) can be tuned to improve the quality of the discrete approximation.     

Optimal control problems with a continuous inequality constraint on the state and the control

We consider an optimal control problem with a nonlinear continuous inequality constraint. Both the state and the control are allowed to appear explicitly in this constraint. By discretizing the control space and applying a novel transformation, a corresponding class of semi-infinite programming problems is derived. A solution of each problem in this class furnishes a suboptimal control for the original problem. Furthermore, we show that such a solution can be computed efficiently using a penalty function method. On the basis of these two ideas, an algorithm that computes a sequence of suboptimal controls for the original problem is proposed. Our main result shows that the cost of these suboptimal controls converges to the minimum cost. For illustration, an example problem is solved.     

Direct scheme in optimal control problems with fast and slow motions

The nonlinear optimal control problems described by singularly perturbed models are studied in order to obtain the ways of decomposition and to construct the suboptimal control sequences. A new scheme, the so-called direct scheme, applying the Vasil'eva boundary layer functions method in control theory is introduced. It consists in directly expanding the problem's conditions into postulated asymptotic series and then in successively solving lower-dimensional approximate decomposition problems of optimal control. By means of the direct scheme a new property of an asymptotic expansion — the relaxation — is obtained, i.e. the decrease of the performance index with each new control approximation. Illustrating examples are given.     

An integrated state space partition and optimal control method of multi-model for nonlinear systems based on hybrid systems

Multilinear model approach turns out to be an ideal candidate for dealing with nonlinear systems control problem. However, how to identify the optimal active state subspace of each linear subsystem is an open problem due to that the closed-loop performance of nonlinear systems interacts with these subspaces ranges. In this paper, a new systematic method of integrated state space partition and optimal control of multi-model for nonlinear systems based on hybrid systems is initially proposed, which can deal with the state space partition and associated optimal control simultaneously and guarantee an overall performance of nonlinear systems consequently. The proposed method is based on the framework of hybrid systems which synthesizes the multilinear model, produced by nonlinear systems, in a unified criterion and poses a two-level structure. At the upper level, the active state subspace of each linear subsystem is determined under the optimal control index of a hybrid system over infinite horizon, which is executed off-line. At the low level, the optimal control is implemented online via solving the optimal control of hybrid system over finite horizon. The finite horizon optimal control problem is numerically computed by simultaneous method for speeding up computation. Meanwhile, the model mismatch produced by simultaneous method is avoided by using the strategy of receding-horizon. Simulations on CSTR (Continuous Stirred Tank Reactor) confirm that a superior performance can be obtained by using the presented method.     

The time-invariant linear-quadratic optimal control problem

This paper provides a review of one of the basic problems of systems theory—the general time-invariant optimal control problem involving linear systems and quadratic costs. The problem includes on one hand the regulator problem of optimal control and on the other, the theory of linear dissipative systems, itself central to network theory and to the stability theory of feedback systems. The theory is developed using simple properties of dynamical systems and involves a minimum of ‘hard’ analysis or algebra. It includes a full existence theory of the matrix quadratic equation, of interest in its own right.     

非线性系统的输入多采样率模糊优化控制

基于多采样率数字控制理论,讨论了非线性连续被控对象和输入多采样率模糊控制器的设计问题.提出用线性矩阵不等式凸优化技术构建非线性系统的输入多采样率T-S模糊模型,并相应地研究了基于优化区域极点配置的PDC状态反馈控制.通过解代数Riccati方程得到控制器的参数,给出了优化数字控制器的设计算法和闭环系统的稳定性条件.计算机仿真表明了所提出方法的有效性.     

Asymptotic optimal control of uncertain nonlinear Euler–Lagrange systems

A sufficient condition to solve an optimal control problem is to solve the Hamilton–Jacobi–Bellman (HJB) equation. However, finding a value function that satisfies the HJB equation for a nonlinear system is challenging. For an optimal control problem when a cost function is provided a priori, previous efforts have utilized feedback linearization methods which assume exact model knowledge, or have developed neural network (NN) approximations of the HJB value function. The result in this paper uses the implicit learning capabilities of the RISE control structure to learn the dynamics asymptotically. Specifically, a Lyapunov stability analysis is performed to show that the RISE feedback term asymptotically identifies the unknown dynamics, yielding semi-global asymptotic tracking. In addition, it is shown that the system converges to a state space system that has a quadratic performance index which has been optimized by an additional control element. An extension is included to illustrate how a NN can be combined with the previous results. Experimental results are given to demonstrate the proposed controllers.     

Efficient symmetric Hessian propagation for direct optimal control

Direct optimal control algorithms first discretize the continuous-time optimal control problem and then solve the resulting finite dimensional optimization problem. If Newton type optimization algorithms are used for solving the discretized problem, accurate first as well as second order sensitivity information needs to be computed. This article develops a novel approach for computing Hessian matrices which is tailored for optimal control. Algorithmic differentiation based schemes are proposed for both discrete- and continuous-time sensitivity propagation, including explicit as well as implicit systems of equations. The presented method exploits the symmetry of Hessian matrices, which typically results in a computational speedup of about factor 2 over standard differentiation techniques. These symmetric sensitivity equations additionally allow for a three-sweep propagation technique that can significantly reduce the memory requirements, by avoiding the need to store a trajectory of forward sensitivities. The performance of this symmetric sensitivity propagation is demonstrated for the benchmark case study of the economic optimal control of a nonlinear biochemical reactor, based on the open-source software implementation in the ACADO Toolkit.     

A unified framework for the numerical solution of optimal control problems using pseudospectral methods

A unified framework is presented for the numerical solution of optimal control problems using collocation at Legendre-Gauss (LG), Legendre-Gauss-Radau (LGR), and Legendre-Gauss-Lobatto (LGL) points. It is shown that the LG and LGR differentiation matrices are rectangular and full rank whereas the LGL differentiation matrix is square and singular. Consequently, the LG and LGR schemes can be expressed equivalently in either differential or integral form, while the LGL differential and integral forms are not equivalent. Transformations are developed that relate the Lagrange multipliers of the discrete nonlinear programming problem to the costates of the continuous optimal control problem. The LG and LGR discrete costate systems are full rank while the LGL discrete costate system is rank-deficient. The LGL costate approximation is found to have an error that oscillates about the true solution and this error is shown by example to be due to the null space in the LGL discrete costate system. An example is considered to assess the accuracy and features of each collocation scheme.