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.     

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.     

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.     

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.     

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.     

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.     

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.     

Suboptimal control for nonlinear systems: a successive approximation approach

This paper presents a successive approximation approach (SAA) designing optimal controllers for a class of nonlinear systems with a quadratic performance index. By using the SAA, the nonlinear optimal control problem is transformed into a sequence of nonhomogeneous linear two-point boundary value (TPBV) problems. The optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoint vector sequence. By using the finite-step iteration of the nonlinear compensation sequence, we can obtain a suboptimal control law. Simulation examples are employed to test the validity of the SAA.     

Unconstrained optimal control of regular languages

This paper formulates an unconstrained optimal policy for control of regular languages realized as deterministic finite state automata (DFSA). A signed real measure quantifies the behavior of controlled sublanguages based on a state transition cost matrix and a characteristic vector as reported in an earlier publication. The state-based optimal control policy is obtained by selectively disabling controllable events to maximize the measure of the controlled plant language without any further constraints. Synthesis of the optimal control policy requires at most n iterations, where n is the number of states of the DFSA model. Each iteration solves a set of n simultaneous linear algebraic equations. As such, computational complexity of the control synthesis is polynomial in n.     

A new approach to fuzzy estimation of Takagi-Sugeno model and its applications to optimal control for nonlinear systems

An efficient approach is presented to improve the local and global approximation and modelling capability of Takagi-Sugeno (T-S) fuzzy model. The main aim is obtaining high function approximation accuracy. The main problem is that T-S identification method cannot be applied when the membership functions are overlapped by pairs. This restricts the use of the T-S method because this type of membership function has been widely used during the last two decades in the stability, controller design and are popular in industrial control applications. The approach developed here can be considered as a generalized version of T-S method with optimized performance in approximating nonlinear functions. A simple approach with few computational effort, based on the well known parameters’ weighting method is suggested for tuning T-S parameters to improve the choice of the performance index and minimize it. A global fuzzy controller (FC) based Linear Quadratic Regulator (LQR) is proposed in order to show the effectiveness of the estimation method developed here in control applications. Illustrative examples of an inverted pendulum and Van der Pol system are chosen to evaluate the robustness and remarkable performance of the proposed method and the high accuracy obtained in approximating nonlinear and unstable systems locally and globally in comparison with the original T-S model. Simulation results indicate the potential, simplicity and generality of the algorithm.     

Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria

Optimal control of general nonlinear nonaffine controlled systems with nonquadratic performance criteria (that permit state- and control-dependent time-varying weighting parameters), is solved classically using a sequence of linear- quadratic and time-varying problems. The proposed method introduces an “approximating sequence of Riccati equations” (ASRE) to explicitly construct nonlinear time-varying optimal state-feedback controllers for such nonlinear systems. Under very mild conditions of local Lipschitz continuity, the sequences converge (globally) to nonlinear optimal stabilizing feedback controls. The computational simplicity and effectiveness of the ASRE algorithm is an appealing alternative to the tedious and laborious task of solving the Hamilton–Jacobi–Bellman partial differential equation. So the optimality of the ASRE control is studied by considering the original nonlinear-nonquadratic optimization problem and the corresponding necessary conditions for optimality, derived from Pontryagin's maximum principle. Global optimal stabilizing state-feedback control laws are then constructed. This is compared with the optimality of the ASRE control by considering a nonlinear fighter aircraft control system, which is nonaffine in the control. Numerical simulations are used to illustrate the application of the ASRE methodology, which demonstrate its superior performance and optimality.     

Analysis and synthesis of networked control systems: Topological entropy, observability, robustness and optimal control

This paper extends the concept of topological to the case of uncertain dynamical systems. We address problems of observability and optimal control via limited capacity digital communication channels. The main results are given in terms of inequalities between date rate of the communication channel and topological entropy of the open-loop system. Topological entropy is calculated for some classes of uncertain dynamical systems.