On L1‐minimization in optimal control and applications to robotics

In this paper, we analyze optimal control problems with control variables appearing linearly in the dynamics. We discuss different cost functionals involving the L^p‐norm of the control. The case p = 0 represents the time‐optimal control, the case p > 1 yields a standard smooth optimal control problem, whereas the case p = 1 leads to a nonsmooth cost functional. Several techniques are developed to deal with the nonsmooth case p = 1. We present a thorough theoretical discussion of the necessary conditions. Two types of numerical methods are developed: either a regularization technique is used or an augmentation approach is applied in which the number of control variables is doubled. We show the precise relations between the L¹‐minimal control and the bang–bang or singular controls in the augmented problem. Using second‐order sufficient conditions (SSC) for bang–bang controls, we obtain SSC for L¹‐minimal controls. The different techniques and results are illustrated with an example of the optimal control for a free‐flying robot which is taken from Sakawa. Copyright © 2006 John Wiley & Sons, Ltd.     

Assessing blocks after spinal anaesthesia for elective caesarean section: how different questions affect findings from the same stimulus

BackgroundA block to touch to T5 is widely used to indicate an adequate level of block for caesarean section with spinal anaesthesia. However, two studies using a “block to light touch” to T5 as their end-point, had a high requirement for intraoperative analgesia and their results cast doubt on the adequacy of a block to touch to T5. On enquiry, these two papers did not assess complete block to touch, but asked mothers when the touch sensation “was the same as” a control stimulus. The difference between these two assessment methods is unknown. The current study presents prospectively collected sensory block data which included both block to touch and the level when touch was the same as a control stimulus.MethodsThe levels of block were assessed using a Neurotip®. The mother was asked four questions to assess the block: first touch level, first sharp level, touch same as control and sharp same as control.ResultsThe first touch level was a median of two dermatomes lower than the touch same as a control level [IQR 0–3, range 0–6]. Block level assessment methods using first sharp and touch same as control were equivalent.ConclusionWhen describing a sensory block, not only is it necessary to indicate the exact stimulus used, but it is important to define the actual question asked of the patient. Clinically, block assessment using the first sharp level and touch same as control are equivalent.     

Sequential conjugate gradient-restoration algorithm for optimal control problems with non-differential constraints and general boundary conditions,part I

In this paper, a new member of the family of sequential gradient-restoration algorithms for the solution of optimal control problems is presented. This is an algorithm of the conjugate gradient type, which is designed to solve two classes of optimal control problems, called Problem P1 and Problem P2 for easy indentification. Problem P1 involves minimizing a functional I subject to differential constraints and general boundary conditions. It consists of finding the state x (t), the control u (t), and the parameter pi so that the functional I is minimized, while the constraints and the boundary conditions are satisfied to a predetermined accuracy. Problem P2 extends Problem P1 to include non-differential constraints to be satisfied everywhere along the interval of integration. The approach taken is a sequence of two-phase cycles, composed of a conjugate gradient phase and a restoration phase. The conjugate gradient phase involves one iteration and is designed to decrease the value of the functional, while the constraints are satisfied to first order. The restoration phase involves one or more iterations; each restorative iteration is designed to force constraint satisfaction to first order, while the norm squared of the variations of the control, the parameter, and the missing components of the initial state is minimized. The resulting algorithm has several properties: (i) it produces a sequence of feasible solutions; (ii) each feasible solution is characterized by a value of the functional I which is smaller than that associated with any previous feasible solution; and (iii) for the special case of a quadratic functional subject to linear constraints, the variations of the state, control, and parameter produced by the sequence of conjugate gradient phases satisfy various orthogonality and conjugacy conditions. The algorithm presented here differs from those of References 1-4, in that it is not required that the state vector be given at the initial point. Instead, the initial conditions can be absolutely general. In analogy with References 1-4, the present algorithm is capable of handling general final conditions; therefore, it is suitable for the solution of optimal control problems with general boundary conditions. The importance of the present algorithm lies in that many optimal control problems either arise naturally in the present format or can be brought to such a format by means of suitable transformations.⁵ Therefore, a great variety of optimal control problems can be handled. This includes: (i) problems with control equality constraints, (ii) problems with state equality constraints, (iii) problems with state-derivative equality constraints, (iv) problems with-control inequality constraints, (v) problems with state inequality constraints, (vi) problems with state-derivative inequality constraints, and (vii) Chebyshev minimax problems. Several numerical examples are presented in Part 2 (Reference 6) in order to illustrate the performance of the algorithm associated with Problem P1 and Problem P2. The numerical results show the feasibility as well as the convergence characteristics of the present algorithm.     

A novel approach for the numerical investigation of optimal control problems containing multiple delays

This paper deals with the numerical solution of optimal control problems with multiple delays in both state and control variables. A direct approach based on a hybrid of block‐pulse functions and Lagrange interpolating polynomials is used to convert the original problem into a mathematical programming one. The resulting optimization problem is then solved numerically by the Lagrange multipliers method. The operational matrix of delay for the presented framework is derived. This matrix plays an imperative role to transfer information between 2 consecutive switching points. Furthermore, 2 upper bounds on the error with respect to the L²‐norm and infinity norm are established. Several optimal control problems containing multiple delays are carried out to illustrate the various aspects of the proposed approach. The simulation results are compared with either analytical or numerical solutions available in the literature.     

The root loci of H∞ optimal control: A polynomial approach

The root loci patterns which result from changing the balance between the terms in optimal H_∞ stability and performance robustness problems are investigated. The steps for a polynomial analysis are presented. These result in two theorems covering the finite starting and the (finite and infinite) terminal points for the root loci of the two types of H_∞ problems. Four examples are given with a root loci analysis for each of the H_∞ optimal control problems.     

Optimal trajectories for angular systems on the projective line

We analyze two optimal problems for a class of nonlinear system on the real projective line Popf¹ induced by a class of bilinear control system: the angular system. Two functional costs are considered: time‐optimal and quadratic. According to the Pontryagin Maximum Principle, in the time‐optimal case we show that if the angle system ?Σ satisfies the controllability property, then there exists a minimal time bang‐bang trajectory connecting any two points on ?¹, the noncontrollable case was discussed in closed form in (SIAM J. Control Optim. 2009; 48 (4):2636–2650). On the other hand, in the quadratic cost, the optimal control is a continuous function (Proyecciones J. Math. 2010; 29 (2):145–164). A comparison is also established between the structure of the solutions for the two optimal problems: time‐optimal and quadratic in the controllable and noncontrollable cases. The extremals are obtained from the adjoint system given by the Pontryagin Maximum Principle onto ?¹ via radial projection. An example is given. Copyright © 2011 John Wiley & Sons, Ltd.     

Linear multi‐model time‐optimization

A linear optimization problem with unknown parameters from a given finite set is tackled. The problem is to find the robust time‐optimal control transferring a given initial point to a convex terminal compact set M for all unknown parameters in a shortest time. The robust maximum principle for this minimax problem is formulated. It gives a necessary and sufficient condition of robust optimality. Under natural conditions, the existence and uniqueness of robust optimal controls are proven when the resource set is a convex polytope. Several illustrating examples, including a bang–bang robust optimal control, are considered in detail. Copyright © 2002 John Wiley & Sons, Ltd.     

A comparison of a Neuropen monofilament and ethyl chloride for assessing loss of touch sensation during combined spinal–epidural anaesthesia for caesarean section

BackgroundBefore caesarean section is performed under regional anaesthesia the block should be assessed, preferably using a touch stimulus. What constitutes a touch stimulus remains unclear. The aim of this study was to compare a Neuropen monofilament with ethyl chloride in the assessment of touch.MethodsForty women undergoing elective caesarean section received combined spinal–epidural anaesthesia. The upper dermatome spread was assessed using touch to a monofilament and ethyl chloride and cold to ethyl chloride at 5, 10, 15 and 20 min after intrathecal injection and again at the end of surgery. Visual analogue pain scores and Apgar scores were collected.ResultsTwo one-sided test analysis demonstrated equivalence for Neuropen touch and ethyl chloride touch within one dermatome (P < 0.0001). Wilcoxon post tests showed that Neuropen touch was marginally lower than ethyl chloride touch (P = 0.0056). The median level of block to touch using both stimuli was below T5 at all time points. Pain scores had a median value of 0 cm and Apgar scores were 10 in all infants at 10 min.ConclusionData from this study suggest that a Neuropen monofilament and ethyl chloride are equivalent when used to assess a block to touch. However, subtle differences in the level of block to touch indicate that sensory level assessments should state the stimulus used. As the block to touch was below T5 at all time points, when opioids are added to local anaesthetics, T5 might no longer represent a necessary goal to ensure the absence of pain during caesarean section.     

Adjoints estimation methods for impulsive Moon‐to‐Earth trajectories in the restricted three‐body problem

A study of optimal impulsive Moon‐to‐Earth trajectories is presented in a planar circular restricted three‐body framework. Two‐dimensional return trajectories from circular lunar orbits are considered, and the optimization criterion is the total velocity change. The optimal conditions are provided by the optimal control theory. The boundary value problem that arises from the application of the theory of optimal control is solved using a procedure based on Newton's method. Motivated by the difficulty of obtaining convergence, the search for the initial adjoints is carried out by means of two different techniques: homotopic approach and adjoint control transformation. Numerical results demonstrate that both initial adjoints estimation methods are effective and efficient to find the optimal solution and allow exploring the fundamental tradeoff between the time of flight and required ΔV. Copyright © 2014 John Wiley & Sons, Ltd.     

Computational optimal control of the terminal bunt manoeuvre—Part 2: minimum‐time case

This is the second part of a paper studies trajectory shaping of a generic cruise missile attacking a fixed target from above. The problem is reinterpreted using optimal control theory resulting in a minimum flight time problem; in the first part the performance index was time‐integrated altitude. The formulation entails non‐linear, two‐dimensional (vertical plane) missile flight dynamics, boundary conditions and path constraints, including pure state constraints. The focus here is on informed use of the tools of computational optimal control, rather than their development. The formulation is solved using a three‐stage approach. In stage 1, the problem is discretized, effectively transforming it into a non‐linear programming problem, and hence suitable for approximate solution with DIRCOL and NUDOCCCS. The results are used to discern the structure of the optimal solution, i.e. type of constraints active, time of their activation, switching and jump points. This qualitative analysis, employing the results of stage 1 and optimal control theory, constitutes stage 2. Finally, in stage 3, the insights of stage 2 are made precise by rigorous mathematical formulation of the relevant two‐point boundary value problems (TPBVPs), using the appropriate theorems of optimal control theory. The TPBVPs obtained from this indirect approach are then solved using BNDSCO and the results compared with the appropriate solutions of stage 1. The influence of boundary conditions on the structure of the optimal solution and the performance index is investigated. The results are then interpreted from the operational and computational perspectives. Copyright © 2007 John Wiley & Sons, Ltd.     

The solution of finite-time optimal control problems with control time delays

The design of deterministic LQP optimal control systems is considered for plants with control delays and a finite optimization interval. First, a solution is obtained to the fixed end-point optimal control problem. An expression for the open-loop optimal control is derived in the s-domain. The closed-loop optimal time-varying gain matrix is then calculated from the s-domain results. The open-loop and closed-loop solutions to the free end-point optimal control problem are also given. The constant gain, infinite-time, feedback control law is obtained as a limiting case of these results. The receding-horizon optimal control problem for plants with control signal delays is also considered. The solution to this problem yields a constant feedback gain matrix which has obvious advantages for implementation. This gain matrix is not the same as the constant solution to the infinite-time problem. The receding-horizon control laws are derived for both the fixed and free end-point problems, and these are shown to produce asymptotically stable closed-loop systems.     

Finite-time optimal pinning control and synchronization for partially interdependent networks

For partially interdependent networks composed of two subnetworks, the finite-time optimal pinning control problem is investigated. Among them, only a part of the nodes between the two subnetworks are interdependent on each other. In the network, the coupling relationship between any two nodes of the network is a continuous nonlinear function. Based on the pinning control, the optimal control theory, Kalman's controllability rank conditions, and introducing the Lagrange function and applying controllers to the partial nodes of the network, we propose some characterization indicators which ensure that the two subnetworks of partially interdependent networks can synchronize to the equilibrium points of their isolate systems, separately. Finally, some numerical simulations are performed on the partially interdependent networks including two small world subnetworks. The results show that the optimal pinning control method proposed in this article can greatly reduce the control costs and achieve the ideal synchronous status quickly.     

Real-time computation of feedback controls for constrained optimal control problems. part 1: Neighbouring extremals

A numerical method is developed for the real-time computation of neighbouring optimal feedback controls for constrained optimal control problems. The first part of this paper presents the theory of neighbouring extremals. Besides a survey of the theory of neighbouring extremals, special emphasis is laid on the inclusion of complex constraints, e.g. state and control variable inequality constraints and discontinuities of the system equations at interior points. The numerical treatment of these constraints is particularly emphasized. The linearization of all necessary conditions of optimal control theory leads to a linear, mulitpoint, boundary value problem with linear jump conditions that is especially well suited for numerical treatment.     

H∞ output tracking control over networked control systems with Markovian jumping parameters

The problem of H_∞ output tracking control over networked control systems (NCSs) with communication limits and environmental disturbances is studied in this paper. A wide range of time‐varying stochastic problem arising in networked tracking control system is reduced to a standard convex optimization problem involving linear matrix inequalities (LMIs). The closed‐loop hybrid NCS is modeled as a Markov jump linear system in which random time delays and packet dropouts are described as two stochastic Markov chains. Gridding approach is introduced to guarantee the finite value of the sequences of transmission delays from sensor to actuator. Sufficient conditions for the stochastic stabilization of the hybrid NCS tracking system are derived by the LMI‐based approach through the computation of the optimal H_∞ performance. The mode‐dependent robust H_∞ output tracking controller is obtained by the optimal iteration method. Numerical examples are given to demonstrate the effectiveness of the proposed robust output tracking controller for NCS. Copyright © 2016 John Wiley & Sons, Ltd.     

Theoretical basis for camera control in teleoperating

Background: The triangle paralaxis method for camera control in teleoperating is presented. Methods: For orientation in the 3D space of the corporic cavity there are three points necessary for the creation of the paralaxis triangle. This triangle is then imagined and compared with topography during surgery. The first and second points are created in one's mind at the locus of the entry of the instruments into the viewing field of the camera. The third apex of the triangle is the area of dissection—the point in which the instruments converge. The fourth point to be viewed determines the course of dissection. Triangle paralaxis may be applied in dissection with only one instrument as well as in the zooming technique, closely viewing a part of the dissecting instrument. Results: Using this technique a 7.78% rate of conversion and 2.15% rate of reoperation could be achieved in 334 evaluated laparoscopic cholecystectomies performed in a small public hospital. Conclusions: Triangle paralaxis seems to be a simple method for ensuring an optimal camera view during laparoscopic surgery.     

Distributed optimal control of viscous Burgers' equation via a high-order,linearization, integral,nodal discontinuous Gegenbauer-Galerkin method

We developed a novel direct optimization method to solve distributed optimal control of viscous Burgers' equation over a finite-time horizon by minimizing the distance between the state function and a desired target state profile along with the energy of the control. Through a novel linearization strategy, well-conditioned integral reformulations, optimal Gegenbauer barycentric quadratures, and nodal discontinuous Galerkin discretizations, the method reduces such optimal control problems into finite-dimensional, nonlinear programming problems subject to linear algebraic system of equations and discrete mixed path inequality constraints that can be solved easily using standard optimization software. The proposed method produces “an auxiliary control function” that provides a useful model to explicitly define the optimal controller of the state variable. We present an error analysis of the semidiscretization and full discretization of the weak form of the reduced equality constraint system equations to demonstrate the exponential convergence of the method. The accuracy of the proposed method is examined using two numerical examples for various target state functions in the existence/absence of control bounds. The proposed method is exponentially convergent in both space and time, thus producing highly accurate approximations using a significantly small number of collocation points.     

Optimal control of the atmospheric reentry of a space shuttle by an homotopy method

This paper deals with the optimal control problem of the atmospheric reentry of a space shuttle with a second‐order state constraint on the thermal flux. We solve the problem using the shooting algorithm combined with an homotopy method, which automatically determines the structure of the optimal trajectory (composed of one boundary arc and one touch point). Copyright © 2010 John Wiley & Sons, Ltd.     

综合性医院护士站品牌接触点的研究

目的 挖掘患者认为的护士站内医院关键接触点,为开展优质护理服务提供依据.方法 采用自行设计的护士站接触点问卷对213例患者进行调查.结果 排在前3位的关键接触点为：护士服务态度（72.8％）、护士各项操作熟练程度与规范性（58.2％）,以及配药室环境（31.9％）;不同学历、职业的患者认为的关键接触点略有不同.结论 护理管理者在做好接触点管理的同时,要结合患者的学历、职业,抓住关键接触点,以提供个性化服务,提高患者满意度.     

Optimal control of parametrically excited linear delay differential systems via Chebyshev polynomials

The use of Chebyshev polynomials in solving finite horizon optimal control problems associated with general linear time‐varying systems with constant delay is well known in the literature. The technique is modified in the present paper for the finite horizon control of dynamical systems with time periodic coefficients and constant delay. The governing differential equations of motion are converted into an algebraic recursive relationship in terms of the Chebyshev coefficients of the system matrices, delayed and present state vectors, and the input vector. Three different approaches are considered. The first approach computes the Chebyshev coefficients of the control vector by minimizing a quadratic cost function over a finite horizon or a finite sequence of time intervals. Then two convergence conditions are presented to improve the performance of the optimized trajectories in terms of the oscillation of controlled states within intervals. The second approach computes the Chebyshev coefficients of the control vector by maximizing a quadratic decay rate of the L₂ norm of Chebyshev coefficients of the state subject to linear matching and quadratic convergence conditions. The control vector in each interval is computed by formulating a non‐linear optimization programme. The third approach computes the Chebyshev coefficients of the control vector by maximizing a linear decay rate of the L_∞ norm of Chebyshev coefficients of the state subject to linear matching and linear convergence conditions. The proposed techniques are illustrated by designing regulation controllers for a delayed Mathieu equation over a finite control horizon. Copyright © 2005 John Wiley & Sons, Ltd.     

Costate approximation in optimal control using integral Gaussian quadrature orthogonal collocation methods

Two methods are presented for approximating the costate of optimal control problems in integral form using orthogonal collocation at Legendre–Gauss (LG) and Legendre–Gauss–Radau (LGR) points. It is shown that the derivative of the costate of the continuous‐time optimal control problem is equal to the negative of the costate of the integral form of the continuous‐time optimal control problem. Using this continuous‐time relationship between the differential and integral costate, it is shown that the discrete approximations of the differential costate using LG and LGR collocation are related to the corresponding discrete approximations of the integral costate via integration matrices. The approach developed in this paper provides a way to approximate the costate of the original optimal control problem using the Lagrange multipliers of the integral form of the LG and LGR collocation methods. The methods are demonstrated on two examples where it is shown that both the differential and integral costate converge exponentially as a function of the number of LG or LGR points. Copyright © 2014 John Wiley & Sons, Ltd.