Switching Time Optimization for Bang-Bang and Singular Controls

Optimal control problems with the control variable appearing linearly are studied. A method for optimization with respect to the switching times of controls containing both bang-bang and singular arcs is presented. This method is based on the transformation of the control problem into a finite-dimensional optimization problem. Therein, first and second-order optimality conditions are thoroughly discussed. Explicit representations of first and second-order variational derivatives of the state trajectory with respect to the switching times are given. These formulas are used to prove that the second-order sufficient conditions can be verified on the basis of only first-order variational derivatives of the state trajectory. The effectiveness of the proposed method is tested with two numerical examples.     

Robust reliable stabilization of uncertain switched neutral systems with delayed switching

This paper investigates the problem of robust reliable control for a class of uncertain switched neutral systems under asynchronous switching, where the switching instants of the controller experience delays with respect to those of the system and the parameter uncertainties are assumed to be norm-bounded. A state feedback controller is proposed to guarantee exponential stability and reliability for switched neutral systems, and the dwell time approach is utilized for the stability analysis and controller design. A numerical example is given to illustrate the effectiveness of the proposed method.     

Continuity in the parameter of the minimum value of an integral functional over the solutions of an evolution control system

The problem of minimization of an integral functional with an integrand that is nonconvex with respect to the control is considered. We minimize our functional over the solution set of a nonlinear evolution control system with a time-dependent subdifferential operator in a Hilbert space. The control constraint is given by a nonconvex closed bounded set. The integrand, the control constraint, the initial conditions and the operators in the equation describing the control system all depend on a parameter. We consider, along with the original problem, the problem of minimizing an integral functional with an integrand convexified with respect to the control over the solution set of the same system, but now subject to the convexified control constraint. By a solution of the control system we mean a “trajectory–control” pair. We prove that for each value of the parameter the convexified problem has a solution, which is the limit of a minimizing sequence of the original problem, and the minimum value of the functional of the convexified problem is a continuous function of the parameter.     

Suboptimal target control for hybrid automata using model predictive control

In this paper, a computationally efficient controller is proposed for the target control problem when the system is modelled by hybrid automata. The design is carried out in two stages. First, we compute off-line the shortest switching path which has the minimum discrete cost from an initial set to the given target set. Next, a controller is derived which successfully drives the system from any given initial state in the initial set to the target set while minimizing a cost function. The model predictive control (MPC) technique is used when the current state is not within a guard set, otherwise the mixed-integer predictive control (MIPC) technique is employed. An on-line, semi-explicit control algorithm is derived by combining these two techniques. When the system is subject to additive bounded disturbance, it is shown that the proposed on-line control algorithm holds if tighter constraints on the original nominal state and controller are imposed. Finally, as an application of the proposed control procedure, the high-speed and energy-saving control problem of the CPU processing is considered.     

Optimal control for systems described by hyperbolic equations with strong nonlinearity

An optimization control problem for a hyperbolic equation is considered. The system is nonlinear with respect to the state derivative. The regularization technique for the state equation is applied. The necessary conditions of optimality for the regularized control problem are proved. It uses the extended differentiability of the control-state mapping for the regularized equation. The convergence of the regularization method is proved. Thus the optimal control for the regularized problem with a small enough regularization parameter can be chosen as an approximate solution of the initial optimization problem.     

Sufficient conditions for relative minima of broken extremals in optimal control theory

We formulate a version of the method of characteristics based on parametrizations of extremals by their terminal values. Sufficient conditions are given for imbedding a reference trajectory into a local field of broken extremals. For a problem with terminal manifold of codimension 1 it is shown that a broken extremal is a relative minimum if (i) the restrictions of the flow to intervals where the control is continuous have nonsingular partial derivatives with respect to the parameter and (ii) the switching surfaces are crossed transversally.     

On the numerical solution of certain boundary control problems for vibrating media in one space-dimension

This paper is concerned with the numerical solution of control problems which consist of minimizing certain quadratic functionals depending on control functions in L ₂[0,1] for some given time T > 0 and bounded with respect to the maximum norm. These control functions act upon the boundary conditions of a vibrating system in one space-dimension which is governed by a wave equation of spatial order 2n They are to be chosen in such a way that a given initial state of vibration at time zero is transferred into the state of rest. This requirement can be expressed by an infinite system of moment equations to be satisfied by the control functions

The control problem is approximated by replacing this infinite system by finitely many, say N, equations (truncation) and by choosing piecewise constant functions as controls (discretization). The resulting problem is a quadratic optimization problem which is solved very efficiently by a multiplier method

Convergence of the solutions of the approximating problems to the solution of the control problem, as N tends to infinity and the discretization is infinitely refined, is shown under mild assumptions. Numerical results are presented for a vibrating beam     

Hybrid Impulsive Control of Stochastic Systems with Multiplicative Noise Under Markovian Switching

A problem of state output feedback stabilization of discrete-time stochastic systems with multiplicative noise under Markovian switching is considered. Under some appropriate assumptions, the stability of this system under pure impulsive control is given. Further under hybrid impulsive control, the output feedback stabilization problem is investigated. The hybrid control action is formulated as a combination of the regular control along with an impulsive control action. The jump Markovian switching is modeled by a discrete-time Markov chain. The control input is simultaneously applied to both the stochastic and the deterministic terms. Sufficient conditions based on stochastic semi-definite programming and linear matrix inequalities (LMIs) for both stochastic stability and stabilization are obtained. Such a nonconvex problem is solved using the existing optimization algorithms and the nonconvex CVX package. The robustness of the stability and stabilization concepts against all admissible uncertainties are also investigated. The parameter uncertainties we consider here are norm bounded. Two examples are given to demonstrate the obtained results.     

Singular solution to the problem of minimizing resource consumption

An iterative method of finding a singular solution to the problem of minimizing resource consumption has been developed. This method is based on the information about the finite control structure. A condition for existence of a singular solution is obtained. The limit value for transferring the time between the normal and the singular solutions is found. A relation between the variations of the control switching instants and the variations of the initial conditions of the adjoint system is found. A system of linear algebraic equations relating the variations of the initial conditions of the adjoint system to the deviations of the phase coordinates from a given final state of the system is obtained. The calculation algorithm and the results of modeling and numerical calculations are presented.     

A switching-based Moving Target Defense against sensor attacks in control systems

Moving Target Defense (MTD) prevents adversaries from being able to predict the effect of their attacks by adding uncertainty in the state of a system during runtime. In this paper, we present an MTD algorithm that randomly changes the availability of the sensor data, so that it is difficult for adversaries to tailor stealthy attacks while, at the same time, minimizing the impact of false-data injection attacks. Using tools from the design of state estimators, namely, observers, and switched systems, we formulate an optimization problem to find the probability of the switching signals that increase the visibility of stealthy attacks while decreasing the deviation caused by false data injection attacks. We show that the proposed MTD algorithm can be designed to guarantee the stability of the closed-loop system with desired performance. In addition, we formulate an optimization problem for the design of the parameters so as to minimize the impact of the attacks. The results are illustrated in two case studies, one about a generic linear time-invariant system and another about a vehicular platooning problem.     

Polynomial motion of non‐holonomic mechanical systems of chained form

In this study a simple general motion planning approach for non‐holonomic mechanical systems of chained form is proposed, in which the intricate motion planning problem (steering the system from an initial state to a final state) is converted to a simple curve‐fitting problem (satisfying a set of end‐point conditions). By means of this approach, other geometric constraints, such as passing a channel and avoiding collision with obstacles, and minimizing some cost functions, such as the minimum path length, can be easily handled, while the control inputs can be derived directly from the smooth path planned. For verifying the effectiveness of the proposed approach, a number of simulations are conducted for various task require ments and environments, with respect to a four‐wheel mobile chart (one‐chain system) and a three‐input firetruck (two‐chain system). Copyright © 1999 John Wiley & Sons, Ltd.     

Sequential stable Kuhn-Tucker theorem in nonlinear programming

A general parametric nonlinear mathematical programming problem with an operator equality constraint and a finite number of functional inequality constraints is considered in a Hilbert space. Elements of a minimizing sequence for this problem are formally constructed from elements of minimizing sequences for its augmented Lagrangian with values of dual variables chosen by applying the Tikhonov stabilization method in the course of solving the corresponding modified dual problem. A sequential Kuhn-Tucker theorem in nondifferential form is proved in terms of minimizing sequences and augmented Lagrangians. The theorem is stable with respect to errors in the initial data and provides a necessary and sufficient condition on the elements of a minimizing sequence. It is shown that the structure of the augmented Lagrangian is a direct consequence of the generalized differentiability properties of the value function in the problem. The proof is based on a “nonlinear” version of the dual regularization method, which is substantiated in this paper. An example is given illustrating that the formal construction of a minimizing sequence is unstable without regularizing the solution of the modified dual problem.     

Optimal parameter selection for nonlinear multistage systems with time-delays

In this paper, we consider a novel dynamic optimization problem for nonlinear multistage systems with time-delays. Such systems evolve over multiple stages, with the dynamics in each stage depending on both the current state of the system and the state at delayed times. The optimization problem involves choosing the values of the time-delays, as well as the values of additional parameters that influence the system dynamics, to minimize a given cost functional. We first show that the partial derivatives of the system state with respect to the time-delays and system parameters can be computed by solving a set of auxiliary dynamic systems in conjunction with the governing multistage system. On this basis, a gradient-based optimization algorithm is proposed to determine the optimal values of the delays and system parameters. Finally, two example problems, one of which involves parameter identification for a realistic fed-batch fermentation process, are solved to demonstrate the algorithm’s effectiveness.     

An Exact Penalty Function Method for Continuous Inequality Constrained Optimal Control Problem

In this paper, we consider a class of optimal control problems subject to equality terminal state constraints and continuous state and control inequality constraints. By using the control parametrization technique and a time scaling transformation, the constrained optimal control problem is approximated by a sequence of optimal parameter selection problems with equality terminal state constraints and continuous state inequality constraints. Each of these constrained optimal parameter selection problems can be regarded as an optimization problem subject to equality constraints and continuous inequality constraints. On this basis, an exact penalty function method is used to devise a computational method to solve these optimization problems with equality constraints and continuous inequality constraints. The main idea is to augment the exact penalty function constructed from the equality constraints and continuous inequality constraints to the objective function, forming a new one. This gives rise to a sequence of unconstrained optimization problems. It is shown that, for sufficiently large penalty parameter value, any local minimizer of the unconstrained optimization problem is a local minimizer of the optimization problem with equality constraints and continuous inequality constraints. The convergent properties of the optimal parameter selection problems with equality constraints and continuous inequality constraints to the original optimal control problem are also discussed. For illustration, three examples are solved showing the effectiveness and applicability of the approach proposed.     

Parametric optimization for a hyperbolic equation in divergence form with a pointwise state constraint: II

On the basis of the results of the first part of the paper, we consider necessary conditions for minimizing sequences in an optimal control problem with a pointwise state constraint of inequality type and with dynamics described by a linear hyperbolic equation in divergence form with the homogeneous Dirichlet boundary condition. The state constraint contains a function parameter that belongs to the class of continuous functions and occurs as an additive term. For the parametric optimization problem, we also consider regularity and normality conditions stipulated by the differential properties of its value function.     

Optimal control for steady flows of the Jeffreys fluids with slip boundary condition

Some optimal control problem is studied for the stationary motion equations for the Jeffreys medium with the slip condition of the Navier type on the boundary. The control parameter is the external force. The existence of a weak solution is proven minimizing a given cost functional. Some properties of the solution set of the optimization problem are established.     

Path planning for robots under stochastic uncertainty *

In order to reduce large online measurement and correction expenses, the a priori informations on the random variations of the model parameters of a robot and its working environment are taken into account already at the planning stage. Thus, instead of solving a deterministic path planning problem with a fixed nominal parameter vector, here, the optimal velocity profile along a given trajectory in work space is determined by using a stochastic optimization approach. Especially, the standard polygon of constrained motion-depending on the nominal parameter vector-is replaced by a more general set of admissible motion determined by chance constraints or more general risk constraints. Robust values (with respect to stochastic parameter variations) of the maximum, minimum velocity, acceleration, deceleration, resp., can be obtained then by solving a univariate stochastic optimization problem Considering the fields of extremal trajectories, the minimum-time path planning problem under stochastic uncertainty can be solved now by standard optimal deterministic path planning methods     

Quantile-based robust optimization of a supersonic nozzle for organic rankine cycle turbines

Organic Rankine Cycle (ORC) turbines usually operate in thermodynamic regions characterized by high-pressure ratios and strong non-ideal gas effects, complicating the aerodynamic design significantly. Systematic optimization methods accounting for multiple uncertainties due to variable operating conditions, referred to as Robust Optimization may benefit to ORC turbines aerodynamic design. This study presents an original and fast robust shape optimization approach to overcome the limitation of a deterministic optimization that neglects operating conditions variability, applied to a well-known supersonic turbine nozzle for ORC applications. The flow around the blade is assumed inviscid and adiabatic and it is reconstructed using the open-source SU2 code. The non-ideal gasdynamics is modeled through the Peng-Robinson-Stryjek-Vera equation of state. We propose here a mono-objective formulation which consists in minimizing the α-quantile of the targeted Quantity of Interest (QoI) under a probabilistic constraint, at a low computational cost. This problem is solved by using an efficient robust optimization approach, coupling a state-of-the-art quantile estimation and a classical Bayesian optimization method. First, the advantages of a quantile-based formulation are illustrated with respect to a conventional mean-based robust optimization. Secondly, we demonstrate the effectiveness of applying this robust optimization framework with a low-fidelity inviscid solver by comparing the resulting optimal design with the ones obtained with a deterministic optimization using a fully turbulent solver.     

Switching Time and Parameter Optimization in Nonlinear Switched Systems with Multiple Time-Delays

In this paper, we consider a dynamic optimization problem involving a general switched system that evolves by switching between several subsystems of nonlinear delay-differential equations. The optimization variables in this system consist of: (1) the times at which the subsystem switches occur; and (2) a set of system parameters that influence the subsystem dynamics. We first establish the existence of the partial derivatives of the system state with respect to both the switching times and the system parameters. Then, on the basis of this result, we show that the gradient of the cost function can be computed by solving the state system forward in time followed by a costate system backward in time. This gradient computation procedure can be combined with any gradient-based optimization method to determine the optimal switching times and parameters. We propose an effective optimization algorithm based on this idea. Finally, we consider three numerical examples, one involving the 1,3-propanediol fed-batch production process, to illustrate the effectiveness and applicability of the proposed algorithm.