Control Parameterization Research Articles

A Computational Approach to Optimal Control Problems with Almost Smooth Controls

In this paper, we consider a class of optimal control problems involving continuous control and state inequality constraints where the control is almost smooth. We first employ the control parametrization technique via approximating the control signal by a piecewise linear function. Then, we develop a time scaling transformation procedure for transforming the approximate problem into an equivalent problem that can be solved readily using conventional methods. On this basis, a novel exact penalty function method is constructed by appending penalized constraint violations to the cost function. The gradient formulas and convergent properties ensure that the transformed unconstrained optimal parameter selection problems can be solved by existing optimization algorithms or software packages. Finally, an example is solved showing the effectiveness and applicability of the approach proposed.

ИССЛЕДОВАНИЕ ПАРАМЕТРОВ ЗАДАЧИ СПЛАЙНОВОЙ АППРОКСИМАЦИИ ЗАШУМЛЕННЫХ ДАННЫХ ЧИСЛЕННЫМИ МЕТОДАМИ ОПТИМАЛЬНОГО УПРАВЛЕНИЯ

Currently, the problems of distortion of measurement data by noise and the appearance of un-certainties in quality criteria have caused increased interest in research in the field of spline approx-imation. At the same time, existing methods of minimizing empirical risk, assuming that the noise is a uniform distribution with heavier tails than Gaussian, limit the scope of application of these studies. The problem of estimating noise-distorted data is usually based on solving an optimi-zation problem with a function containing uncertainty arising from the problem of finding optimal parameters. In this regard, the estimation of distorted noise cannot be solved by classical methods. Aim. This study is aimed at solving and analyzing the problem of spline approximation of data under uncertainty conditions based on the parametrization of control and the gradient projec-tion algorithm. Methods. The study of the problem of spline approximation of noisy data is carried out by the method of approximation of the piecewise constant control function. In this case, para-metrization of the control is possible only for a finite number of break points of the first kind. In the framework of the experimental study, the gradient projection algorithm is used for the numerical solution of the spline approximation problem. The proposed methods are used to study the parameters of the problem of spline approximation of data under conditions of uncertain-ty. Results. The numerical study of the control parametrization approach and the gradient projec-tion algorithm is based on the developed software and algorithmic tool for solving the problem of the spline approximation model under uncertainty. To evaluate the noise-distorted data, numerical experiments were conducted to study the model parameters and it was found that increasing the value of the parameter α leads to an increase in accuracy, but a loss of smoothness. In addition, the analysis showed that the considered distribution laws did not change the accuracy and convergence rate of the algorithm. Conclusion. The proposed approach for solving the problem of spline approx-imation under uncertainty conditions allows us to determine the problems of distortion of measure-ment data by noise and the appearance of uncertainties in the quality criteria. The study of the model parameters showed that the constructed system is stable to the error of the initial approxima-tion, and the distribution laws do not significantly affect the accuracy and convergence of the gra-dient projection method.

Parameterization of All Output-Rectifying Retrofit Controllers

This study investigates a parameterization of all output-rectifying retrofit controllers for distributed design of a structured controller. It has been discovered that all retrofit controllers can be characterized as a constrained Youla parameterization, which is difficult to solve analytically. For synthesis, a tractable and insightful class of retrofit controllers, referred to as output-rectifying retrofit controllers, has been introduced. An unconstrained parameterization of all output-rectifying retrofit controllers can be derived under a technical assumption on measurability of particular signals. The aim of this note is to reveal the structure of all output-rectifying retrofit controllers in the general output-feedback case. It is found out that the existing developments can be generalized based on the notions of state projection and an inverse system. The result leads to the conclusion that output-rectifying retrofit controllers can readily be designed even in the general case.

Time-optimal control problem for a linear parameter varying system with nonlinear item

In this paper, we considered a time-optimal control problem for a new type of linear parameter varying (LPV) system which is obtained through data identification in the process of dealing with actual problems. The addition of non-linear terms is compensation for the method that does not require linear expansion at the equilibrium point. Since the objective function is the terminal time which is an implicit function concerning decision variables, it is a non-standard optimal control problem with uncertain terminal time. To find the global optimal solution to this problem, firstly, the control parameterization method is used to transform it into a nonlinear optimization problem of parameter selection, and then the modifed particle swarm optimization (PSO) algorithm is combined to solve the equivalent nonlinear programming problem. Numerical examples are used to illustrate the effectiveness of the proposed algorithm.

Stabilising quasi-time-optimal nonlinear model predictive control with variable discretisation

This paper deals with novel time-optimal point-to-point model predictive control concepts for nonlinear systems. Recent approaches in the literature apply a time transformation, however, which do not maintain recursive feasibility for a piecewise constant control parameterisation. The key idea in this paper is to introduce uniform grids with variable discretisation. A shrinking-horizon grid adaptation scheme ensures convergence to a specific region around the target state and recursive feasibility. The size of the region is configurable by design parameters. This facilitates the systematic dual-mode design for quasi-time-optimal control to restore asymptotic stability and establish a smooth stabilisation. Two nonlinear programme formulations with different sparsity patterns are introduced to realise and implement the underlying optimal control problem. For a class of numerical integration schemes, even nominal asymptotic stability and true time-optimality are achieved without dual-mode. A comparative analysis as well as experimental results demonstrate the effectiveness of the proposed techniques.

Fast control parameterization optimal control with improved Polak–Ribière–Polyak conjugate gradient implementation for industrial dynamic processes

This paper proposes a fast control parameterization optimal control algorithm for industrial dynamic process with constraints. Derived from the frame of control variable parameterization (CVP) technique, the proposed method combines an efficient gradient computation strategy with an improved nonlinear optimization computation approach to overcome the challenge of computation efficiency caused by gradients and bounds in optimal control problems. Firstly, a fast gradient computation method based on the costate system of Hamiltonian function is developed to decrease the computational expense of gradients by employing approximate treatments and numerical integration strategy. Then, a trigonometric function transformation scheme is presented to tackle the boundary constraints so that the original optimal control problem is further converted into an unconstrained one. On this basis, an improved restricted Polak–Ribière–Polyak (PRP) conjugate gradient approach is introduced to solve the nonlinear optimization problem by using conjugate gradient iterations and strong Wolfe line search. Meanwhile, to enhance the convergence, a restricting condition is imposed in strong Wolfe line search to create iteration step-length. Finally, the proposed algorithm is implemented on three dynamic processes. The detailed comparison among the classical CVP method, literature results and the proposed method are carried out. Simulation studies show that the proposed fast approach averagely saves more than 90% computation time in contrast to the classical CVP method, demonstrating the effectiveness of the proposed fast optimal control approach.

Controller Design via Experimental Exploration With Robustness Guarantees

For a partially unknown linear systems, we present a systematic control design approach based on generated data from measurements of closed-loop experiments with suitable test controllers. These experiments are used to improve the achieved performance and to reduce the uncertainty about the unknown parts of the system. This is achieved through a parametrization of auspicious controllers with convex relaxation techniques from robust control, which guarantees that their implementation on the unknown plant is safe. This approach permits to systematically incorporate available prior knowledge about the system by employing the framework of linear fractional representations.

A new gradient computational formula for optimal control problems with time-delay

<p style='text-indent:20px;'>In this paper, we consider a class of time-delay optimal control problem (TDOCP) with canonical equality and inequality constraints. By applying control parameterization method together with time-scaling transformation, a TDOCP can be readily solved by gradient-based optimization methods. The partial derivative of the cost as well as the constraint functions with respect to the decision variables are obtained by variational approach, which is inefficient when the discretization for the control function is relatively dense. For general optimal control problem without time-delay, co-state approach is an effective way to compute the gradients, however, when time-delay is involved in the dynamic system, the co-state system is not known. In this paper, we derive the co-state system for TDOCP to compute the gradients of the cost and constraints. Numerical results show that the computational efficiency is much higher when compared with the traditional variational approach.</p>

Multistage optimal control for microbial fed-batch fermentation process

<p style='text-indent:20px;'>In this paper, we consider multistage optimal control of bioconversion glycerol to 1, 3-propanediol(1, 3-PD) in fed-batch fermentation process. To maximize the productivity of 1, 3-PD, the whole fermentation process is divided into three stages according to the characteristics of microbial growth. Stages 2 and 3 are discussed mainly. The main aim of stage 2 is to restrict accumulation of 3-hydroxypropionaldehyde and maximize the biomass in the shortest time, and the purpose of stage 3 is to get high productivity of 1, 3-PD. With these different objectives, multi-objective optimal control problems are proposed in stages 2 and 3. In order to solve the above optimal control problems, the multi-objective problems are transformed to the corresponding single-objective problems using the mass balance equation of biomass and normalization of the objective. Furthermore, the single-objective optimal control problems are transformed to two-level optimization problems by the control parametrization technique. Finally, numerical solution methods combined an improved Particle Swarm Optimization with penalty function method are developed to solve the resulting optimization problems. Numerical results show that the productivity of 1, 3-PD is higher than the reported results.</p>

The method of parameterization in the quadratic optimal control problem

In the framework of control parameterization method the optimization problem with respect to linear phase system with quadratic functional is considered. Approximation of the control is obtained in the class of piecewise constant functions. It is formed as a linear combination of a special set of support functions. Coefficients of this combination are variables of the finite-dimensional problem. To effectively solve this problem explicit expressions for the functional with respect to parameters of approximations are obtained. As a result the quadratic mathematical programming problem is formulated.

Optimal Switching Time Control for Suppressing Residual Vibration in a High-Speed Macro-Micro Manipulator System

This brief studies a novel residual vibration suppression problem arising in a high-speed macromicromanipulator system designed for various precision positioning tasks. First, the dynamic of the macromicrosystem is mathematically modeled by a nonlinear partial differential equation (PDE). Subsequently, a new nonlinear vibration suppression and trajectory planning problem governed by a couple of ordinary differential equations (ODEs) are successfully formulated by using the assumed mode method (AMM) and Hamilton’s principle. Based on the developed model, an optimal switching time control strategy combined with control parameterization is proposed to realize the residual vibration suppression by trajectory planning. The gradients of the cost functional with respect to the unknown control variables and switching time variables are derived analytically by the sensitivity analysis method. Both numerical simulations and real experiments in the laboratory verify that the proposed method is superior to the classical fifth-order polynomial (FOP) approach for the vibration suppression problem.

The quasidifferential descent method in a control problem with nonsmooth objective functional

The paper is devoted to the problem of optimal control of an object described by a system with a continuously differentiable right-hand side and a nondifferentiable (but only quasidifferentiable) quality functional. We consider a problem in the form of Mayer with both a free and a fixed right end. Admissible controls are piecewise continuous (with a finite number of discontinuity points) and bounded vector-functions, which belong to certain polyhedron at each moment of time. Standard discretization of the initial system and control parameterization are performed, and theorems on the convergence of the discrete system solution obtained to the desired solution of the continuous problem are presented. Further, for the discrete system obtained, the necessary and, in some cases, sufficient minimum conditions are written in terms of quasidifferential. The quasidifferential descent method is applied to this problem. The algorithm developed is demonstrated by examples.

Integrated optimal control strategies for freeway traffic mixed with connected automated vehicles: A model-based reinforcement learning approach

This paper proposes an integrated freeway traffic flow control framework that aims to minimize the total travel cost, improve greenness and safety for freeway traffic mixed with connected automated vehicles (CAVs) and regular human-piloted vehicles (RHVs). The proposed framework devises an integrated action of several control strategies such as ramp metering, lane changing control (LCC) for CAVs and lane changing recommendation (LCR) for RHVs, variable speed limit control (VSLC) for CAVs and variable speed limit recommendation (VSLR) for RHVs with minimum safety gap control measures for lane changing and merging maneuvers. The CAVs are assumed to follow the system control instructions fully and immediately. In contrast, the RHVs would make decisions in response to the recommendations disseminated and also the behaviors of CAVs. The compliance rate of drivers to the LCR is captured by the underlying traffic flow model. A set of constraints is imposed to restrict VSLC/VSLR and LCC/LCR measures from changing too frequently or too sharply on both temporal and spatial dimensions to avoid excessive nuisance to passengers and traffic flow instability. A reinforcement learning based solution algorithm is proposed. First, a control parameterization technique is adopted to reduce the dimension of the original optimal control problem to increase computational efficiency. Then, a gradient-free Cross-Entropy-Method based algorithm is used to search the optimal parameters to circumvent the non-differentiability of the traffic flow model. The feasibility and effectiveness of the proposed framework are illustrated via numerical examples for a variety of penetration rates of CAVs under various traffic conditions. A sensitivity analysis is conducted to demonstrate the impacts of several important parameters such as the reaction time of the CAVs. It is found that the integrated control strategy can reduce the total travel cost by reducing the lane changing maneuvers and vehicles queuing at the bottleneck meanwhile smooth the traffic flow and suppress the adverse impact of shockwaves. The effect of ramp metering is not significant when the penetration rate of CAVs is high enough. Speed harmonization (with minimum gap control) in conjunction with LCC/LCR would be a better integrated control strategy under high penetration rate of CAVs.

The uncoupled microbial fed-batch fermentation optimization based on state-dependent switched system

In the actual microbial fermentation process, excessive or insufficient substrate can produce inhibitory effects on cells growth. The artificial substrate feeding rules by past experiences have great blindness to keep substrate concentration in a given appropriate range. This paper considers that alkali feed depends on pH value of the solution and glycerol feed depends on glycerol concentration of the solution in the uncoupled microbial fed-batch fermentation process, and establishes a state-dependent switched system in which the flow rates of glycerol and alkali, the number of mode switches, the mode sequence and the switching times are prior unknown. To maximize the yield of target product 1,3-Propanediol (1,3-PD), we formulate a switching optimal control problem with the flow rates of glycerol and alkali, the number of mode switches, the mode sequence and the switching times as decision variables, which is a mixed-integer dynamic programming problem. For solving the mixed-integer dynamic programming problem, the control parametrization technique, the time scaling transformation and the embedded system technology are used to obtain an approximate parameter optimization problem. By using a parallel optimization algorithm, we obtain the optimal control strategies. Under the obtained optimal control strategies, the 1,3-PD yield at the terminal time is increased significantly compared with the previous results.

On the optimal control problems with characteristic time control constraints

<p style='text-indent:20px;'>In this paper, we consider a class of optimal control problems with control constraints on a set of characteristic time instants. By applying the control parameterization technique, these constraints are imposed on the subdomains that contain the characteristic time points. The values of the control functions as well as the lengths for their corresponding subdomains become decision variables. Time-scaling transformation is an effective technique to optimize the length of each subdomain in a new time horizon. However, the characteristic time instants in the original time horizon become variable time instants in the new time horizon, and hence the control constraints imposed on these characteristic time points are difficult to be formulated in the new time horizon. We propose a surrogate condition and show that the characteristic time control constraints will be satisfied once the surrogate condition holds. Moreover, this surrogate condition is easy to formulate in the new time horizon. The resulting approximate problem can be readily solved by many existing computational methods for solving constrained optimal control problems. Finally, we conclude this paper by solving two examples.</p>

A control parametrization based path planning method for the quad-rotor uavs

<p style='text-indent:20px;'>A time optimal path planning problem for the Quad-rotor unmanned aerial vehicles (UAVs) is investigated in this paper. A 3D environment with obstacles is considered, which makes the problem more challenging. To tackle this challenge, the problem is formulated as a nonlinear optimal control problem with continuous state inequality constraints and terminal equality constraints. A control parametrization based method is proposed. Particularly, the constraint transcription method together with a local smoothing technique is utilized to handle the continuous inequality constraints. The original problem is then transformed into a nonlinear program. The corresponding gradient formulas for both of the cost function and the constraints are derived, respectively. Simulation results show that the proposed path planning method has less tracking error than that of the rapid-exploring random tree (RRT) algorithm and that of the A star algorithm. In addition, the motor speed has less changes for the proposed algorithm than that of the other two algorithms.</p>

The codifferential descent method in the problem of finding the global minimum of a piecewise affine objective functional in linear control systems

The article considers the problem of optimal control of an object described by a linear nonstationary system and with a piecewise affine quality functional. The problem is examined in Mayer’s form with both free and partially fixed right endpoints. Piecewise continuous and bounded controls that lie in some parallelepiped at each moment of time are admissible. The standard discretization of the original system and the control parametrization are used, some convergence theorems of the discrete problem solution to the continuous problem solution are presented. Further, for the obtained discrete system, the necessary and sufficient minimum conditions are written out in terms of the codifferential, the method of the modified codifferential descent is applied to it, which guarantees to find the global minimum of this problem in a finite number of steps. The proposed algorithm is illustrated with examples.

An efficient numerical solution for time switching optimal control problems

In this paper, an efficient computational algorithm for the solution of Hamiltonian boundary value problems arising from bang-bang optimal control problems is presented. For this purpose, at first, based on the Pontryagin’s minimum principle, the first order necessary conditions of optimality are derived. Then, an indirect shooting method with control parameterization, in which the control function is replaced with a piecewise constant function with values and switching points taken as unknown parameters, is presented. Thereby, the problem is converted to the solution of the shooting equation, in which the values of the control function and the switching points as well the initial values of the costate variables are unknown parameters. The important advantages of this method are that, the obtained solution satisfies the first order optimality conditions, further the switching points can be captured accurately which is led to an accurate solution of the bangbang problem. However, solving the shooting equation is nearly impossible without a very good initial guess. So, in order to cope with the difficulty of the initial guess, a homotopic approach is combined with the presented method. Consequently, no priori assumptions are made on the optimal control structure and number of the switching points, and sensitivity to the initial guess for the unknown parameters is resolved too. Illustrative examples are included at the end and efficiency of the method is reported.

Virtual Point Control for Step-Down Perturbations and Downhill Slopes in Bipedal Running.

Bipedal running is a difficult task to realize in robots, since the trunk is underactuated and control is limited by intermittent ground contacts. Stabilizing the trunk becomes even more challenging if the terrain is uneven and causes perturbations. One bio-inspired method to achieve postural stability is the virtual point (VP) control, which is able to generate natural motion. However, so far it has only been studied for level running. In this work, we investigate whether the VP control method can accommodate single step-down perturbations and downhill terrains. We provide guidelines on the model and controller parameterizations for handling varying terrain conditions. Next, we show that the VP method is able to stabilize single step-down perturbations up to 40 cm, and downhill grades up to 20–40° corresponding to running speeds of 2–5 ms−1. Our results show that the VP approach leads to asymmetrically bounded ground reaction forces for downhill running, unlike the commonly-used symmetric friction cone constraints. Overall, VP control is a promising candidate for terrain-adaptive running control of bipedal robots.

A quasi-sequential algorithm for PDE-constrained optimization based on space–time orthogonal collocation on finite elements

Orthogonal collocation on finite elements (OCFE) has been used universally to approximate ODEs to date. For PDEs, this contribution presents a novel discretization scheme applying the methodology of OCFE to discretize both space and time domain simultaneously, named as space–time orthogonal collocation on finite elements (ST-OCFE). Due to the existence of boundary conditions, the selection of discrete points and the constitution of discretized algebraic equations are different in space and time domain. Furthermore, for solving optimal control problems constrained by PDEs, a discretize-then-optimize algorithm based on ST-OCFE and quasi-sequential approach is proposed. The formulation of discretized optimization problems and the procedure of sensitivity computation are deduced. In the algorithm, diverse types of PDEs, state constraints, and general control parameterization are considered. The proposed method has the advantages of generality, higher numerical accuracy, and easy handling of state constraints, as demonstrated by three examples.