Continuous Control Function Research Articles

Relative controllability of nonlinear switched fractional systems

AbstractThis article investigates relative controllability of nonlinear switched fractional systems (SFSs). With the aid of Mittag‐Leffler functions and invariant subspace theory, two sufficient and necessary conditions for corresponding linear SFSs are first established. Then, piecewise continuous control functions and a novel nonlinear operator are constructed to overcome the difficulties arising from switching rules, nonlinearity, and fractional derivatives. Under different constraints on nonlinear functions, two controllability conditions depending on system parameters for nonlinear SFSs are proposed by applying Schauder's and Banach's fixed point theorems, respectively. The obtained criteria may well show the influence of coefficient matrices, fractional derivatives, and switching rules on relative controllability. In addition, our proposed method is also applicable for integer‐order switched systems. Finally, two simulated examples are worked out to verify the theoretical results.

Neural Network Trajectory Tracking Control on Electromagnetic Suspension Systems

A new adaptive-like neural control strategy for motion reference trajectory tracking for a nonlinear electromagnetic suspension dynamic system is introduced. Artificial neural networks, differential flatness and sliding modes are strategically integrated in the presented adaptive neural network control design approach. The robustness and efficiency of the magnetic suspension control system on desired smooth position reference profile tracking can be improved in this fashion. A single levitation control parameter is tuned on-line from a neural adaptive perspective by using information of the reference trajectory tracking error signal only. The sliding mode discontinuous control action is approximated by a neural network-based adaptive continuous control function. Control design is firstly developed from theoretical modelling of the nonlinear physical system. Next, dependency on theoretical modelling of the nonlinear dynamic system is substantially reduced by integrating B-spline neural networks and sliding modes in the electromagnetic levitation control technique. On-line accurate estimation of uncertainty, unmeasured external disturbances and uncertain nonlinearities are conveniently evaded. The effective performance of the robust trajectory tracking levitation control approach is depicted for multiple simulation operating scenarios. The capability of active disturbance suppression is furthermore evidenced. The presented B-spline neural network trajectory tracking control design approach based on sliding modes and differential flatness can be extended to other controllable complex uncertain nonlinear dynamic systems where internal and external disturbances represent a relevant issue. Computer simulations and analytical results demonstrate the effective performance of the new adaptive neural control method.

Final state observability estimates and cost-uniform approximate null-controllability for bi-continuous semigroups

We consider final state observability estimates for bi-continuous semigroups on Banach spaces, i.e. for every initial value, estimating the state at a final time T>0 by taking into account the orbit of the initial value under the semigroup for tin [0,T], measured in a suitable norm. We state a sufficient criterion based on an uncertainty relation and a dissipation estimate and provide two examples of bi-continuous semigroups which share a final state observability estimate, namely the Gauß–Weierstraß semigroup and the Ornstein–Uhlenbeck semigroup on the space of bounded continuous functions. Moreover, we generalise the duality between cost-uniform approximate null-controllability and final state observability estimates to the setting of locally convex spaces for the case of bounded and continuous control functions, which seems to be new even for the case of Banach spaces.

A Generalization of Michel’s Result on the Pontryagin Maximum Principle

We provide an improvement of the maximum principle of Pontryagin of the optimal control problems, for a system governed by an ordinary differential equation, in the presence of final constraints, in the setting of the piecewise continuously differentiable state functions (valued in a Banach space) and of piecewise continuous control functions (valued in a metric space). As Michel, we use the needlelike variations, but we introduce tools of functional analysis and a recent multiplier rule of the static optimization to make our proofs.

Adaptive asymptotic stabilization of switched parametric strict‐feedback systems with switched control

SummaryThis paper is concerned with the problem of adaptive stabilization for a class of switched linear‐parametric nonlinear systems under arbitrary switching. The traditional adaptive backstepping control is successfully extended to switched systems from nonswitched ones where the asymptotic regulation of system state is not destroyed due to rapid or abrupt changes of switching parameters. A new switched adaptive controller is designed by exploiting a common high‐order Lyapunov function with a σ‐modification mechanism, which can reflect sufficiently the changes of plant by designing different adaptive laws and control laws for different subsystems. An explicit formula for constructing a continuous and piecewise virtual control function is given to remove the restriction where some bound functions have to be constructed blindly by designers in the existing results, which may be somewhat too strict to be applied. A numerical example is provided to validate the proposed approach.

Existence of Mild Solutions and Controllability of Fractional Impulsive Integrodifferential Systems with Nonlocal Conditions

This paper is concerned with the existence results of nonlocal problems for a class of fractional impulsive integrodifferential equations in Banach spaces. We define a piecewise continuous control function to obtain the results on controllability of the corresponding fractional impulsive integrodifferential control systems. The results are obtained by means of fixed point methods. An example to illustrate the applications of our main results is given.

Concept and Application of PID Control and Implementation of Continuous PID Controller in Siemens PLCs

Background/Objectives: The main objectives are to define the concept and application of PID control in parallel structure of PID controller and to describe implementation of the continuous PID controller in SIMATIC STEP 7. Methods/Statistical Analysis: Various designs of PID controllers in SIMATIC STEP 7 are available for different kinds of feedback control systems. Parallel structure is used in the algorithm of PID controllers in Siemens PLCs. In this study PID controller with the continuous control function block FB41 (CONT_C) is analyzed. Programming with PID controller in Siemens PLCs with continuous Manipulated Variable output and the option of influencing the Manipulated Value manually is thoroughly discussed. Findings: PID controller is the most common form of feedback control and it can be found in all the areas where control loops exist. The simplicity of understanding of the PID algorithm is one of its advantages but due to a wide range of systems and different manufacturers of control systems with various design and tuning methods of PID controllers in their products, the implementation process gets complicated. This difficulty is reduced by using several kinds of PID controllers for different types of systems in Siemens control systems. FB41 is a function block of continuous PID controller which was designed for Siemens PLCs to provide control performance with analog signals; it can function both in a manual and automatic mode.The most important part of FB41 functioning is addressing of the analog Input and Output in this function block. The results of P, I, D, PI, PD and PID controller implementation in FB41 have been compared. Applications/Improvements: This continuous control function block can be used in different kinds of control loop systems where analog input is sensed and analog output is required.

Optimal control of anthracnose using mixed strategies

In this paper we propose and study a spatial diffusion model for the control of anthracnose disease in a bounded domain. The model is a generalization of the one previously developed in [15]. We use the model to simulate two different types of control strategies against anthracnose disease. Strategies that employ chemical fungicides are modeled using a continuous control function; while strategies that rely on cultivational practices (such as pruning and removal of mummified fruits) are modeled with a control function which is discrete in time (though not in space). For comparative purposes, we perform our analyses for a spatially-averaged model as well as the space-dependent diffusion model. Under weak smoothness conditions on parameters we demonstrate the well-posedness of both models by verifying existence and uniqueness of the solution for the growth inhibition rate for given initial conditions. We also show that the set [0, 1] is positively invariant. We first study control by impulsive strategies, then analyze the simultaneous use of mixed continuous and pulse strategies. In each case we specify a cost functional to be minimized, and we demonstrate the existence of optimal control strategies. In the case of pulse-only strategies, we provide explicit algorithms for finding the optimal control strategies for both the spatially-averaged model and the space-dependent model. We verify the algorithms for both models via simulation, and discuss properties of the optimal solutions.

On the approximation of the set of trajectories of control system described by a Volterra integral equation

In this paper, the set of trajectories of the control system described by a nonlinear Volterra integral equation is studied. It is assumed that the set of admissible control functions is the closed ball of the space Lp, p > 1, with radius µ and centered at the origin. It is shown that the sections of the set of trajectories can be approximated by the sections of trajectories, generated by the mixed constrained and Lipschitz continuous control functions, the Lipschitz constant of which is bounded.

Note on local quadratic growth estimates in bang–bang optimal control problems

Strong second-order conditions in mathematical programming play an important role not only as optimality tests but also as an intrinsic feature in stability and convergence theory of related numerical methods. Besides of appropriate firstorder regularity conditions, the crucial point consists in local growth estimation for the objective which yields inverse stability information on the solution. In optimal control, similar results are known in case of continuous control functions, and for bang–bang optimal controls when the state system is linear. The paper provides a generalization of the latter result to bang–bang optimal control problems for systems which are affine-linear w.r.t. the control but depend nonlinearly on the state. Local quadratic growth in terms of L1 norms of the control variation are obtained under appropriate structural and second-order sufficient optimality conditions.

Common Fixed Point Results in Cone Metric Spaces Using Altering Distance Function

Cone metric space was introduced by Huang Long-Guang et al. (2007) which generalized the concept of metric space. Several fixed point results have been proved in such spaces which generalized and extended the analogous results in metric spaces by different authors. In the present paper two common fixed point results for a sequence of self maps of a complete cone metric space, using altering distance function between the points under a certain continuous control function, are obtained, which generalize the results of Sastry et al. (2001) and Pandhare et al. (1998). Two examples are given in support of our results.

Optimality Condition‐Based Sensitivity Analysis of Optimal Control for Hybrid Systems and Its Application

Gradient‐based algorithms are efficient to compute numerical solutions of optimal control problems for hybrid systems (OCPHS), and the key point is how to get the sensitivity analysis of the optimal control problems. In this paper, optimality condition‐based sensitivity analysis of optimal control for hybrid systems with mode invariants and control constraints is addressed under a priori fixed mode transition order. The decision variables are the mode transition instant sequence and admissible continuous control functions. After equivalent transformation of the original problem, the derivatives of the objective functional with respect to control variables are established based on optimal necessary conditions. By using the obtained derivatives, a control vector parametrization method is implemented to obtain the numerical solution to the OCPHS. Examples are given to illustrate the results.

An ecologically inspired direct search method for solving optimal control problems with Bézier parameterization

An optimal control problem can be formulated through a set of differential equations describing the trajectory of the control variables that minimize the cost functional (related to both state and control variables). Direct solution methods for optimal control problems treat them from the perspective of global optimization: i.e. perform a global search for the control function that optimizes the required objective. In this article we use a recently developed ecologically inspired optimization technique called Invasive Weed Optimization (IWO) for solving such optimal control problems. Usually the direct solution method operates on discrete n-dimensional vectors and not on continuous functions. Consequently it can become computationally expensive for large values of n. Thus, a parameterization technique is required to represent the control functions using a small number of real-valued parameters. Typically, direct methods based on evolutionary computing techniques parameterize control functions with a piecewise constant approximation. This has obvious limitations both for accuracy in representing arbitrary functions, and for optimization efficiency. In this paper a new parameterization is introduced using Bézier curves, which can accurately represent continuous control functions with only a few parameters. It is combined with IWO into a new evolutionary direct method for optimal control. The effectiveness of the new method is demonstrated by solving a wide variety of optimal control problems.

Optimal switching control of a fed-batch fermentation process

Considering the hybrid nature in fed-batch culture of glycerol biconversion to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae, we propose a state-based switching dynamical system to describe the fermentation process. To maximize the concentration of 1,3-PD at the terminal time, an optimal switching control model subject to our proposed switching system and constraints of continuous state inequality and control function is presented. Because the number of the switchings is not known a priori, we reformulate the above optimal control problem as a two-level optimization problem. An optimization algorithm is developed to seek the optimal solution on the basis of a heuristic approach and control parametrization technique. Numerical results show that, by employing the obtained optimal control strategy, 1,3-PD concentration at the terminal time can be increased considerably.

Genetic algorithms with filters for optimal control problems in fed-batch bioreactors

When using a genetic algorithm (GA) to solve optimal control problems that can arise in a fed-batch bioreactor, the most obvious direct approach is to rely on a finite dimensional discretization of the optimal control problem into a nonlinear programming problem. Usually only the control function is discretized, and the continuous control function is approximated by a series of piecewise constant functions. Even though the piecewise discretized controls that the GA produces for the optimal control problem may give good performances, the control policies often show very high activity and differ considerably from those obtained using a continuous optimization strategy. The present study introduces a few filters into a real-coded genetic algorithm as additional operators and investigates the smoothing capabilities of the filters employed. It is observed that inclusion of a filter significantly smoothens the optimal control profile and often encourages the convergence of the algorithm. The applicability of the technique is illustrated by solving two previously reported optimal control problems in fed-batch bioreactors that are known to have singular arcs.

Generalization of fixed point theorems relating to
 the diameter of orbits by using a control function

The main purpose of this paper is to obtain fixed points for a selfmap $T$ of a metric space which is $T$-orbitally complete under a more general contraction type condition by using a certain continuous control function. Further generalization relating to the diameter of orbits is given.

Adaptive ABR congestion control with active buffer control

AbstractATM Forum has defined that the ABR service is designed mainly for data traffic. The design of a simple and efficient congestion control scheme is a problem that network managers have to face. In this paper, we propose a control scheme for ABR service for better scalability and response. The scheme uses both the load level and an increase in queue length to detect the presence of congestion. Continuous control functions are designed for the bandwidth allocation. A better buffer control is achieved by incorporating the percentage of buffer occupancy into the control function. All of the control functions are designed in such a way that they are sensitive to the change of the system states when they are far from the optimal level to get a fast response, but insensitive and change slowly when the states are close to the optimal level to keep the system around the steady state. Our simulation results show that the algorithm is fair to all the connections and it converges fast when the network's condition changes. Copyright © 2001 John Wiley & Sons, Ltd.

A Finite Algorithm for Solving Infinite Dimensional Optimization Problems

We consider the general optimization problem (P) of selecting a continuous function x over a σ-compact Hausdorff space T to a metric space A, from a feasible region X of such functions, so as to minimize a functional c on X. We require that X consist of a closed equicontinuous family of functions lying in the product (over T) of compact subsets Y t of A. (An important special case is the optimal control problem of finding a continuous time control function x that minimizes its associated discounted cost c(x) over the infinite horizon.) Relative to the uniform-on-compacta topology on the function space C(T,A) of continuous functions from T to A, the feasible region X is compact. Thus optimal solutions x * to (P) exist under the assumption that c is continuous. We wish to approximate such an x * by optimal solutions to a net {P i }, i∈I, of approximating problems of the form min x∈X i c i(x) for each i∈I, where (1) the net of sets {X i } I converges to X in the sense of Kuratowski and (2) the net {c i } I of functions converges to c uniformly on X. We show that for large i, any optimal solution x * i to the approximating problem (P i ) arbitrarily well approximates some optimal solution x * to (P). It follows that if (P) is well-posed, i.e., lim sup X i * is a singleton {x *}, then any net {x i *} I of (P i )-optimal solutions converges in C(T,A) to x *. For this case, we construct a finite algorithm with the following property: given any prespecified error δ and any compact subset Q of T, our algorithm computes an i in I and an associated x i * in X i * which is within δ of x * on Q. We illustrate the theory and algorithm with a problem in continuous time production control over an infinite horizon.

An approach to risk sensitive adaptive control

Abstract Adaptive control of discrete time Markov processes with an infinite time horizon risk sensitive cost functional is investigated. The transition probability for the Markov process depends on an unknown parameter. It is shown that the optimal risk sensitive cost is a continuous function of the parameter. Two almost optimal adaptive procedures that are based on the large deviations of the cost functional and discretized maximum likelihood estimates are described. A finite family of continuous control functions, where one control function is fixed after a nonrandom time that is explicitly given for each of the adaptive procedures, provides an almost optimal adaptive control.

Nonlinear bounded control for norm-bounded uncertain systems

In this paper, a method to derive a bounded continuous feedback control function which stabilizes a norm bounded uncertain linear system in a certain region is proposed. The aim is to ensure a certain performance level in a neighborhood (ellipsoid) of the origin, through a guaranteed cost state feedback control. Outside this neighborhood, a linear-like control with state-dependent gains is obtained from the solutions of a family of parameter dependent algebraic Riccati equation similar to the one arising in H/sub /spl infin// control.