Feedback control of linear systems with distributed delay

The problem of optimal control of linear systems containing lumped delay, given by differential-difference equations, has been pursued by several authors. However, transportation-lags are better described by distributed delays giving systems that are described by a set of coupled partial and ordinary differential equations. The lumped part of the system is described by ordinary differential equations and the distributed part of the system is described by partial differential equations. The lumped as well as distributed parts are subject to control. The present paper discusses the control of such systems with quadratic performance measures. Riccati-like equations are derived and a technique for their numerical solution is presented.     

Weighted residual methods in optimal control

Weighted residual methods (WRM) afford a viable approach to the numerical solution of differential equations. Application of WRM results in the transformation of differential equations into systems of algebraic equations in the modal coefficients. This suggests that WRM can be used as a tool for reducing optimal control problems to mathematical programming problems. Thereby, the optimal control problem is replaced by the minimization of a cost function of static coefficients subject to algebraic constraints. The motivation for this approach lies in the profusion of sophisticated computational algorithms and digital computer codes for the solution of mathematical programming problems. In this note the solution of optimal control problems as mathematical programming problems via WRM is illustrated. The example presented indicates that reasonable accuracy is obtained for modest computational effort. While the simplest types of modes-polynomials and piecewise constants-are employed in this note, the ideas delineated can be applied in conjunction with cubic splines for the generation of computational algorithms of enhanced efficiency.     

Boundary control for an industrial under-actuated tubular chemical reactor

Several control strategies are presented and studied for an industrial under-actuated tubular chemical reactor. This work presents a case-study of the performance of a decentralized versus centralized control strategy. The tubular reactor under consideration is characterized by nonlinear kinetic laws, and it has some structural constraints on the location of the heat exchangers and of the sensors. For this system, a set of PI controllers is considered and a multivariable LQR controller is constructed to optimally choose the gains. The performance of these control strategies is studied. Finally, a direct numerical treatment of optimal control of the partial differential equations is presented. Industrial results are given for the linear controllers. Simulations emphasize the possible relevance of a direct numerical treatment of the nonlinear partial differential equations.     

Numerical integration of partial differential equations using cubic splines

This paper presents techniques for the numerical solution of partial differential equations using cubic spline collocation.

The main spline relations are presented and incorporated into solution procedures for partial differential equations. The computational algorithm in every case is a tridiagonal matrix system amenable to efficient inversion methods. Truncation errors and stability are briefly discussed. Finally, some examples of their application to parabolic and hyperbolic systems with mixed boundary conditions are presented.

The results obtained are encouraging and justify further research in this field.     

A numerical algorithm for optimal control of a class of hybrid systems: differential transformation based approach

A novel numerical algorithm based on differential transformation is proposed for optimal control of a class of hybrid systems with a predefined mode sequence. From the necessary conditions for optimality of hybrid systems, the hybrid optimal control problem is first converted into a two-point boundary value problem (TPBVP) with additional transverse conditions at the switching times. Then we propose a differential transformation algorithm for solving the TPBVP which may have discontinuities in the state and/or control input at the switching times. Using differential transformation, the hybrid optimal control problem reduces to a problem of solving a system of algebraic equations. The numerical solution is obtained in the form of a truncated Taylor series. By taking advantage of the special properties of the linear subsystems and a quadratic cost functional, the differential transformation algorithm can be further simplified for the switched linear quadratic optimal control problem. We analyse the error of the numerical solution computed by the differential transformation algorithm and some computational aspects are also discussed. The performance of the differential transformation algorithm is demonstrated through illustrative examples. The differential transformation algorithm has been shown to be simple to be implemented and computationally efficient.     

Partial eigenstructure assignment and its application to largescale systems

The notion of partial eigenstructure assignment (PEA) via linear state feedback control in linear multivariable systems is introduced. This notion is a natural extension of eigenstructure assignment and partial eigenvalue assignment. Some theoretical basis for PEA is provided, and a parametric expression for feedback gain matrices achieving PEA is derived. An effective numerical algorithm for PEA tailored to large-scale systems is presented. As an extension of the algorithm, a recursive algorithm for eigenstructure assignment is presented. These algorithms possess the following desired properties: (1) compared to existing methods, the presented algorithms significantly reduce the required computation time via transforming high-dimensional matrix computations into low-dimensional matrix computations; (2) they can be implemented in a parallel fashion. The proposed algorithm for PEA is applied to modal control of large flexible space structure systems     

Optimal control of linear time-delay systems

A method is presented whereby an optimal control may be obtained for a linear time-varying system with time delay. The performance criterion is quadratic with a fixed, finite upper limit, and results in a set of differential equations with boundary conditions whose solution yields an optimal feedback control. A numerical technique is developed for the solution of the differential equations, and two examples are worked.     

Corporate dynamics of systems of logistic delay equations with large delay control

A system of two logistic equations with delay coupled by delayed control has been considered. It has been shown that, in the case of a fairly large delay control coefficient, the problem of the dynamics of the initial systems has been reduced to investigating the nonlocal dynamics of special families of partial differential equations that do not contain small and large parameters. New interesting dynamic phenomena are discovered based on the results of numerical analysis. Systems of three logistic delay equations with two types of diffusion relations have been considered. Special families of partial differential equations that do not contain small and large parameters have also been constructed for each of these systems. The research results for the dynamic properties of the original equations have been presented. It has been shown that the difference in the dynamics of the considered systems of three equations may be of a fundamental nature.     

分布参数系统边界控制的Haar小波算法

由于分布参数系统通常由偏微分方程描述,采用解析法求解分布参数系统最优边界控制问题,是非常难以解决的.正交函数逼近的方法在分布参数系统控制方面,已经取得了较好的效果.Haar小波作为正交基函数,利用小波的一些运算及变换矩阵,将分布参数系统转化为集总参数系统,再求其逼近解.仿真示例验证了所提出的算法是非常有效的.该方法为分布参数系统的控制算法提出了一条新的解决方案.     

An optimal control method for linear systems with time delay

An optimal control method for linear systems with time delay is developed in this paper. In the proposed control method, the differential equation with time delay of the system dynamics is first rewritten into a form without any time delay through a particular transformation. Then, the optimal controller is designed by using the classical optimal control theory. A numerical algorithm for control implementation is presented, since the obtained expression of the optimal controller contains an integral term that is not convenient for online calculation. The time delay is considered at the very beginning of the control design, and no approximation and estimation are made in the control system. Thus the system performance and stability are prone to be guaranteed. Instability in responses might occur only if a system with time delay is controlled by the optimal controller that was designed with no consideration of time delay. The effectiveness of the proposed optimal controller is demonstrated by simulation studies.     

A type insensitive code for delay differential equations basing on adaptive and explicit Runge-Kutta interpolation methods

For the numerical solution of initial value problems for delay differential equations with constant delay a partitioned Runge-Kutta interpolation method is studied which integrates the whole system either as a stiff or as a nonstiff one in subintervals. This algorithm is based on an adaptive Runge-Kutta interpolation method for stiff delay equations and on an explicit Runge-Kutta interpolation method for nonstiff delay equations. The retarded argument is approximated by appropriate Lagrange or Hermite interpolation. The algorithm takes advantage of the knowledge of the first points of jump discontinuities. An automatic stiffness detection and a stepsize control are presented. Finally, numerical tests and comparisons with other methods are made on a great number of problems including real-life problems.     

Self-tuning controller design for systems with arbitrary time delays Part 2. Algorithms and simulation examples

Recent theoretical developments in the self-tuning control of multivariable systems with arbitrary time delays have been considered in Part 1 of this work (Tade et al. 1988) and a simple time delay extraction procedure has been proposed. In this paper, that procedure is used to derive an adaptive controller based on the generalized minimum variance criterion. The computation count for the proposed self-tuning control algorithm is compared with those for two other algorithms, one proposed by Prager and Wellstead (1981) and the other by Dugard et al. (1984). Assuming that all computation requirements for other stages of the implementation of these algorithms are equal, the results indicate that the proposed algorithm requires less computational effort for updating the control action at each sampling interval. Simulation examples are presented to demonstrate the performance of the proposed algorithm.     

Sequential synthesis of the time-optimal control in real time

The paper considers the sequential synthesis method for time-optimal control of linear systems. The method is based on piecewise constant finite controls that ensure approximate solutions for time-optimal control problems. A sequence of finite controls is thereafter transformed into the optimal control. The appropriate computations are reduced to a sequence of linear algebraic equation problems and the integration of a matrix differential equation over the intervals of control switching points and final point change. It is proven that the sequence of finite controls converge to the optimal control. The sliding mode conditions are obtained, as well as the control structure modifications for the motions on switching manifolds. The initial approximations reducing considerably computational complexity are considered. The computational algorithm, together with the modelling and numerical results, are presented.     

PDE-based event-triggered containment control of multi-agent systems with input delay under spatial boundary communication

This article studies the containment control problem for multi-agent systems with input delay under spatial boundary communication by employing an event-based approach. Firstly, the collective dynamics of multi-agent systems are described as parabolic partial differential equations (PDEs). Applying the integral transformation method developed in PDEs, the delayed parabolic PDEs are transformed into a series of new coupled PDE-PDE systems. Then, by taking the spatial boundary communication scheme into account and using the local boundary information, two boundary containment control protocols together with a dynamic event-triggered scheme (DETS) are designed, such that the states of all followers converge to a convex hull formed by multiple leader agents with and without input delay. The optimal protocols are given by minimizing the 2-norm of the designed control gain matrix, and the exclusion of the Zeno behavior is analyzed. Finally, a numerical simulation example is provided to support the main results.     

Suboptimal sensors location in the state estimation problem for stochastic non-linear distributed parameter systems

The present paper deals with the minimal number sensor choice and their optimal location for the estimation in non-linear stochastic distributed parameter systems described by parabolic and hyperbolic partial differential equations. The necessary condition for the optimal sensor location by using the matrix minimum principle was obtained. In turn, the computational algorithm of the sensors location was determined on the basis of the necessary condition, applying the optimal control theory. The computational efficiency of this algorithm is defined by the suboptimal filtering algorithm which does not require solving of the matrix Riccati equation for the filter error covariance. Finally, one example is given to demonstrate the effectiveness of the present approach.     

Application of the hybrid functions to solve neutral delay functional differential equations

This paper presents a computational technique for the solution of the neutral delay differential equations with state-dependent and time-dependant delays. The properties of the hybrid functions which consist of block-pulse functions plus Legendre polynomials are presented. The approach uses these properties together with the collocation points to reduce the main problems to systems of nonlinear algebraic equations. An estimation of the error is given in the sense of Sobolev norms. The efficiency and accuracy of the proposed method are illustrated by several numerical examples.     

Efficient and Generic Algorithm for Rigorous Integration Forward in Time of dPDEs: Part I

We propose an efficient and generic algorithm for rigorous integration forward in time of partial differential equations written in the Fourier basis. By rigorous integration we mean a procedure which operates on sets and return sets which are guaranteed to contain the exact solution. The presented algorithm generates, in an efficient way, normalized derivatives which are used by the Lohner algorithm to produce a rigorous bound. The algorithm has been successfully tested on several partial differential equations (PDEs) including the Burgers equation, the Kuramoto-Sivashinsky equation, and the Swift-Hohenberg equation. The problem of rigorous integration in time of partial differential equations is a problem of large computational complexity and efficient algorithms are required to deal with PDEs on higher dimensional domains, like the Navier-Stokes equation. Technicalities regarding the various optimization techniques implemented in the software used in this paper will be reported elsewhere.     

Convex Control Systems and Convex Optimal Control Problems With Constraints

This note discusses the concepts of convex control systems and convex optimal control problems. We study control systems governed by ordinary differential equations in the presence of state and target constraints. Our note is devoted to the following main question: under which additional assumptions is a "sophisticated" constrained optimal control problem equivalent to a "simple" convex minimization problem in a related Hilbert space. We determine some classes of convex control systems and show that, for suitable cost functionals and constraints, optimal control problems for these classes of systems correspond to convex optimization problems. The latter can be reliably solved using standard numerical algorithms and effective regularization schemes. In particular, we propose a conceptual computational approach based on gradient-type methods and proximal point techniques.     

Minimum energy control of time delay systems via piecewise linear polynomial functions

The piecewise linear polynomial function approach to the minimum energy control of linear systems with time delay, is presented in this paper. The concepts of a delay shift matrix and an operational matrix for integration are employed in solving the related state and costate equations containing terms with advanced and delayed arguments. An attractive feature of the present method is its ultimate simplicity and convenience. The differential equations with delay and advance terms are converted into a set of linear algebraic equations using a recurrence algorithm. An example demonstrates the accuracy of the method.     

A variable step-size selective partial update LMS algorithm

Selective partial update of the adaptive filter coefficients has been a popular method for reducing the computational complexity of least mean-square (LMS)-type adaptive algorithms. These algorithms use a fixed step-size that forces a performance compromise between fast convergence speed and small steady state misadjustment. This paper proposes a variable step-size (VSS) selective partial update LMS algorithm, where the VSS is an approximation of an optimal derived one. The VSS equations are controlled by only one parameter, and do not require any a priori information about the statistics of the system environment. Mean-square performance analysis will be provided for independent and identically distributed (i.i.d.) input signals, and an expression for the algorithm steady state excess mean-square error (MSE) will be presented. Simulation experiments are conducted to compare the proposed algorithm with existing full-update VSS LMS algorithms, which indicate that the proposed algorithm performs as well as these algorithms while requiring less computational complexity.