Asymptotic series solution of singularly perturbed optimal control problems

An optimal control problem of some nonlinear differential equations containing a small parameter is considered. This problem leads to a two-point boundary value problem (TPBVP) whose differential order can be reduced by neglecting the small parameter. In this paper an explicit method of constructing an asymptotic power series solution of the TPBVP as the small parameter tends to zero is presented. The method allows a separation of slow and fast dynamics in the problem while reducing the differential order of the equations. The determination of asymptotic series terms is computationally simple since one needs to solve only initial or final value problems instead of TPBVP's. The method has computational similarities with quasilinearization and second variation techniques.     

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.     

Polynomial solution to stabilization problem for multivariable sampled-data systems with time delay

Necessary and sufficient stabilizability conditions are derived in a polynomial form for a multivariable sampled-data systems with scalar delays. A polynomial representation of the set of stabilizing controllers is given.     

Asymptotic behaviour for some large-scale systems with infinite delay

The aggregation-decomposition method is used to derive sufficient conditions for stability of large-scale systems with infinite delays under structural perturbations described by functional differential equations with finite lags appearing only in the interconnections. Electrical networks consisting of transmission lines interconnected with lumped linear and memoryless nonlinear elements can be represented by the dynamical systems considered in this paper.     

Computation of optimal control for linear systems with delay

A numerical algorithm is presented to calculate an optimal control recursively for linear multivariable systems with delay. The algorithm is based on the method of steepest descent in Hilbert space. The optimal control of a multivariable system with delay for a quadratic criterion function is given by the Riccati partial differential equations. These are simultaneous partial differential equations which are difficult to solve numerically. In most computational algorithms, errors are inevitable, a vast memory is required, and a lot of computational time is needed. This makes it impractical to use available computers for these algorithms. The algorithm presented here gives optimal control effectively without a large memory requirement. It also provides a practical computational method for obtaining the optimal control of multivariable systems with delay. Several numerical computations are performed to show how the effectiveness of the algorithm compares with other methods.     

PI stabilization of first-order systems with time delay

This paper considers the problem of stabilizing a first-order plant with dead time using a PI controller. Using a version of the Hermite–Biehler Theorem applicable to quasipolynomials, a complete and constructive characterization of all stabilizing PI gain values is obtained.     

Asymptotic invariance and stabilization of uncertain delay systems

This paper is concerned with stabilization of uncertain delay systems by continuous control. The parameters are assumed to take on unknown values from specified ranges. Asymptotic invariance methods are employed for parametric design of linear high-gain algorithms and regularized VSS algorithms which provide asymptotic stability of a closed-loop control system.     

Minimum energy control of systems with time delay

The problem of minimizing the cost functional corresponding to minimum control energyI = intmin{0}max{infty}u^{2} dtfor the linear system with delaydot{x}(t) = Ax(t) + Bx(t - h) + cu(t)is considered. From sufficient conditions for optimality, one obtains a set of algebraic equations. Numerical solution of these equations yields the optimal control law.     

Asymptotic solution of a discrete optimal control problem for one class of weakly controllable systems

An asymptotic expansion in integer nonnegative powers of small parameter for a solution of a discrete optimal control problem for one class of weakly controllable systems is constructed by substituting an assumed asymptotic expansion into the problem conditions and obtaining a series of problems in the coefficients of the asymptotics. Conditions of existence of a solution to the perturbed problem for sufficiently small values of the parameter are found. Estimates of closeness of the approximate and exact solutions in terms of trajectory, control, and functional are obtained. The values of the minimized functional are proven not to increase when higher-order asymptotic approximations of the optimal control are used. The discussion is illustrated by examples.     

A neural network solution for fixed-final time optimal control of nonlinear systems

In this paper, fixed-final time optimal control laws using neural networks and HJB equations for general affine in the input nonlinear systems are proposed. The method utilizes Kronecker matrix methods along with neural network approximation over a compact set to solve a time-varying HJB equation. The result is a neural network feedback controller that has time-varying coefficients found by a priori offline tuning. Convergence results are shown. The results of this paper are demonstrated on an example.     

Error analysis of the Chebyshev series solution of linear time invariant systems

The first part of this paper calculates the error between a function and its m-term Chebyshev series expansion. The second part calculates the error between the integral of a function and its m-term Chebyshev series obtained using the usual operational matrix of integration. The final part derives the error between the exact solution of a state equation and the m-term Chebyshev series solution obtained by the Chebyshev polynomial method.     

Multivariable singularly perturbed feedback systems with time delay

Multivariable singularly perturbed systems with both small and large time delays in the feedback loop are considered. Conditions for stability of the full-order closed-loop system are given in terms of properties of the slow and fast subsystems.     

Adaptive PID control of systems with unknown time delay

A new algorithm for the adaptive control of systems with unknown time delay is presented in which the underlying control law has the familiar PID structure. The design is based on a continuous-time approach. This algorithm can be applied to a class of high-order systems which can be modelled by either a second-order system with time delay or a first-order system with time delay. The parameters of the system including time delay are estimated recursively by a derivative-free least-squares algorithm. The controller settings are selected according to a set of analytical formulae derived from mean-square error criteria. The algorithm is tested on simulated examples as well as a laboratory-scale coupled-tank apparatus.     

Robust stability of uncertain large-scale systems with time delay

In this paper, the Lyapunov functional has been used to study the robust stability of uncertain large-scale systems with time delay; some new delay-independent conditions for stability of these uncertain large-scale systems are obtained. Finally, some examples are given to illustrate our results and to compare with previous results.     

Online identification of continuous-time systems with unknown time delay

In this note, we present a recursive algorithm for online identification of systems with unknown time delay. The proposed algorithm can be interpreted as an approximate nonlinear least-squares, corresponding to modified normalized or un-normalized least-squares when the normalizing factor /spl gamma/>0 or /spl gamma/=0, respectively. Both algorithms are essentially extensions to general least-squares methodology. Simulation studies demonstrate the characteristics and performance of these algorithms.     

Stability theory of hybrid dynamical systems with time delay

By defining the bounded time lag space and other related concepts, this note formulates the notion of hybrid dynamical systems with time delay, which provides a unified framework for describing a large class of systems and is suitable for qualitative analysis. It also investigates stability properties of invariant sets by the method of Lyapunov functions and the Razumihin techniques. Criteria on uniform stability and uniform asymptotic stability are established for hybrid dynamical systems with time delay. Those criteria are then applied to conduct stability analysis for impulsive delay differential systems and nonlinear sampled-data feedback control systems with time delay.     

Application of the SRIF to systems with time delay

Square-root information falter formulations of the discrete-time Kalman filter are sometimes thought to be restricted by a requirement that the state transition matrix be nonsingular. In a recent publication [1], Boncelet, Jr. and Dickinson have shown that this restriction is not necessary. For problems where both formulations can be applied, the generalized formulation is, however, more computation intensive than the direct formulation described by Bierman [2]. Also, it is shown here that transition singularities associated with time delays are removable and can be efficiently handled in the SRIF.     

Disturbance decoupling for time delay systems

In this paper the disturbance decoupling problem associated with a class of linear and nonlinear time delay systems is considered. The solution to this problem is given in terms of the relative degrees associated with the input and the disturbance of the corresponding time delay systems. A stability study using some results due to Pontryagin is presented for the resultant closed-loop system when the delay system is linear. Also, an approximate causal solution is proposed for the nonlinear time delay system based on its linearization around an equilibrium point. © 1997 by John Wiley & Sons, Ltd.