Achievable PID performance using sums of squares programming

In addition to process elements like time delay, the PID structure of the controller can pose fundamental limitations on the achievable control performance. A key difficulty in characterizing the limitations due to controller structure is the non-convexity of the resulting optimization problem. In this paper, we present a global lower bound on the achievable PID performance, defined in terms of output variance, by solving a series of convex programs using sums of squares programming. This result is also extended to minimize the weighted sum of the variances of input rate and output. The tightness of the proposed bounds is demonstrated using several benchmark examples drawn from literature.     

An extended PID type iterative learning control

This paper presents a new iterative learning control (ILC) for discrete-time single-input single-output (SISO) linear time-invariant (LTI) systems. To establish this ILC, the input of the controlled system is modified by using a novel four-parametric algorithm. This algorithm is called the extended proportional plus integral and derivative (EPID) type, since by eliminating the fourth parameter of it one would get to the PID type ILC, therefore PID type ILC is a special case of it. The convergence of the proposed ILC is analyzed and an optimal method is presented to determine its parameters. It is shown that the given ILC has a better performance than the PID-type one. Three illustrative examples are included to demonstrate the effectiveness and the preference of the presented ILC.     

迭代学习在PID控制中的应用

利用迭代学习过程记忆的期望控制、期望轨迹以及跟踪误差,提出了拟合控制系统的PID控制器参数的方法。用这种方法实现的PID控制器结构简单,作用于系统可获得较佳的动态特性和较强的鲁棒性。仿真实例表明了这种方法具有很好的可行性和实用性。     

An iterative procedure for solving convex optimal control problems

A doubly iterative procedure for computing optimal controls in linear systems with convex cost functionals is presented. The procedure is based on an algorithm due to Gilbert [3] for minimizing a quadratic form on a convex set. Each step of the procedure makes use of an algorithm due to Neustadt and Paiewonsky [1] to solve a strictly linear optimal control problem.     

Optimal control of batch reactors by iterative dynamic programming

Four batch reactor systems are chosen to examine the viability of using iterative dynamic programming (IDP) for highly nonlinear systems encountered by chemical engineers. The first system is mildly nonlinear and rapid convergence resulted with the use of only a single state grid point. The use of piecewise linear continuous control with 40 stages yielded better results that the use of 80 stages with piecewise constant control. The need for more than a single grid point for the other three systems led to a systematic study of the effects of the number of grid points, of the number of allowable values for control and of the region contraction factor on the convergence of IDP. In every case the global optimum could be obtained with reasonable computational effort, and no difficulties were encountered even with systems exhibiting several local optima. The use of stages of different length allowed a refined solution to be obtained with a reasonably small number of stages in the last example.     

Relay auto-tuning of PID controllers using iterative feedback tuning

In this paper, ideas from iterative feedback tuning (IFT) are incorporated into relay auto-tuning of the proportional-plus-integral-plus-derivative (PID) controller. The PID controller is auto-tuned to give specified phase margin and bandwidth. Good tuning performance according to the specified bandwidth and phase margin can be obtained and the limitation of the standard relay auto-tuning technique using a version of Ziegler-Nichols formula can be eliminated. Furthermore, by using common modelling assumptions for the relay system, some of the required derivatives in the IFT algorithm can be derived analytically. The algorithm was tested in the laboratory on a coupled tank and good tuning result was demonstrated.     

First-order sequential convex programming using approximate diagonal QP subproblems

Optimization algorithms based on convex separable approximations for optimal structural design often use reciprocal-like approximations in a dual setting; CONLIN and the method of moving asymptotes (MMA) are well-known examples of such sequential convex programming (SCP) algorithms. We have previously demonstrated that replacement of these nonlinear (reciprocal) approximations by their own second order Taylor series expansion provides a powerful new algorithmic option within the SCP class of algorithms. This note shows that the quadratic treatment of the original nonlinear approximations also enables the restatement of the SCP as a series of Lagrange-Newton QP subproblems. This results in a diagonal trust-region SQP type of algorithm, in which the second order diagonal terms are estimated from the nonlinear (reciprocal) intervening variables, rather than from historic information using an exact or a quasi-Newton Hessian approach. The QP formulation seems particularly attractive for problems with far more constraints than variables (when pure dual methods are at a disadvantage), or when both the number of design variables and the number of (active) constraints is very large.     

On the application of iterative dynamic programming to singularoptimal control problems

The convergence properties of an iterative dynamic programming algorithm are examined by considering a singular optimal control problem involving five differential equations. Even with a relatively coarse grid, convergence to the optimal control policy is rapid. The procedure is easy to program, and the computations can be easily done on a personal computer     

Robust iterative learning controller design using the performance weighting function of feedback control systems

Iterative learning controllers combined with existing feedback controllers have prominent capability of improving tracking performance in repeated tasks. However, the iterative learning controller has been designed without utilizing effective information such as the performance weighting function to design a feedback controller. In this paper, we deal with a robust iterative learning controller design problem for an uncertain feedback control system using its explicit performance information. We first propose a robust convergence condition in the ?₂-norm sense for an iterative learning control (ILC) scheme. We present a method to design an iterative learning controller using the information on the performance of the existing feedback control system such as performance weighting functions and frequency ranges of desired trajectories. From the obtained results, several design criteria for iterative learning controller are provided. Through analysis on the remaining error, the loop properties before and after learning are compared. We also show that, in the ?₂-norm sense, the remaining error can be less than the initial error under certain conditions. Finally, to show the validity of the proposed method, simulation studies are performed.     

多扰动动态下的PID控制器性能评价

对多扰动动态下PID控制器的性能评价进行了研究。当系统具有多个扰动动态时，PID控制器并不能使对多个扰动动态的控制性能都达到最佳，此时需根据对各扰动的重视程度来折中选择对其的控制性能。为了对此情况下系统的控制性能进行评价，文中构造了一种带加权的综合性能评价函数，其权值与对各扰动的重视程度相对应。通过选取适当的权值并对综合性能评价函数求最优，得到了系统对各扰动动态的最佳折中控制性能。以此最佳折中控制性能作为参照基准与系统实际性能相对比，实现了上述性能评价的目的。仿真例子表明了该方法的有效性。     

Pareto iterative learning control: Optimized control for multiple performance objectives

Iterative learning control (ILC) is a 2-degree-of-freedom technique that seeks to improve system performance along the time and iteration domains. Traditionally, ILC has been implemented to minimize trajectory-tracking errors across an entire cycle period. However, there are applications in which the necessity for improved tracking performance can be limited to a few specific locations. For such systems, a modified learning controller focused on improved tracking at the selected points can be leveraged to address multiple performance metrics, resulting in systems that exhibit significantly improved behaviors across a wide variety of performance metrics. This paper presents a pareto learning control framework that incorporates multiple objectives into a single design architecture.     

Performance assessment for iterative learning control of batch units

A new method is developed to estimate the minimum variance bounds and the achievable variance bounds for the assessment of the batch control system when the iterative learning control is applied. Unlike continuous processes, the performance assessment of batch processes requires particular attention to both disturbance changes and setpoint changes. Because of the intrinsically dynamic operations and the non-linear behavior of batch processes, the conventional approach of controller assessment cannot be directly applied. In this paper, a linear time-variant system for batch processes is used to derive the performance bounds from the routine operating batch data. The bounds at each time point computed from the deterministic setpoint and the stochastic disturbance for the controlled output variance can help create simple monitoring charts. They are used to track the progress easily in each batch run, to monitor the occurrence of the observable upsets, and to accordingly improve the current performance. The applications are discussed through simulation cases to demonstrate the advantages of the proposed strategies.     

Dynamic programming and convex clustering

In this paper the notion of convexity of clusterings for the given ordering of units is introduced. In the case when at least one (optimal) solution of the clustering problem is convex, dynamic programming leads to a polynomial algorithm with complexityO(kn ³). We prove that, for several criterion functions, convex optimal clusterings exist when dissimilarity is pyramidal for a given ordering of units.This research was supported in part by the Research Council of Slovenia.     

Stochastic iterative dynamic programming: a Monte Carlo approach to dual control

Practical exploitation of optimal dual control (ODC) theory continues to be hindered by the difficulties involved in numerically solving the associated stochastic dynamic programming (SDPs) problems. In particular, high-dimensional hyper-states coupled with the nesting of optimizations and integrations within these SDP problems render their exact numerical solution computationally prohibitive. This paper presents a new stochastic dynamic programming algorithm that uses a Monte Carlo approach to circumvent the need for numerical integration, thereby dramatically reducing computational requirements. Also, being a generalization of iterative dynamic programming (IDP) to the stochastic domain, the new algorithm exhibits reduced sensitivity to the hyper-state dimension and, consequently, is particularly well suited to solution of ODC problems. A convergence analysis of the new algorithm is provided, and its benefits are illustrated on the problem of ODC of an integrator with unknown gain, originally presented by Åström and Helmersson (Computers and Mathematics with Applications 12A (1986) 653-662).     

A practical iterative PID tuning method for mechanical systems using parameter chart

In this paper, we propose a method of iterative proportional-integral-derivative parameter tuning for mechanical systems that possibly possess hidden mechanical resonances, using a parameter chart which visualises the closed-loop characteristics in a 2D parameter space. We employ a hypothetical assumption that the considered mechanical systems have their upper limit of the derivative feedback gain, from which the feasible region in the parameter chart becomes fairly reduced and thus the gain selection can be extremely simplified. Then, a two-directional parameter search is carried out within the feasible region in order to find the best set of parameters. Experimental results show the validity of the assumption used and the proposed parameter tuning method.     

Refinery-wide control loop performance assessment

This paper discusses the application of control loop performance assessment (CLPA) in a refinery setting. The CLPA algorithm has several parameters that have to be adjusted correctly to give the best results. Procedures are described for selecting these parameters which make it feasible to implement the algorithm on a refinery-wide scale. We report practical experiences with the use of the techniques.