Brazilian Society of Simulation

Estimation of the parameters of a reducible (inflated common denominator) model for the transfer function matrix of MIMO systems is well known. However, the reduction of the model to the minimal form by pole-zero cancellation is possible only in the noise-free case. This paper presents an algorithm for the estimation of the minimal continuous-time transfer function matrix model. Monte Carlo simulation results are presented for discrete-time and continuous-time models. Least-squares and generalized least-squares methods have been used in both cases. An asymptotic analysis of convergence has also been provided for these models in the noise-free case. The computation times and space complexities of different variants of the algorithm are compared. The results show that in noisy situations, obtaining a discrete-time model by discretizing an estimated continuous-time model may be a viable proposition     

An EM-based identification algorithm for a class of hybrid systems with application to power electronics

In this paper we present an identification algorithm for a class of continuous-time hybrid systems. In such systems, both continuous-time and discrete-time dynamics are involved. We apply the expectation-maximisation algorithm to obtain the maximum likelihood estimate of the parameters of a discrete-time model expressed in incremental form. The main advantage of this approach is that the continuous-time parameters can directly be recovered. The technique is particularly well suited to fast-sampling rates. As an application, we focus on a standard identification problem in power electronics. In this field, our proposed algorithm is of importance since accurate modelling of power converters is required in high- performance applications and for fault diagnosis. As an illustrative example, and to verify the performance of our proposed algorithm, we apply our results to a flying capacitor multicell converter.     

Discrete-time minimal control synthesis adaptive algorithm

This article proposes a discrete-time Minimal Control Synthesis (MCS) algorithm for a class of single-input single-output discrete-time systems written in controllable canonical form. As it happens with the continuous-time MCS strategy, the algorithm arises from the family of hyperstability-based discrete-time model reference adaptive controllers introduced in (Landau, Y. (1979), Adaptive Control: The Model Reference Approach, New York: Marcel Dekker, Inc.) and is able to ensure tracking of the states of a given reference model with minimal knowledge about the plant. The control design shows robustness to parameter uncertainties, slow parameter variation and matched disturbances. Furthermore, it is proved that the proposed discrete-time MCS algorithm can be used to control discretised continuous-time plants with the same performance features. Contrary to previous discrete-time implementations of the continuous-time MCS algorithm, here a formal proof of asymptotic stability is given for generic n-dimensional plants in controllable canonical form. The theoretical approach is validated by means of simulation results.     

基于即时交货的离散时间模型及其在炼油过程调度优化中的应用

针对炼油过程调度优化的离散时间与连续时间建模方法在应用中的局限性,提出一种基于即时交货的离散时间建模方法.该方法总体上采用离散时间表示,通过对成品油交货环节相关部分更加细致的描述,使得交货活动能够在时间片段内部发生,既可保证对实际问题的准确刻画,又能实现对模型规模的有效控制.以华北地区某炼油厂的改质及调合过程为例,分别建立离散时间模型、连续时间模型以及所提出的基于即时交货的离散时间模型,并结合对3个案例的求解,从不同的角度对3种模型进行对比研究.研究结果表明,所提出的模型兼具原离散时间模型和连续时间模型的优势,能够在不同的情形下保持稳定且优良的性能.     

A novel hybrid metaheuristic algorithm for model order reduction in the delta domain: a unified approach

Delta operator parameterization provides a unified framework in modeling, analysis and design of discrete-time systems, in which the resultant model converges to its continuous-time counterpart at high sampling limit. Capitalizing this unique property of delta operator, a new hybrid algorithm combining gray wolf optimizer and firefly algorithm has been proposed for model order reduction of high-dimensional linear discrete-time system. It has been shown that the reduced discrete-time model inherits all the dominant characteristics of the higher-order discrete-time model and with the increase in sampling frequency it converges to the continuous-time reduced model. The effectiveness of the proposed method is illustrated with the help of an example.

     

Rapprochement between continuous and discrete model reference adaptive control

In the literature to date there is a dichotomy between results on continuous- and discrete-time model reference control. This is highlighted in the case of continuous-time systems having relative degree greater than one. It is known that, for rapid sampling, these systems always give rise to a non-stably invertible discrete-time system and thus discrete model reference control is ruled out. On the other hand, there are many results pertaining to continuous-time model reference control of such systems. This apparent paradox can be resolved by a slight modification to the discrete-time model format as shown later. This alternative model is used to develop a new discrete model reference adaptive control law and a convergence analysis for the algorithm is presented.     

On a general optimal algorithm for multirate output feedbackcontrollers for linear stochastic periodic systems

A modified optimal algorithm for multirate output feedback controllers of linear stochastic periodic systems is developed. By combining the discrete-time linear quadratic regulation (LQR) control problem and the discrete-time stochastic linear quadratic regulation (SLQR) control problem to obtain an extended linear quadratic regulation (ELQR) control problem, one derives a general optimal algorithm to balance the advantages of the optimal transient response of the LQR control problem and the optimal steady-state regulation of the SLQR control problem. In general, the solution of this algorithm is obtained by solving a set of coupled matrix equations. Special cases for which the coupled matrix equations can be reduced to a discrete-time algebraic Riccati equation are discussed. A reducable case is the optimal algorithm derived by H.M. Al-Rahmani and G.F. Franklin (1990), where the system has complete state information and the discrete-time quadratic performance index is transformed from a continuous-time one     

Stability analysis and synthesis of fuzzy singularly perturbed systems

In this paper, we investigate the stability analysis and synthesis problems for both continuous-time and discrete-time fuzzy singularly perturbed systems. For continuous-time case, both the stability analysis and synthesis can be parameterized in terms of a set of linear matrix inequalities (LMIs). For discrete-time case, only the analysis problem can be cast in LMIs, while the derived stability conditions for controller design are nonlinear matrix inequalities (NMIs). Furthermore, a two-stage algorithm based on LMI and iterative LMI (ILMI) techniques is developed to solve the resulting NMIs and the stabilizing feedback controller gains can be obtained. For both continuous-time and discrete-time cases, the reduced-control law, which is only dependent on the slow variables, is also discussed. Finally, an illustrated example based on the flexible joint inverted pendulum model is given to illustrate the design procedures.     

Observers for state-affine systems

A system whose discrete-time model is linear in state variables and nonlinear in control variables is considered. Under the assumption of generic observability (under single experiment) an exact observer algorithm is suggested. Also, it is shown that the existence of a general nonlinear asymptotic observer implies the existence of a linear one. Two design procedures for asymptotic observers are presented. Surprisingly, one of them extends to a rather complete solution for continuous-time systems with piecewise constant controls provided discrete output measurements are more frequent than switches of control values.     

Solving for time-varying and static cube roots in real and complex domains via discrete-time ZD models

Different from conventional gradient-based neural dynamics, a special type of neural dynamics has been proposed by Zhang et al. for online solution of time-varying and/or static (or termed, time-invariant) problems. The design of Zhang dynamics (ZD) is based on the elimination of an indefinite error function, instead of the elimination of a square-based positive (or at least lower-bounded) energy function usually associated with gradient dynamics (GD). In this paper, we generalize, propose and investigate the continuous-time ZD model and its discrete-time models in two situations (i.e., the time-derivative of the coefficient being known or unknown) for time-varying cube root finding, including the complex-valued continuous-time ZD model for finding cube roots in complex domain. In addition, to find the static scalar-valued cube root, a simplified continuous-time ZD model and its discrete-time model are generated. By focusing on such a static problem solving, Newton-Raphson iteration is found to be a special case of the discrete-time ZD model by utilizing the linear activation function and fixing the step-size value to be 1. Computer-simulation and testing results demonstrate the efficacy of the proposed ZD models (including real-valued ZD models and complex-valued ZD models) for time-varying and static cube root finding, as well as the link and new explanation to Newton-Raphson iteration.     

Difference algebra and system identification

The framework of differential algebra, especially Ritt’s algorithm, has turned out to be a useful tool when analyzing the identifiability of certain nonlinear continuous-time model structures. This framework provides conceptually interesting means to analyze complex nonlinear model structures via the much simpler linear regression models. One difficulty when working with continuous-time signals is dealing with white noise in nonlinear systems. In this paper, difference algebraic techniques, which mimic the differential-algebraic techniques, are presented. Besides making it possible to analyze discrete-time model structures, this opens up the possibility of dealing with noise. Unfortunately, the corresponding discrete-time identifiability results are not as conclusive as in continuous time. In addition, an alternative elimination scheme to Ritt’s algorithm will be formalized and the resulting algorithm is analyzed when applied to a special form of the nfir model structure.     

Markowitz's mean-variance portfolio selection with regime switching: from discrete-time models to their continuous-time limits

We study a discrete-time version of Markowitz's mean-variance portfolio selection problem where the market parameters depend on the market mode (regime) that jumps among a finite number of states. The random regime switching is delineated by a finite-state Markov chain, based on which a discrete-time Markov modulated portfolio selection model is presented. Such models either arise from multiperiod portfolio selections or result from numerical solution of continuous-time problems. The natural connections between discrete-time models and their continuous-time counterpart are revealed. Since the Markov chain frequently has a large state space, to reduce the complexity, an aggregated process with smaller state-space is introduced and the underlying portfolio selection is formulated as a two-time-scale problem. We prove that the process of interest yields a switching diffusion limit using weak convergence methods. Next, based on the optimal control of the limit process obtained from our recent work, we devise portfolio selection strategies for the original problem and demonstrate their asymptotic optimality.     

An efficient sequential linear quadratic algorithm for solving nonlinear optimal control problems

We develop a numerically efficient algorithm for computing controls for nonlinear systems that minimize a quadratic performance measure. We formulate the optimal control problem in discrete-time, but many continuous-time problems can be also solved after discretization. Our approach is similar to sequential quadratic programming for finite-dimensional optimization problems in that we solve the nonlinear optimal control problem using sequence of linear quadratic subproblems. Each subproblem is solved efficiently using the Riccati difference equation. We show that each iteration produces a descent direction for the performance measure, and that the sequence of controls converges to a solution that satisfies the well-known necessary conditions for the optimal control.     

Linearization of discrete-time systems via restricted dynamic feedback

We extend the results from a previous paper of ours to multiple-input systems, and utilize these to obtain necessary and sufficient conditions for the linearization of discrete-time nonlinear systems via restricted dynamic feedback. We observe that for discrete-time nonlinear systems, the bound on the number of delays (or integrators) needed to synthesize the linearizing dynamic feedback differs from the continuous-time analogue.     

Infinite horizon LQR for systems with multiple delays in a single input channel

This paper is concerned with the linear quadratic regulation (LQR) problem for both linear discrete-time systems and linear continuous-time systems with multiple delays in a single input channel. Our solution is given in terms of the solution to a two-dimensional Riccati difference equation for the discrete-time case and a Riccati partial differential equation for the continuous-time case. The conditions for convergence and stability are provided.     

Optimal H∞ model reduction via linear matrix inequalities: continuous- and discrete-time cases

Necessary and sufficient conditions are derived for the existence of a solution to the continuous-time and discrete-time H_∞ model reduction problems. These conditions are expressed in terms of linear matrix inequalities (LMIs) and a coupling non-convex rank constraint set. In addition, an explicit parametrization of all reduced-order models that correspond to a feasible solution is provided in terms of a contractive matrix. These results follow from the recent solution of the H_∞ control design problem using LMIs. Particularly simple conditions and a simple parametrization of all solutions are obtained for the zeroth-order H_∞ approximation problem, and the convexity of this problem is demonstrated. Computational issues are discussed and an iterative procedure is proposed to solve the H_∞ model reduction problem using alternating projections, although global convergence of the algorithm is not guaranteed.     

Delta-operator formulated discrete-time approximations ofcontinuous-time systems

Given a continuous-time system, a technique to directly obtain an approximate delta-operator formulated discrete-time system (δ-system) is presented. For this purpose, the analog of the well known Boxer-Thaler integrators (q-forms) applicable to shift-operator formulated discrete-time systems (q-systems) are derived for δ-systems. Next, using these δ-forms, a method to obtain an approximate δ-system of a given continuous-time system is derived. This algorithm is easily implementable in a computer with little computational burden. It is shown that, as sampling time decreases, the δ-system thus obtained yields the given continuous-time system further verifying the close equivalence between this formulation and continuous-time systems. Two examples illustrating advantages that may be gained by utilizing these δ-forms in digitizing analog systems are also included     

Solving time-varying nonlinear inequalities using continuous and discrete-time Zhang dynamics

A special kind of neural dynamics, termed Zhang dynamics (ZD), is proposed, generalized and investigated for the online solution of time-varying scalar-valued nonlinear inequalities by following Zhang et al.’s design method. The continuous-time ZD (CTZD) model based on an exponent-type design formula can be guaranteed to exponentially converge to the time-varying solution set of the problem in an error-free manner. For potential hardware implementation on digital circuits, the corresponding discrete-time ZD (DTZD) model is generated through the well-known Euler forward difference rule. Newton-type algorithm is also developed for comparison purposes. In addition, the simplified CTZD (S-CTZD) and DTZD (S-DTZD) models are developed for solving static scalar-valued nonlinear inequalities. Numerical simulative examples further demonstrate and verify the efficacy of the ZD models for solving time-varying and static scalar-valued nonlinear inequalities. Besides, the DTZD model possesses the lower complexity and higher accuracy, as compared with the Newton-type algorithm.     

Realization of continuous-time positive linear systems

Positive linear systems are used in biomathematics, economics, and other research areas. For discrete-time positive linear systems, part of the realization problem has been solved. In this paper the solution of the corresponding problem for continuous-time positive linear systems will be presented, which can be deduced from that of the discrete-time case by a transformation. Sufficient and necessary conditions for the existence of a positive realization are presented. To solve the problem of minimality, the solution of the factorization of positive matrices is needed.     

On uniform asymptotic stability of time-varying parameterized discrete-time cascades

Recently, a framework for controller design of sampled-data nonlinear systems via their approximate discrete-time models has been proposed in the literature. In this paper, we develop novel tools that can be used within this framework and that are useful for tracking problems. In particular, results for stability analysis of parameterized time-varying discrete-time cascaded systems are given. This class of models arises naturally when one uses an approximate discrete-time model to design a stabilizing or tracking controller for a sampled-data plant. While some of our results parallel their continuous-time counterparts, the stability properties that are considered, the conditions that are imposed, and the the proof techniques that are used, are tailored for approximate discrete-time systems and are technically different from those in the continuous-time context. A result on constructing strict Lyapunov functions from nonstrict ones that is of independent interest, is also presented. We illustrate the utility of our results in the case study of the tracking control of a mobile robot. This application is fairly illustrative of the technical differences and obstacles encountered in the analysis of discrete-time parameterized systems.