State-space forms for higher-order discrete-time models

This paper presents a fundamental study of the connection between continuous- and discrete-time systems. Provided is a definition for discrete-time models, that is discrete-time systems with a continuous-time counterpart, whose order can be higher than that of the continuous-time system. This definition is based on a comparison in a certain sense on the time responses of continuous- and discrete-time systems. A theorem is presented for relating the higher-order discrete-time models to their continuous-time counterparts, which is an extension of a previous theorem for models with order equal to that of the continuous-time system. State-space forms are derived for models obtained through the use of a certain class of hold elements and through the use of mapping models, and these discrete-time systems are shown to be valid according to the definition. Special cases are models obtained using first-order and slewer hold devices, whose convergence to a continuous-time counterpart has not been shown mathematically before, and mapping models corresponding to two-step linear multi-step methods, which have not previously presented in the state-space form. The derived state-space forms provide a convenient way to implement these models for purposes of analysis, design, and implementation of discrete-time systems and finds applications in such areas as digital signal processing, digital simulation, and digital control.     

A computational method for simultaneous LQ optimal control designvia piecewise constant output feedback

The paper is concerned with simultaneous linear-quadratic (LQ) optimal control design for a set of LTI systems via piecewise constant output feedback. First, the discrete-time simultaneous LQ optimal control design problem is reduced to solving a set of coupled matrix inequalities and an iterative LMI algorithm is presented to compute the feedback gain. Then, simultaneous stabilization and simultaneous LQ optimal control design of a set of LTI continuous-time systems are considered via periodic piecewise constant feedback gain. It is shown that the design of a periodic piecewise constant feedback gain simultaneously minimizing a set of given continuous-time performance indexes can be reduced to that of a constant feedback gain minimizing a set of equivalent discrete-time performance indexes. Explicit formulas for computing the equivalent discrete-time systems and performance indexes are derived. Examples are used to demonstrate the effectiveness of the proposed method.     

Stability of sampled-data piecewise affine systems: A time-delay approach

This paper addresses the stability analysis of sampled-data piecewise-affine (PWA) systems consisting of a continuous-time plant in feedback connection with a discrete-time emulation of a continuous-time state feedback controller. The sampled-data system is first considered as a continuous-time system with a variable time delay. Conditions under which the trajectories of the sampled-data closed-loop system will converge to an attracting invariant set are then presented. It is also shown that when the sampling period converges to zero, these conditions coincide with sufficient conditions for non-fragility of the stabilizing continuous-time PWA state feedback controller. The results are successfully applied to a helicopter example.     

A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models

A unified framework for design of stabilizing controllers for sampled-data differential inclusions via their approximate discrete-time models is presented. Both fixed and fast sampling are considered. In each case, sufficient conditions are presented which guarantee that the controller that stabilizes a family of approximate discrete-time plant models also stabilizes the exact discrete-time plant model for sufficiently small integration and/or sampling periods. Previous results in the literature are extended to cover: 1) continuous-time plants modeled as differential inclusions; 2) general approximate discrete-time plant models; 3) dynamical discontinuous controllers modeled as difference inclusions; and 4) stability with respect to closed arbitrary (not necessarily compact) sets.     

Computation of all stabilizing PID gains for digital controlsystems

This note determines the set of all proportional-integral-derivative (PID) gains that can stabilize a given discrete-time plant of arbitrary order. Utilizing earlier results obtained for the continuous-time case and the bilinear transformation, it is shown that a complete solution can be obtained using linear programming. Several numerical examples are presented to illustrate the procedure     

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.     

Subspace identification of continuous time models for process fault detection and isolation

This paper proposes a novel subspace approach towards identification of optimal residual models for process fault detection and isolation (PFDI) in a multivariate continuous-time system. We formulate the problem in terms of the state space model of the continuous-time system. The motivation for such a formulation is that the fault gain matrix, which links the process faults to the state variables of the system under consideration, is always available no matter how the faults vary with time. However, in the discrete-time state space model, the fault gain matrix is only available when the faults follow some known function of time within each sampling interval. To isolate faults, the fault gain matrix is essential. We develop subspace algorithms in the continuous-time domain to directly identify the residual models from sampled noisy data without separate identification of the system matrices. Furthermore, the proposed approach can also be extended towards the identification of the system matrices if they are needed. The newly proposed approach is applied to a simulated four-tank system, where a small leak from any tank is successfully detected and isolated. To make a comparison, we also apply the discrete time residual models to the tank system for detection and isolation of leaks. It is demonstrated that the continuous-time PFDI approach is practical and has better performance than the discrete-time PFDI approach.     

Matrosov theorem for parameterized families of discrete-time systems

A version of Matrosov's theorem for parameterized discrete-time time-varying systems is presented. The theorem is a discrete-time version of the continuous-time result in Loria et al., 2002 (δ-persistency of excitation: a necessary and sufficient condition for uniform attractivity, 2002, submitted for publication). Our result facilitates controller design for sampled-data nonlinear systems via their approximate discrete-time models. An application of the theorem to establishing uniform asymptotic stability of systems controlled by model reference adaptive controllers designed via approximate discrete-time plant models is presented.     

A series of discrete-time models of a continuous-time system based on fluency signal approximation

A discrete-time system model based on the discretization of a continuous-time system has been called a sampled-data system. But, in such discretization of the continuous-time system, it has been assumed that input signal u(t) is a staircase signal, that is, u(τ) has a constant value of u(kh) = u_k over the integration interval. The present paper derives a series of discrete-time models of a continuous-time system based on m-order fluency signal approximation. It is revealed that the series of models includes and generalizes the conventional system model based on the assumption of staircase signal input (m = 1). Furthermore, the adaptive discretization is obtained by selecting the appropriate order m according to the characteristic (continuous differentiability) of the input signal of the continuous-time system we are dealing with. Thus, this concept provides a better family of the relationship between the discrete-time system model and the continuous-time system     

A simplified approach to Bode's theorem for continuous-time anddiscrete-time systems

A simplified approach to W.H. Bode's (1945) theorem for both continuous-time and discrete-time systems, along with some generalization, are presented. For continuous-time systems, the constraints of open-loop stability and roll-off at s=∝ are removed. A counterexample shows that when the excess poles/zeros vanishes, the Bode integral drops from infinite to finite value when the open-loop gain crosses a critical value. A revised result is also developed. The salient feature of this approach is that at no stage are either Cauchy's theorem or the Poisson integral invoked; the simplified proof relies only on elementary analysis. This approach carries over to the discrete-time cases in a straightforward manner     

Parameter estimation of excitation systems from sampled data

The parameter estimation of excitation systems using a proposed indirect stochastic method is presented. In this method, the discrete-time ARMA model corresponding to each block of the excitation system is first identified from sampled input/output data. Then, the proper reduced order of the continuous-time model is determined using the concept of dominant energy modes. Finally, the parameters of the continuous-time model are found by matching its frequency response to that of the identified ARMA model. The proposed method is used to estimate the parameters of an excitation system in a pumped storage power plant. The results show that the estimated continuous-time models are rather close to those supplied by the vender     

Fault detection for nonlinear discrete-time systems via deterministic learning

Recently, an approach for the rapid detection of small oscillation faults based on deterministic learning theory was proposed for continuous-time systems. In this paper, a fault detection scheme is proposed for a class of nonlinear discrete-time systems via deterministic learning. By using a discrete-time extension of deterministic learning algorithm, the general fault functions (i.e., the internal dynamics) underlying normal and fault modes of nonlinear discrete-time systems are locally-accurately approximated by discrete-time dynamical radial basis function (RBF) networks. Then, a bank of estimators with the obtained knowledge of system dynamics embedded is constructed, and a set of residuals are obtained and used to measure the differences between the dynamics of the monitored system and the dynamics of the trained systems. A fault detection decision scheme is presented according to the smallest residual principle, i.e., the occurrence of a fault can be detected in a discrete-time setting by comparing the magnitude of residuals. The fault detectability analysis is carried out and the upper bound of detection time is derived. A simulation example is given to illustrate the effectiveness of the proposed scheme.     

On the discrete-time dynamics of the basic Hebbian neural network node

In this paper, the dynamical behavior of the basic node used for constructing Hebbian artificial neural networks (NNs) is analyzed. Hebbian NNs are employed in communications and signal processing applications, among others. They have been traditionally studied on a continuous-time formulation whose validity is justified via some analytical procedures that presume, among other hypotheses, a specific asymptotic behavior of the learning gain. The main contribution of this paper is the study of a deterministic discrete-time (DDT) formulation that characterizes the average evolution of the node, preserving the discrete-time form of the original network and gathering a more realistic behavior of the learning gain. The new deterministic discrete-time model provides some unstability results (critical for the case of large similar variance signals) which are drastically different to the ones known for the continuous-time formulation. Simulation examples support the presented results, illustrating the practical limitations of the basic Hebbian model.     

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.     

Intelligent Control of Discrete Linear Systems Based on a Supervised Adaptive Multiestimation Scheme

A pole-placement based adaptive controller synthesised from a multiestimation scheme is designed for linear plants. A higher level switching structure between the various estimation schemes is used to supervise the reparameterisation of the adaptive controller in real time. The basic usefulness of the proposed scheme is to improve the transient response so that the closed-loop stability is guaranteed. The switching process is subject to a minimum dwelling or residence time within which the supervisor is not allowed to switch between the multiple estimation schemes. The high level supervision is based on the multiestimation identification scheme. The residence time condition guarantees the closed-loop stability. The above higher level switching structure is on-line supervised by a closed-loop tracking error based algorithm. This second supervision on-line tunes the free design parameters which appear as time varying weights in the loss function of the above switching structure. Thus, the closed-loop behaviour, compared to the constant parameter case one, is improved when the design parameter is not tightly initialised. Both supervisors are hierarchically organised in the sense that they act on the system at different rates. Furthermore, a projection algorithm has been considered in the estimation scheme in order to include a possible a priori knowledge of the estimates parameter vector value in the estimation algorithm.     

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     

Adaptive pole placement without excitation probing signals

This paper presents an indirect adaptive control scheme for linear systems which may possibly be a nonminimum phase. The control scheme achieves asymptotical pole placement without either introducing persistent excitation probing signals into the systems or assuming any a priori knowledge on the plant parameters. The system order is the only a priori knowledge required on the plant. The adaptive control law is free from singularities in the sense that the estimated plant model is always controllable. The singularities are overcome by a suitable parameter estimates modification which is based upon standard least squares covariance matrix properties. The analysis of the stability and the global convergence of a closed-loop system is given in detail for both discrete-time and continuous-time systems     

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.     

Constrained controllability of discrete-time systems†

The constrained controllability of the discrete-time system x_k+1=A(k)x_k+B(k)u,_k is considered where the control uk is termed admissible if it satisfies specified magnitude constraints. Constrained controllability is concerned with the existence of an admissible control which steers the state x to a given target set from a specified initial state

Conditions for checking constrained controllability to a given target set from a specified initial state are presented. These conditions involve solving finite-dimensional optimization problems and can be checked via numerical computation. In addition, conditions for checking global constrained controllability to a given target set are presented. A system is globally constrained controllable if for every initial state, there exists an admissible control that steers the system to the target.

If a given discrete-time system is constrained controllable, it may be desirable to obtain an admissible control that steers the system to the target from a specified initial state. Such a control is called a steering control. Results for computing steering controls are also presented

This paper is concluded with a numerical example. In this example, it is shown that the constrained controllability of a continuous-time system which has been discretized is dependent on the discretization time. The set of states which can be steered to the target changes as the discretization time changes. Furthermore, the example shows that a discrete-time steering control cannot always be obtained by discretizing a continuous-time steering control; the steering control for the discrete-time system must be obtained directly from the discrete-time model.