Generalized Sampled-data Hold Functions Research Articles

A sufficient and necessary condition for stabilization of zeros in discrete-time multirate sampled systems

It is well known that the presence of unstable zeros limits the control performance, which can be achieved, and some control schemes cannot be directly applied. This manuscript investigates a sufficient condition for stabilization of zeros in the discrete-time multirate sampled systems. It is shown that the discretization zeros, especially sampling zeros, can be arbitrarily placed inside the unit circle in the case of relative degree being greater than or equal to three and multirate input and hold, such as generalized sampled-data hold function (GSHF), for the linear continuous-time systems when the sampling period T tends to zero. Moreover, the authors further consider the multivariable case, which has a similar asymptotic behavior. Finally, the simulation proves the validity of the method, and we also propose a hold design that places the sampling zeros asymptotically to the origin for fast sampling rates.

Design for Networked Control Systems Based on Multi-rate Technique

This paper is concerned with the design of networked control systems based on multi-rata technique. Firstly, based on the Fourier series of periodic signal, a novel nonlinear two part periodic function (NTPPF) method is proposed to construct the generalized sampled-data hold function (GSHF). Secondly, a hybrid nonlinear delay system model is con- structed by establishing the relationship between the network-induced delay and the frequency of the NTPPF. Further- more, by using the Lyapunov theory, the stability condition can be obtained. Finally, a numerical example is provided to demonstrate the effectiveness and less conservativeness of the proposed methods.

Generalized sampled-data holds to reduce energy consumption in resonant systems

This paper presents an exhaustive experimental study on the performance of a new design of generalized sampled-data hold function (GSHF) introduced in Ugalde, Bárcena, and Basterretxea (2012). First a simple tuning procedure is developed for the GSHF to improve the intersample response while not damaging the plant output response. Next the reconstruction algorithm is enhanced so that the GSHF can remove the steady-state errors that dry friction causes in positioning systems. The experiments conducted on a small-scale lightly damped resonant system show that significant energy savings can be achieved in regulation with no extra hardware, and that no additional computational load results, except having to call the D/A converter more than once per sampling period.

Pole Assignment With Improved Control Performance by Means of Periodic Feedback

This technical note is concerned with the pole placement of continuous-time linear time-invariant (LTI) systems by means of LQ suboptimal periodic feedback. It is well-known that there exist infinitely many generalized sampled-data hold functions (GSHF) for any controllable LTI system to place the modes of its discrete-time equivalent model at prescribed locations. Among all such GSHFs, this technical note aims to find the one which also minimizes a given LQ performance index. To this end, the GSHF being sought is written as the sum of a particular GSHF and a homogeneous one. The particular GSHF can be readily obtained using the conventional pole-placement techniques. The homogeneous GSHF, on the other hand, is expressed as a linear combination of a finite number of functions such as polynomials, sinusoidals, etc. The problem of finding the optimal coefficients of this linear combination is then formulated as a linear matrix inequality (LMI) optimization. The procedure is illustrated by a numerical example.

Discrete-time control of continuous systems with approximate decentralized fixed modes

In this paper, the discrete-time control of decentralized continuous-time systems, which have approximate decentralized fixed modes, is studied. It is shown that under certain conditions, discrete-time controllers can improve the overall performance of the decentralized control system, when a linear time-invariant continuous-time controller is ineffective. In order to obtain these conditions, a quantitative measure for different types of approximate fixed modes in a decentralized system is given. In this case, it is shown that discrete-time zero-order hold (ZOH) controllers, and in particular, that generalized sampled-data hold functions (GSHF), can significantly improve the overall performance of the resultant closed-loop system. The proposed sampled-data controller is, in fact, a linear time-varying controller for the continuous-time system.

Optimal periodic feedback design for continuous-time LTI systems with constrained control structure

This paper aims to design a high-performance controller with any predefined structure for continuous-time LTI systems. The control law employed is the generalized sampled-data hold function (GSHF), which can have any special form, e.g. polynomial, exponential, piecewise constant, etc. The GSHF is first written as a linear combination of a set of basis functions obtained in accordance with its desired form and structure. The objective is to find the coefficients of this linear combination, such that a prespecified linear-quadratic performance index is minimized. A necessary and sufficient condition for the existence of such GSHF is first obtained in the form of matrix inequality, which can be solved by using the existing methods to obtain a set of stabilizing initial values for the coefficients or to conclude the non-existence of such structurally constrained GSHF. An efficient algorithm is then presented to compute the optimal coefficients from their initial values, so that the performance index is minimized. The paper utilizes the latest developments in the area of sum-of-square polynomials. The effectiveness of the proposed method is demonstrated in two numerical examples.

Robustness of linear uncertain sampled‐data control systems with generalized sampled‐data hold functions

AbstractThis paper investigates robust stability of linear time‐invariant (LTI) uncertain sampled‐data control systems with generalized sampled‐data hold functions (GSHFs). A new sufficient condition for robust stability of such systems is developed. Unlike that of most of the previous works, it directly uses the data of the continuous‐time plant and therefore it is less conservative. The condition is expressed in terms of the spectral radius of a certain matrix and is shown to be a unimodal function of a free parameter. This property enabled us to use standard one‐dimensional optimization algorithm to perform the proposed test. Copyright © 2006 John Wiley & Sons, Ltd.

Structural modification of systems using discretization and generalized sampled-data hold functions

This paper investigates the decentralized control of LTI continuous-time plants using generalized sampled-data hold functions (GSHF). GSHFs can be used to modify the structure of the digraph of the resultant discrete plant, by removing certain interconnections in the discrete-time equivalent model to form a hierarchical system model of the plant. This is a new application of discretization and has, as its motivation, the design of decentralized controllers using centralized methods.

Multi-layer Switching Control using Generalized Sampled-data Hold Functions

In this paper, a novel switching control architecture for linear time-invariant (LTI) systems using generalized sampled-data hold functions (GSHF) is proposed to reduce the magnitude of the transient response. It is assumed that the plant model belongs to a finite set of known models. The output of the system is periodically sampled and a control signal is being generated by using a suitable hold function from a set of GSHFs. A control architecture consisting of a layer of high-performance GSHFs(one for each plant model) and some other layers of simultaneous stabilizing GSHFs is introduced. It is shown that using the above sets of GSHFs and a proper switching path,onecanreduce thenumberof switchings to destabilizing GSHFs. As a result, the transient performance which is the main shortcoming of most switching control schemes is improved using the proposed strategy. Furthermore, it is shown that using GSHFs instead of continuous-time controllers reduces the complexity ofonline computations required to obtain the upper-bound signals. Simulation results show the effectiveness of the proposed method in improving the transient performance.

APPLICATION OF GENERALIZED SAMPLED-DATA HOLD FUNCTIONS TO DECENTRALIZED CONTROL STRUCTURE MODIFICATION

This paper investigates the decentralized control of LTI continuous-time plants using Generalized Sampled-data Hold Functions (GSHF). GSHFs can be used to modify the structure of the digraph of the resultant discrete plant, by removing certain interconnections in the equivalent discrete-time model to form a hierarchical system model of the plant. This is a new application of discretization, and has as its motivation, the design of decentralized controllers using centralized methods.

Tuning of controllers and generalized hold functions in sampled-data systems using iterative feedback tuning

In this paper it is shown that Iterative Feedback Tuning can be applied to multi-input multi-output linear sampled-data systems where the sensed outputs may be sampled at different rates and where the controllers operate synchronously at one sampling rate which is a multiple of the sampling rates of the controlled variables. It is also shown that generalized sampled-data hold functions (GSDHF) can be tuned. By way of an example, it is shown that one case where GSDHF's may improve the performance significantly is when the continuous time system is minimum phase but the sampled counterpart is non-minimum phase.

Gain/phase margin improvement using static generalized sampled-data hold functions

This paper considers the robust control of a finite-dimensional linear time-invariant (FDLTI) continuous-time plant by a static generalized sampled-data hold function (GSHF) controller. It is shown that it is possible to design a static GSHF controller to provide a gain margin as large as desired, and a phase margin of up to 90°. The design is straight-forward, and we illustrate it with an example.

Robustness of zero shifting via generalized sampled-data hold functions

In this paper we study robustness and sensitivity properties of a sampled-data feedback system with a generalized sampled-data hold function (GSHF). We argue that shifting non-minimum phase zeros using GSHF control can lead to difficulties unless the zero is outside the closed-loop bandwidth.

Non-pathological sampling for generalized sampled-data hold functions

Given a generalized sampled-data hold function (GSHF), we define a frequency dependent response function having many properties analogous to those of a transfer function. In particular, the zeros of the response function have transmission blocking properties. We then study the problem of non-pathological sampling with a GSHF. Sampling is said to be pathological if the discretized version of a stabilizable/detectable continuous time plant is not itself stabilizable and detectable. Sufficient conditions for nonpathological sampling with a zero order hold have long been known. We extend these to the case of a GSHF, and describe the role in non-pathological sampling played by right half plane zeros of the response function. The results are presented for square multivariable linear systems and include a generalization to allow for a time delay.

Assignment of nonlinear sampled‐data dynamics using generalized hold function control

This paper considers the problem of assigning the dynamics of a nonlinear analytic system using nonlinear generalized sampled‐data hold function (GSHF) control, in the neighborhood of an equilibrium point. On every sampling interval, the control input consists of a nonlinear time‐periodic function applied to the sampled value of the state vector. The nonlinear monodromy map is the state transition map from one sample time to the next. It is shown that this map is arbitrarily assignable by GSHF feedback if and only if the linear part of the system is controllable. Two approaches are proposed to construct a GSHF controller that performs the assignment. The first approach matches the coefficients of the Taylor series expansion of the monodromy map around the equilibrium. The second approach interpolates an optimal control law at several points in the vicinity of the equilibrium. These approaches are illustrated on an example.

Gain margin improvement using generalized sampled-data hold function based multirate output compensator

For a SISO, strictly proper, nonminimum-phase, continuous-time, linear time-invariant (LTI) plant, it was shown in Yan, W., B. D. O. Anderson and R. R. Bitmead (1993). IEEE Trans. Automat. Contr., that the closed-loop gain margin obtained via a generalized sampled-data hold function (GSHF) based dynamic compensator is significantly improved over that achieved via a conventional periodic controller used in Francis, B. A. and T. T. Georgiou (1988). IEEE Trans. Automat. Contr., AC-33, 820–832. Nevertheless, the compensator so designed is not necessarily strictly proper. It is well-known that it is practically difficult and sometimes impossible to implement a nonstrictly proper compensator due to the requirement of zero computation time. Further, as has been shown in Vidyasagar, M. (1985). Systems and Control Letters, 5, 413–418, stabilization by a nonstrictly proper controller is never robust against singular perturbations. In this paper, we propose a new type of GSHF based compensator which employs multirate sampling of the plant output. Using the proposed compensator, not only the same level of gain margin as in Yan, W., B. D. O. Anderson and R. R. Bitmead (1993). IEEE Trans. Automat. Contr., can be achieved, but also, more importantly, the compensator is strictly proper.

Multi-channel output gain margin improvement using generalized sampled-data hold functions

This paper considers the multichannel gain margin problem for linear time-invariant continuous-time multi-input multi-output plants controlled by memoryless generalized sampled-data hold functions (GSHF). It is proved that arbitrarily large output gain margins can be achieved by suitable selection of the sampling time and the hold function. We use state space methods and give a constructive design procedure. The controller we use is memoryless in the sense that the discrete-time closed-loop system has the same McMillan degree as the open-loop continuous-time plant. >

On the gain margin improvement using dynamic compensation based on generalized sampled-data hold functions

This paper shows that the use of dynamic compensation based on generalized sampled-data hold functions (GSHF) can arbitrarily improve the gain margin for continuous-time nonminimum phase linear systems. The GSHF compensator is a particular periodic digital controller. The effect of sampling period on the gain margin is analyzed. Furthermore, it is proved that under a mild condition, the gain margin improvement can be achieved without forcing the sampling period small. An important advantage of periodic compensation over LTI compensation is found to be the capability of reducing conflict between gain margin and sensitivity. >

Discrete-time loop transfer recovery via generalized sampled-data hold functions based compensator

Loop transfer recovery (LTR) techniques are known to enhance the input or output robustness properties of linear quadratic gaussian (LQG) designs. One restriction of the existing discrete-time LQG/LTR methods is that they can obtain arbitrarily good recovery only for minimum-phase plants. A number of researchers have attempted to devise new techniques to cope with non-minimum-phase plants and have achieved some degrees of success.6-9 Nevertheless, their methods only work for a restricted class of non-minimum-phase systems. Here, we explore the zero placement capability of generalized sampled-data hold functions (GSHF) developed in Reference 14 and show that using the arbitrary zero placement capability of GSHF, the discretized plant can always be made minimum-phase. As a consequence, we are able to achieve discrete-time perfect recovery using a GSHF-based compensator irrespective of whether the underlying continuous-time plant is minimum-phase or not.

A correspondence between LTI controller and generalized hold function

Generalized sampled data hold function (GSHF) has been studied in a number of works and it is known to be a nonstandard sampled data control scheme. In this paper, we show that some linear time-invariant controllers become GSHF when a standard zero-order sample and hold is applied, thus establishing a correspondence between GSHF and a standard sampled data control scheme.