A Realizable Reconstruction Filter for Sampled data Systems

Muwahida Liaquatl *, Mohammad Bilal Malikl 1 Department of Electrical Engineering, College of Electrical and Mechanical Engineering

National University of Sciences and Technology

Islamabad, Pakistan * Email:muwahida@ee.ceme.edu.pk

Abstract

A realizable reconstruction filter is presented. Since the standard reconstruction filters are not realizable, our aim is to produce a reconstruction filter as an impulsive system that contains the dynamics of the reference signal. The proposed design is a memory less system and is able to overcome the intersampling behavior when applied in the closed loop system. As compared to other techniques such as generalized sampled data hold devices being used this design does not require the input and output of the reconstruction filter to be of different dimensions.

1. Introduction

The signals of interest in control systems such as command inputs, tracking errors etc are usually continuous signals. Whenever, there is a need to control these continuous systems via digital controller the signals that describes the systems behavior must be sampled. Such systems are referred to as sampled-data control systems.

To examine the signals at arbitrary times, estimates must be computed from the available information. The traditional approach to solve sample data control assumes that the discrete time plant model is obtained from the continuous time system by using prespecified hold devices- zero or first order hold [17]. A control signal is then generated by a digital controller. As the frequency spectrum of sample and hold is a sinc function, many high frequency components remain after interpolation. Though very simple to implement it is not a very effective interpolator.

Linear interpolation is a method of curve fitting using linear polynomials. It is heavily employed in mathematics and control theory ([18] and the references therein). It is a better filter than sample and hold as less frequency components can pass through it; however there is still some low frequency distortion.

A common and well researched way of interpolation is to interpolate the sequence of the sampled data by fitting spline curves to the samples. It is preferred over the linear polynomial interpolation since the interpolation error can be made very small even when splines of low degree polynomials are utilized [1].

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Although this method has been utilized to perform real time interpolation, it is usually preferred for applications where it can be utilized offline e.g. in image processing algorithms. A comparison study based on error interpolation among various interpolation methods such as bilinear, cubic splines and fuzzy is discussed in [9].

The problem of identifying discrete-time systems with missing output data is well understood and has been studied extensively in time series analysis and system identification ([2-3], and the references therein). If not properly taken into account, the missing measurements can seriously deteriorate the quality of the estimates. Under periodic sampling, the sampled-data system is time varying but also periodic, and thus it may be modeled by a discrete-time system obtained by discretizing the plant. However, this discrete model does not capture the inter sample behavior of the real system, which may be critical in a number of applications. A comparison based on time domain error measurements discussed in [8] concludes that the simple resampling methods such as sample and hold is robust and provide unbiased estimate of variance as compared to other mentioned schemes. Moreover the discussed interpolations schemes or reconstruction filters are not very effective in real time and fail to encounter inter sample behavior. In the conventional sampled-data control technique the subsequent intersampling behavior of the controlled continuous system is a well researched area [12-13]. However, all these studies assume that hold functions are not part of the design problem.

The idea of using generalized sampled-data hold functions (GSHF) instead of a simple zero-order hold (or first-order hold) for the reconstruction in control systems was first introduced by Chammas and Leondes [4]. Kabamba showed that by using GSHFs advantages of state feed back controllers can be achieved without using state estimation techniques; in particular, it was shown that GSHFs can significantly improve the performance of a closed-loop system [5].

A class of system models involving continuous-time dynamics with periodically occurring jumps was used in [12] to solve sampled-data H� control problem. Such models have also been

referred to as systems with impulse effects [13]. A sampled data output regulation problem with jumps view

point is investigated in [14]. It is established that if output regulation's internal stability is achievable via continuous-time error feedback then the same is true using sampled error feedback, discrete-time compensation, and a generalized hold device (GHD) that generates the continuous-time control signal. GHD is a memory-less time-varying (periodic) gain. When combined with a suitable discrete-time compensator the tandem act as a continuous-time internal model of the exosystem so that dynamic continuous-time pre-compensation is not required, which makes it different from the compensation schemes discussed in [10-11].

Motivated by the GHD presented in [14] a realizable reconstruction filter is presented here. The idea is to design an impulsive discrete-time compensator that contains the continuous time internal modal of the system. The proposed system requires a discrete input signal which is being controlled by a discrete controller. Output of the compensator is a continuous signal which contains the characteristics of the dynamics of the exosystem and does not require pre-compensation. The continuous control signal generated can then be used as a control variable to overcome the intersampling behavior of the closed loop system. Unlike [14] the dimensions of input and output signals remain same using the proposed filter.

The notation used in the paper is standard. Function time, f(·) , is continuous from the right but possibly discontinuous at some time t, fer) denotes lime� f(t - e) assuming that the limit exists.

2. Realizable reconstruction filter

We propose the reconstruction filter to behave like a system with jumps of the form

1i(t) = Ah17(t) 17(k+l) = Aj17[k]+Bhu[k] u(t) = Ch17(t)

(1)

Where 17(t) is the continuous-time state that undergoes

jumps at the integer multiples of a fixed period h, u[k] is the discrete-time control input that needs to be

interpolated and u(t) is the continuous-time control input which is also the output of the filter.

Given an initial time 10 and a final time If> 10, for notational simplicity we denote the subset of impulse

times 'irl(/o, lf) by {'i1''i2' •••• ''ik}. We then define

00 = 'i1 -/0 q = 'i;+l -'i; i = 1, ... , k -1

Ok = If -'ik In terms of this the state transition matrix for (1) is given by

Where it maybe understood that certain terms may not

appear depending on the relation between 10 and If.

Then the state response of (1) for IE ['ik, 'ik+1) can be

compactly given as

k x(t) = <I>(t, to)x(/o) + L <I>(t, 'i)(Bhu[jD (3) j=l

The first term is the zero-input state response while the last term is the zero state response resulting from the discrete-time input.

Consider an exogenous system of the form

wet) =Sw(t) Yref(t) = Qw(t) (3)

To compensate the internal model characteristics of

the exosystem the following matrices are assumption

to be same.

(4)

If this is the case then all we need to do is to design a state feedback controller (since for the known exosystem model there is no need for state estimation). The output regulation problem via full information feedback is to find, if possible, a state feedback law of the form

u[k] = Fj17[k]+ Kw[k] For which the closed loop system satisfies

1. For w(to) = 0 and any 17(to) ,

2.

17(t) � Oas t � 00

For w(to) and any 17(to),

e(t) � Oast �oo

Where the error signal e(t) is given by

e(t)=[Ch Q] [17(t) ] wet)

(5)

(6)

The solution of these problems is based on the

following assumptions

AI: 0"( S) C C+ := { A E C I Re[ A] � O}

is controllable. Then the following theorem due to francis [24], describes the necessary and sufficient conditions for the existence of solutions to the above mentioned problems.

2.1. Proposition 1: Suppose the assumptions Al and A2 hold then the output regulation problem via state feedback is solvable if and only if there exist matrices TI and r which solve the linear matrix equations

nSd = Ahdn + Bhd o =Chdn+Qd (7)

A AhhA B AhhB C Ah hC hd = e j' hd = e h and hd = e h . While S d and Qd represent the discrete exosystem

realized through a zero-order hold. Moreover, if this condition is satisfied then a suitable feedback is given by

u[k] = F/1][k]-TIw[k])+ rw[k] (8)

Thus for a system with jumps (1) that is controllable

via state feedback, a stabilizing discrete-time

controller can be constructed as such that

(Ahd + BhdF) is stable. Where Fj is the feedback

gain. Figure (1) shows the working of the interpolator.

,..-----..,1 u[k] u(t) 1-1 __ ....,�o� Realizable reconstruction filter

I)[k] 1------------1

w�l h ZOH r-JL(_(t) __ ] Discrete Controller Exosystem

Figure 1. Block diagram of the closed loop system with realizable reconstruction filter

3. Example

The proposed interpolator (1) was applied to an exosystem which generates a sine wave. System matrices for the sine wave are given as

wet) = [� -� ]W(t)

Yrej = [I 0] wet)

Using (4) the we can now write (1) as

iJ(t) = [� -� ]1](t)

1](k+l) = Aj1][k]+BhU[k] u(t) = Ch1](t)

WhereAj=I,Bh= [�] and Ch =[1 0] .

(9)

(10)

The discrete-time control input is generated by (8) where the feedback gain Fj is obtained by

placing the eigenvalues at [0.8 0.8] is

Fj =[0.36 -0.4648]

Then the regulator equation (7) is used to obtain u[ k] . This discrete control input when passes through the real time interpolator (10) converts the signal u[ k] into a continuous-time

signal u(t) which is a sine wave of the same frequency as (9). The results are shown in the following figure.

4,-----�------�._�----��� I- Output u(t) of the filter I- Reference sine wave

�O�---- �O�.5------�1------�1.5�----�2 TIme x 1�

Figure 3. Comparison between the output of the filter and reference sine wave

2!,-----�----�----�-----[-input u[k] �1 A

- 10::------:'50.,------- 1--'-OO---1-'--50------J200

Time Figure 2. Controlled discete input u[k] to the filter

stabilizing the closed loop system

4. Conclusion

A realizable reconstruction filter is presented in this paper. Unlike the standard compensation filters which are not realizable we present a real time impulsive interpolator to generate continuous-time control signal from the discrete counter part. The resultant filter is realizable and contains the distinctiveness of the exosystem. The digital to analog conversion is carried out without increasing the signal dimensions which reduces the computational complexity. Since the continuous-signal contains internal modal of the reference signal it can be used as a control variable to overcome the intersampling behavior of the closed loop system it does not pre-compensation. Unlike It can be incorporated in the output regulation problem of sampled-data systems.

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