Research Article  Open Access
Improvement of the Asymptotic Properties of Zero Dynamics for SampledData Systems in the Case of a Time Delay
Abstract
It is well known that the existence of unstable zero dynamics is recognized as a major barrier in many control systems, and deeply limits the achievable control performance. When a continuoustime system with relative degree greater than or equal to three is discretized using a zeroorder hold (ZOH), at least one of the zero dynamics of the resulting sampleddata model is obviously unstable for sufficiently small sampling periods, irrespective of whether they involve time delay or not. Thus, attention is here focused on continuoustime systems with time delay and relative degree two. This paper analyzes the asymptotic behavior of zero dynamics for the sampleddata models corresponding to the continuoustime systems mentioned above, and further gives an approximate expression of the zero dynamics in the form of a power series expansion up to the third order term of sampling period. Meanwhile, the stability of the zero dynamics is discussed for sufficiently small sampling periods and a new stability condition is also derived. The ideas presented here generalize wellknown results from the delayfree control system to timedelay case.
1. Introduction
It is well known that unstable zero dynamics limit the control performance that can be achieved. Some techniques based on zero dynamics cancellation for control system design are hard to be applied when a plant has unstable zero dynamics [1, 2]. When a continuoustime system is discretized by a sampler and a zeroorder hold (ZOH), stable poles are transformed into the unit circle. The transformations of zero dynamics, however, are much more complicated and the stability of zero dynamics is not preserved in the discretization process in some cases [1]. It is generally impossible to derive a closedform expression that relates the continuoustime zero dynamics with the discretetime ones because the zero dynamics of discretetime systems depend on sampling period . Therefore, the analysis of zero dynamics has a great deal of interest [1–7].
Perhaps the first attempt to study discrete system zero dynamics was given by Åström and coworkers [1], who described the asymptotic behavior of the discretetime zero dynamics for fast sampling rate when the original continuoustime plant is discretized with ZOH. More research for discrete zero dynamics has been shown in the past decade, and these known results present the asymptotic characterization of zero dynamics for the discretized systems without time delay [8–15]. However, time delay of controlled systems inherently exists in many mechanical engineering applications and therefore must be integrated into system models. In fact, it is inevitable that the influence of time delay on digital control system must be considered because it also occurs frequently in information transmission between elements or systems, transport of controls and sensors, data computation, and so forth. Hence, it is important to investigate the asymptotic properties of zero dynamics in the sampleddata models corresponding to the continuoustime plants with time delay for the digital control system design.
Anyway, one would reasonably expect similar results in the case of delayfree systems to hold for timedelay plants. However, the situation for the timedelay case is more complex than for delayfree models. Indeed, to the best of our knowledge, an explicit characterization of the zero dynamics for timedelay systems has previously remained unresolved, although an implicit characterization has been given in [16–18].
On the other hand, from the view point of the stability of the discrete zero dynamics, attention is here focused on continuoustime systems with relative degree equal to two. Furthermore, when the relative degree of a continuoustime transfer function with time delay is two, discrete zero dynamics, especially sampling zero dynamics, are located just on the unit circle, that is, in the marginal case of the stability when the sampling period tends to zero. Thus, the asymptotic behavior of the sampling zero dynamics is an interesting issue because it is stable for sufficiently small sampling periods if it approaches the unit circle from inside as the sampling period tends to zero.
Nevertheless, many problems remain unsolved in spite of the constant efforts. It is still interesting to find out the asymptotic properties on the zero dynamics of the sampleddata models with time delay and to derive new stability criteria of the zero dynamics though the impact of the stability of zero dynamics in sampleddata timedelay models may not have been strong in control engineering applications. In this paper, we present an approximate sampleddata model for a continuoustimedelay system, which is accurate to some order in the sampling period. We also show how a particular strategy can be used to approximate the system output and state variable and their derivatives in such a way as to obtain an approximate expression of discrete zero dynamics by Taylor expansion with respect to a sampling period up to the third order term. An insightful interpretation of the proposed results can be made in terms of an explicit characterization of these zero dynamics. Moreover, the zero dynamics turn out to be identical to those found in the delayfree case. Thus, the current paper extends the wellknown notion of zero dynamics from the delayfree case to timedelay system.
The layout of the paper is structured as follows. In Section 2, we review the results of time delay and basic concepts. Section 3 presents the main result of this paper, namely, the asymptotic behavior of zero dynamics in the case of time delay and relative degree two. The numerical simulation is represented in Section 4. Finally, conclusions are presented in Section 5.
2. Preliminaries
In the present study, we consider a class of the following singleinput singleoutput thorder continuoustime control systems with time delay and statespace representations of the form where is the vector of the states evolving in an open subset , is the input variable, and is the system constant time delay that directly affects the input. It is assumed that the vector fields and and the output function are analytic.
We are interested in the sampleddata model for the linear systems when the input is a piecewise constant signal generated by a ZOH. Thus, for a sampling period ,
Furthermore, we suppose the time delay and mesh are related as follows: where and . Equivalently, the time delay is customarily represented as an integer multiple of the sampling period plus a fractional part of [19, 20]. Under the ZOH assumption and the above notation, it is rather straightforward to verify that the “delayed” input variable attains the following two distinct values within the sampling interval [21]:
When the relative degree of a continuoustime system with time delay is two, we consider the continuoustime transfer function of a control system (1): where , and , are real coefficients and is the time delay. As it can be easily inferred from (5), the input variable remains constant within the two subintervals: , . One readily obtains from which in which the final equation of (7) is supported from the Appendix, and notice here that and and further, where
Remark 1. Expression (7) represents the sampleddata representation of the original continuoustime system (5) with time delay . Note that the value of the state vector at is a linear combination of the states evaluated at and the past values of the input variable at and .
The paper treats systems with relative degree two because at least one of the zero dynamics is unstable when the relative degree is greater than or equal to three though it is slightly a limitation.
First, the following assumptions are introduced.
Assumption 1. The unique equilibrium point lies on the origin.
Assumption 2. The continuoustime linear system (1) has the uniform relative degree and is minimum phase in the open subset , where the state evolves.
These assumptions ensure that there is a coordinate transformation such that we express the system in its normal form. The normal form of (5) with relative degree two, , is represented with an input and an output as [22, 23] where and the scalars and are obtained from where
When a ZOH is used and the relations are taken notice of inputoutput relations, the normal form (9) yields the derivatives of the output Further, the derivatives of are also represented as Hence, substitute (13)–(14) into the right hand side of and define the state variables , then discretetime state equations are obtained.
Now, the zero dynamics of the discretetime system (15) are analyzed using the explicit expressions of and as follows:
3. Main Results
In this section, an approximate expression of zero dynamics of a discretetime system with time delay and relative degree two is obtained from (15). The main result is given by Theorem 2.
Theorem 2. The zero dynamics of a discretetime system with time delay and relative degree two for a continuoustime transfer function (5) are approximately given for by the roots of where
Proof. Zero dynamics of a discretetime system with time delay and relative degree two (15), equivalently (16), are given by substituting into (16) as follows:
where , , , and are the transforms of , , , and , respectively, and the matrix is defined by
with
Thus, the zero dynamics are derived from
where
Here, consider matrices , , and which are defined by neglecting the higherorder terms with respect to in the block matrices , and because the interests lie in the case of .
Multiplying and by
from the right hand side and by
where
from the left hand side and further multiplying the result by
where
from the left hand side yield
where
Noting here that , leads to
Hence, the approximate values of the zero dynamics of the discretetime system with time delay and relative degree two are obtained as the roots of (17).
As a result, the proof is complete.
Remark 3. The approximate values of the sampling zero dynamics are expressed as and the approximate values of the intrinsic zero dynamics are derived from
Remark 4. An insightful observation in Theorem 2 is that it clearly gives the approximate values with higher order of accuracy than the previous result [16, 17] in the form of a power term of .
Remark 5. When the ZOH signal reconstruction device is used, notice here that the delayfree discretetime systems can be considered as a particular case of the discretized plants with time delay when and .
In particular, consider the simplest possible system of relative degree two, namely, . Then we have the corresponding sampling zero dynamics in the case of a time delay
It is easy to obtain the following corollary and theorem from Theorem 2. The following corollary shows the asymptotic behavior of the sampling zero dynamics when a continuoustime system has a time delay and relative degree two, and the sum of the zero dynamics of a continuoustime system is equal to the sum of the poles; that is, . A new stability condition of the zero dynamics, which is shown in the following theorem, for discretetime system with time delay and relative degree two (5) is also presented by analyzing (32).
Corollary 6. Assume that the relative degree of a continuoustime system is two, and . The sampling zero dynamics of a discretetime system corresponding to a continuoustime transfer function with time delay (5) are expressed as
Theorem 7. Consider a transfer function with time delay in the case of relative degree two (5). Then, for sufficiently small sampling periods, all the zero dynamics of the corresponding sampleddata model (15) are stable if all the zero dynamics of (5) are stable and
Proof. First, we have
Thus
Next, it is obvious that all of the zero dynamics are stable, that is, located strictly inside the unit circle when satisfying by selecting the suitable parameters , , sampling period , and time delay . Simple straightforward calculation can verify that all the discretized zero dynamics are stable if (36) holds.
As a result, the proof is complete.
Remark 8. When a continuoustime system with time delay reduces to a delayfree model, stability of the corresponding discretetime zero dynamics can be decided by , that is, the difference between the sum of the zero dynamics and that of the poles for a continuoustime plant. In particular, when , then the corresponding discretized system has the stable zero dynamics if .
Remark 9. A key interpretation is given in terms of an explicit characterization of the discretized zero dynamics of the obtained model. Of particular interest is that the contribution of this paper on the linear singleinput singleoutput cases with time delay can be extended the multivariable timedelay systems and further will be generalized the nonlinear plants in the case of a time delay.
4. Numerical Simulation
In this section, we present an interesting example to show the stability of discrete zero dynamics with time delay and relative degree two. It has been also shown that the stability of zero dynamics, in the case of a time delay and relative degree two, will be improved by comparing with the previous results. Both kinds of the zero dynamics are calculated by applying MATLAB, and in the simulation figures, the solid line (blue) and dotted line (red) indicate the exact values and approximate values, respectively.
Consider a transfer function with time delay and relative degree two, and the sum of the zero dynamics is equal to the sum of the poles as follows: It is obvious that The intrinsic zero dynamics and the sampling zero dynamics , are expressed approximately as, respectively,
We can easily determine that the stability of the sampling zero dynamics can be satisfied by choosing design parameters , , , and of (36). Without loss of generality, let the time delay for a sufficiently small sampling period and select the suitable design parameters and ; then the approximate values (17) and the exact values of zero dynamics of the discretetime system with time delay and relative degree two for the transfer function (39) are shown in Table 1 and corresponding figures, where the sampling zero dynamics and intrinsic zero dynamics are, respectively, shown in Figures 1 and 2.

As you can see in these table and figures, (17) gives good approximation and the sampling zero dynamics can lie inside the unit circle for a sufficiently small sampling period by satisfying condition (36).
5. Conclusions
This paper analyzes the asymptotic behavior of zero dynamics for a discretetime system when a continuoustime system with a time delay is explicitly discretized in the case of a ZOH and relative degree two. Moreover, we also give an approximate expression of the zero dynamics for the discrete systems with time delay as power series expansions up to the third order term with respect to sampling periods. Further, in this case, the stability of discrete zero dynamics is discussed for sufficiently small sampling periods and a new stability condition is also derived. The idea of this paper is a further extension from the delayfree cases to the timedelay systems.
Appendix
Assumptions 1 and 2 ensure the existence of the normal form for the linear timedelay system. When the input signal is generated by a ZOH, we first apply the Taylor formula with remainder to the system output and its derivatives with respect to a sampling period up to the order term at any point . However, owing to the existence of time delay, it is difficult to derive directly the corresponding discretetime models. Therefore, we present an approximate sampleddata model for linear system with time delay by means of multiple step approach. Namely, apply the higherorder Taylor expansion formula in the sampling periods such as
Without loss of generality, assume that and ; then (, is an integer). In the time interval , when the input is a , one can obtain for sufficiently small sampling periods that where
Denote then where
Similarly, in the other time interval , we can also give approximate asymptotic expressions of the outputs as power series expansions with respect to a sufficiently small sampling period when the input is the constant value . Deleting leads to the approximate expressions of : where in which
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors gratefully acknowledge the financial support of the National Basic Research Program of China (“973” Grant no. 2013CB328903), the National Natural Science Foundation of China under Grant no. 60574003, and Natural Science Foundation Project of CQ CSTC under Project no. cstc2012jjA40026 and Project no. CDJXS12170006 supported by the Fundamental Research Funds for the Central Universities. And the authors also gratefully acknowledge the helpful comments and suggestions of the viewers, which have improved the presentation.
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Copyright © 2014 Cheng Zeng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.