2007 International Symposium on Information Technology Convergence
Second Order Hold Based Discretization Method of Input Time-delay Systems Zheng Zhang1,2, Kil To Chong1 1 Division of Electronics and Information Engineering, Chonbuk National University, Duckjin-Dong, Duckjin-Gu, Jeonju 561-756, Korea 2 School of Mechanical Engineering, Xi’an Jiaotong University, 28 Xianning West Street, Xi’an 710049 P.R.China
that can solve a system with time delays are necessary [1][2]. The proposed discretization scheme is based on the Taylor-Lie series and uses a similar mathematical framework previously developed for delay-free nonlinear systems [3]. The traditional approaches, such as Euler, Runge-Kutta, require a “small” time step in order to be deemed accurate, and this may not be the case in control applications where large sampling periods are inevitably introduced due to physical and technical limitations. The performance of previous method is significantly affected by the selected discretization method and the selected sampling interval. In certain cases, however, the sampling rate is constrained by either the computational speed of the microprocessor for digital control or by the measurement scheme, and it has to be selected low [4]. In these large sampling period systems, Taylor series method was used to improved the performance of the controller [5]. However, in the previous paper zero-order hold (ZOH) assumption was used in the discretization method. The performance of ZOH assumption is seriously depended on the input signal and the sampling time should be short enough for a certain control precision. A high-order method is a method that provides extra digits of accuracy with only a modest increase in computational cost [6]. Therefore, second-order hold (SOH) assumption is introduced in this paper to enhance the performance under the situation that large sampling interval is inevitable. The present study aims at the development of a new method for the time discretization of nonlinear inputdriven dynamic systems with time delay based Taylor
Abstract Second order hold is a method can provide a high precision for discretization of input-driven nonlinear systems. A new discretization scheme combined second order hold with Taylor-series is proposed. The sampled-data representation and the mathematical structure of the new discretization scheme are explored. Both exact sampled-data representation and approximate sampled-data representation are described in detail. The performance of the proposed discretization procedure is evaluated by simulation studies. Various sampling rates, time-delay values and truncation order of Taylor-series are considered to investigate the proposed method. The results demonstrate that the proposed scheme is practical and is easy to use for time-delay systems. The comparison between second order with first order and zero order is given to show the characteristic of the proposed method.
1. Introduction Control systems with time delays exhibit complex behaviors because of their infinite dimensionality. Even in the case of linear time-invariant systems that have constant time delays in their inputs or states have infinite dimensionality if expressed in the continuous time domain. It is therefore difficult to apply the controller design techniques that have been developed during the last several decades for finite-dimensional systems to systems with any time delays in the variables. Thus, new control system design methods
0-7695-3045-1/07 $25.00 © 2007 IEEE DOI 10.1109/ISITC.2007.35
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series and second-order hold assumption. This kind of discretization method inherits some of the system theoretic properties of the original continuous-time system (such as equilibrium and stability properties). And most importantly, it is a finite dimensional representation, which allowing the direct application of existing nonlinear control system design techniques. Secondly, performance evaluation of the proposed algorithm is presented using several illustrative case studies. The paper is organized as follows: Section 2 contains some mathematical preliminaries and Sec. 3 gives the description of the proposed method. Section 4 demonstrating the effectiveness of the proposed discretization scheme by simulation, whereas Sec. 5 provides a few concluding remarks drawn from this study.
u ( k ) u ( k 1)
s ( k ) a(k )
T s ( k ) s(k 1)
(4)
(5) T Where s( k ) represents the derivation at time kT , a( k ) represents the second order derivation at time kT .
Equation (3) is shortly represented as, 1 u (t ) u (k ) s( k )( t kT ) a( k )( t kT ) 2 . (6) 2 This compact form will be used in the following part of this paper.
2.2 SOH for time-delay system The time-delay D can also be expressed as, D (q )T qT
(7)
Where q {0,1, 2,...} , (0,1) and 0 T . From
(), we can get
2. Preliminaries
T .
(8)
Equivalently, the time-delay D is customarily represented as an integer multiple of the sampling period and plus a time interval , and that is less than
In the present study single-input nonlinear continuous-time control systems are considered with a state-space representation of the form: dx(t ) f ( x(t )) g ( x(t ))u(t D) dt (1) n Where x X R is the vector of the states and an open and connected set, u R is the input variable and D is the system’s constant time-delay (dead-time) that directly affects the input. It is assumed that f ( x)
sampling period. Based on the SOH assumption and the above notation, the expressions of SOH can be derived for time-delay systems step by step. Because of the existence of , in the procedure of deductive method, it should be divided into two time intervals within a certain sampling period, which are 1 [kT , kT ) , (9)
2 [kT , kT T ) .
g ( x) are real analytic vector fields on X .
(10)
therefore,
1 (t ); 2 (t );
An equidistant grid on the time axis with mesh is considered, where T tk 1 t k 0
u (t D)
[tk , tk 1 ) [kT , ( k 1)T ) is the sampling interval, T is
t I 1 t I 2
(11)
Where,
1 (t ) u ( k q 1) s( k q 1)[t D (k q 1)T ]
the sampling period. It is also assumed that system (1) is driven by an input that is piecewise quadratic over the sampling interval, i.e. the Second-Order Hold (SOH) assumption holds true.
1
a (k q 1)[t D ( k q 1)T ]2 ,
2 2 (t ) u (k q) s (k q)[t D ( k q)T ]
2.1 SOH for delay free system For the SOH, while D 0 , and kT t kT T ,
1
a(k q)[t D ( k q)T ]2 .
u ( kT ) u[( k 1)T ] (t KT ) u (t ) u ( kT ) T 1 u ( kT ) 2u[( k 1)T ] u[( k 2)T ] (t kT ) 2 (2) 2 2 T It can also be written as, u ( k ) u (k 1) u (t ) u ( k ) (t KT ) T 1 u( k ) 2u (k 1) u ( k 2) (3) (t kT ) 2 2 2 T Furthermore, assume that
2
(12)
(13)
3. Second-order hold based discretization 3.1 Delay free nonlinear systems Initially, delay-free ( D 0 ) nolinear control systems are considered with a state-space representation of the form, dx(t ) (14) f ( x( t )) g ( x(t ))u( t ) . dt
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Under the SOH assumption and within the sampling interval, the solution of (14) is expanded in a uniformly convergent Taylor series and the resulting coefficients can be easily computed by taking successive partial derivatives of the right hand-side of (14):
x(k 1) x( k )
As shown in (11) to (13), under SOH assumption, the input variable expressions are different within the two subintervals [kT , kT ) and [kT , kT T ) . Successively applying formula (20), we readily obtain, x(kT ) e A x (kT )
t k
kT T
x(k ) A ( x( k ), u( k )) []
T
1
!
e A (T ) x(kT )
kT
(15)
(21)
A[ ] ( x, u ) ( f ( x) ug ( x)) x
(16) here 1,2,3... . Therefore, an exact sampled-data representation (ESDR) of (14) can be derived by retaining the full infinite series of (15), x(k 1) T ( x( k ), u (k ))
1
T
.
!
b 2 ( )d
(22)
Motivated by the linear approach described above, a similar line of thinking is adopted for the nonlinear case as well. Indeed, by applying the Taylor series discretization method for nonlinear systems presented before to the [kT , kT ) subinterval one immediately obtains the state vector evaluated at kT , (23) x(kT ) ( x( kT ), 1( kT ))
A[1] ( x, u ) f ( x ) ug ( x )
A ( kT T )
3.3 Nonlinear systems with time-delay
x (k ) A[ ] ( x( k ), u (k ))
e
where 1 ( ) and 2 ( ) are defined by equation (11).
where x(k ) is the value of the state vector x at time t t k kT and A[ ] ( x, u ) are determined recursively by: A[ 1] ( x, u )
e A ( kT )b1 ( )d
x( kT T )
1
kT
and
T d x
! dt
kT
Where the map can be derived through a direct application of formula (15) and the subsequent calculation of the corresponding Taylor coefficients can be realized through the recursive formulas (16). x(kT ) and 1 ( kT ) are the instantaneous state vector and
(17)
input value respectively at time kT . Furthermore, it can be derived from (12) that, 1 (kT ) u ( k q 1) s( k q 1)[( q 1)T ]
Simultaneously, an approximate sampled-data representation (ASDR) of equation (20) is resulted from a truncation of the Taylor series order N, x(k 1) N T ( x ( k ), u (k ))
1
a(k q 1)[( q 1)T ]2
the dependence on the sampling period T , and the superscript N denotes the finite series truncation order associated with the ASDR of equation (18).
(24) 2 Similarly, the Taylor discretization method applied to the [kT , kT T ) subinterval yields the state vector evaluated at (k 1)T as a function of x( kT ) and the input value at time kT , (25) x( kT T ) T ( x (kT ), 2 (kT ))
3.2 Linear systems with time-delay
and,
N
x( k ) A[ ] ( x( k ), u( k ))
1
T
!
(18)
where the subscript T of the mapping N T denotes
It is now feasible to extend the aforementioned Taylor discretization method to nonlinear continuoustime systems with a constant time-delay ( D 0 ) in the input. In order to motivate the development of the proposed discretization procedure and draw the appropriate analogies from the field of linear systems, let us first begin the exposition of the paper’s main results by briefly reviewing the ones available in the case of linear systems, dx (t ) dt
Ax (t ) bu (t D)
2 (kT ) u ( k q) s(k q)( qT ) 1
a(k q)(qT ) 2
(26) 2 Based on (17), the above equation (23) and (25) can be rewritten as follows, x(kT ) x( kT )
A ( t f ti )
x(ti )
t f
e t i
A (t f )
( x( kT ), 1 ( kT ))
!
(27)
x(kT T ) x( kT )
(19)
bu( ) d .
[]
1
A[ ] ( x(kT ), 2( kT ))
1
where A, b are constant matrices of appropriate dimensions. It is known that for any time interval I [ti , t f ) , the following formula holds true, x(t f ) e
A
(T ) !
(28)
And furthermore, according to (18), the approximate sampled-data representation (ASDR) of equation (27) and (28) are resulted from a truncation of the Taylor series order N, as shown below,
(20)
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u(t D ) 0.9sin(1.6 (t D ))
x (kT ) N ( x ( kT ), 1 ( kT )) N
x(kT ) A[ ] ( x( kT), 1( kT ))
1
x(kT T )
N T
!
(29)
( x( kT ), 2 ( kT ))
N
x(kT ) A[ ] ( x( kT ), 2 ( kT ))
1
(T ) !
(30)
It should be emphasized, that the functional representation of the A[ ] -coefficients of the map T remains exactly the same subpart as for the
subinterval [kT , kT ) , and it is only need to reuse the same part with the aid of a symbolic software package such as MAPLE. For the consecutive subintervals, combing equations (23) and (25), the desired sampled-data representation of the original system (1) is obtained, x(kT T ) D T ( x (kT ), 1 (kT ), 2 ( kT )) T ( ( x( kT ), 1 ( kT )), 2 (kT ))
.
(37) Different sampling rates, different time-delay and different truncation order of Taylor-series are studied. Simultaneously, MATLAB 7.0 is used to calculate the accurate value. Two of the different cases are shown as follows. 4.1 Case 1 While truncation order N=3, sampling time T=0.05s, time delay D=0.07s, the state response for SOH is shown in fig.1.
(31)
Notice, that a finite series truncation o rder N for the above series would naturally produce an ASDR, x( kT T ) NT , D ( x (kT ), 1 (kT ), 2 (kT )) (32) or x( k ) NT , D ( x( k ), 1 (k ), 2 (T )) .
(33)
4. Simulation
Fig.1. State response (N=3)
A simple chemical process system is considered in simulation. The system can be described as follow, dx f ( x ) g ( x )u (1 2a ) x au ux ax 2 . dt (34) In the simulation, a=0.3 is used. The initial system state was assumed that x (0) 0 .
Within the sampling interval, the solution of (34) is obtained using uniformly convergent Taylor series. According to the methodology described in earlier sections, the sampled-data representation of the system is shown as (29) and (30). In this system, f ( x ) (1 2a ) x ax 2 g ( x) (a x) .
Fig.2. Errors comparison
(35)
At the same time, First-Order Hold (FOH) and ZeroOrder Hold (ZOH) are used to make a comparison. Since MATLAB is used to calculate the exact values, the response errors comparison is shown in fig.2. It is obvious to find that the maximum error of ZOH is decreased by 41.17% from 0.0066 to 0.0039, and that the maximum error of FOH is decreased by 12.1% from 0.0039 to 0.0034.
So that, the partial derivative terms A[ ] ( x, u ) are determined recursively by (16). The following sine-wave input is applied to the system, u (t ) 0.9sin(1.6 t ) . (36) Therefore, the time-delay input applied to the system is as follow,
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results show that the second order hold method is practical. Simultaneously, the precision can be improved obviously compared with first order hold and zero order hold schemes.
4.2 Case 2 While truncation order N=4, sampling time T=0.06s, time delay D=0.03s, the state response for SOH is shown in fig.3.
6. Acknowledgements This work was supported by the grant of the Second stage of Brain Korea 21.
7. References [1] Park Ji Hyang, Chong Kil To, Kazantzis Nikolaos and Parlos Alexander G, 2004, “Time-Discretization of Nonlinear Systems with Delayed Multi-Input Using Taylor Series,” KSME International Journal, Vol. 18 No. 7, pp. 1107~1120. [2] Park Ji Hyang, Chong Kil To, Kazantzis Nikolaos and Parlos Alexander G, 2004, “Time-Discretization of Nonaffine Nonlinear System with Delayed Input Using TaylorSeries,” KSME International Journal, Vol. 18 No. 8, pp. 1297~1305.
Fig.3 State response (N=4)
At the same time, FOH and ZOH are used to make a comparison. The response errors comparison is shown in fig.4. It is obvious to find that the maximum error of ZOH is decreased by 41.97% from 0.0079 to 0.0046, and that the maximum error of FOH is decreased by 6.32% from 0.0046 to 0.0043.
[3] Kazantzis, N., and Kravaris, C., 1999, “TimeDiscretization of Nonlinear Control Systems via Taylor Methods,” Comput. Chem. Eng., 23, pp. 763–784. [4] Yuping Gu and Masayoshi Tomizuka, 1999 .“Digital Redesign of Continuous Time Controller by Multirate Sampling and High Order Holds”.Proceedings of the 38"Conference on Decision & Control Phoenix, Arizona USA December, pp. 3422-3427. [5] Nikolaos Kazantzis, K.T.Chong, J.H.Park, Alexander G. Parlos 2005.”Control-Relevant Discretization of Nonlinear Systems With Time-Delay Using Taylor-Lie Series”. Journal of Dynamic Systems, Measurement, and Control, Vol.127 No.3, pp.153~159. [6] Albert Lozano, Javier Rosell, Ram6n Pallas-Areny. 1992. “On the Zero- and First-Order Interpolation in Synthesized Sine Waves for Testing Purposes”. IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 41, NO. 6, DECEMBER 1992. pp.820-823.
Fig.4 Errors comparison
5. Conclusions A second order hold based discretization scheme is proposed. The expressions are described in detail. By a simulation, the method of how to use the proposed discretization scheme is explained. The simulation
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