Estimation of Non-Stationary Delay for a Linear ... - Springer Link

c Pleiades Publishing, Ltd., 2009. ISSN 0005-1179, Automation and Remote Control, 2009, Vol. 70, No. 7, pp. 1140–1152.  c A.Yu. Torgashov, 2009, published in Avtomatika i Telemekhanika, 2009, No. 7, pp. 58–72. Original Russian Text 

DISCRETE SYSTEMS

Estimation of Non-Stationary Delay for a Linear Discrete Dynamic Plant1 A. Yu. Torgashov Institute of Automation and Control Processes, Far-Eastern Branch, Russian Academy of Sciences, Vladivostok, Russia Received June 23, 2008

Abstract—A nonlinear recurrent gradient-type algorithm is proposed for estimation of nonstationary delay arising in finite impulse response of a linear discrete dynamic plant. Convergence of the algorithm is investigated. Using linear matrix inequalities, its optimal functioning is investigated in order to provide for the maximal rate of convergence of the estimation error to zero. PACS number: 02.30.Yy DOI: 10.1134/S0005117909070066

1. INTRODUCTION Investigation of systems with non-stationary delay in finite impulse response (FIR) is dictated by design requirements for control systems for modern chemical-engineering processes (CEP), for instance, chemical reactors, in which a heterogeneous mixture of liquid with grinded hard raw stock (suspension) is loaded. The time, when reaction starts on the surface of each hard particle, is not constant and depends on a multitude of uncontrollable factors. This time is expressed in the form of a non-stationary delay in FIR, because such an approach makes it possible to describe actual situation, when for constant input flow discharges (some of them may be control actions) the output variables of the plant (temperature, concentration of a target component and so on) change incessantly. Below we demonstrate the difference between dynamic properties of systems with a delay in FIR and those of systems with well-known input, output, and state delays [1]. We note that widespread autoregression model with moving average and independent input can be represented in the form of state-space models with input and output delays. Let us overview principal approaches to estimation of delay in linear plants. For continuous-time systems with input and state delays, in [2] it was proposed to use observers with variable structure; in [3], and adaptive identifier is designed using an improved Lyapunov–Razumikhin method. Nevertheless, the assumption on accessibility of the state vector for measurement is not always true, and an identifier should be considered in applied problems as a discrete dynamic system. Utilization of correlation analysis for input delay estimation in closed control loop is presented in [4]. But in this case there are significant drawbacks, namely, necessity to simultaneously estimate auto- and cross-correlation functions of signals, and, correspondingly, difficulties in generalization of the correlation approach onto multivariable systems. Another alternative approach is based on application of distribution theory [5]. In this case, differentiation of output variable is required, and original problem of identifying input or output delays reduces to an eigenvalue problem. The authors of [5] themselves 1

This work was supported in part by the Russian Foundation for Basic Research, project no. 08-08-00004, and Council of the President of the Russian Federation for young scientists supporting grants, project no. MK-2034.2008.8. 1140

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note in the conclusions the following shortcoming of the method: it is necessary to develop further methods of robustization of the obtained identifier, because its work is unstable in the presence of noise, and it has singular properties. Application of genetic algorithm for identification of variable input delay is considered in [6] without convergence analysis; such parameters as probabilities of cross-over and mutation are chosen heuristically. In the present work we develop, on the basis of a proposed gradient-type recurrent algorithm, a new solution of the problem of delay estimation in FIR for a linear discrete dynamic plant. As distinct from the aforementioned methods, convergence of the estimation algorithm is proved, and, moreover, results of optimizing parameters of the algorithm that provide for the maximal convergence rate of estimation error to zero, are presented. Such an optimization is based on the use of linear matrix inequalities. 2. STATE-SPACE DESCRIPTION OF SYSTEM WITH NON-STATIONARY DELAY IN FIR AND STATEMENT OF THE PROBLEM Let a scalar linear discrete dynamic delay-free plant have the following FIR: ˜= h



0 hT

T

,

(1)

h = (h1 . . . hN )T ;

(2)

where

˜ is a hq = 0 for q  N + 1, . . . , ∞; the superscript T denotes the transposition. The vector h component of the operator that transforms plant input u into its output y using the convolution sum y(k) =

N +1 

˜ h(i)u(k − i + 1),

(3)

i=1

where k denotes discrete time. It is assumed that we know elements of the vector h and bounds of delay variation d(k) ∈ [dmin ; dmax ], but the rule of delay variation inside this interval is unknown. Then, on the kth step, the vector FIR of dimension Q = N + 1 + dmax has the following structure: h∗ (k) = f (d(k)) =

( 0 ... 0 



dmin

0 ... 0

 





˜T | |h

dmin +1,...,d(k)

0 . . . 0 )T ,







(4)

d(k)+1,...,dmax

and from Eq. (3) we arrive, with account for (4), to the following expression: 

y(k) =

Q 

h∗ (k − i + 1, i)u(k − i + 1),

(5)

i=1

which is a discrete-time counterpart of integral equation with unknown elements of two-dimensional kernel h∗ (j, i). From (4), only bounds of h∗ (j, i) can be found. Equation (5) can be represented as state-space equation of system model. With this aim in view, we introduce a finite set of time-independent Toeplitz convolution matrices (H0, . . . , Hdq , . . . , Hdmax ), where 

Hdq = h(dq )(Q×1) ; (0 . . . 0)(1×(Q−1)) AUTOMATION AND REMOTE CONTROL

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0.94

8000

0.92

7000

0.90

6000

0.88

5000

0.86

4000

0.84

3000 0

10

20

λ

30

40

0.82 30

50

(c)

5500

40

50 λ

60

70

(d)

0.96

ar = 0; br = 5

ar = 0; br = 5

5000

0.94

4500

0.92

4000

α

Mean square estimation error

ar = 0; br = 1

ar = 0; br = 1

9000

2000

(b)

0.96

α

Mean square estimation error

10 000

0.90

3500 0.88

3000

0.86

2500 2000 1.0

1.5

2.0 λ

2.5

3.0

0.84 1.0

1.5

2.0

3.0

2.5 λ

3.5

4.0

Fig. 1. On the choice of parameters of estimation algorithm that provide for its optimal (maximal) convergence: (a) and (c) determination of λn opt by simulation of the nonlinear estimation algorithm; (b) and (d) computation I using linear matrix inequalities. of λLM opt

has dimensions Q × Q; Q = Q − 1. The Toeplitz matrices are specified by their first column and first row [7]. Equation (6) means that the first column of the matrix is h(dq ), while all the elements of the first row, starting from the second one, are given by [0 . . . 0], i.e., are zero. For instance, if N = 3, dmax = 2, and dmin = 0, then ⎛

⎜ ⎜ ⎜ H0 = ⎜ ⎜ ⎝

h1 0 0 0 0 h2 h1 0 0 0 h3 h2 h1 0 0 0 h3 h2 h1 0 0 0 h3 h2 h1

⎞

⎟ ⎟ ⎟ ⎟; ⎟ ⎠

⎛

⎜ ⎜ ⎜ H1 = ⎜ ⎜ ⎝

0 0 0 0 h1 0 0 0 h2 h1 0 0 h3 h2 h1 0 0 h3 h2 h1

0 0 0 0 0

⎞

⎟ ⎟ ⎟ ⎟; ⎟ ⎠

⎛

⎜ ⎜ ⎜ H2 = ⎜ ⎜ ⎝

0 0 0 0 0 0 h1 0 0 h2 h1 0 h3 h2 h1

0 0 0 0 0

0 0 0 0 0

⎞

⎟ ⎟ ⎟ ⎟. ⎟ ⎠

¯ d ,j , we obtain the following finite set of (dmax +1)×Q Denoting each column of the matrix (6) by h q vectors: 



¯ 0,1 , . . . , h ¯ 0,j , . . . , h ¯ 0,Q , . . . , h ¯ d ,1 , . . . , h ¯ d ,j , . . . , h ¯ d ,Q , . . . , h ¯ d ,1 , . . . , h ¯ d ,j , . . . , h ¯ d ,Q . (7) ¯= h H q q q max max max Using the set (7), we construct the block row-matrices: 

Wj =

¯ 0,j . . . h ¯ d ,j . . . h ¯ d ,j h q max



,

j = 1, . . . , Q.

(8)

In order to take account of non-stationarity of the delay, we introduce the vector ϕ¯ (d(k − j)) = (δ (d(k − j), 0) , . . . , δ (d(k − j), dmax ))T , AUTOMATION AND REMOTE CONTROL

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where δ(i, j) is the Kroneker symbol. As a result, we represent the non-stationary system with delay in FIR (5) as a stationary system with nonlinear output, unknown state vector xd (k), and input d(k): xd (k + 1) = An xd (k) + bn d(k),

xn (k + 1) = An xn (k) + bn u(k),

y(k) = Cn (xd (k)) xn (k),

(10)

where xd (k) = (d(k − 1) . . . d(k − Q))T ; ⎛

⎞

... 0 ⎜ .. ⎜ 1 0 . ⎜ An = ⎜ .. .. ⎜ . . ⎝ 0 1 0 

Cn (xd (k)) =

xn (k) = (u(k − 1) . . . u(k − Q))T ;

0

⎟ ⎟ ⎟ ⎟ ⎟ ⎠



;

bn =

1 0 ... 0

(Q×Q)

S1 W1 ϕ(d(k ¯ − 1)) . . . S1 WQ ϕ(d(k ¯ − Q))

T (Q×1)

;



;

S1 = (0, . . . , 1)

is a row matrix of dimension Q. Let us represent linear discrete systems with input, output and state delays in the canonical controllability form as delay-free systems with extended state vectors. Model of a plant with input delay by d steps. 

x(k + 1) xu (k + 1)





= A1 

y(k) =

cT

x(k) xu (k) 0





+

0 bu

  x(k) 

xu (k)



u(k),

,

(11)

where xu (k) = (xu1 (k) . . . xud (k))T ; xu1 (k) = u(k − d); . . . ; xuj (k) = u(k − d − 1 + j); . . . ; xud (k) = u(k − 1); b = (0 . . . 0 1)T (N ×1) ;

bu = (0 . . . 0 1)T (d×1) ; ⎛

0 .. .

cT = (bN . . . b2 b1 ) ; 1

0 ...

0

⎜ ⎜ ⎜ ⎜ 0 0 0 ... 1   ⎜ ⎜ ⎜ A bcT −a . . . −a1 −a N N −1 u =⎜ A1 = ⎜ 0 Au ... 0 ⎜ 0 ⎜ . ⎜ . ⎜ . ⎜ ⎝

0

...

0

cT u = (1 0 . . . 0)(1×d) ; ⎞

0 0 ... 0 ⎟ .. ⎟ . ⎟

0 0 ... 0 ⎟ ⎟ ⎟ 1 0 ... 0 ⎟ ⎟. 0 1 ... 0 ⎟ ⎟ ⎟ .. ⎟ . ⎟ ⎟

0 0 ... 1 ⎠ 0 0 ... 0

Model of a plant with output delay by d steps. 

x(k + 1) xy (k + 1)





= A2 

y(k) = AUTOMATION AND REMOTE CONTROL

0

x(k) xy (k)

cT y

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+

x(k) xy (k)

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,

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where by = (0 . . . 0 1)T (d×1) ;

cT y = (1 0 . . . 0)(1×d) ;

xy (k) = (xy1 (k) . . . xyd (k))T ;

xy1 (k) = y(k); . . . ; xyj (k) = y(k + j − 1); . . . ; xyd (k) = y(k + d − 1); ⎛

0 1 0 ... 0 .. . 0 0 0 ... 1 . . . −a1 −aN −aN −1 0 ... 0 .. .. . . 0 0 bN −1 . . . b1 bN

⎜ ⎜ ⎜ ⎜   ⎜ ⎜ A 0 ⎜ = A2 = ⎜ ⎜ by cT Ay ⎜ ⎜ ⎜ ⎜ ⎝

Model of a plant with state delay by d steps. 

x(k + 1) Λ(k + 1)



= A3 

y(k) =

A3 = ⎛

A Ad cΛ bΛ AΛ

⎛



;

⎛ ⎜ ⎜ As = ⎜ ⎝

⎛ ⎜ ⎜ bs = ⎜ ⎝

0 0 .. . 1

;

⎜ ⎜ ⎜ bΛ = ⎜ ⎜ ⎝

(d×1)

⎛ ⎜ ⎜ ⎜ ⎜ Ad cΛ = ⎜ ⎜ ⎜ ⎜ ⎝

0

0 .. .

bs

0 0 .. .. . . 0 0  T  aN cs aN −1 cT s

Λ(k)

,

(13) ⎞

⎞

⎜ ⎜ ⎜ AΛ = ⎜ ⎜ ⎝

As

0

0 .. .

As

0

...

... 0 .. ⎟ . ⎟

0

⎟ ⎟ ⎟ 0 ⎠

. 0 bs ...

0 ... 1 0 ... 0

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

0 1 0 ... 0 .. ⎟ ⎟ . ; ⎟ 0 0 0 ... 1 ⎠ . . . a1 (N ×N ) aN aN −1

⎞

0 ... 0

  x(k) 

⎛

..

0 ... 0 0 ... 0 1 ... 0

⎞

b u(k), 0

xj (k − 1)

bs

cT s

+

xj (k − d) ⎜ ⎟ ⎜ xj (k − d + 1) ⎟ ⎟ xsj (k) = ⎜ .. ⎜ ⎟ ⎝ ⎠ .

⎞

0 ... 0

 

⎛

0 1 ... 0 .. ⎟ ⎟ . ; ⎟ 0 0 ... 1 ⎠ 0 0 . . . 0 (d×d) ⎛

⎞ ⎟ ⎟ ⎟ ⎠

⎜ ⎜ ⎝

Ad = ⎜

⎞

xs1 (k) ⎜ ⎟ .. ⎜ ⎟ ⎜ ⎟ . ⎜ ⎟ ⎟ x (k) ; Λ(k) = ⎜ ⎜ sj ⎟ ⎜ ⎟ . ⎜ ⎟ .. ⎝ ⎠ xsN (k) (N d×1)



x(k) Λ(k)

cT 0

where 



0 .. . 0 0 0 .. . 0 0

;

(d×1)

... ..

cT s .. .. . . 0 ... 0 cT s  T  . . . a2 cs a1 cT s

⎞

0 .. ⎟ . ⎟ ⎟ ⎟ ⎟ 0 ⎠

. 0 As

;

(N d×N d)

⎛ T cs 0 . . . ⎜ ⎜ 0 cT ⎜ s cΛ = ⎜ ⎜ .. .. ⎝ . .

0 ...

(N d×N )

0 .. .

; j = 1, . . . , N ;

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

;

0

⎞

0 .. ⎟ . ⎟ ⎟ ⎟ ⎟ 0 ⎠

cT s

;

(N ×N d)

cT s = (1 0 . . . 0)(1×d) .

(N ×N d)

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I Fig. 2. Functioning of the nonlinear estimation algorithm with λLM opt and various statistical parameters (mathLM I I ematical expectation, variance) of the input sequence: (a) λopt = 48; (b) λLM opt = 2.

Dynamics of the nonlinear system (10) differs from (11)–(13) by output and, first of all, at the moments k  N1 + Q when delay disturbance arise, if the input remained constant on a time interval (N1 ; N2 ), where N1 + Q  N2 . Assertion 1. Let for each system (11)–(13) the following conditions hold: (a) dmax  d(k)  dmin ; (b) ρ(A1 ) < 1; ρ(A2 ) < 1; ρ(A3 ) < 1; (c) u(k) = const, N1 < k < N2 . Then, any disturbances of delay on the interval (a) do not affect the output of the systems (11)–(13). Here ρ is the spectral radius of a matrix. Assertion 2. Let, for the system (10), the FIR vector be given, and the conditions (a) and (c) hold. Then, disturbance of delay cause a change of output of the system (10). If disturbance of delay and variation of input u(k) take place simultaneously, the output of the system (10) will differ from the outputs of the systems (11)–(13) as well. The feature of the system (10) formulated in Assertion 2 is practically significant in description of delay disturbances for transportation of reagents in reaction zones of chemical-engineering processes. Statement of the problem. It is required to construct an algorithm for estimation of the state vector in (10) that ensure, at each step, the maximal rate of convergence of the estimation error ˆd (k+1) to zero for unknown variations of d(k) inside the interval [dmin ; dmax ]; e(k+1) = xd (k+1)− x here x ˆd (k + 1) is an estimate of the state vector. Thus, original problem of estimating delay in FIR is transformed to a problem of estimating the state vector of the nonlinear delay-free system (10). It is important to note that the problem of estimation of non-stationary delay can be considered as a problem of constructing a state observer with unknown input for the system (10). Nevertheless, in this case the necessary condition for existence of such a solution (observer) is violated, because for some k we have rank (C (xd (k)) bn ) = rank(bn ) [8]. 3. ALGORITHM FOR DELAY ESTIMATION To minimize the estimation criterion J(k) = 12 (y(k) − yˆ(k))2 , for each step we have the following gradient-type algorithm: yˆ(k) = Cˆ (ˆ xd (k)) xn (k), ∂ yˆ(k) , ˆ ∂ d(k − dmax − 1) ˆ ˆd (k) + bn d(k), x ˆd (k + 1) = An x

ˆ = d(k ˆ − 1) + λ(k) (y(k) − yˆ(k)) d(k)

where λ(k) is a parameter that affects the rate of convergence of the algorithm; AUTOMATION AND REMOTE CONTROL

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TORGASHOV  





ˆ − 1) . . . S1 Wj ϕ d(k ˆ − j) . . . S1 WQ ϕ d(k ˆ − Q) ˆ xd (k)) = S1 W1 ϕ d(k C(ˆ 



ˆ − j) = ϕ d(k linear functions









ˆ − j) . . . fi d(k ˆ − j) . . . fd ˆ f0 d(k max d(k − j)

;

is a vector of piecewise-

⎧ ˆ ⎪ ⎨ d(k − j) − i + 1,

ˆ − j)  i i − 1  d(k ˆ − j) = −d(k ˆ − j) + i + 1, i < d(k ˆ − j)  i + 1 d(k ⎪ ⎩ ˆ ˆ − j); 0, d(k − j) < i − 1; i + 1 < d(k



fi

T





ˆ − j) ∂fi d(k





ˆ − j) ∂fi−1 d(k

(15)



ˆ − j) = i = 1 and = −1, if d(k ˆ − j) ˆ − j) ∂ d(k ∂ d(k and i > 0. Introduction of functions (15) is caused by the necessity to ensure proper operation of ˆ − j) is not an integer number. the algorithm (14) for the case when d(k ˆ − dmax − 1) is performed in order to avoid zero values of the derivaDifferentiating by d(k tive in (14). Owing to properties of the functions (15), after differentiation of ϕ only the last (dmax + 1)-element of the vector of derivatives is not zero for any current value of the delay in the interval [dmin ; dmax ]. For the input sequence {u(k)}∞ −Q the persistent excitation (PE) condition holds, so that its covariance matrix is positive definite [9, 10]. This condition can be true for various types of signals, including random “white noise” with covariance matrix σu2 I(Q×Q) [10, p. 341], where σu2 is the variance of the input sequence. i is an integer. Moreover, we take

4. CONVERGENCE OF THE ESTIMATION ALGORITHM Let us prove that the estimation error tends to zero for a system with unknown stationary and slowly changing delay with the limit lim d(k) = dmax . Consider functioning of the algorithm for k→∞

ˆ 0 − q0 ) and q0 = 0, . . . , Q. Then, the range of variation of the initial conditions d(k0 − q0 ) > d(k following nonlinear function is determined from the inequality:   ˆ ˆ − j))  a d(k − j) − d(k − j) , ¯ − j)) − ϕ(d(k 0  S1 Wj ϕ(d(k j dmax

(16)

where aj = max {S1 Wj } − min {S1 Wj }. In the given case, the max/min operation applied to a vector denotes the choice of its element with the maximal/mininal value. Slow variations of the ˆ delay do not violate positiveness of the right-hand side of (16), i.e., d(k)  d(k). Let us represent the system (14) as an interval linear system. Using (16), we obtain ˆ e(k + 1) = A(k)e(k), where



⎛ ⎜ ⎜ ⎜ A0 = ⎜ ⎜ ⎜ ⎝

ˆ A(k) = AT 0 − 1 0 0 1 0 0 0 1 0 .. .. . . 0 0 0 ⎧ ⎨

··· ··· ··· .. .

0 0 0 .. .

1

0

λ dmax ⎞

⎩

diag (˜ a(k)) xn (k)Eu xn (k)bT n

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

a ¯j = aj max S1 Wdmax +1

(17)

;

T

;

a ˜(k) = (˜ a1 (k) . . . a ˜j (k) . . . a ˜Q (k))T ;

(Q×Q)



⎫

ˆ − dmax − 1) ⎬ ∂ϕ d(k ˆ − dmax − 1) ∂ d(k

⎭

;

a ˜j (k) ∈ [0; a ¯j ] .

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Then, we find the mathematical expectation (17), taking into account noncorrelatedness of the factors in the right-hand side: 



ˆ M {e(k + 1)} = M A(k) M {e(k)} ,

(18)



where M {xn (k)Eu xn (k)} = m2u . . . m2u ; (m2u + σu2 )(dmax +1),1 ; m2u . . . m2u

T Q×1

; mu is the mathemat-

ical expectation of the input sequence; Eu = [0 . . . 01(1,dmax +1 ) 0 . . . 0](1×Q) . For the case of uniform distribution, we have mu = (ar + br )/2, σu2 = (br − ar )2 /12, where ar and br are parameters of the ˜j (k). uniform distribution; a ˜m is the vector of average values of a In the general case, ar and br are time-dependent; therefore, the matrix Aˆm (k) in (18) remains dependent on time and λ: ⎛

β0 (k) −β1 (k) · · · −βQ−1 (k) ⎜ 0 ··· 0 ⎜ 1 Aˆm (k) = ⎜ .. .. ⎜ . . ⎝ 0 0 0 1 0

⎞ ⎟ ⎟ ⎟, ⎟ ⎠

(19)

where βi (k) ∈ [β i ; β¯i ], i = 0, . . . , Q − 1, and their elements are related by the following inequalities:   0.9 < β 0 ; β¯0 < 1; 0 < β i ; β¯i  0.1; Q max β¯1 , . . . , β¯Q−1  β 0 ;

Q−1 

  β¯i  > 1.

(20)

i=0

Now we prove that the zero solution of the system (18) is, with account for (19), (20), asymptotically stable, i.e., lim Φ(k) = 0,

(21)

k→∞

where Φ(k) is the fundamental matrix; Φ(k) =

k−1  j=k0

Aˆm (j); Φ(k0 ) = I. The last inequality in (20)

does not make it possible to prove validity of (21) by Theorem 4.12 in [11]. Therefore, we start with finding the first element of Φ(k) for increasing k. k = 1 : z1,1 = β0 (k), k = 2 : z2,1 = β02 (k) − β1 (k), k = 3 : z3,1 = z2,1 β0 (k) − β0 (k)β1 (k) − β2 (k), k = 4 : z4,1 = z3,1 β0 (k) − z2,1 β1 (k) − β0 (k)β2 (k) − β3 (k), k = Q : zQ,1 = zQ−1,1 β0 (k) − zQ−2,1 β1 (k) − . . . − βQ−1 (k), k = j : zj,1 = zj−1,1 β0 (k) −

Q 

(22)

zj−i,1 βi−1 (k).

i=2

Since βi (k) are inside the intervals (20), for k = 1, . . . , Q, . . . it is evident that z1,1 > z2,1 > Q

. . . > zQ−1,1 > 0, and for k > Q, lim

k→∞ i

zk−i,1 βi−1 (k) → 0, i.e., lim zk,1 = 0. k→∞

A special feature of the matrix Φ(k) is the sequential shift of its rows downward as k increases. Moreover, the remaining elements of the first row also tend to zero, because they depend on the AUTOMATION AND REMOTE CONTROL

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 first element zk,i i=1 = f (zk−1,1 ): ⎛

⎜ zk−1,1 β0 (k − 1) + zk−1,2 −zk−1,1 β1 (k − 1) + zk−1,3 . . .

Φ(k) = ⎝

.. .

.. .

.. .

. . . −zk−1,1 βQ−2 (k − 1) + zk−1,Q −zk−1,1 βQ−1 (k − 1) 

.. .

.. .



zk,Q

.. .

⎞

 ⎟ ⎟. ⎠

At the kth step, the first row of the fundamental matrix is less than the previous one (elementwise), so that we arrive at equality (21). It should be noted that Φ(k)1 will grow for k < Q, and will vanish for k > Q, because the unit elements of (19) are changed for smaller values, which converge to zero. Maximal rate of convergence. Let us find the value of the parameter λopt that provide for the maximal rate of convergence (decay) of the estimation error to zero at each step k. A definition of decay rate for a continuous-time non-stationary system is given in [12, p. 66]. Here we consider the discrete case and use the following quadratic Lyapunov function: V (em (k)) = eT m (k)P em (k)

(23)

with a symmetric positive definite matrix P , which hereinafter will be denoted as P > 0; em (k) = M {e(k)}. The rate of convergence of the trajectory to zero is expressed in terms of increment of the quadratic function (23) ΔV (em (k)) = V (em (k + 1)) − V (em (k)) < −μV (em (k)).

(24)

The more the coefficient μ (Eq. (24) holds for all trajectories of the system), the faster response has the system. In our case, dynamic properties of (19) are determined by λ, so that we arrive at the problem of finding the λopt such that the maximal value μmax takes place, and the rate of convergence of the estimation error to zero is also maximal. Equation (24) can be written in the form V (em (k + 1)) < αV (em (k)),

(25)

and further we can obtain the inequality V (em (k)) < αk V (em (0)), where α = 1 − μ is the rate of convergence of the trajectory to zero for the discrete-time case (0 < α < 1) [13, p. 29]. Therefore, the problem of maximizing μ (calculation of μmax ) is, according to (25), equivalent to the problem of minimizing α. Substituting (23) into (25), we have T ˆT ˆ eT m (k)Am (λ, k)P Am (λ, k)em (k) − em (k)αP em (k) < 0.

(26)

Hence, we formulate the optimization problem in terms of linear matrix inequalities: min α ˆ AˆT m (λ, k)P Am (λ, k) − αP < 0, ¯ P > 0, λ  λ  λ, AUTOMATION AND REMOTE CONTROL

(27)

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¯ are lower and upper bounds of λ, which are taken a priori. Solution of (27) makes it where λ and λ possible to find λopt (k), for which the maximal rate of convergence of the estimation error to zero takes place for variations of d(k) in the interval [dmin ; dmax ]. In the case of fast delay variation, oscillation of the error near the zero solution will, in the average, remain, because the rate of convergence of the algorithm is bounded. We should note practical aspects of solving the problem (27). Such problems can be solved using existing effective program packages [14]. As a rule, such a solution takes several seconds at modern computers, and this makes it possible to employ (27) at each step, because the control period for industrial CEP lie in the interval from 0.5 to 1 minute [15]. 5. EXAMPLE h = (0.1813 0.1484 0.1215 0.0995 0.0814)T ; N = 5; dmax = 5; dmin = 1; Q = 10. The matrices Wj are formed according to (6)–(8) for the model (10) in such a way that we have ⎛

⎛ ⎜ ⎜ ⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎜ S 1 W1 ⎜ ⎜ ⎟ .. ⎟ =⎜ ⎜ . ⎠ ⎜ S1 WQ Q×(d +1) ⎜ ⎜ max ⎜ ⎜ ⎜ ⎝

0.1813 0.1484 0.1215 0.0995 0.0814 0 0 0 0 0

0 0.1813 0.1484 0.1215 0.0995 0.0814 0 0 0 0

0 0 0.1813 0.1484 0.1215 0.0995 0.0814 0 0 0

0 0 0 0.1813 0.1484 0.1215 0.0995 0.0814 0 0

0 0 0 0 0.1813 0.1484 0.1215 0.0995 0.0814 0

0 0 0 0 0 0.1813 0.1484 0.1215 0.0995 0.0814

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

6. CONCLUSIONS In the present work, the problem of estimating system delay, arising in FIR, is solved. It is shown that dynamics of such system differs from that of systems with input, output and state delays, and can be represented in the form of a nonlinear (with respect to the output) state-space model (10). The algorithm of delay estimation, developed for (10), is nonlinear. Its representation in the form of a linear non-stationary system is found. It is shown that the algorithm can reduce the estimation error to zero (zero solution of the system will be asymptotically stable). I Using linear matrix inequalities, we solved the problem of finding λLM opt that provides for the maximal rate of decay (convergence) of the error to zero. Long-time (taking several hours) and cumbersome computational experiments with the nonlinear system (14) performed in off-line mode with the aim of computing the optimal parameter value λnopt have shown that this value almost I coincide with λLM opt , which was found as a result of a much faster (several seconds) solution of the problem (27) (see table). Results of computational experiments on ensuring optimal functioning of the estimation algorithm no. ar br λnopt λLMI opt 1 0 1 44 48 2 0 2 15 12 0 3 5.25 5.5 3 4 0 4 2.5 3 5 0 5 2.25 2

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APPENDIX Proof of Assertion 1. System with input delay. Let at the moment k  N1 + Q the state of the system be such that x∞ (k + 1) = x∞ (k) = [α . . . α]T (N ×1) and d(k) = d1 , then x∞ (k + 1) = Ax∞ (k) + bcT u1 xu1 (k). At the moment k + 1 the delay changes as dk+1 = d2 : x(k + 2) = Ax∞ (k + 1) + bcT u2 xu2 (k + 1). Find increment of the state vector: T x(k + 2) − x∞ (k + 1) = Ax∞ (k + 1) − Ax∞ (k) +bcT u2 xu2 (k + 1) − bcu1 xu1 (k)







⎛

= b⎝ ⎛



0

T = b cT u2 xu2 (k + 1) − cu1 xu1 (k) d2 

d1 

c2j u(k − d2 + 1 − j) −

j=1

 ⎞

c1j u(k − d1 + 1 − j)⎠

j=1

⎞

⎜ ⎟ d2 d1 ⎜ ⎟   ⎜ ⎟ = b ⎜c21 u(k − d2 ) − c11 u(k − d1 ) + c2j u(k − d2 + 1 − j) − c2j u(k − d1 + 1 − j)⎟ ⎜ ⎟   j=2 j=1 ⎝ ⎠ 0      

=

0 T (0 . . . 0)(N ×1) ,

0

where c2j = c1j = 1 for j = 1 and c2j = c1j = 0 for j = 1. System with output delay. Let us have, at the moment k  N1 +Q for d(k) = d1 and d(k+1) = d2 , ∞ x∞ y1 (k + 1) = xy1 (k),

∞ x∞ y2 (k + 1) = xy2 (k),

T ∞ ∞ x∞ y1 (k + 1) = by1 c x (k) + Ay1 xy1 (k),

xy2 (k + 2) = by2 cT x∞ (k + 1) + Ay2 x∞ y2 (k + 1), T x∞ y1 (k) = [β . . . β](d1 ×1) ;

T x∞ y2 (k) = [β . . . β](d2 ×1) ;

x∞ (k) = [α . . . α]T (N ×1) under the constraints imposed on the delay 0 < dr  dmax , r = 1, 2. Each ith (i = 1, . . . , d2 ) element of the vector xy2 (k + 2) is defined by the following formula: xiy2 (k

+ 2) =

N  

T



by2 c

ij

j=1

α+

d2 

(Ay2 )iq β.

(A.1)

q=1

Hence, xiy2 (k + 2) = β, xiy2 (k

+ 2) = α

i = 1, . . . , d2 − 1, N 

bj ,

i = d2 .

j=1

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Notice that the output of the system, which is represented in the canonical controllability form, is the sum of weighted state variables xi (i = 1, . . . , N ) with the corresponding elements of the vector c, i.e., β = α

N j=1

bj . Then, T xy2 (k + 2) = x∞ y2 (k + 1) = [β . . . β](d2 ×1) .

Thus, variation of the output delay by the value |d2 − d1 | does not affect the state vector, and manifests itself only in changing zero terms in Eq. (A.1). System with state delay. Assume that, for a system with state vectors Λ∞ (k) = [γ . . . γ]T (N d1 ×1) , ∞ T ∞ ∞ T Λ (k+1) = [γ . . . γ](N d2 ×1) , x (k) = x (k+1) = [γ . . . γ](N ×1) , the delay changes at the moment k: x∞ (k + 1) = Ax∞ (k) + Ad cΛ1 Λ∞ 1 (k) + bu(k),

x(k + 2) = Ax∞ (k + 1) + Ad cΛ2 Λ∞ 2 (k + 1) + bu(k). We find increment of the vector x ∞ Δx(k + 2) = x(k + 2) − x∞ (k + 1) = Ad [cΛ2 Λ∞ 2 (k + 1) − cΛ1 Λ1 (k + 1)] .

Elements of the vector (A.2) have the form i

Δx (k + 2) =

d2  j2 =1

because

d2 j2 =1

(cΛ2 )i,j2 =

d1 j1 =1

(cΛ2 )i,j2 γ −

d1  j1 =1

(cΛ1 )i,j1 γ = 0,

i = 1, . . . , N

(cΛ1 )i,j1 owing to (cΛ1 )i,(i−1)d1 +1 = (cΛ2 )i,(i−1)d2 +1 = 1, (cΛ1 )i,j1 = 0,

(cΛ2 )i,j2 = 0,

j1 = 1, . . . , N d1 ,

j2 = 1, . . . , N d2 ,

i = 1, . . . , N.

Proof of Assertion 2. Let a disturbance d(k) = β2 act on a system having state vector T x∞ d (k) = [β1 . . . β1 ](N ×1)

for k  N1 + Q. Then, ⎛

0

... 0 ⎜ .. ⎜ 1 0 . ⎜ xd (k + 1) = ⎜ . . ⎜ .. .. ⎝ 0 1 0

⎞⎛ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

β1 β1 .. .

⎞

⎛

⎟ ⎜ ⎟ ⎜ ⎟+⎜ ⎟ ⎜ ⎠ ⎝

β1

1 0 .. .

⎞

⎛

⎟ ⎜ ⎟ ⎜ ⎟ d(k) = ⎜ ⎟ ⎜ ⎠ ⎝

0

β2 β1 .. .

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

β1

. (N ×1)

Increment of the output is not zero: Δy(k + 1) = y(k + 1) − y(k) =



S1 W1 (ϕ(β ¯ 2 ) − ϕ(β ¯ 1 )) 0 . . . 0

¯ 1 ). because ϕ(β ¯ 2 ) = ϕ(β AUTOMATION AND REMOTE CONTROL

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x∞ (k) = 0,

(A.2)

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REFERENCES 1. Richard, J.-P., Time-Delay Systems: An Overview of Some Recent Advances and Open Problems, Automatica, 2003, vol. 39, pp. 1667–1694. 2. Drakunov, S.V., Perruquetti, W., Richard, J.P., and Belkoura, L., Delay Identification in Time-Delay Systems Using Variable Structure Observers, Ann. Rev. Control , 2006, vol. 30, pp. 143–158. 3. Orlov, Y., Belkoura, L., Richard, J.P., and Dambrine, M., Adaptive Identification of Linear Time-Delay Systems, Int. J. Robust Nonlin. Control , 2003, vol 13, no. 9, pp. 857–872. 4. Thomassin, M., Bastogne, Th., Richard, A., and Garnier, H., Generalization of a Correlation Method for Time-Delay Estimation with Application to a River Reach, in Proc. 14 IFAC Symp. Syst. Identificat., Newcastle, Australia, 2006, pp. 891–896. 5. Belkoura, L. and Richard, J.P., A Distribution Framework for the Fast Identification of Linear Systems with Delays, in Proc. 6 IFAC Workshop on Time Delay Syst., L’Aquila, Italy, 2006, pp. 243–248. 6. Pan, F., Han, R.-C., and Feng, D.-M., An Identification Method of Time-Varying Delay Based on Genetic Algorithm, in Proc. 2 Int. Conf. Machine Learning and Cybernet., Xian, China, 2003, pp. 781–783. 7. Horn, R.A. and Johnson, C.R., Matrix Analysis, Cambridge: Cambridge Univ. Press, 1990. 8. Korbicz, J., Witczak, M., and Puig, V., LMI-based Strategies for Designing Observers and Unknown Input Observers for Non-Linear Discrete-Time Systems, Bull. Polish Acad. Sci., Techn. Sci., 2007, vol. 55, no. 1, pp. 31–42 (http://bulletin.pan.pl/(55-1)31.pdf). 9. Ljung, L., System Identification—Theory for the User, New York: Prentice-Hall, 1999. 10. Katayama, T., Subspace Methods for System Identification, London: Springer-Verlag, 2005. 11. Elaydi, S., An Introduction to Difference Equations, New York: Springer, 2005. 12. Boyd, S., Ghaoui, L.E., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 1994, vol. 15. 13. Rubensson, M., Stability Properties of Switched Dynamical Systems. A Linear Matrix Inequality Approach, PhD Dissertation, Sweden: Chalmers Univ. of Technology, 2003. 14. Churilov, A.N. and Gessen, A.V., Issledovanie lineinykh matrichnykh neravenstv. Putevoditel’ po programmnym paketam (Investigation of Linear Matrix Inequalities. A Toolbox Guide), St.-Petersburg: S.-Peterburg. Gos. Univ., 2004. 15. Tatjewski, P., Advanced Control of Industrial Processes: Structures and Algorithms, London: Springer, 2007.

This paper was recommended for publication by B.T. Polyak, a member of the Editorial Board

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