Adaptive unit-vector law with time-varying gain for finite-time parameter estimation in LTI systems

Journal Pre-proof Adaptive unit-vector law with time-varying gain for finite-time parameter estimation in LTI systems

M.N. Kapetina, A. Pisano, M.R. Rapai´c, E. Usai

PII:

S0168-9274(19)30252-1

DOI:

https://doi.org/10.1016/j.apnum.2019.09.013

Reference:

APNUM 3663

To appear in:

Applied Numerical Mathematics

Received date:

30 April 2019

Revised date:

16 September 2019

Accepted date:

19 September 2019

Please cite this article as: M.N. Kapetina et al., Adaptive unit-vector law with time-varying gain for finite-time parameter estimation in LTI systems, Appl. Numer. Math. (2019), doi: https://doi.org/10.1016/j.apnum.2019.09.013.

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Adaptive unit-vector law with time-varying gain for finite-time parameter estimation in LTI systems M. N. Kapetinaa , A. Pisano b,, M. R. Rapai´ca , E. Usaib a University

of Novi Sad, Faculty of Technical Sciences, Department of Computing and Control Engineering, Trg Dositeja Obradovi´ ca 6, 21000 Novi Sad, Serbia b Department of Electrical and Electronic Engineering, University of Cagliari, Via Marengo, 09123 Cagliari

Abstract A continuation of previous authors’ work on adaptive parameter estimation for linear dynamical systems having irrational transfer function is presented in this work. An original modification of the gradient algorithm, inspired by the variable structure control techniques and additionally featuring a time-varying adaptation gain, is presented and analyzed using Lyapunov techniques. The exposition is illustrated by several numerical examples which illustrate the effectiveness of the proposed algorithm. Keywords: Adaptive Parameter Estimation, Fractional Order Systems, Distributed Parameter Systems, Delayed Systems, Gradient Algorithm, Lyapunov Stability

1. Introduction This paper deals with the problem of adaptively estimating unknown parameters in a wide class of linear systems. Classical parameter estimation techniques, such as those based on the celebrated least-squares (LS) algorithm, as5

sume that the model can be expressed as a linear regression with respect to all $ This work has been partially supported by Serbian Ministry of Education and Science, grant no. TR32018 (M.N.K., M.R.R.) and grant no. TR33013 (M.R.R.), and by ”Fondazione di Sardegna”, CUP n. F72F16003170002 (A.P., E.U.) Email address: [email protected] (A. Pisano )

Preprint submitted to Journal of Applied Numerical Mathematics

October 21, 2019

unknown parameters. This requirement implies that the transfer function of the system must be rational, i.e. represented by the ratio between two polynomials. In the present study we consider a more general class of linear systems having transfer functions which are not necessarily rational. Whenever a delay appears 10

somewhere in the system structure, or when there are spatially or otherwise distributed components, or additionally when differential operators of non-integer order appear in the system differential equation, the overall transfer function is irrational [1, 2]. In such cases, alternative identification techniques are required in order to solve the problem. This is especially difficult in cases when adaptive

15

identification is required, i.e. when parameter values may change in time, either continuously or abruptly, and the parameter estimation algorithm must be able to track these changes. In this paper we present continuation of our recent work on adaptive parameter estimation in linear systems having irrational transfer function. To

20

the best of our knowledge, the first work addressing this problem in its full generality was published in [3], where a two-stage algorithm was proposed combining least-squares with gradient-based optimization techniques. There were, off course, numerous contributions in literature prior to [3], but they all tackle just specific classes of systems. For example, delay systems were considered

25

in [4, 5, 6, 7, 8], distributed parameter systems in [2, 9], and fractional-order systems in [10, 11, 12, 13]. In our recent publication [14] the classical gradient adaptation algorithm, based on quadratic instantaneous cost function, was extended to this kind of problems, and conditions under which the algorithm converge were investigated in detail.

30

An attempt to develop more robust schemes with superior performance as compared to the scheme proposed in [14] is made in the present work. Particularly, we consider a novel gradient identification algorithm, inspired by unitvector control techniques [15, 16, 17]. In addition, a time-varying adaptation gain is considered in the present work, and a particular gain-adjustment mech-

35

anism is designed, ensuring large gain values during the convergence transient and smaller gain values in steady state in order to speed up the convergence of 2

the algorithm while limiting at the same time too large steady-state gain values, that could yield performance degradation in the presence of measurement noise. The paper is illustrated by several examples, in each of which a comparison with 40

the linear gradient scheme presented in [14] is also made. The outline of the paper is as follows. This introductory section ends with an overview of the basic notation. Section 2 contains problem formulation. Detailed explanation of the proposed algorithm along with the corresponding Lyapunov-based proof of convergence is given in Section 3. Numerical studies

45

of several carefully selected examples are presented in Section 4. Concluding remarks are given in the final Section 5. 1.1. Notation and preliminaries The set of real numbers will be denoted by R. The Euclidean norm of a

50

vector a ∈ Rn is denoted by a. The induced 2-norm of matrix A ∈ Rm×n  is denoted as A := λmax (AT A) = σmax (A) where λmax (·) (respectively λmin (·)) denotes the maximum (respectively minimum) eigenvalue, σmax (·) is the largest singular value, whereas aT and AT denote transposes of vector a or matrix A. AS = 12 (A + AT ) denotes the symmetric part of the square matrix A. If c ∈ Rn , then the set B(c, r) = {y ∈ Rn : c − y ≤ r} is referred to as the

55

closed ball B(c, r) with center c and radius r. Convolution operator will be denoted by  for any time dependent signals g and u, 

t

g(t)  u(t) = 0

g(t − τ )u(τ )dτ .

A linear, time invariant system is completely specified by its transfer function, G(s), such that Y (s) = G(s)U (s) , where U (s) and Y (s) are Laplace transforms of the input and output, respectively. As usual, we denote the Laplace transform variable by s, and adopt convention by which signals are denoted by lower-case letters and their Laplace

3

transforms by the corresponding upper-case ones. In multiple-input multipleoutput (MIMO) systems input and output are vector-valued signals, and the transfer function G(s) is actually a matrix. In time domain, the equivalent process model is y(t) = g(t)  u(t) where g(t) is the impulse response of the system, i.e. the inverse Laplace transform of its transfer function. In order to simplify notation, we assume that the convolution behaves like matrix product when applied to vector-valued signals, so that in MIMO case g(t) is actually matrix-valued. 60

Vector of unknown process parameters will be denoted by θ. Estimated ˆ values will be denoted by “hat”, so that for example θ(t) will stand for the estimated value of the unknown parameters at the time instant t. Similarly, ˜ = θ − θ(t) ˆ will be estimation errors will be expressed with “tildes”, e.g. θ(t) used to de note estimation error at time t. Explicit time dependence of different

65

quantities will be omitted whenever meaning is clear from the context.

2. Problem formulation Consider the following class of linear time-invariant systems y(t) = g(t; θ)  u(t)

(1)

where y(t) ∈ Rp is the measurable output vector, u(t) ∈ Rm is the control input vector, g(t; θ) is the impulse response of the system and θ ∈ Rq is the vector of unknown constant parameters. The structure of g(t; θ) is assumed to be known. 70

The corresponding transfer matrix is denoted with G(s, θ). The aim of this paper is to provide a finite-time converging estimation of the parameter vector θ, using the real-time measurement of the output and input vectors y(t) and u(t) and assuming that the structure of g(t; θ) (and therefore G(s, θ)) is known.

4

The adaptive plant model, which is part of the overall parameter estimation algorithm, is given by ˆ  u(t) yˆ(t) = g(t; θ)

(2)

where yˆ ∈ Rp is the estimated output vector and θˆ is the estimated parameter vector. The output estimation error is defined as y˜(t) = y(t) − yˆ(t)

(3)

and parameter estimation error is expressed as ˜ = θ − θ(t) ˆ θ(t)

75

(4)

Let system (1) be satisfying the following assumptions. Assumption 1. There exist Λ > 0 and ε1 > 0 such that    ∂ yˆ    ≤ Λ < ∞ , ∀θˆ ∈ B(θ, ε1 ) , ∀t > 0  ∂ θˆ 

ˆ Assumption 2. There exist m > 0 and ε2 > 0 such that for every ξ ∈ (θ, θ), 

the matrix ˆ ξ) = M(θ,

∂ yˆ ∂ θˆ

T 

 ∂ yˆ ˆ ∂ θˆ θ=ξ

is such that

   S ˆ ≥m>0, min σmin M θ(t), ξ t≥0

∀θˆ ∈ B(θ, ε2 ),

(5)

∀t > 0 .

(6)

Note that the number p of independent outputs must not be smaller than the number q of uncertain parameters, otherwise the Assumption 2 cannot be satisfied. Note also that both assumption constrain the permitted profiles of the input excitation signal u(t). In fact, these assumptions can be seen as 80

generalizations of the classical notion of ”persistence of excitation”. The values of Λ and m can be evaluated numerically during experiments.

5

3. Parameter Identification Algorithm The estimation problem may be regarded as a minimization problem. In this scenario, a common approach is to define a cost function of the output estimation error to subsequently be minimized. We choose the following standard structure for the cost function J=

1 T y˜ y˜ . 2

(7)

The algorithm proposed in the present work considers the following normalized adaptation law

∂J

˙ ∂ θˆ θˆ = −k(t)   ∂J  .  ∂ θˆ 

(8)

2

where k(t) > 0 is an adaptive gain which is governed by the following differential equation ˙ y (t) , k(t) = −γ1 (k(t) − kmin ) + γ2 ˜

k(0) ≥ kmin ,

(9)

where γ1 > 0, γ2 > 0 and kmin > 0 are constant parameters. Differentiating the cost function (7) and substituting it into the adaptation law (8) yields



∂ yˆ ∂ θˆ

T

˙ θˆ = k(t)   ∂ yˆ T   ∂ θˆ

y˜ ,  y˜ 

(10)

The form of the proposed adaptation rule (8) is inspired to a robust control 85

methodology for nonlinear multivariable uncertain systems referred to as unitvector control law [15], and due to this the proposed adaptation law (8) is denoted as unit vector adaptation with adaptive gain. The block diagram of this adaptive system is shown in Fig. 1. The performance of the proposed parameter estimation algorithm is estab-

90

lished in the following Theorem. Theorem 1. Consider the process model (1) with vector of unknown parameters θ. Let Assumption 1 and 2 hold and assume that the unknown param˜ eters are estimated using the adaptation law (10). If θ(0) < ε, where ε = 6

d(t) u(t)

y(t, θ)

G(s, θ)

+ −

ˆ yˆ(t, θ)

ˆ G(s, θ)

ˆ y˜(t, θ)

∂ ∂ θˆ 1 s

×

k

× 1 ·2

kmin

− +

1 s

+

γ2 ·2

γ1

Figure 1: The overall block structure of the proposed parameter identification algorithm.

min{ε1 , ε2 }, then the unknown parameters are identified after a finite transient process, i.e. ˆ − θ = 0, θ(t)

∀t ≥ T .

An upper bound of the reaching time is T ≤

Λ2 ε , m kmin

(11)

Proof. Let us consider the following candidate Lyapunov function: V =

1 ˜T ˜ θ θ. 2

In order to simplify the notation, denote

∂ yˆ ∂ θˆ

(12) as Φ. By virtue of (10), the time

derivative of V (t) is ΦT y˜ ˙ ˙ . V˙ = θ˜T θ˜ = −θ˜T θˆ = −k(t)θ˜T ΦT y˜

7

(13)

By using McLeod’s Mean Value Theorem [18], output y(t) can be expressed as y(t) = yˆ(t; θ) ˆ + θ(t)) ˜ = yˆ(t; θ(t) ˆ = yˆ(t; θ(t)) +

p  i=1

ˆ ∂ yˆ(t; θ(t)) ri (t) ˆ ∂ θˆ

˜ , θ(t)

(14)

θ(t)=ξi (t)

 ˆ < β < 1 and ri (t) are nonnegative scalar where ξi (t) ∈ βθ(t) + (1 − β)θ(t)|0 p functions such that i=1 ri (t) = 1. It follows that the output error signal can be expressed as y˜(t) =

p  i=1

ˆ ∂ yˆ(t; θ(t)) ri (t) ˆ ∂ θˆ

˜ . θ(t)

(15)

θ(t)=ξi (t)

By considering (14) and (15) one can manipulate (13) as follows   p ∂ yˆ ˜ ΦT r θ i=1 i ∂ θˆ ˆ θ=ξi V˙ = −k(t)θ˜T ΦT y˜  p

 k(t) T ˆ ξi )θ˜ ri θ˜ M(θ, =− T Φ y˜ i=1

(16) (17)

˜ Note that θ(0) < ε by assumption. Thus, there exists t¯ > 0 such that the ˜ ≤ ε for all t ∈ [0, t¯). Due to parameter error vector is confined in the ball θ Assumption 2, the next estimation holds p 

ˆ ξi )θ˜ ≥ ri θ˜T M(θ,

i=1

p 

˜ 2, ri mθ˜T θ˜ = mθ

t ∈ [0, t¯)

(18)

i=1

Let us now derive an upper estimate of ΦT y˜. By means of (15) one derives that ⎛ p  ˆ ∂ yˆ(t; θ(t)) T T ⎝ Φ y˜ = Φ ri (t) ˆ ∂ θˆ i=1

=

p 

⎞ ˜ ⎠ θ(t)

θ(t)=ξi (t)

ˆ ξi (t))θ(t) ˜ . ri (t)M(θ,

i=1

8

(19)

Due to basic properties of norms relation (19) yields  p      ˜  ˆ ξi (t)) ΦT y˜ ≤  ri (t)M(θ,  θ(t)    i=1

≤

p 

ˆ ξi (t))θ(t) ˜ ri (t)M(θ,

(20)

i=1

ˆ and By Assumption 1, the next chain of inequalities is in force for all ξi ∈ (θ, θ) ∀t ∈ [0, t¯) ˆ ξi (t)) = ΦT (t, θ)Φ(t, ˆ M(θ, ξi ) ˆ ξi ) ≤ Λ2 , ≤ ΦT (t, θ)Φ(t,

(21)

which, in combination with (20), implies an upper bound ˜ ΦT y˜ ≤ Λ2 θ(t),

∀t ∈ [0, t¯) .

(22)

Finally, by taking into account inequalities (18) and (22), the time-derivative of the Lyapunov function candidate is further estimated as m ˜ m V˙ ≤ −k(t) 2 θ = −k(t) 2 Λ Λ

√ √ 2 V,

∀t ∈ [0, t¯) .

(23)

The differential equation governing the time evolution of the adaptation gain k(t) takes the form ˙ k(t) = −γ1 k(t) + γ1 kmin + γ2 ˜ y (t) .

(24)

Solving (24) one gets k(t) = e−γ1 t k(0) + e−γ1 t  (γ1 kmin ) + e−γ1 t  (γ2 ˜ y (t)) ,

(25)

and choosing k(0) = kmin the next expression hold k(t) = kmin + ϕ(˜ y (t)) ,

(26)

where ϕ(˜ y (t)) = e−γ1 t  (γ2 ˜ y (t)). It turns out that ϕ(˜ y (t)) ≥ 0. By (23) and (26), we conclude that V˙ is negative definite. Since V˙ is nonpos˜ ≤ ε is guaranteed to be invariant, whence t¯ = ∞. itive then the domain θ 9

Therefore, from (26) and (23) m√ y )) 2 2V V˙ ≤ −(kmin + ϕ(˜ Λ m√ m√ y ) 2 2V . ≤ −kmin 2 2V − ϕ(˜ Λ Λ

(27)

Using the Comparison Lemma (which can be found in [19], Lemma 3.4) an upper bound for V (t) is found V (t) ≤ (

 m V (0) − (kmin + ϕ(˜ y )) √ (t − t0 ))2 . 2Λ2

Then V (t) reaches zero at most when √ 2  2Λ 1 t≤ V (0) + t0 m kmin + ϕ(˜ y (t)) 1 Λ2 ˜ θ(0) + t0 . = m kmin + ϕ(˜ y)

(28)

(29)

From this, the reaching time can be estimated as follows T ≤

Λ2 1 ˜ θ(0) . m kmin + ϕ(˜ y)

(30)

and an upper bound for the reaching time can finally be derived as T ≤

Λ2 ε Λ2 ε ≤ . m kmin + ϕ(˜ y) m kmin

(31)

Remark 1. For practical implementation proposed adaptation law is replaced by the smooth approximation ∂J

˙ ∂ θˆ  , θˆ = −k(t)   ∂J   ∂ θˆ  + δ 2

where δ is a small positive constant. This approximation is used in order to avoid division by a small number in the steady state.

95

4. Numerical examples In the following examples we compare the adaptation law  T ∂ yˆ ˙ ˆ y˜ , θ=k ∂ θˆ 10

(32)

proposed in [14] with the unit-vector adaptation law (10) proposed in the present work. Tests with the adaptation gain k(t) being kept to a constant value are also presented to highlight the superior performance of the method proposing the time-varying adaptation gain. 100

Note 1. In all following examples, whenever the process is described with nonrational transfer function, the response of the system is directly calculated as convolution of input signal and system’s kernel. 4.1. Example 1 Consider the first-order delay system described by transfer function G0 (s) = Kp

1 −sτ e , s+1

(33) T

where both the process gain Kp and delay τ are unknown, so that θ = [Kp τ ] . Assume that output and its first derivative are directly accessible for measurements. In order to apply the proposed scheme, let us use the input u(t) = cos(ωt)

(34)

where ω > 0 is fixed, and let us consider the transfer function matrix ⎤ ⎡ ⎤ ⎡ 1 −sτ (s) e G K 0 p ⎥ ⎢ ⎥ ⎢ s+1 G(s; θ) = ⎣ ⎦ . ⎦=⎣ s Kp s+1 e−sτ sG0 (s) The adaptation law (10) reduces to ⎡

∂ yˆ1 ⎣ ∂ Kˆ p

⎡

⎤

⎣

⎦ = k

˙ Kˆp τˆ˙



∂ yˆ1 ∂ τˆ

∂ yˆ1 ˜1 ˆp y ∂K

+

∂ yˆ2 ˜2 ˆp y ∂K

⎤⎡ ⎤

∂ yˆ2 ˜1 ˆ p ⎦ ⎣y ∂K ⎦ ∂ yˆ2 ∂ τˆ

2

+

y˜2 

∂ yˆ1 ˜1 ∂ τˆ y

+

The estimated output signals can be rewritten as ˆ p e−(t−ˆτ )  cos(ωt) yˆ1 (t) = K yˆ2 (t) =

d ˆ −(t−ˆτ ) (Kp e  cos(ωt)) , dt 11

∂ yˆ2 ˜2 ∂ τˆ y

2

(35)

ˆp ≡ K ˆ p (t) is used for brevity. where shorthand notation τˆ ≡ τˆ(t) and K 105

Actual values of unknown parameters are assumed to be Kp = 10 and τ = 0.8 and the input signal is u(t) = cos(t).

    Figure 2a shows the temporal evolution of  ∂∂yθˆˆ  evaluated along the trajec-

tory of the adaptation process and it is not hard to verify that Assumption 1 is satisfied. Also, it is possible to directly evaluate the upper bound of the ad110

ˆ ξ), missible initial parameter errors. To do so, the symmetrized matrix MS (θ, defined in Assumption 2 for ξ = [Kξ τξ ], is computed together with its eigenvalues. The computations are cumbersome and we choose to omit them. Let us introduce ΔKp = Kξ − Kp and Δτ = τξ − τ , which can be interpreted as conservative upper bounds for the maximal allowable initial estimation errors. The

115

ˆ ξ) is plotted in Fig 2b. Please note that the plot smallest eigenvalue of MS (θ, is translated so that θˆ is indicated in the origin, leaving ξ as the only variable. The axes are therefore labeled as ΔKp and Δτ , in order to indicate allowable initial errors. If one chooses the initial estimates error within region shown on Fig 2c, MS is guaranteed to be positive definite, and it is always possible to

120

find a positive m such that Assumption 2 holds. In this light, Figure 2c defines a region in the Δτ - ΔKp plane within which the convergence is guaranteed. The simulation results are presented in Fig. 3. We assume that initial estiˆ p (0) = 8 and τˆ(0) = 0.2. Parameter values abruptly change mates values are K to Kp = 15 and Kp = 8 at t = 15 and t = 30, respectively. In addition, assume

125

that delay changes to τ = 0.4 at t = 25. The actual values of the unknown parameters are presented in blue line, simulation results obtained with proposed unit-vector adaptation law (10) with adaptive gain are presented in black dotted line, while with constant gain in magenta dashed line and results obtained with linear gradient adaptation law (32) in red dotdashed line. Unit-vector adapta-

130

tion law with constant gain and linear gradient adaptation algorithms were run with k = 1, while Fig. 4 shows the time profiles of the adaptive gain k(t) in proposed algorithm applied with kmin = 1, γ1 = 1 and γ2 = 1. It can be noticed that all algorithms achieve a correct parameter estimation, the proposed unitvector adaptation being characterized by a finite-time converging transient. In 12

12

10

8

6

4

2

0

0

5

10

15

20

25

30

35

40

45

50

55

(a)
































(b)

(c)

    Figure 2: Convergence region obtained in Example 1. Plot(a) shows the evolution of  ∂ yˆˆ  ∂θ

defined in Assumption 1. Plot (b) shows the smallest eigenvalue of MS for different values of initial errors. Plot (c) represents the region of allowable value of Δτ w.r.t. ΔKp .

135

the estimation of the delay (lower plot of Fig. 3) the linear gradient scheme exhibits faster reaction and shorter transient after the abrupt parameter change at t = 25. This is due to the fact that in the unit-vector adaptation (10) the magnitude of the right-hand side cannot exceed the magnitude of the gain k(t), being the magnitude of the “unit-vector” term less or equal than 1, thereby lim-

140

iting the transient speed of the algorithm. On the contrary, the linear gradient  T adaptation law, the right-hand side of which depends linearly on ∂∂yθˆˆ y˜, may have larger transient adaptation speed.

13

Process gain estimation 15

10

5

0

5

10

15

20

25

30

35

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55

45

50

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Delay estimation 1.2

0.8

1

0.6

0.8

0.8 0.6

0.4

0.6

0.4

0.2

0.4 0.2 0

0

1

5

10

2

25 15

20

25

30

25.5 35

40

Figure 3: Comparisons of the actual and estimated parameter values (Example 1). Suffix “UVconst” refers to the unit-vector algorithm with constant adaptation gain. “UVadapt” refers to the unit-vector algorithm with time-varying adaptation gain proposed in the present work, whereas “LG” refers to the “linear gradient” algorithm (32) proposed in [14].

4

3.5

3

2.5

2

1.5

1 0

5

10

15

20

25

30

35

40

45

50

55

Figure 4: Temporal evolution of the adaptation gain k(t) using (10)-(9) during Example 1.

4.2. Example 2 Supercapacitors are an important class of devices storing electrical energy. They are used to power both various mobile low-power devices, as well as highpower ones, such as electric vehicles. Supercapacitors dynamics can be described by fractional order models [20], and particularly by means of the transfer func-

14

tion G0 (s) = Kp

1 e−sτ sα + 1

(36)

where the process gain is denoted with Kp and term e−sτ represent transport delay in the model. Assume that parameters Kp and τ are unknown and α has a known, non integer, fixed value. The proposed estimation scheme will be utilized to identify both Kp and τ using input signal u(t) = cos(ωt)

(37)

where ω > 0 is fixed and known. Assume that output and its first derivative are directly accessible for measurements. The adaptation law (10) reduces to ⎤⎡ ⎤ ⎡ ∂ yˆ1 ∂ yˆ2 y˜ ⎣ ∂ Kˆ p ∂ Kˆ p ⎦ ⎣ 1 ⎦ ⎡ ⎤ ∂ yˆ2 ∂ yˆ1 ˙ y˜2 Kˆ ∂ τˆ ∂ τˆ ⎣ p⎦ = k     2 . 2 τˆ˙ ∂ yˆ1 ∂ yˆ2 ∂ yˆ1 ∂ yˆ2 ˜1 + ∂ Kˆ y˜2 + ∂ τˆ y˜1 + ∂ τˆ y˜2 ˆ y ∂K p

p

One has (see [21]) that L−1 t



sαγ−β (sα − z)γ



γ (ztα ) = tβ−1 Eα,β

(38)

γ where Eα,β is the three parameters Mittag-Leffler function γ Eα,β (z) :=

∞ 1  Γ (γ + n) zn , Γ (γ) n=0 n!Γ (αn + β)

Re (α) > 0, Re (β) > 0, γ > 0, (39)

and as usual Γ (·) denotes the Euler Gamma function. Estimated output signals can be rewritten as ˆ p g(t − τˆ)  cos(ωt) , yˆ1 (t) = K yˆ2 (t) =

d ˆ (Kp g(t − τˆ)  cos(ωt)) . dt

The function g(t) is inverse Laplace transform of

1 sα +1

and its expression is

obtained specializing (38) for γ = 1, z = −1, β = α 1 g(t) = tα−1 Eα,α (−tα ) .

15

(40)

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

0

5

10

15

20

25

30

35

40

45

50

55

    Figure 5: Plot shows the evolution of  ∂ yˆˆ  defined in Assumption 1 obtained in Example 2. ∂θ

Actual values of parameters are chosen as Kp = 10 and τ = 1, and these val145

ues abruptly change to τ = 0.5 and Kp = 13 at t = 20 and t = 30, respectively. ˆ p (0) = 8 and τˆ(0) = 0.6. The initial values of the unknown parameters are K The input signal is u(t) = cos(0.5t). Figure 5 shows the time evolution of    ∂ yˆ   ∂ θˆ  evaluated along the trajectory of the adaptation process, confirming that Assumption 1 holds.

150

The simulation results presented in Figure 6 show that the parameter errors vanish in finite time. The actual values of the unknown parameters are presented in blue line, simulation results obtained with proposed adaptive gain unit-vector adaptation law (10) are presented in dotted black line, results with constant gain in the same adaptation law are presented in magenta dashed line, whereas results

155

obtained with linear gradient adaptation law (32) are presented in red dotdashed line. Unit-vector adaptation law with constant gain and linear gradient adaptation algorithms were run with k = 1, whereas Fig. 7 shows the time profiles of the adaptive gain k(t) within the proposed algorithm applied for kmin = 1, γ1 = 1 and γ2 = 1.

16

Process gain estimation 15

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Delay estimation 1.2 1 0.8 0.6 0.4 0.2 0

5

10

15

20

25

30

35

Figure 6: Comparisons of the actual and estimated parameter values (Example 2). Suffix “UVconst” refers to the unit-vector algorithm with constant gain adaptation “UVadapt” refers to the unit-vector algorithm with adaptive gain in adaptation law (10) proposed in the present work, whereas “LG” refers to the “linear gradient” algorithm (32) proposed in [14].

2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0

5

10

15

20

25

30

35

40

45

50

55

Figure 7: The evolution of adaptive gain k(t) defined in proposed adaptation law (10) with (9) obtained in Example 2.

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4.3. Example 3 Consider a process described by the partial differential equation ∂2Q ∂Q =α 2 , ∂t ∂x 17

(41)

where t is time variable, and x space variable. Assuming that (41) represents one dimensional heat diffusion process within semi-infinite insulated thin solid rod, Q(x, t) could be interpreted as temperature distribution within the rod, and α is the thermal diffusivity of the material. 165

Assume that the process is initially at rest, and that temperature at terminal cross-sections is specified externally, so that Q(0, t) = u1 (t). Assume also that temperature measurements are taken at cross-section x = x0 ∈ (0, ∞), so that the output is y(t) = Q(x0 , t). The transfer function from u1 (t) to y(t) is [2] G(x0 , s) = e−

√s

α x0

.

Let the temperature measurements be taken at the position x0 = 1. The proposed estimation scheme (10) will be utilized to identify α using function u = sin(t) as input signal and estimated output response which is rewritten as  x0

x0

ˆ e− 4αt (42) yˆ(t) = ( √ 3/2 )  sin(t) 2 πt     Temporal evolution of  ∂∂yθˆˆ , plotted in Figure 8, shows that Assumption

α ˆ

170

1 holds. The numerical simulations are made considering the actual thermal diffusion α = 0.6 and using α ˆ (0) = 0.2. At t = 30, the thermal diffusion abruptly changes to α = 0.2. Simulation results are shown in Figure 9, where the performance of the proposed unit-vector adaptation law (10) with constant and adaptive gain is compared with linear algorithm (32). The adaptation

175

process has been initiated, at t = 0 for kmin = 1, γ1 = 1 and γ2 = 1. The better convergence features of the unit-vector estimation law, especially its finite-time convergence, are apparent from the obtained comparative results. Figure 10 shows the time profiles of the adaptive gain k(t) in proposed algorithm. Due to fast convergence and vanishing of output error, adaptation gain used in unit

180

vector adaptation law deviates only slightly from the fixed value used by the unit vector algorithm with constant gain, as shown on Figure 10. For this reason, the behaviour of these two algorithms are almost identical.

18

1.2

1

0.8

0.6

0.4

0.2

0

0

5

10

15

20

25

30

35

40

45

    Figure 8: Temporal evolution of  ∂ yˆˆ  defined in Assumption 1 obtained in Example 3. ∂θ

Thermal diffusivity estimation

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

5

10

15

20

25

30

35

40

45

Figure 9: Actual and estimated parameter values (Example 3). Suffix “UVconst” refers to the unit-vector algorithm with constant gain, “UVadapt” refers to the unit-vector algorithm (10) with time-varying gain proposed in the present work, whereas “LG” refers to the “linear gradient” algorithm (32) proposed in [14].

19

1.3 1.25 1.2 1.15 1.1 1.05 1 0.95 0.9

0

5

10

15

20

25

30

35

40

45

Figure 10: The evolution of adaptive gain k(t) defined in proposed adaptation law (10) with (9) obtained in Example 3.

5. Conclusion Adaptive parameter estimation algorithm applicable to linear systems hav185

ing irrational transfer functions of arbitrary form is presented. The technique is an original modification of the gradient algorithm, inspired by the variable structure control technique. Another feature of the proposed scheme is that it has a time-varying gain which speeds up the transient. Effectiveness of the proposed algorithm is illustrated by several examples, in which it has been compared to

190

similar technique recently proposed in [14]. Among the several possible interesting extensions of the present results, addressing the time-varying parameter case and including certain types of nonlinearities in the process model, appear worth to be investigated.

Acknowledgements 195

This work partially supported by Serbian Ministry of Education and Science, grant no. TR32018 (M.N.K., M.R.R.) and grant no. TR33013 (M.R.R.), and by ”Fondazione di Sardegna”, CUP n.F72F16003170002 (A.P., E.U.)

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