Adaptive Control with Unknown Parameters in Reference Input

MoA06-7

Proceedings of the 13th Mediterranean Conference on Control and Automation Limassol, Cyprus, June 27-29, 2005

Adaptive Control with Unknown Parameters in Reference Input Chengyu Cao, Jiang Wang and Naira Hovakimyan and it vanishes as the parameter error converges to a neighborhood of the origin. The paper is organized as follows. Section II presents the problem formulation for the adaptive control with reference input dependent upon the unknown parameters of the system. Intelligent excitation is introduced in Section III. The convergence of the regulated output to the desired value is shown in Section IV. In Section V, we illustrate the simulation results, and Section VI concludes the paper.

Abstract— In this paper, Model Reference Adaptive Control problem is considered with reference input dependent upon the unknown parameters of the system. Due to the uncertainty in the reference input, conventional linear-inparameters adaptive controller cannot achieve the control objective without enforcing persistent excitation (PE). A new technique is presented for introducing an excitation signal and regulating its magnitude, dependent upon the convergence of the output tracking and parameter errors. Intelligent excitation ensures parameter convergence similar to conventional PE, but initiates only when necessary. As a result, the regulated output tracks the unknown reference input. Simulations illustrate the theoretical findings.

II. P ROBLEM F ORMULATION Consider the following single-input single-output system dynamics:

I. I NTRODUCTION

x(t) ˙ = Ax(t) + bλu(t) y(t) = c
x(t)

Conventional Model Reference Adaptive Controller (MRAC) is designed to track a given reference trajectory. However, in realistic applications, very often the reference trajectory is not specified apriori and depends upon the unknown parameters of system dynamics. One such example is the target tracking problem with visual sensors, where the characteristic length of the target is unknown and the relative range is not observable from available measurements [1]. Another example is the autopilot design problem for a trailing aircraft that follows a lead aircraft in a closed-coupled formation, where the trailing aircraft must constantly seek an optimal position relative to the leader to minimize the unknown drag effects introduced by the wing tip vortices of the lead aircraft [2]. Motivated by practical applications, in this paper we generalize the conventional model reference adaptive control framework for a reference input, dependent upon the unknown parameters of the system dynamics. Following the methodologies in [3], [4], one can enforce PE condition to ensure parameter convergence, however a constant PE deteriorates the output tracking and the control objective cannot be met. A new technique, named intelligent excitation, is presented to solve the output tracking problem while ensuring parameter convergence. The main feature of it is that it initiates excitation only when necessary,

where x ∈ IRn is the system state vector (measurable), u ∈ IR is the control signal, c ∈ IRn is a given constant vector, A is an unknown n × n matrix, b ∈ IRn is a known constant vector, λ ∈ IR is an unknown constant with known sign, y ∈ IR is the regulated output. The control objective is to enforce the output y(t) to track r(A, λ), where r is a known map r : IRn×n × IR → IR, dependent upon unknown parameters of the system A and λ. As in classical model reference adaptive control (MRAC), we introduce the following assumption: Assumption 1: There exist a Hurwitz matrix Am ∈ IRn×n , a column vector bm ∈ IRn , ideal constant parameters θx∗ ∈ IRn , θr∗ ∈ IR such that • (Am , bm ) is controllable • Am − A = λ b (θx∗ )


• λ b θr∗ = bm . In addition, we assume that λ belongs to a compact set: λ ∈ [λmin , λmax ] , where λmin and λmax have the same sign. Hence, the region, where the unknown parameter θr∗ lies, can be characterized as: θr∗ ∈ [Θrmin , Θrmax ], ||bm || ||bm || , Θrmax = . Θrmin = λmax ||b|| λmin ||b||

This material is based upon work supported by the United States Air Force under Contract No. FA9550-05-1-0157, and MURI subcontract F49620-03-1-0401. Chengyu Cao is with Aerospace & Ocean Engineering, Virginia Polytechnic Institute & State University, Blacksburg, VA 24061-0203, e-mail: [email protected], Member IEEE Jiang Wang is with Aerospace & Ocean Engineering, Virginia Polytechnic Institute & State University, Blacksburg, VA 24061-0203, e-mail: [email protected] Naira Hovakimyan is with Aerospace & Ocean Engineering, Virginia Polytechnic Institute & State University, Blacksburg, VA 24061-0203, e-mail: [email protected], Senior Member IEEE, Corresponding author

0-7803-8936-0/05/$20.00 ©2005 IEEE

(1)

(2)

In case of a known reference signal r, conventional MRAC provides a solution to the above problem formulation. The challenge here is that the reference signal r(A, λ) depends upon unknown parameters A, λ and is therefore unknown, even if the map r is given.

225

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where kvx and kvr are positive design gains, Θrmin and Θrmax are defined in (2). Define the Lyapunov function candidate

III. A DAPTIVE CONTROLLER U SING I NTELLIGENT E XCITATION In this section, we will propose a new intelligent excitation technique and incorporate it into conventional MRAC to solve the tracking problem in the presence of unknown r. In section III-A, we will recall the traditional MRAC for the system in (1). In section III-B, we will introduce the concept of intelligent excitation to achieve the control objective.

V = e
(t)P e(t) + |λ|θ˜x
(t)θ˜x (t)/kvx + |λ|θ˜r2 (t)/kvr . (12) It can be verified easily that V˙ (t) ≤ −e
(t)Qe(t) ≤ 0.

Hence all error signals are bounded. Application of the Barbalat’s lemma further implies asymptotic convergence of the tracking error to zero, i.e.

A. Model Reference Adaptive Controller In a traditional MRAC scheme, for any bounded reference input signal rˆ(t), a reference model is given by:

lim e(t) = 0.

t→∞

x˙ m (t) = Am xm (t) + bm kg rˆ(t) ym (t) = c
xm (t)

lim y(t) = r¯.

t→∞

lim ym (t) = r¯.

(4)

(5)

Direct adaptive model reference feedback/feedforward control signal is defined as u(t) = θx
(t)x(t) + θr (t)kg rˆ(t)

B. Adaptive Controller with Intelligent Excitation Let

(6)

ˆ = λ(t)

where θx (t) ∈ IRn and θr (t) ∈ IR are the adaptive gains. Substituting (6) into (1), the closed-loop system dynamics are: x(t) ˙ = (A + λbθx (t))x(t) + bλθr (t)kg rˆ(t)

If

||bm ||
ˆ ˆ = Am − bλ(t)θ , A(t) x (t) . θr (t)||b|| lim θ˜x (t) = 0, lim θ˜r (t) = 0,

t→∞

(7)

t→∞

(16)

(17)

then it can be verified easily that

It follows from Assumption 1 that the system in (7) can be rewritten:
 x˙ = Am − λb(θx∗ )
+ λbθx
(t) x + bλθr (t)kg rˆ(t) (8)

ˆ = λ, lim A(t) ˆ = A. lim λ(t) t→∞

t→∞

(18)

Let the reference signal rˆ(t) be

and the reference model in (3) can be rewritten: x˙ m (t) = Am xm (t) + bλθr∗ kg rˆ(t)

(15)

However, nothing can be claimed for the convergence of parameter errors to zero. Since (A, λ) are unknown, then the reference input r(A, λ) is not available. We ˆ ˆ can use A(t), λ(t), the estimates of (A, λ), to construct ˆ ˆ r(A(t), λ(t)). However without parameter convergence, ˆ ˆ there is no guarantee of r(A(t), λ(t)) → r(A, λ) as t → ˆ ˆ ∞. Next, we augment r(A(t), λ(t)) with an intelligent excitation signal to redefine the reference input rˆ(t) to meet the control objective.

In case if rˆ(t) = r¯, where r¯ is a constant, one can substitute (4) into the transfer function between rˆ and ym ym (s)
−1 bm and apply the final value rˆ(s) = kg c (sI − Am ) theorem to arrive at t→∞

(14)

If rˆ(t) = r¯ = const, it follows from (5) and (14) that

(3)

where Am and bm are defined in Assumption 1, c is the same as in (1) and kg ∈ IR is a design gain, defined by 1 kg = lim
s→0 c (sI − Am )−1 bm

(13)

rˆ(t)

(9)

= rˆ0 (t) + Ex (t)

(19)

ˆ ˆ are defined in (16), and ˆ ˆ λ where rˆ0 (t)
r(A(t), λ(t)), A, Ex (t) is the intelligent excitation signal, defined as:

Let e(t) = x(t) − xm (t) be the tracking error. Then the tracking error dynamics are: e(t) ˙ = Am e(t) + λb((θ˜x (t))
x(t) + θ˜r (t)kg rˆ(t)) , (10)

Ex (t) = kx (t)

where θ˜x (t) = θx (t) − θx∗ and θ˜r (t) = θr (t) − θr∗ denote parameter errors. Let P = P
be the solution of the algebraic equation A
m P + P Am = −Q for arbitrary Q > 0. Consider the following adaptive laws, by invoking the projection operator [7]:

m 

sin(ωi t)

(20)

i=1

⎧ t ∈ [0, kx2 ) ⎨ kx3 , t min{kx1 t−kx e
(τ )Qe(τ )dτ, kx (t) = 2 ⎩ kx3 − kx4 } + kx4 , t ∈ [kx2 , ∞) where kx1 , kx2 , kx3 , kx4 , m and ωi , i = 1, .., m are positive design gains to be determined. In what follows, we will show how to set these design gains.

(11) θ˙x (t) = −kvx x(t)e
(t)P b sgn(λ)

θ˙r (t) = Proj −kvr kg rˆ(t)e (t)P b sgn(λ), θr (t)

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It follows from (3) that the transfer function from rˆ(t) to xm (t) is: xm (s) = (sI − Am )−1 bm kg . rˆ(s) Let x ¯(t) = [ˆ r(t) xm (t)
]
, H(s) =

independent polynomials. We can rewrite the transfer function in (23) as: 1
[d(s) n1 (s) . . . nn (s)] . (30) H(s) = d(s)

(21)

x ¯(s) . rˆ(s)

Since d(s) is a nth order polynomial, and ni (s) are (n − 1)th order polynomials, then d(s) and ni (s) are linearly independent. Hence, H(s) contains n + 1 linearly independent functions of s, and therefore there exist ω1 , .., ωm , which ensure that Ω has full row rank.

(22)

It follows from (21) that

H(s) = [1 (sI − Am )−1 bm kg ]
.

(23)

IV. C ONVERGENCE R ESULT OF A DAPTIVE C ONTROLLER WITH I NTELLIGENT E XCITATION

We define a (n + 1) × 2m matrix Ω with its pth row, q th column element as  Re(Hp (jω q2  )) if q is odd Ωpq = (24) Im(Hp (jω q2  )) if q is even,

Theorem 1: For the system in (1) and the adaptive controller with intelligent excitation in (3), (6), (11), (20), we have (31) lim |y(t) − r(A, λ)| ≤ γ(kx4 ) t→∞

where  2q  denotes the smallest integer that is greater than q th element of H(s) defined in (23). 2 and Hp (s) is the p We choose m and ω1 , .., ωm to ensure that Ω has full row rank. The design gain kx2 is chosen as kx2 =

where γ(kx4 ) = kx4

m 

|G(jωi )|,

(32)

i=1

and

2π mini=1,..,m ωi

.

G(s) = c
(sI − A)bkg . (33) Proof of Theorem 1: We prove the theorem in two steps. At first, we prove the asymptotic convergence of parameter errors θ˜x (t) and θ˜r (t) to zero. Then, we prove the convergence of y(t) to the desired reference signal r(A, λ) with error bounds proportional to kx4 . Step 1: In this step, we will prove

(25)

Details on how to chose the constants kx1 , kx3 , kx4 , will be discussed in section IV. The transfer function (sI − Am )−1 bm kg can be expressed as: (sI − Am )−1 bm kg = n(s) d(s) where d(s) = th det(sI − Am ) is a n order polynomial, and n(s) is a n × 1 vector with its ith element being a polynomial function: n  nij sj−1 . (26) ni (s) =

lim θ˜x (t) = 0, lim θ˜r (t) = 0.

t→∞

t→∞

It follows from equation (14) that lim e(t + τ ) = 0, ∀τ ∈ [0, kx2 ],

j=1

t→∞

Lemma 1: If (Am , bm ) is controllable, the matrix N of the entries nij defined in (26) is full rank. Proof of Lemma 1: Controllability of (Am , bm ) for the time-invariant linear system in (3) implies that for arbitrary x ¯ ∈ IRn , xm (t0 ) = 0 and arbitrary t1 , there exists u(τ ), τ ∈ [t0 , t1 ] such that xm (t1 ) = x ¯.

(34)

and hence

lim

t→∞

t+kx2

t

e
(τ )Qe(τ )dτ = 0,

(35)

(36)

which combined with (20) implies that lim kx (t) = kx4 .

t→∞

(27)

(37)

It follows from (35) and the adaptive law (11) that for τ ∈ [0, kx2 ] there exist θ¯x ∈ IRn and θ¯r ∈ IR such that

n

If N is not full rank, then there exists k = 0, k ∈ IR such that (28) k
n(s) = 0.

lim θx (t + τ ) = θ¯x , lim θr (t + τ ) = θ¯r .

t→∞

t→∞

(38)

Equation (38) further implies that

If xm (t0 ) = 0, for any u(τ ), τ > t0 , it follows from (28) that ∀τ > t0 (29) k
xm (τ ) = 0,

lim rˆ0 (t + τ ) = const,

t→∞

τ ∈ [0, kx2 ],

(39)

while equation (37) implies that for τ ∈ [0, kx2 ]

n

This contradicts (27), in which x ¯ ∈ IR is assumed to be an arbitrary point. Therefore, N must be full rank. Remark 1: The existence of m and ω1 , .., ωm can be verified easily. It follows from Assumption 1 and Lemma 1 that N is full rank, and hence n(s) contains n linearly

lim Ex (t + τ ) = kx4

t→∞

m 

sin(ωi (t + τ )) .

(40)

i=1

From (22) it follows that x ¯(t+τ ) is the output response of the linear system, defined by H(s), to rˆ(t + τ ) = rˆ0 (t +

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¯3 (t + τ ) exponential functions. Therefore x ¯2 (t + τ ) + x cannot be expressed by linear combination of sinusoidal functions x ¯1 (t + τ ) over τ ∈ [0, kx2 ]. Hence, all the elements of x ¯(t + τ ) are linearly independent over the time interval [0, kx2 ] as t → ∞. Let θ¯ = [θ¯r θ¯x
]
where θ¯r and θ¯x are defined in (38). In what follows we will prove by contradiction that

τ ) + Ex (t + τ ) over the interval τ ∈ [0, kx2 ]. Therefore, x ¯(t+τ ) can be expressed as the sum of three components: x ¯(t + τ ) = x ¯1 (t + τ ) + x ¯2 (t + τ ) + x ¯3 (t + τ ),

(41)

¯1 (τ ) and x ¯2 (τ ) are the particular for τ ∈ [0, kx2 ] where x solutions of the linear system (22) corresponding to Ex (τ ) and rˆ0 (τ ) respectively, while x ¯3 (τ ) is the general solution of the linear system (22) for the initial value at t and can therefore be expressed by a set of exponential functions of τ . It follows from (39) that lim x ¯2 (t + τ ) = const, τ ∈ [0, kx2 ].

t→∞

θ¯ = 0

¯ since the elements of x For a nonzero θ, ¯(τ ) are linearly independent over the interval [t, t+kx2 ], there exists some t ∈ [0, kx2 ] such that

(42)

¯(t + t ) = 0 θ¯
lim x

Next we use Fourier transformation to construct the particular solutions of the system (22) in response to Ex (t + τ ) defined in (40). If the input signal to the linear system H(s) is: m  kx4 sin(ωi (t)), (43)

t→∞

(50)

Since x ¯(t) is Lipschitz continuous, it follows from (50) that there exist τ1 , τ2 ∈ [0, kx2 ], where τ1 < τ2 , such that t+τ2 lim ¯(τ )dτ = 0. (51) θ¯
x

i=1

with its Fourier transformation m

 π j (δ(ω + ωi ) − δ(ω − ωi )) , FEx (ω) = 2 i=1

(49)

t→∞

t+τ1

It follows from (10), (35) and (51) that lim (e(t + τ2 ) − e(t + τ1 )) t+τ2 = lim ( θ¯
x ¯(τ )dτ ) = 0.

(44)

t→∞

(52) the Fourier transformation of the particular solution, cort→∞ t+τ 1 responding to the input (43), is:

Since (52) contradicts (35), (49) must be true, i.e. m π  Fx¯1 = j (H(−jωi )δ(ω + ωi ) − H(jωi )δ(ω − ωi )). lim θx (t+τ ) = 0, lim θr (t+τ ) = 0, τ ∈ [0, kx2 ]. (53) 2 i=1 t→∞ t→∞ (45) From (35) and (53), we have limt→∞ V (t) = 0. Since Hence, the particular solution corresponding to the input V (t) is non-increasing, then (34) holds. signal in (43) will be Proof of Step 2: Since ym (t) is the response of the m  linear system (3) to the input signal rˆ(t) = rˆ0 (t) + Ex (t), x ¯1 (t) = (Re(H(jωi )) sin(ωi t) + Im(H(jωi )) cos(ωi t)). it can be presented as ym (t) = y1 (t) + y2 (t), where y1 (t) i=1 and y2 (t) are the responses to the input signal components (46) (t) and Ex (t), respectively. It follows from (18), (17) r ˆ Therefore, the particular solution corresponding to Ex (t + 0 and (34) that τ ) defined in (40) satisfies: (54) lim rˆ0 (t) = r(A, λ) m  t→∞ lim x (Re(H(jωi )) sin(ωi (t + τ )) + ¯1 (t + τ ) = t→∞ which along with (5) implies that i=1

Im(H(jωi )) cos(ωi (t + τ ))), τ ∈ [0, kx2 ] .(47)

lim y1 (t) = r(A, λ).

t→∞

Equation (47) can be rewritten as x ¯1 (t + τ ) = Ω¯b(t + τ ), τ ∈ [0, kx2 ]

(55)

Since (3) is a linear system, then a sinusoidal input sin(ωi t) results in a sinusoidal output with the magnitude bounded by |G(jωi )|, where G(jωi ) is defined in (33). From (37) it follows that as t → ∞,the steady state m response y2 (t) to the input signal kx4 i=1 sin(ωi t), is bounded by γ(kx4 ):

(48)

where Ω is defined in (24) and ¯b is a 2m × 1 vector with its q th element being:  sin(ωq (t + τ )) if q is odd ¯bq (t + τ ) = cos(ωq (t + τ )) if q is even.

lim |y2 (t)| ≤ γ(kx4 ).

t→∞

where τ ∈ [0, kx2 ]. Since Ω has full row rank and the elements of ¯b are linearly independent functions, then the elements of x ¯1 (t + τ ) are linearly independent over τ ∈ [0, kx2 ]. Notice that as t → ∞, x ¯2 (t+τ ) tends to a constant (see eq.(42)). The general solution x ¯3 (t + τ ) is a sum of

(56)

It follows from (55) and (56) that lim |ym (t) − r(A, λ)| ≤ γ(kx4 ) ,

t→∞

which along with (14) yields (31).

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(57)

20 y rideal

Theorem 1 demonstrates that the intelligent excitation signal ensures asymptotic convergence of parameter errors to zero. It also proves that the upper bound of the output error between y(t) and r(A, λ) is proportional to kx4 , which defines the magnitude of the excitation signal as t → ∞. To reduce the output error, we need to set kx4 extremely small. However small excitation signal can cause extremely slow parameter convergence, which is not acceptable in practice. The rationale of the intelligent excitation is its ability to keep the magnitude of the excitation signal at a reasonable value kx3 for improving the parameter convergence rate and automatically reduce it to extremely small kx4 upon parameter convergence. The ability of the intelligent excitation to keep the magnitude of excitation signal at kx3 before the parameter error converges to zero is demonstrated in the following Lemma. Lemma 2: For the system in (1) and the adaptive controller with intelligent excitation in (3), (6), (11), (20), if kx1 → ∞, we have kx (t) = kx3

50 60 time (seconds)

70

80

90

100

90

100

Trajectories of y(t) and r(t)

kx(t) 4

2

0

0

10

20

30

Fig. 2.

40

50 60 time (seconds)

70

80

Trajectory of kx (t)

from above that as V (t) > 0, kx (t) = kx3 , which concludes the proof of Lemma 2. Upon the parameter error convergence to zero, for any reference input signal rˆ(t), no matter it has an excitation in it or not, we have t e(τ )
Qe(τ )dτ = 0, (66)

(59)

t−kx2

and hence, from (20) it follows that kx (t) = kx4 . Intelligent excitation adjusts the magnitude of the excitation signal through “a feedback mechanism” determined by the integral of the tracking error e(t) over a finite time interval.

(61)

Remark 2: For practical implementation, due to the presence of noise and transient errors, we can choose kx4 = 0 without worrying about the premature disappearance of the intelligent excitation. The constant gain kx1 is inverse proportional to the bound of the parameter tracking error, so setting it large will increase the accuracy of parameter estimates. The gain kx3 is the amplitude of the excitation signal, which controls the rate of convergence.

(63)

and hence (64)

for constant θ¯x ∈ IRn and θ¯r ∈ IR. Since V (t − kx2 ) ≥ V (t) > 0, we conclude from (12) and (63) that θ¯ = [θ¯r θ¯x
]
= 0.

40

6

then, since e(τ ) is Lipschitz continuous, we have

τ ∈ [t − kx2 , t)

30

8

t−kx2

θ˜x (τ ) = θ¯x , θ˜r (τ ) = θ¯r

20

10

from (20) it follows that kx (τ ) = kx3 as kx1 → ∞. If (61) is not true, i.e. t e(τ )
Qe(τ )dτ = 0, (62)

∀τ ∈ [t − kx2 , t),

10

12

t−kx2

e(τ ) = 0,

0

Fig. 1.

For any time instant t ≥ kx2 such that V (t) > 0, we will prove that kx (t) = kx3 (60) if kx1 → ∞. Indeed, if t e(τ )
Qe(τ )dτ = 0,

0

−10

(58)

τ < t.

5

−5

unless V (t) → 0. Proof of Lemma 2: We will prove this by mathematical induction principle. Assume that kx (τ ) = kx3

10

y and reference signal r

ideal

15

V. S IMULATION R ESULT We consider the SISO dynamic system in (1) with: ⎤ ⎡ 1 −1 ⎦ ⎣ A= , b = [1 1]
, c = [1 0]
, and −24 −9.0711 λ(t) = −3.3333 when t ≤ 50 and λ(t) = −16.6667 when t > 50. The reference signal is r(A, λ(t)) = −[1 0]A[1 0]
− λ(t) and the objective is to design control signal u(t) to let system output y(t) track the reference input r(A, λ(t)).

(65)

Following the same arguments as in the proof of Theorem 1, it can be shown that (63) and consequnetly (62) contradict to (61). Since (61) is true if V (t) > 0 and kx (τ ) = kx3 , τ < t, we conclude that kx (t) = kx3 as kx1 → ∞. Since kx (t) = kx3 , ∀t ∈ [0, kx2 ], it follows

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shows the convergence of θˆx1 (t), θˆx2 (t), θˆr (t) to θx∗1 (t), θx∗2 (t), θr∗ (t), respectively, under intelligent excitation. Figures 1, 2, and 3 demonstrate that i) intelligent excitation vanishes as parameter convergence takes place, ii) intelligent excitation initiates automatically when parameter change occurs. Finally we compare simulations with kx1 = 2000 and kx1 = 1. In the second case, intelligent excitation disappears earlier (Fig. 4), which consequently stalls the decrease of Lyapunov function at a larger value compared to kx1 = 2000 case. When kx1 = 2000, the final value of Lyapunov function is 1.8861 × 10−4 ; when kx1 = 1, the final value of Lyapunov function is 2.3764 × 10−4 . This demonstrates that the convergence criterion of the parameter error depends upon the choice of kx1 . Larger kx1 implies better precision. A proof on this is provided in [1].

4 *

θx

1

θx

3.5

1

3

1

2.5

θx

2

1.5

1

0.5

0

0

10

20

30

40

50 60 time (seconds)

70

80

90

100

0 *

θx θx

2 2

θx

2

−0.5

−1

−1.5

0

10

20

30

40

50 60 time (seconds)

70

80

90

100

−1 *

θr θ r

−2

−3

θ

r

−4

VI. C ONCLUSION

−5

−6

In this paper, we augment the traditional MRAC with an intelligent excitation signal to solve the output tracking for a reference input that depends upon the unknown parameters of the system. The main feature of the new technique is that it initiates excitation only when necessary. It is shown that the intelligent excitation is a general technique for the situations where parameter convergence is needed in adaptive control architecture. It can also be used to increase robustness of the adaptive controller, where parameter drift may cause instability. Extension to nonlinear systems will be reported in a forthcoming publication.

−7

−8

0

10

Fig. 3.

20

30

40

50 60 time (seconds)

70

80

90

100

Trajectories of θˆx1 (t), θˆx2 (t), θˆr (t) k =2000 x1 k =1

12

x1

kx(t) for different values of k

x1

10

8

6

4

2

0

Fig. 4.

0

10

20

30

40

50 60 time (seconds)

70

80

90

R EFERENCES

100

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Trajectories of kx (t) for kx1 = 1 and kx1 = 2000

We choose ⎡a reference model with the following param⎤ 0 1 ⎦ , bm = [25 25]
, kg = eters: Am = ⎣ −25 −7.0711 0.1239. Lyapunov equation is solved with the following 0.1 0 Q matrix: Q = . The design gains in the 0 0.1   5 0 intelligent excitation are set to kvx = , kvr = 5, 0 5 kx1 = 2000, kx2 = 2π/15, kx3 = 12, kx4 = 0. We choose ωi (i = 1, 2) as ω1 = 15, ω2 = 35 rad/sec. It can be verified that Ω has full row rank with the chosen ωi . Simulation results are given in Figs. 1, 2, and 3. Fig. 1 plots the time histories of y(t) and ideal reference signal r(A, λ(t)). It demonstrates that with the change in unknown parameter λ(t), the output y(t) converges to r(A, λ(t)) with the help of intelligent excitation. The trajectory of kx (t), which defines the amplitude of the intelligent excitation, is plotted in Fig. 2. The time histories of parameter estimates are demonstrated in Fig. 3, which

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