A globally convergent estimator for n-frequencies - Semantic Scholar
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 5, MAY 2002
A Globally Convergent Estimator for -Frequencies
857
which we consider to be generated by the linear model
w_ 1 =w2
G. Obregón-Pulido, B. Castillo-Toledo, and A. Loukianov
Abstract—In this note, we propose a solution to the well-know problem of ensuring a simultaneous globally convergent online estimation of the state and the frequencies of a sinusoid signal composed of sinusoidal terms. We present an estimator which guarantees global boundedness and convergence of the state and frequencies estimation for all initial conditions and frequencies values. Index Terms—Adaptive signal processing, frequency estimation, nonlinear systems.
I. INTRODUCTION In this note, we address the problem of simultaneous online estimation of the state and the frequency of a measurable signal composed of the sum of n sinusoidal terms given by
y (t) =
n i=1
A1i sin ( i t + 1 i ) + A2i cos ( i t + 2 i )
w_ 2 = 0 2 w1 y(t) = k1 w1 + k2 w2 in which the parameter 2 is unknown, k1 , k2 , = 6 0 and the initial conditions
w1 (0) w2 (0)
=
k1 k2 01 A1 sin(1 ) + A2 cos(2 ) 2 0 k2 k1 1 A1 cos(1 ) 0 A2 sin(2 )
are also unknown. The parameter is introduced here to scale the magnitude of the signals. It is not difficult to see that this system is immersed into the system
z_1 =z2 2 z_2 = 0 z1 _ =0 y(t) = k1 z1 + k2 z2 :
(1)
where the amplitudes A1i ; A2i 6= 0, the frequencies i 6= j ; i 6= j and the phases 1 i ; 2 i are all unknown. The frequency estimation is a very important issue in control theory, due to the number of practical applications in rotational mechanical processes like induction motors, disk drivers, helicopters or vibration control among others ([2], [4], [6], [7]). The problem of estimation of the frequency of the signal has been studied extensively by means of differents technics both in the offline case, for example the work of [11] and the online estimation [8], but only recently, a globally convergent estimator was proposed in [1] on the basis of an adaptive notch (AN) filter first proposed in the discrete-time version in [8] and adapted in [2] for the continuous-time case. A key feature of this AN filter was the scaling of the forcing term to normalize the parameters, which does not affect stability and ensures the positivity for all time of the estimate. The problem of simultaneous online globally convergent estimation of the frequency and the state is, as pointed out in [1], a well-known open problem in system theory. In this note, we propose a solution to this long-standing problem and then extend the results naturally to the case of n unknown frequencies. More specifically, we present a new estimator which ensures a continuous-time online simultaneous frequency and state estimation, ensuring that all signals are globally bounded and the estimation of the frequencies and the states are asymptotically correct for all initial conditions and frequency values. Recently, the same problem was addressed in [9] using a set of adaptive observers first developed in [10]. We briefly discuss this results and the main difference with the approach presented here. II. ESTIMATION OF A SINGLE FREQUENCY
We are interested in the estimation of the state (w1 , w2 ) and the square of the frequency 2 . Let us denote by x = (x1 ; x2 ; x3 )T the states of the estimator and define the estimation error vector e = (e1 ; e2 ; e3 )T , with e1 = w1 0 x1 , e2 = w2 0 x2 and e3 = x3 0 2 . Now, the structure of (2) allows us to derive the estimator, described in the following result. Proposition 1: The estimator
x_ 1 =x2 + (y 0 y~) k2 x x x_ 2 = 0 1 3 + (y 0 y~) x_ 3 = 0 x1 (y 0 y~) y~ = k1 x1 + k2 x2
Manuscript received May 22, 2000; revised May 14, 2001. Recommended by Associate Editor B. Bernhardsson. This work was supported by CONACYT under Grant 26358-A. The authors are with the Centro de Investigación y de Estudios Avanzados del IPN, Unidad Guadalajara 44550, Guadalajara, México. Publisher Item Identifier S 0018-9286(02)04750-5.
(3) (4) (5) (6)
where , , > 0, k1 , k2 6= 0, ensures that limt!1 e = 0. Proof: The error dynamics take the form
e_ 1 = 0 k1 e1 k2 e_ 2 = 0 2 + k1 e1 0 k2 e2 + x1 e3 e_ 3 = 0 x1 (k1 e1 + k2 e2 ) : Now, let us take the Lyapunov candidate function
Let us consider a sinusoidal signal
y(t) = A1 sin ( t + 1 ) + A2 cos ( t + 2 )
(2)
T
T
V (e) =e Me = e =
c 2 k 2 0
k 2 k 2 0
0 0 2
e
c1 e2 + k2 e2 + e2 + k e e : 1 1 2 1 2 3 2 2 2
(7)
(8)
Since M is Hermitian, a standard calculation shows that M is definite–positive if c1 > 0 and c1 k2 > k12 from which k2 must be positive.
0018-9286/02$17.00 © 2002 IEEE
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 5, MAY 2002
Fig. 1. Signals x , x , and x for = 1.
The derivative of V (e) is now calculated as
namics, we verify that the solution of the error dynamics plus estimator in is given by
V_ (e) = 0 c1 k1 e12 + k2 e2 (0 1 e1 0 k2 e2 + x1 e3 ) k2 + k1 e1 (0 1 e1 0 k2 e2 + x1 e3 ) + k1 e2 0 kk1 e1 + e3 0 x1 (k1 e1 + k2 e2 ) 2 = 0 c1 kk1 + k1 2 + k12 e12 0 k22 e22 2 2 0 k2 2 + 2k1 k2 + kk12 e1 e2
x_ 1 =x2 x_ 2 = 0 x1 x3 x_ 3 =0 e_ 2 =0 = x1 e3 e_ 3 =0:
To ensure that V_ is semidefinite–negative, we need to ensure that
k2 2 + 2k1 k2 + kk k 1 2 2 c1 + k1 + k1 > k2 4k22 so that, to satisfy both conditions c1 k2 equation that
c1 > max
; k2 k2 k1
k12
2
(9)
> k12 and (9), we impose the
k2 2 +2k1 k2 + kk 4k22
2
0 k1 20 k12
:
Now, since V_ is not definite–negative, we see that V_ = 0 in the set = f(e; x) j e1 = e2 = 0g : To prove the stability of the error dy-
From this, one can observe that the fourth equation imposes that e3 must be zero, since e1 = 0 implies that x1 = w1 and this is different from zero except in the trivial situation, so e3 = 0 and the error dynamics tend asymptotically to zero, this is x3 tends asymptotically to the value of 2 . So, in the invariant set, the only solution for the error dynamics is the trivial solution and for the observer dynamics, the only solution is a limit cycle and the output is precisely the exact output of the original system. So, since V (e) is radially unbounded for all values of e, invoking the Lasalle–Krasovskii theorem [3], we conclude that the observer guarantees global convergence of the estimation error. Remark 2: Note that if the signal have some constant bias then we can introduce one integrator and make the same procedure. III. THE n-FREQUENCIES CASE In this section, we present the extension to the case of a signal containing n frequencies and given by
y(t) =
n i=1
A1 i sin ( i t + 1 i ) + A2 i cos ( i t + 2 i ) :
This signal may be viewed as the output of the dynamical system
w_ (t) = Sw(t)
(10)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 5, MAY 2002
859
Fig. 2. Signals x , x , and x for = 100.
where
Si
S= y (t) =
n
where 0
0
.. .
..
.. .
0
0
i=1
.
;
Sn
0
Si =
0 i2
1
a0 =
0
w_ 1 =w2 w_ 2 =w3 .. .
w_ 2n01 =w2n w_ 2n = 0 a0 w1 0 a2 w3 0 1 1 1 0 a2n02 w2n01 1
i=1
z_ (t) =
(11)
0
1
0
0
.. .
.. .
0
0
0
0
0
0
0
y(t) =
2n01
i=1
0
111 111
0
2 .. .
0
ki z + k z i 2n 2n 2n01 j
j =i
n
n j =1
2i ; . . . ; a2n02 =
n i=1
2i
gi >0 for i = 2 . . . n + 2; g1 = 2n01 k2n 2n01 k i y~ = xi + k2n x2n 2n01 i=1 j =i j
2n
ki wi 2n01 j =1j j i=1
2i ; a2 =
are the coefficients of the characteristic polynomial p(s) = s2n + a2n02 s2n02 + :: + a2 s2 + a0 and j are design parameters to conveniently scale the magnitude of the estimator signals. As in the case of a single frequency, we may verify that (11) is immersed into the system (12), as shown at the bottom of the page, for every i 6= 0 and ki 6= 0. The structure of (12) allows us to derive the estimator, which is given in the following result. Theorem 3: The estimator is shown in the equation at the bottom of the next page, with
c1 i w1 i (t) + c2 i w2 i (t)
for some constants c1 i = 6 0; c2 i =6 0 which may be rewritten as
y=
n
..
.
0
0
0
0
.. .
.. .
z (t) 1 1 1 2n02 0 111 0 2n01 111 0 0 (12)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 5, MAY 2002
Fig. 3.
Signals x , x , a and a .
and ki chosen such that polynomial
e_ 2n01 =w2n 0
k2n01 2n02 k2n02 2n03 k2 k p(s) = s + s + s + 111 + s+ 1 k2n k2n k2n k2n is stable with all its roots distinct, is such that y~ ! y , x2n+i ! a2i02 for i = 1 . . . n when t ! 1. 2n01
=
Proof: We first consider the error system
e_ 2n02 =e2n01 = w2n01 0
2n02 i=1
0x
0
1
0
0
.. .
.. .
0
0
0
0
0
111 111
0
2 .. .
..
0
0x
x_ 2n+1 = 0 g3 x1 (y 0 y~) x_ 2n+2 = 0 g4 x3 (y 0 y~) .. .
i=1i
(2n01 x2n + g1 (y
i x2n
0
2n02 i=1i
0 y~))
i g1 (y 0 y~) :
e_ i =ei+1 for i = 1; ::; 2n 0 2 2n01 1 e_ 2n01 = ke k2n i=1 i i
i x2n01
x_ (t) =
i
i=1i 2n01
2n01 Now, taking e2n = w2n 0 i=1 i x2n we have the equation shown at the bottom of the next page. Choosing g1 = 2n01 =k2n , we have the system
e1 =w1 0 x1 e_ 1 =e2 = w2 0 1 x2 e_ 2 =e3 = w3 0 1 2 x3 .. .
w2n 0
2n02
x_ 2n+n = 0 gn+2 x2n01 (y 0 y~)
0
.
0
0
0
0
.. . 2n02
.. .
111 1 1 1 0x 0 111
0
2n01 0
0 0
x(t) +
.. .
0
g1 g2
(y
0 y~)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 5, MAY 2002
861
Fig. 4. Signals y and y~.
which characteristic polynomial is
and the second equation at the bottom of the next page holds true. Since matrix A has distinct eigenvalues, then there exists a matrix T such that
k k k k p(s) = s2n01 + 2n01 s2n02 + 2n02 s2n03 + 1 1 1 + 2 s + 1 k2n k2n k2n k2n
A~ =T 01 AT = diag ( 1 e1 =T e~1
and is stable by hypothesis. Now, we continue with the first scalar equation shown at the bottom of the next page. If we define e2n+i = x2n+i 0 a2i02 for i = 1::n, then the error dynamics may be rewritten in the form
e~_ 1 =A~e~1 e_ 2n = 0 ~1T e~1 0 2 e2n + xT1 e3 x1 k~1T e~1 + k2n e2n e_ 3 = 0 0 where
~1T = 1T T; k~1T = k1T T:
where A is a Hurwitz matrix given by
A=
1 0 0
..
0 kk
0 kk
e_ 2n01 =e2n 0
111
2n02
=e2n 0 g1
i=1
.
0
Since A~ is diagonal, there exists a Lyapunov function Vo (~ e1 ) satisfying
.. .
Vo (~e1 ) =~e1T P e~1 V_ o (~e1 ) = 0 e~1T Qe~1
1 1 1 k1 111 0 k
i g1
1 2n k w 2n01 i=1 i i
2n =e2n 0 g1 ke : 2n01 i=1 i i
2n01 )
and the error dynamics take the form
e_ 1 =Ae1 e_ 2n = 0 1T e1 0 2 e2n + xT1 e3 x1 k1T e1 + k2n e2n e_ 3 = 0 0
0 0 0
111
2n 1 2n01 j i=1 ki wi
j =1
0 2n101
0
k1 x1 +
2n01 i=1
2n01 i=1
i
k
i 2n01 j xi + k2n x2n j =i 2n01 1 k2n x2n 0 2n01 i=2
i01 j =1
j ki xi
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 5, MAY 2002
for some P , Q diagonal and positive–definite. Let us now consider the Lyapunov candidate function T e~ e + c e2n V (~e1 ; e2n ; e3 ) = Vo (~e1 ) + C12 1 2n 2n 2 with 2
C12 = [ c1;2n c2;2n C3 =diag ( c13 c23
1 1 1 c n0 ; n 1 1 1 cn ) 2
2 2
Choosing
c2n = k2n ; C12 = k~1 ; and C3 = 001
+ 1 e3T C3 e3 2
c2n01;2n
we finally get that
]T
V_ (~e1 ; e2n ; e3 ) = 0e~1T Q + k~1 ~1T e~1 0 k2n 2 e22n
3
0 k~T 0 k~T A~ + ~T k n e~ e n
which is positive–definite for some P and c2n > 0. From this, we have the third equation shown at the bottom of the page, where the fact that C3 is diagonal has been used. Since 0 is also diagonal, we have that
1
T 0 CT A ~ + ~1T c2n e~1 e2n C12 2 12
+
e~1T C12
0 e~T k~
xT1 e3
xT1 0C3 e3
1
1
+c2n xT1 e3 e2n 0 k2n xT1 0C3e3 e2n:
01
2n
e_ 2n =w_ 2n 0
01
2n
i=1
1
1
i x_ 2n = 0a0 w1 0 a2 w3 0 1 1 1 0 a2n02 w2n01
x2n+2 0x2n+n 0 x1 0 2n01 i x3 0 1 1 1 0 2n01 x2n01 + g2 (y 0 y~) i=3 i=1 2n02 = 0 a0 (e1 + x1 ) 0 a2 (e3 + 1 2 x3 ) 0 1 1 1 0 a2n02 e2n01 + i x2n01
0
0x n
2 +1 2n 1 i i=1
i
+ x1 x2n+1 + 1 2 x3 x2n+2 + 1 1 1 +
02
2n
i=1
i=1
2n
i x2n01 x3n 0 g2
i=1
ki ei
= 0 (a0 + g2 k1 ) e1 0 g2 k2 e2 0 (a2 + g2 k3 ) e3 0 1 1 1 0 (a2n02 + g2 k2n01 ) e2n01 0 g2 k2n e2n + x1 (x2n+1 0 a0 ) + 1 2 x3 (x2n+2 0 a2 ) + 1 1 1 2n02 i x2n01 (x3n 0 a2n02 ) : + i=1
e1 = [ e1 1 1 1 e2n01 ]T e3 = [ e2n+1 1 1 1 e3n ]T 1 = [ a0 + g2 k1 g2 k2 a2 + g2 k3 2 =g2 k2n
0 =diag k1 = [ k1
g
111
g2 k2n02 a2n02 + g2 k2n01 ]T
0 j x2i01
111
x1 = x1 1 2 x3
111
2i 2 j =1
g
k2n01 ]T :
(
)
T 2n02 j =1 j x2n01 g
111 111
T e~ V_ (~e1 ; e2n ; e3 ) = 0 e~1T Qe~1 + C12 1
111
g
0 ~T e~ 0 e n + xT e + C T A~e~ e n + c n e n 0 ~T e~ 0 e n + xT e T 1 + 1 00x k~T e~ + k n e n C e + eT C 00 x k~T e~ + k n e n 2 2 = 0 e~T Q + C ~T e~ 0 c n e n 0 C T 0 C T A~ + ~T c n e~ e n T + e~T C xT e + c n xT e e n 0 0 x k~T e~ + k n e n C e 12
1
2
1
1
2
1
12 1
1
1
12
1
3
1
1
2 2
2
2
1
2
1
2
2
2
1 2
which is negative–semidefinite for some Q. We therefore use the same analysis made in the single frequency case, namely, in this case the invariant set is given by e~1 = 0, e2n = 0 and then the vector e3 must be zero because the supposition that the frequencies are distinct implies that the signals x2i01 are linearly independent for i = 1::n. Remark 4: Recently, a frequency estimator was presented in [9], based in the use of an adaptive estimator of dimension 5n 0 1. In contrast, the order of the estimator presented in this work is 3n.
V_ (~e1 ; e2n ; e3 ) = 0 e~1T (Q + C12 ~1T )~e1 0 c2n 2 e22n
0
2
1
1
2 2
3 3
2 2 2
1
3
3
1
3
3
12 2
2
3
1
12
1
1
1
1
1
1
2
2
2
2
2
1 2
3 3
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 5, MAY 2002
863
IV. SIMULATIONS RESULTS
On Hybrid Systems and Closed-Loop MPC Systems
The performance of the estimator was tested by extensive simulations, some of which are presented in the following figures. First, we show in Fig. 1 the behavior of the estimator for a single frequency and = 1, = 10, = 1, k1 = 1, k2 = 1, = 1 and y (t) = sin( t). We stress the fact that if the frequency is increased, then the magnitude of w2 increase too, and therefore, we may increase to reduce the magnitude of x2 and reduce k2 and k1 for which the dynamics of x1 are faster. In this situation, we can increase and thus we have faster dynamics of x3 . The response of the estimator for = 100, where the parameters are k1 = 0:5, k2 = :25, = 100, = 1000, = 1000 and y (t) = 10 sin(100t) is shown in the Fig. 2. We observe the good behavior of the estimator. Taking now two frequencies, let us say 1 = 1, 2 = 2, we get a0 = 4 and a2 = 5 and the performance of the filter with parameters k1 = 6, k2 = 11, k3 = 6, k4 = 1, i = 1 for i = 1; ::; 3 and y(t) = 10(sin(t) + sin(2t)), are shown in Figs. 3 and 4. As it may be observed, the estimator exhibites a good convergence properties, so this results suggest the validity of the proposed estimator.
Alberto Bemporad, W. P. Maurice H. Heemels, and Bart De Schutter
Abstract—The following five classes of hybrid systems were recently proven to be equivalent: linear complementarity, extended linear complementarity, mixed logical dynamical, piecewise affine, and max-min-plus-scaling systems. Some of the equivalences were obtained under additional assumptions, such as boundedness of certain system variables. In this note, for linear or hybrid plants in closed-loop with a model predictive control (MPC) controller based on a linear model, fulfilling linear constraints on input and state variables, and utilizing a quadratic cost criterion, we provide a simple and direct proof that the closed-loop system is a subclass of any of the former five classes of hybrid systems. This result is of extreme importance as it opens up the use of tools developed for the mentioned hybrid model classes, such as (robust) stability and safety analysis tools, to study closed-loop properties of MPC. Index Terms—Complementarity systems, hybrid systems, mixed logical dynamical systems, model predictive control (MPC), piecewise affine systems.
I. INTRODUCTION
V. CONCLUSION In this note the problem of global state and frequency simultaneous estimation is addressed. We propose a new simple estimator which provides a solution to this important problem in system theory. This estimator is globally convergent for all initial conditions and frequency values and its dimension is 3n, which is, as far as we know, the lower dimensional estimator for this problem. The extensive performed simulations allows us to state the validity of the proposed solution. REFERENCES [1] L. Hsu, R. Ortega, and G. Damm, “A globally convergent frequency estimator,” IEEE Trans. Automat. Contr., vol. 44, pp. 698–713, Apr. 1999. [2] M. Bodson and S. Douglas, “Adaptive algorithms for the rejection of sinusoidal disturbances with unknown frequency,” in Proc. 13th IFAC World Conf., San Francisco, CA, July 1–5, 1996. [3] H. Khalil, Nonlinear Systems, 2nd ed. NJ: Prentice-Hall, 1996. [4] C. Fuller and A. von Flotow, “Active control of sound and vibration,” IEEE Contr. Syst. Mag., vol. 15, pp. 9–19, Dec. 1996. [5] P. Regalia, IIR Filtering in Signal Processing and Control. New York: Marcel Dekker, 1995. [6] S. Hall and N. Wereley, “Performance of Higher harmonic control algorithms for helicopter vibration reduction,” J. Guid. Control Dyn., vol. 116, no. 4, pp. 793–797, 1993. [7] R. Herzog, P. Buhler, C. Gahler, and R. Larsonneur, “Unbalance compensation using generalized notch filters in the multivariable feedback of magnetic bearings,” IEEE Trans. Contr. Syst. Technol., vol. 4, pp. 580–586, Sept. 1996. [8] P. Regalia, “An improved lattice-based adaptive IIR notch filter,” IEEE Trans. Signal Processing, vol. 39, pp. 2124–2128, Sept. 1991. [9] R. Marino and P. Tomei, “Global estimation of n unknown frequencies,” in 39th Conf. Decision Control, Sydney, Australia, Dec. 12–15, 2000. [10] , “Global adaptive observers for nonlinear systems via filtered transformations,” IEEE Trans. Automat. Contr., vol. 37, pp. 1239–1245, Aug. 1992. [11] O. Besson and P. Stoica, “Estimation of the parameters of a random amplitude sinusoid by correlation fitting,” IEEE Trans. Signal Processing, vol. 44, pp. 2911–2916, Nov. 1996.
Hybrid dynamical models describe systems where both analog (continuous) and logical (discrete) components are relevant and interacting [1]. Recently, hybrid systems received a lot of attention from both the computer science and the control community, but general analysis and control design methods for hybrid systems are not yet available. For this reason, several authors have focused on special subclasses of hybrid systems for which analysis and synthesis techniques are currently being developed. Some examples of such subclasses are: linear complementarity (LC) systems [2], [3], extended linear complementarity (ELC) systems [4], mixed logical dynamical (MLD) systems [5], [6], piecewise affine (PWA) systems [7], and max-min-plus-scaling (MMPS) systems [8]. In [9], we showed that the previous five subclasses of hybrid systems are equivalent. Some of the equivalences were obtained under additional assumptions related to well-posedness (i.e., existence and uniqueness of solution trajectories) and boundedness of (some) system variables. These results are extremely important, as they allow to transfer all the analysis and synthesis tools developed for one particular class to any of the other equivalent subclasses of hybrid systems. The main result of this note will show that all these hybrid tools can be used for the analysis of closed-loop model predictive control (cl-MPC) systems as well. Indeed, as we will prove that cl-MPC systems can be written as LC and MLD systems, the transfer of the machinery is immediate. Related results were obtained in [10], where the authors showed that MPC control is equal to a piecewise affine control law that can be computed offline by using multiparametric quadratic programming solvers (and, therefore, that the closed-loop system is a PWA system). Rather than exploiting the equivalence results of [9] in
Manuscript received June 7, 2001; revised November 2, 2001. Recommended by Associate Editor G. De Nicolao. A. Bemporad is with the Dipartimento Ingegneria dell’Informazione, Università di Siena, 53100 Siena, Italy (e-mail: [email protected]). W. P. M. H. Heemels is with the Department of Electrical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands (e-mail: [email protected]). B. De Schutter is with the Control Systems Engineering Group, Delft University of Technology, 2600 GA Delft, The Netherlands (e-mail: [email protected]). Publisher Item Identifier S 0018-9286(02)04749-9.
0018-9286/02$17.00 © 2002 IEEE
A Globally Convergent Estimator for -Frequencies
857
which we consider to be generated by the linear model
w_ 1 =w2
G. Obregón-Pulido, B. Castillo-Toledo, and A. Loukianov
Abstract—In this note, we propose a solution to the well-know problem of ensuring a simultaneous globally convergent online estimation of the state and the frequencies of a sinusoid signal composed of sinusoidal terms. We present an estimator which guarantees global boundedness and convergence of the state and frequencies estimation for all initial conditions and frequencies values. Index Terms—Adaptive signal processing, frequency estimation, nonlinear systems.
I. INTRODUCTION In this note, we address the problem of simultaneous online estimation of the state and the frequency of a measurable signal composed of the sum of n sinusoidal terms given by
y (t) =
n i=1
A1i sin ( i t + 1 i ) + A2i cos ( i t + 2 i )
w_ 2 = 0 2 w1 y(t) = k1 w1 + k2 w2 in which the parameter 2 is unknown, k1 , k2 , = 6 0 and the initial conditions
w1 (0) w2 (0)
=
k1 k2 01 A1 sin(1 ) + A2 cos(2 ) 2 0 k2 k1 1 A1 cos(1 ) 0 A2 sin(2 )
are also unknown. The parameter is introduced here to scale the magnitude of the signals. It is not difficult to see that this system is immersed into the system
z_1 =z2 2 z_2 = 0 z1 _ =0 y(t) = k1 z1 + k2 z2 :
(1)
where the amplitudes A1i ; A2i 6= 0, the frequencies i 6= j ; i 6= j and the phases 1 i ; 2 i are all unknown. The frequency estimation is a very important issue in control theory, due to the number of practical applications in rotational mechanical processes like induction motors, disk drivers, helicopters or vibration control among others ([2], [4], [6], [7]). The problem of estimation of the frequency of the signal has been studied extensively by means of differents technics both in the offline case, for example the work of [11] and the online estimation [8], but only recently, a globally convergent estimator was proposed in [1] on the basis of an adaptive notch (AN) filter first proposed in the discrete-time version in [8] and adapted in [2] for the continuous-time case. A key feature of this AN filter was the scaling of the forcing term to normalize the parameters, which does not affect stability and ensures the positivity for all time of the estimate. The problem of simultaneous online globally convergent estimation of the frequency and the state is, as pointed out in [1], a well-known open problem in system theory. In this note, we propose a solution to this long-standing problem and then extend the results naturally to the case of n unknown frequencies. More specifically, we present a new estimator which ensures a continuous-time online simultaneous frequency and state estimation, ensuring that all signals are globally bounded and the estimation of the frequencies and the states are asymptotically correct for all initial conditions and frequency values. Recently, the same problem was addressed in [9] using a set of adaptive observers first developed in [10]. We briefly discuss this results and the main difference with the approach presented here. II. ESTIMATION OF A SINGLE FREQUENCY
We are interested in the estimation of the state (w1 , w2 ) and the square of the frequency 2 . Let us denote by x = (x1 ; x2 ; x3 )T the states of the estimator and define the estimation error vector e = (e1 ; e2 ; e3 )T , with e1 = w1 0 x1 , e2 = w2 0 x2 and e3 = x3 0 2 . Now, the structure of (2) allows us to derive the estimator, described in the following result. Proposition 1: The estimator
x_ 1 =x2 + (y 0 y~) k2 x x x_ 2 = 0 1 3 + (y 0 y~) x_ 3 = 0 x1 (y 0 y~) y~ = k1 x1 + k2 x2
Manuscript received May 22, 2000; revised May 14, 2001. Recommended by Associate Editor B. Bernhardsson. This work was supported by CONACYT under Grant 26358-A. The authors are with the Centro de Investigación y de Estudios Avanzados del IPN, Unidad Guadalajara 44550, Guadalajara, México. Publisher Item Identifier S 0018-9286(02)04750-5.
(3) (4) (5) (6)
where , , > 0, k1 , k2 6= 0, ensures that limt!1 e = 0. Proof: The error dynamics take the form
e_ 1 = 0 k1 e1 k2 e_ 2 = 0 2 + k1 e1 0 k2 e2 + x1 e3 e_ 3 = 0 x1 (k1 e1 + k2 e2 ) : Now, let us take the Lyapunov candidate function
Let us consider a sinusoidal signal
y(t) = A1 sin ( t + 1 ) + A2 cos ( t + 2 )
(2)
T
T
V (e) =e Me = e =
c 2 k 2 0
k 2 k 2 0
0 0 2
e
c1 e2 + k2 e2 + e2 + k e e : 1 1 2 1 2 3 2 2 2
(7)
(8)
Since M is Hermitian, a standard calculation shows that M is definite–positive if c1 > 0 and c1 k2 > k12 from which k2 must be positive.
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Fig. 1. Signals x , x , and x for = 1.
The derivative of V (e) is now calculated as
namics, we verify that the solution of the error dynamics plus estimator in is given by
V_ (e) = 0 c1 k1 e12 + k2 e2 (0 1 e1 0 k2 e2 + x1 e3 ) k2 + k1 e1 (0 1 e1 0 k2 e2 + x1 e3 ) + k1 e2 0 kk1 e1 + e3 0 x1 (k1 e1 + k2 e2 ) 2 = 0 c1 kk1 + k1 2 + k12 e12 0 k22 e22 2 2 0 k2 2 + 2k1 k2 + kk12 e1 e2
x_ 1 =x2 x_ 2 = 0 x1 x3 x_ 3 =0 e_ 2 =0 = x1 e3 e_ 3 =0:
To ensure that V_ is semidefinite–negative, we need to ensure that
k2 2 + 2k1 k2 + kk k 1 2 2 c1 + k1 + k1 > k2 4k22 so that, to satisfy both conditions c1 k2 equation that
c1 > max
; k2 k2 k1
k12
2
(9)
> k12 and (9), we impose the
k2 2 +2k1 k2 + kk 4k22
2
0 k1 20 k12
:
Now, since V_ is not definite–negative, we see that V_ = 0 in the set = f(e; x) j e1 = e2 = 0g : To prove the stability of the error dy-
From this, one can observe that the fourth equation imposes that e3 must be zero, since e1 = 0 implies that x1 = w1 and this is different from zero except in the trivial situation, so e3 = 0 and the error dynamics tend asymptotically to zero, this is x3 tends asymptotically to the value of 2 . So, in the invariant set, the only solution for the error dynamics is the trivial solution and for the observer dynamics, the only solution is a limit cycle and the output is precisely the exact output of the original system. So, since V (e) is radially unbounded for all values of e, invoking the Lasalle–Krasovskii theorem [3], we conclude that the observer guarantees global convergence of the estimation error. Remark 2: Note that if the signal have some constant bias then we can introduce one integrator and make the same procedure. III. THE n-FREQUENCIES CASE In this section, we present the extension to the case of a signal containing n frequencies and given by
y(t) =
n i=1
A1 i sin ( i t + 1 i ) + A2 i cos ( i t + 2 i ) :
This signal may be viewed as the output of the dynamical system
w_ (t) = Sw(t)
(10)
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Fig. 2. Signals x , x , and x for = 100.
where
Si
S= y (t) =
n
where 0
0
.. .
..
.. .
0
0
i=1
.
;
Sn
0
Si =
0 i2
1
a0 =
0
w_ 1 =w2 w_ 2 =w3 .. .
w_ 2n01 =w2n w_ 2n = 0 a0 w1 0 a2 w3 0 1 1 1 0 a2n02 w2n01 1
i=1
z_ (t) =
(11)
0
1
0
0
.. .
.. .
0
0
0
0
0
0
0
y(t) =
2n01
i=1
0
111 111
0
2 .. .
0
ki z + k z i 2n 2n 2n01 j
j =i
n
n j =1
2i ; . . . ; a2n02 =
n i=1
2i
gi >0 for i = 2 . . . n + 2; g1 = 2n01 k2n 2n01 k i y~ = xi + k2n x2n 2n01 i=1 j =i j
2n
ki wi 2n01 j =1j j i=1
2i ; a2 =
are the coefficients of the characteristic polynomial p(s) = s2n + a2n02 s2n02 + :: + a2 s2 + a0 and j are design parameters to conveniently scale the magnitude of the estimator signals. As in the case of a single frequency, we may verify that (11) is immersed into the system (12), as shown at the bottom of the page, for every i 6= 0 and ki 6= 0. The structure of (12) allows us to derive the estimator, which is given in the following result. Theorem 3: The estimator is shown in the equation at the bottom of the next page, with
c1 i w1 i (t) + c2 i w2 i (t)
for some constants c1 i = 6 0; c2 i =6 0 which may be rewritten as
y=
n
..
.
0
0
0
0
.. .
.. .
z (t) 1 1 1 2n02 0 111 0 2n01 111 0 0 (12)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 5, MAY 2002
Fig. 3.
Signals x , x , a and a .
and ki chosen such that polynomial
e_ 2n01 =w2n 0
k2n01 2n02 k2n02 2n03 k2 k p(s) = s + s + s + 111 + s+ 1 k2n k2n k2n k2n is stable with all its roots distinct, is such that y~ ! y , x2n+i ! a2i02 for i = 1 . . . n when t ! 1. 2n01
=
Proof: We first consider the error system
e_ 2n02 =e2n01 = w2n01 0
2n02 i=1
0x
0
1
0
0
.. .
.. .
0
0
0
0
0
111 111
0
2 .. .
..
0
0x
x_ 2n+1 = 0 g3 x1 (y 0 y~) x_ 2n+2 = 0 g4 x3 (y 0 y~) .. .
i=1i
(2n01 x2n + g1 (y
i x2n
0
2n02 i=1i
0 y~))
i g1 (y 0 y~) :
e_ i =ei+1 for i = 1; ::; 2n 0 2 2n01 1 e_ 2n01 = ke k2n i=1 i i
i x2n01
x_ (t) =
i
i=1i 2n01
2n01 Now, taking e2n = w2n 0 i=1 i x2n we have the equation shown at the bottom of the next page. Choosing g1 = 2n01 =k2n , we have the system
e1 =w1 0 x1 e_ 1 =e2 = w2 0 1 x2 e_ 2 =e3 = w3 0 1 2 x3 .. .
w2n 0
2n02
x_ 2n+n = 0 gn+2 x2n01 (y 0 y~)
0
.
0
0
0
0
.. . 2n02
.. .
111 1 1 1 0x 0 111
0
2n01 0
0 0
x(t) +
.. .
0
g1 g2
(y
0 y~)
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Fig. 4. Signals y and y~.
which characteristic polynomial is
and the second equation at the bottom of the next page holds true. Since matrix A has distinct eigenvalues, then there exists a matrix T such that
k k k k p(s) = s2n01 + 2n01 s2n02 + 2n02 s2n03 + 1 1 1 + 2 s + 1 k2n k2n k2n k2n
A~ =T 01 AT = diag ( 1 e1 =T e~1
and is stable by hypothesis. Now, we continue with the first scalar equation shown at the bottom of the next page. If we define e2n+i = x2n+i 0 a2i02 for i = 1::n, then the error dynamics may be rewritten in the form
e~_ 1 =A~e~1 e_ 2n = 0 ~1T e~1 0 2 e2n + xT1 e3 x1 k~1T e~1 + k2n e2n e_ 3 = 0 0 where
~1T = 1T T; k~1T = k1T T:
where A is a Hurwitz matrix given by
A=
1 0 0
..
0 kk
0 kk
e_ 2n01 =e2n 0
111
2n02
=e2n 0 g1
i=1
.
0
Since A~ is diagonal, there exists a Lyapunov function Vo (~ e1 ) satisfying
.. .
Vo (~e1 ) =~e1T P e~1 V_ o (~e1 ) = 0 e~1T Qe~1
1 1 1 k1 111 0 k
i g1
1 2n k w 2n01 i=1 i i
2n =e2n 0 g1 ke : 2n01 i=1 i i
2n01 )
and the error dynamics take the form
e_ 1 =Ae1 e_ 2n = 0 1T e1 0 2 e2n + xT1 e3 x1 k1T e1 + k2n e2n e_ 3 = 0 0
0 0 0
111
2n 1 2n01 j i=1 ki wi
j =1
0 2n101
0
k1 x1 +
2n01 i=1
2n01 i=1
i
k
i 2n01 j xi + k2n x2n j =i 2n01 1 k2n x2n 0 2n01 i=2
i01 j =1
j ki xi
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for some P , Q diagonal and positive–definite. Let us now consider the Lyapunov candidate function T e~ e + c e2n V (~e1 ; e2n ; e3 ) = Vo (~e1 ) + C12 1 2n 2n 2 with 2
C12 = [ c1;2n c2;2n C3 =diag ( c13 c23
1 1 1 c n0 ; n 1 1 1 cn ) 2
2 2
Choosing
c2n = k2n ; C12 = k~1 ; and C3 = 001
+ 1 e3T C3 e3 2
c2n01;2n
we finally get that
]T
V_ (~e1 ; e2n ; e3 ) = 0e~1T Q + k~1 ~1T e~1 0 k2n 2 e22n
3
0 k~T 0 k~T A~ + ~T k n e~ e n
which is positive–definite for some P and c2n > 0. From this, we have the third equation shown at the bottom of the page, where the fact that C3 is diagonal has been used. Since 0 is also diagonal, we have that
1
T 0 CT A ~ + ~1T c2n e~1 e2n C12 2 12
+
e~1T C12
0 e~T k~
xT1 e3
xT1 0C3 e3
1
1
+c2n xT1 e3 e2n 0 k2n xT1 0C3e3 e2n:
01
2n
e_ 2n =w_ 2n 0
01
2n
i=1
1
1
i x_ 2n = 0a0 w1 0 a2 w3 0 1 1 1 0 a2n02 w2n01
x2n+2 0x2n+n 0 x1 0 2n01 i x3 0 1 1 1 0 2n01 x2n01 + g2 (y 0 y~) i=3 i=1 2n02 = 0 a0 (e1 + x1 ) 0 a2 (e3 + 1 2 x3 ) 0 1 1 1 0 a2n02 e2n01 + i x2n01
0
0x n
2 +1 2n 1 i i=1
i
+ x1 x2n+1 + 1 2 x3 x2n+2 + 1 1 1 +
02
2n
i=1
i=1
2n
i x2n01 x3n 0 g2
i=1
ki ei
= 0 (a0 + g2 k1 ) e1 0 g2 k2 e2 0 (a2 + g2 k3 ) e3 0 1 1 1 0 (a2n02 + g2 k2n01 ) e2n01 0 g2 k2n e2n + x1 (x2n+1 0 a0 ) + 1 2 x3 (x2n+2 0 a2 ) + 1 1 1 2n02 i x2n01 (x3n 0 a2n02 ) : + i=1
e1 = [ e1 1 1 1 e2n01 ]T e3 = [ e2n+1 1 1 1 e3n ]T 1 = [ a0 + g2 k1 g2 k2 a2 + g2 k3 2 =g2 k2n
0 =diag k1 = [ k1
g
111
g2 k2n02 a2n02 + g2 k2n01 ]T
0 j x2i01
111
x1 = x1 1 2 x3
111
2i 2 j =1
g
k2n01 ]T :
(
)
T 2n02 j =1 j x2n01 g
111 111
T e~ V_ (~e1 ; e2n ; e3 ) = 0 e~1T Qe~1 + C12 1
111
g
0 ~T e~ 0 e n + xT e + C T A~e~ e n + c n e n 0 ~T e~ 0 e n + xT e T 1 + 1 00x k~T e~ + k n e n C e + eT C 00 x k~T e~ + k n e n 2 2 = 0 e~T Q + C ~T e~ 0 c n e n 0 C T 0 C T A~ + ~T c n e~ e n T + e~T C xT e + c n xT e e n 0 0 x k~T e~ + k n e n C e 12
1
2
1
1
2
1
12 1
1
1
12
1
3
1
1
2 2
2
2
1
2
1
2
2
2
1 2
which is negative–semidefinite for some Q. We therefore use the same analysis made in the single frequency case, namely, in this case the invariant set is given by e~1 = 0, e2n = 0 and then the vector e3 must be zero because the supposition that the frequencies are distinct implies that the signals x2i01 are linearly independent for i = 1::n. Remark 4: Recently, a frequency estimator was presented in [9], based in the use of an adaptive estimator of dimension 5n 0 1. In contrast, the order of the estimator presented in this work is 3n.
V_ (~e1 ; e2n ; e3 ) = 0 e~1T (Q + C12 ~1T )~e1 0 c2n 2 e22n
0
2
1
1
2 2
3 3
2 2 2
1
3
3
1
3
3
12 2
2
3
1
12
1
1
1
1
1
1
2
2
2
2
2
1 2
3 3
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863
IV. SIMULATIONS RESULTS
On Hybrid Systems and Closed-Loop MPC Systems
The performance of the estimator was tested by extensive simulations, some of which are presented in the following figures. First, we show in Fig. 1 the behavior of the estimator for a single frequency and = 1, = 10, = 1, k1 = 1, k2 = 1, = 1 and y (t) = sin( t). We stress the fact that if the frequency is increased, then the magnitude of w2 increase too, and therefore, we may increase to reduce the magnitude of x2 and reduce k2 and k1 for which the dynamics of x1 are faster. In this situation, we can increase and thus we have faster dynamics of x3 . The response of the estimator for = 100, where the parameters are k1 = 0:5, k2 = :25, = 100, = 1000, = 1000 and y (t) = 10 sin(100t) is shown in the Fig. 2. We observe the good behavior of the estimator. Taking now two frequencies, let us say 1 = 1, 2 = 2, we get a0 = 4 and a2 = 5 and the performance of the filter with parameters k1 = 6, k2 = 11, k3 = 6, k4 = 1, i = 1 for i = 1; ::; 3 and y(t) = 10(sin(t) + sin(2t)), are shown in Figs. 3 and 4. As it may be observed, the estimator exhibites a good convergence properties, so this results suggest the validity of the proposed estimator.
Alberto Bemporad, W. P. Maurice H. Heemels, and Bart De Schutter
Abstract—The following five classes of hybrid systems were recently proven to be equivalent: linear complementarity, extended linear complementarity, mixed logical dynamical, piecewise affine, and max-min-plus-scaling systems. Some of the equivalences were obtained under additional assumptions, such as boundedness of certain system variables. In this note, for linear or hybrid plants in closed-loop with a model predictive control (MPC) controller based on a linear model, fulfilling linear constraints on input and state variables, and utilizing a quadratic cost criterion, we provide a simple and direct proof that the closed-loop system is a subclass of any of the former five classes of hybrid systems. This result is of extreme importance as it opens up the use of tools developed for the mentioned hybrid model classes, such as (robust) stability and safety analysis tools, to study closed-loop properties of MPC. Index Terms—Complementarity systems, hybrid systems, mixed logical dynamical systems, model predictive control (MPC), piecewise affine systems.
I. INTRODUCTION
V. CONCLUSION In this note the problem of global state and frequency simultaneous estimation is addressed. We propose a new simple estimator which provides a solution to this important problem in system theory. This estimator is globally convergent for all initial conditions and frequency values and its dimension is 3n, which is, as far as we know, the lower dimensional estimator for this problem. The extensive performed simulations allows us to state the validity of the proposed solution. REFERENCES [1] L. Hsu, R. Ortega, and G. Damm, “A globally convergent frequency estimator,” IEEE Trans. Automat. Contr., vol. 44, pp. 698–713, Apr. 1999. [2] M. Bodson and S. Douglas, “Adaptive algorithms for the rejection of sinusoidal disturbances with unknown frequency,” in Proc. 13th IFAC World Conf., San Francisco, CA, July 1–5, 1996. [3] H. Khalil, Nonlinear Systems, 2nd ed. NJ: Prentice-Hall, 1996. [4] C. Fuller and A. von Flotow, “Active control of sound and vibration,” IEEE Contr. Syst. Mag., vol. 15, pp. 9–19, Dec. 1996. [5] P. Regalia, IIR Filtering in Signal Processing and Control. New York: Marcel Dekker, 1995. [6] S. Hall and N. Wereley, “Performance of Higher harmonic control algorithms for helicopter vibration reduction,” J. Guid. Control Dyn., vol. 116, no. 4, pp. 793–797, 1993. [7] R. Herzog, P. Buhler, C. Gahler, and R. Larsonneur, “Unbalance compensation using generalized notch filters in the multivariable feedback of magnetic bearings,” IEEE Trans. Contr. Syst. Technol., vol. 4, pp. 580–586, Sept. 1996. [8] P. Regalia, “An improved lattice-based adaptive IIR notch filter,” IEEE Trans. Signal Processing, vol. 39, pp. 2124–2128, Sept. 1991. [9] R. Marino and P. Tomei, “Global estimation of n unknown frequencies,” in 39th Conf. Decision Control, Sydney, Australia, Dec. 12–15, 2000. [10] , “Global adaptive observers for nonlinear systems via filtered transformations,” IEEE Trans. Automat. Contr., vol. 37, pp. 1239–1245, Aug. 1992. [11] O. Besson and P. Stoica, “Estimation of the parameters of a random amplitude sinusoid by correlation fitting,” IEEE Trans. Signal Processing, vol. 44, pp. 2911–2916, Nov. 1996.
Hybrid dynamical models describe systems where both analog (continuous) and logical (discrete) components are relevant and interacting [1]. Recently, hybrid systems received a lot of attention from both the computer science and the control community, but general analysis and control design methods for hybrid systems are not yet available. For this reason, several authors have focused on special subclasses of hybrid systems for which analysis and synthesis techniques are currently being developed. Some examples of such subclasses are: linear complementarity (LC) systems [2], [3], extended linear complementarity (ELC) systems [4], mixed logical dynamical (MLD) systems [5], [6], piecewise affine (PWA) systems [7], and max-min-plus-scaling (MMPS) systems [8]. In [9], we showed that the previous five subclasses of hybrid systems are equivalent. Some of the equivalences were obtained under additional assumptions related to well-posedness (i.e., existence and uniqueness of solution trajectories) and boundedness of (some) system variables. These results are extremely important, as they allow to transfer all the analysis and synthesis tools developed for one particular class to any of the other equivalent subclasses of hybrid systems. The main result of this note will show that all these hybrid tools can be used for the analysis of closed-loop model predictive control (cl-MPC) systems as well. Indeed, as we will prove that cl-MPC systems can be written as LC and MLD systems, the transfer of the machinery is immediate. Related results were obtained in [10], where the authors showed that MPC control is equal to a piecewise affine control law that can be computed offline by using multiparametric quadratic programming solvers (and, therefore, that the closed-loop system is a PWA system). Rather than exploiting the equivalence results of [9] in
Manuscript received June 7, 2001; revised November 2, 2001. Recommended by Associate Editor G. De Nicolao. A. Bemporad is with the Dipartimento Ingegneria dell’Informazione, Università di Siena, 53100 Siena, Italy (e-mail: [email protected]). W. P. M. H. Heemels is with the Department of Electrical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands (e-mail: [email protected]). B. De Schutter is with the Control Systems Engineering Group, Delft University of Technology, 2600 GA Delft, The Netherlands (e-mail: [email protected]). Publisher Item Identifier S 0018-9286(02)04749-9.
0018-9286/02$17.00 © 2002 IEEE
